Finite Group Databasebeta

← #29697320

Subgroups — #29697320

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×#16839760C211yesmaximal, characteristic
2#21448684C211yesmaximal, characteristic, complemented
3#21448658C211yesmaximal, characteristic
4#21446519C211yesmaximal, characteristic, complemented
5#21446499C211yesmaximal, characteristic, complemented
6#21446441C211yesmaximal, characteristic, complemented
7#21446084C211yesmaximal, characteristic, complemented
8#21446076C211yesmaximal, characteristic, complemented
9#21445908C211yesmaximal, characteristic, complemented
10#21445849C211yesmaximal, characteristic, complemented
11#21445756C211yesmaximal, characteristic
12#21445652C211yesmaximal, characteristic, complemented
13#21445638C211yesmaximal, characteristic, complemented
14#21445601C211yesmaximal, characteristic, complemented
15C61×#1926C211yesmaximal, characteristic, complemented
16C22×#11758224C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519734—21no—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×#14519711C2211yescharacteristic
23#16839760C418yescomplemented
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31C2×#14519688C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839662—21no—
35#16839662—21no—
36#16839660C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839646—21no—
39#16839644—21no—
40#16839624—21no—
41#16839620—21no—
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C22⋊C4×Dic61—21no—
46C4×#11758198C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C2×#11758224—22no—
61C2×#11758224—22no—
62C2×#11758224—22no—
63C2×#11758224—41no—
64C2×#11758224—41no—
65C2×#11758224C2×C414yes—
66C22×Dic122C2311yescharacteristic
67C22×C22⋊D61—21no—
68C23×Dic61—21no—
69C22×C4×D61—21no—
70C22×C4×D61C2311yescharacteristic
71C2×#11758189—41no—
72C2×#11758189—41no—
73C2×#11758189C2×C414yes—
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76C4×Dic122—22no—
77C4×Dic122—41no—
78#14519742—22no—
79#14519742—41no—
80#14519734—44no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61C2×C414yes—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198D411yescharacteristic
96C2×#11758198D411yescharacteristic
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61C2311yescharacteristic
101#14519711—41no—
102#14519702—22no—
103#14519702C2×C414yes—
104#14519700—22no—
105#14519700C2×C414yes—
106#14519698—41no—
107#14519698C2×C414yes—
108#14519689—22no—
109#14519688—41no—
110C8×Dic61—41no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684D411yescharacteristic
117#14519684D411yescharacteristic
118C22×D4×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#106×C61—41no—
129#102×C61—41no—
130#102×C61C2×C414yes—
131#95×C61—21no—
132C24×C61—21no—
133C23×D61C2×D411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61C22×C412yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—24no—
147C2×C22⋊D61—42no—
148C2×C22⋊D61—42no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—24no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159#11758224—28no—
160#11758224—44no—
161#11758224—44no—
162#11758224—44no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—24no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61C22×C412yes—
171C2×C4×D61#4112yes—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122C22×C412yes—
176C2×Dic122#4112yes—
177C2×Dic122#4112yes—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61C22×C412yes—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C22×C412yes—
190C22⋊C4×C61—42no—
191C22⋊C4×C61—81no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—42no—
195C22⋊Dic61—42no—
196C22⋊Dic61—81no—
197#11758198—22no—
198#11758198—41no—
199#11758198—42no—
200#11758198—81no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C22×C412yes—
207#11758196—24no—
208#11758196—42no—
209#11758196—42no—
210#11758196—42no—
211#11758196—42no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—42no—
215C4×Dic61—42no—
216C4×Dic61—42no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C22×C412yes—
223#11758189—44no—
224C23×C61—22no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61C22×D411yescharacteristic
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61C4×D412yes—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61—41no—
242D4×C61—41no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
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—21no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—48no—
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×Dic61C4×D412yes—
286C2×Dic61AGL(1,ℤ/8)12yes—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—48no—
292Dic122—22no—
293Dic122—22no—
294Dic122—22no—
295Dic122#11714yescomplemented
296C8×C61—22no—
297C8×C61—41no—
298#92418—22no—
299#92418—41no—
300#22583—611nomaximal, Sylow
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
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—22no—
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—41no—
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—41no—
322C4×C61—41no—
323C4×C61—42no—
324C4×C61—41no—
325C4×C61—81no—
326C4×C61—81no—
327C4×C61C2×#11711yescharacteristic
328C4×C61#51811yescharacteristic
329C4×C61#55111yescharacteristic
330C4×C61#55111yescharacteristic
331Dic61—24no—
332Dic61—22no—
333Dic61—22no—
334Dic61—42no—
335Dic61—41no—
336Dic61—81no—
337Dic61—81no—
338C2×#417—611no—
339#2344—611no—
340#2341—611no—
341#1936—611no—
342#1934—611no—
343#1926D6111yescharacteristic, complemented
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#1753—611no—
353C2×C61—82no—
