Finite Group Databasebeta

← #29697322

Subgroups — #29697322

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839764C211yesmaximal, characteristic
2#21448686C211yesmaximal, characteristic, complemented
3#21448659C211yesmaximal, characteristic
4#21446516C211yesmaximal, characteristic, complemented
5#21446502C211yesmaximal, characteristic, complemented
6#21446447C211yesmaximal, characteristic, complemented
7#21446440C211yesmaximal, characteristic, complemented
8#21446090C211yesmaximal, characteristic, complemented
9#21446076C211yesmaximal, characteristic, complemented
10#21445910C211yesmaximal, characteristic, complemented
11#21445852C211yesmaximal, characteristic, complemented
12#21445760C211yesmaximal, characteristic
13#21445650C211yesmaximal, characteristic, complemented
14C4×#14519702C211yesmaximal, characteristic, complemented
15C61×#1753C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519733—21no—
20C22×C4⋊Dic61C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23#16839764C418yescomplemented
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839662—21no—
35#16839662—21no—
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839647—21no—
39#16839647—21no—
40#16839623—21no—
41#16839622C2211yescharacteristic
42#16839622C2211yescharacteristic
43#16839621—21no—
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46#16839609—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C22×M16×C61C2211yescharacteristic
54C2×C22⋊C8×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C4×C22⋊C4×C61C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C2×#11758222—22no—
61C2×#11758222—22no—
62C2×#11758222—22no—
63C2×#11758222—41no—
64C2×#11758222—41no—
65C2×#11758222C2×C414yes—
66C22×Dic122C2311yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C22⋊D61—21no—
69C23×Dic61—21no—
70C22×C4×D61—21no—
71C4×Dic122—22no—
72C4×Dic122—22no—
73C4×Dic122—41no—
74C4×Dic122—41no—
75#14519733—44no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—22no—
78C2×C4⋊Dic61C2×C414yes—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99#14519711—41no—
100#14519709—41no—
101#14519702—22no—
102#14519702—22no—
103#14519702C2×C414yes—
104#14519702C2×C414yes—
105#14519699—41no—
106#14519699—41no—
107#14519699C2×C414yes—
108#14519699C2×C414yes—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115C23×C4×C61C2311yescharacteristic
116C2×M16×C61—41no—
117C2×M16×C61—41no—
118C2×M16×C61C2×C414yes—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C2×C22⋊C4×C61D411yescharacteristic
124C2×C22⋊C4×C61D411yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C22⋊C8×C61—22no—
128#97×C61—41no—
129C4×C8×C61—41no—
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C24×C61C2×D411yescharacteristic
133C23×D61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61#4112yes—
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—24no—
147C2×C22⋊D61—42no—
148C2×C22⋊D61—42no—
149C22×Dic61—21no—
150C22×Dic61—24no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159#11758222—28no—
160#11758222—44no—
161#11758222—44no—
162#11758222—44no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244C22×C412yes—
169C2×C4×D61—24no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122C22×C412yes—
176C2×Dic122#4112yes—
177C2×Dic122#4112yes—
178M16×C61—44no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61C22×C412yes—
190C2×C8×C61C22×C412yes—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—41no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61—21no—
196C42×C61—21no—
197C22⋊Dic61—42no—
198C22⋊Dic61—81no—
199#11758198—42no—
200#11758198—42no—
201#11758198—81no—
202#11758198—81no—
203C4⋊Dic61—24no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61C22×C412yes—
215C4⋊Dic61C22×C412yes—
216#11758196—42no—
217#11758196—42no—
218#11758196—42no—
219#11758196—42no—
220C4×Dic61—42no—
221C4×Dic61—42no—
222C4×Dic61—42no—
223C4×Dic61—42no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61C4×D412yes—
231C23×C61C22×D411yescharacteristic
232C22×D61—22no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61C4×D412yes—
257C2×C4×C61AGL(1,ℤ/8)12yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C61H(5,𝔽2)11yescharacteristic
264C22⋊D61—48no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—48no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287D244—22no—
288D244—22no—
289D244—41no—
290D244—41no—
291C4×D61—48no—
292Dic122—22no—
293Dic122—22no—
294Dic122—22no—
295Dic122#11714yescomplemented
296C8×C61—22no—
297C8×C61—22no—
298C8×C61—41no—
299C8×C61—41no—
300#22583—611nomaximal, Sylow
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61C2×C4×D411yescharacteristic
309C22×C61#54611yescharacteristic
310C22×C61#54611yescharacteristic
311C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C61#57511yescharacteristic
313C22×C61#57711yescharacteristic
314C22×C61#57811yescharacteristic
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—81no—
319C2×D61—81no—
320C2×D61—81no—
321C2×D61—81no—
322C4×C61—22no—
323C4×C61—22no—
324C4×C61—41no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61C2×#11711yescharacteristic
328C4×C61#51811yescharacteristic
329C4×C61#55111yescharacteristic
330C4×C61#55111yescharacteristic
331Dic61—24no—
332Dic61—42no—
333Dic61—42no—
334Dic61—81no—
335Dic61—81no—
336Dic61—81no—
337Dic61—81no—
338C2×#417—611no—
339#2344—611no—
340#2341—611no—
341#1936—611no—
342#1934—611no—
343#1926—611no—
344#1925—611no—
345#1865—611no—
346#1863—611no—
347#1824—611no—
348#1809—611no—
349#1788—611no—
350#1763—611no—
351C4×#103—611no—
352#1753D6111yescharacteristic, complemented
353C2×C61—42no—
354C2×C61#234211yescharacteristic
355C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
356C2×C61#294711yescharacteristic
357C2×C61#330811yescharacteristic
358C2×C61#331711yescharacteristic
359C2×C61#332711yescharacteristic
360C2×C61#333211yescharacteristic
361D61—82no—
362C22×#41—611no—
363C22×M16C2×D6111yescharacteristic
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×#117—1221no—
367C22×C4⋊C4—611no—
368C2×#107—611no—
369C2×#106—611no—
370#417—618no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×C22⋊C8C2×D6111yescharacteristic
380C2×#97C2×D6111yescharacteristic
381C2×C4×C8C2×D6111yescharacteristic
382#390—611no—
383#390—611no—
384#390—1221no—
385#390—1221no—
386#389—611no—
387#389—611no—
388#387—1221no—
389#387—1221no—
390#381—1221no—
391#380—611no—
392#380—611no—
393#380—1221no—
394C4×C4⋊C4—611no—
395C4×C4⋊C4—611no—
396C4×C22⋊C4—1221no—
397C4×C22⋊C4C2×D6111yescharacteristic
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#336C2×D6111yescharacteristic
405#336C2×D6111yescharacteristic
406C61#2258311yescharacteristic, complemented, Sylow
407C2×#41—614no—
408C2×#41—1222no—
409C2×#41—1222no—
410C2×#41—1222no—
411C2×#41—2441no—
412C2×#41—2441no—
413C22×Q8—611no—
414C22×D4—611no—
415C22×D4—1221no—
416C23×C4—1221no—
417C23×C4—1221no—
418C23×C4C22×D6111yescharacteristic
419C2×M16—41no—
420C2×M16—41no—
421C2×M16C4×D6114yes—
422C22×C8C22×D6111yescharacteristic
423C22×C8C22×D6111yescharacteristic
424C22×C8C22×D6111yescharacteristic
425C22×C8C22×D6111yescharacteristic
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×Q8—2441no—
429C4×Q8—2441no—
430#117—2444no—
431C2×C4⋊C4—614no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1222no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4D24411yescharacteristic
451C2×C22⋊C4D24411yescharacteristic
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42C22×D6111yescharacteristic
457C2×C42C22×D6111yescharacteristic
458#107—2441no—
459#106—2441no—
460#103—614no—
461#103—614no—
462#103—1222no—
463#103—1222no—
464#102—614no—
465#102—614no—
466#102—2441no—
467#102—2441no—
468C22⋊C8—22no—
469#97—41no—
470C4×C8—41no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95D24411yescharacteristic
478#95D24411yescharacteristic
479C24—1221no—
480C24C2×D24411yescharacteristic
481#41—1228no—
482#41—2444no—
483#41—2444no—
484#41—2444no—
485C2×Q8—612no—
486C2×Q8—612no—
487C2×Q8—612no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×D4—612no—
492C2×D4—1224no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—2442no—
499C2×D4—2442no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1224no—
508C22×C4—1224no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4C2×C4×D6112yes—
518C22×C4C2×D24411yescharacteristic
519C22×C4C2×D24411yescharacteristic
520C22×C4C2×D24411yescharacteristic
521C22×C4C2×D24411yescharacteristic
522C22×C4C2×D24411yescharacteristic
523C22×C4#1175822212yes—
524C22×C4C23×D6111yescharacteristic
525M16—44no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8C2×C4×D6112yes—
537C2×C8C2×C4×D6112yes—
538C4⋊C4—612no—
539C4⋊C4—612no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1224no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C4⋊C4—2442no—
554C4⋊C4—2442no—
555C22⋊C4—22no—
556C22⋊C4—41no—
557C22⋊C4—4881no—
558C22⋊C4—4881no—
559C22⋊C4—4881no—
560C22⋊C4—2442no—
561C22⋊C4—2442no—
562C22⋊C4—2442no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—2442no—
568C42—2442no—
569C42—2442no—
570C42—2442no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—1222no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23—2441no—
583C23—2441no—
584C23C4×D24412yes—
585C23C22×D24411yescharacteristic
586Q8—614no—
587Q8—1222no—
588Q8—1222no—
589Q8—1222no—
590D4—1222no—
591D4—1222no—
592D4—2448no—
593D4—2441no—
594D4—2441no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—2448no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2448no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4C4×D24412yes—
636C2×C4#1451980512yes—
637C2×C4C22×C4×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4#41×D6111yescharacteristic
640C2×C4C2×#1175822211yescharacteristic
641C2×C4#1451982211yescharacteristic
642C2×C4#1451982211yescharacteristic
643C8—22no—
644C8—22no—
645C8—41no—
646C8—41no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—4881no—
655C22—4881no—
656C22—4881no—
657C22—4881no—
658C22—2442no—
659C22—2442no—
660C22—2442no—
661C22C2×C4×D24411yescharacteristic
662C22#1684040711yescharacteristic
663C22#1684040711yescharacteristic
664C22C2×#1451980511yescharacteristic
665C22#1684056211yescharacteristic
666C22#1684057411yescharacteristic
667C22#1684057911yescharacteristic
668C4—22no—
669C4—22no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—1224no—
674C4—4881no—
675C4—4881no—
676C4—4881no—
677C4—4881no—
678C4—2442no—
679C4—2442no—
680C4#117×D6111yescharacteristic
681C4#1684023511yescharacteristic
682C4#1684044111yescharacteristic
683C4#1684044111yescharacteristic
684C2—42no—
685C2—4882no—
686C2#2144867111yescharacteristic
687C2C4×#1451980511yescharacteristic
688C2#2145466411yescharacteristic
689C2#2145781211yescharacteristic
690C2#2145794711yescharacteristic
691C2#2145808911yescharacteristic
692C2#2145814711yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1753C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1753D61
hypocenterC61#22583
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21448686C2semidirect
4. #21446516C2semidirect
5. #21446502C2semidirect
6. #21446447C2semidirect
7. #21446440C2semidirect
8. #21446090C2semidirect
9. #21446076C2semidirect
10. #21445910C2semidirect
11. #21445852C2semidirect
13. #21445650C2semidirect
14. C4×#14519702C2semidirect
15. C61×#1753C2semidirect
23. #16839764C4semidirect
295. Dic122#117semidirect
352. #1753D61semidirect
406. C61#22583semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168397642
#214486862
#214486592
#214465162
#214465022
#214464472
#214464402
#214460902
#214460762
#214459102
#214458522
#214457602
#214456502
C4×#145197022
C61×#17532
#2258361