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—42no—
362C22×#41—611no—
363C22×M16—611no—
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×#117—1221no—
367C22×C4⋊C4—611no—
368C2×#107—611no—
369C2×#106C2×D6111yescharacteristic
370#417—618no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102C2×D6111yescharacteristic
378C2×#102C2×D6111yescharacteristic
379C2×C22⋊C8—611no—
380C2×#97—611no—
381C2×C4×C8—611no—
382#390—611no—
383#390—1221no—
384#390—1221no—
385#390C2×D6111yescharacteristic
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⋊C4C2×D6111yescharacteristic
396C4×C22⋊C4—611no—
397C4×C22⋊C4—1221no—
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340C2×D6111yescharacteristic
404#336—611no—
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—1221no—
415C22×D4C22×D6111yescharacteristic
416C23×C4—611no—
417C23×C4—1221no—
418C23×C4—1221no—
419C2×M16—614no—
420C2×M16—2441no—
421C2×M16—2441no—
422C22×C8—611no—
423C22×C8—611no—
424C22×C8C22×D6111yescharacteristic
425C22×C8C22×D6111yescharacteristic
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×Q8—2441no—
429C4×Q8—2441no—
430#117—2444no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—614no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—611no—
447C2×C22⋊C4—611no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C42—21no—
453C2×C42—611no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42—1221no—
457C2×C42C22×D6111yescharacteristic
458#107—2441no—
459#106—41no—
460#103—614no—
461#103—614no—
462#103—1222no—
463#103—1222no—
464#102—41no—
465#102—614no—
466#102—2441no—
467#102C4×D6114yes—
468C22⋊C8—1222no—
469#97—2441no—
470C4×C8—2441no—
471#95—21no—
472#95—611no—
473#95—611no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24—21no—
480C24—611no—
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—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—1224no—
497C2×D4—2442no—
498C2×D4—2442no—
499C2×D4C2×C4×D6112yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—612no—
506C22×C4—612no—
507C22×C4—611no—
508C22×C4—1224no—
509C22×C4—1224no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4D4×D6111yescharacteristic
523C22×C4D4×D6111yescharacteristic
524C22×C4C23×D6111yescharacteristic
525M16—2444no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—612no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8C2×C4×D6112yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—612no—
544C4⋊C4—1224no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—2442no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C4⋊C4—2442no—
554C4⋊C4C2×C4×D6112yes—
555C22⋊C4—42no—
556C22⋊C4—81no—
557C22⋊C4—1222no—
558C22⋊C4—4881no—
559C22⋊C4—4881no—
560C22⋊C4—2441no—
561C22⋊C4—2442no—
562C22⋊C4—2442no—
563C42—21no—
564C42—21no—
565C42—42no—
566C42—1221no—
567C42—1221no—
568C42—2442no—
569C42—2442no—
570C42—2442no—
571C23—22no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—612no—
579C23—1221no—
580C23—1221no—
581C23—1221no—
582C23—1221no—
583C23—1221no—
584C23—1221no—
585C23C2×D4×D6111yescharacteristic
586Q8—614no—
587Q8—1222no—
588Q8—1222no—
589Q8—1222no—
590D4—22no—
591D4—22no—
592D4—41no—
593D4—41no—
594D4—2448no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—1222no—
613C2×C4—1221no—
614C2×C4—1222no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1222no—
618C2×C4—1221no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—2448no—
628C2×C4—2448no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4C22×C4×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4#41×D6111yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4#1451982211yescharacteristic
642C2×C4#1451982311yescharacteristic
643C8—22no—
644C8—41no—
645C8—1222no—
646C8—2441no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—1222no—
655C22—1222no—
656C22—1222no—
657C22—2441no—
658C22—2441no—
659C22—2441no—
660C22—2441no—
661C22C4×D4×D6111yescharacteristic
662C22#1684040411yescharacteristic
663C22#1684040911yescharacteristic
664C22AGL(1,ℤ/8)×D6111yescharacteristic
665C22#1684055811yescharacteristic
666C22#1684056811yescharacteristic
667C22#1684057611yescharacteristic
668C4—42no—
669C4—41no—
670C4—41no—
671C4—81no—
672C4—81no—
673C4—1222no—
674C4—1222no—
675C4—1224no—
676C4—4881no—
677C4—4881no—
678C4—2441no—
679C4—2442no—
680C4#117×D6111yescharacteristic
681C4#1684023411yescharacteristic
682C4#1684043711yescharacteristic
683C4#1684044211yescharacteristic
684C2—82no—
685C2—2442no—
686C2#2144866311yescharacteristic
687C2#2145463011yescharacteristic
688C2#2145465511yescharacteristic
689C2#2145781711yescharacteristic
690C2#2145795211yescharacteristic
691C2#2145808511yescharacteristic
692C2#2145814411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1926C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
cosocleC22×C4×C61C24
hypercenter#1926D61
hypocenterC61#22583
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1926

Complements

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

NH ≅ G/Ntype
2. #21448684C2semidirect
4. #21446519C2semidirect
5. #21446499C2semidirect
6. #21446441C2semidirect
7. #21446084C2semidirect
8. #21446076C2semidirect
9. #21445908C2semidirect
10. #21445849C2semidirect
12. #21445652C2semidirect
13. #21445638C2semidirect
14. #21445601C2semidirect
15. C61×#1926C2semidirect
23. #16839760C4semidirect
295. Dic122#117semidirect
343. #1926D61semidirect
406. C61#22583semidirect

Maximal subgroups

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

subgroupindex
C2×#168397602
#214486842
#214486582
#214465192
#214464992
#214464412
#214460842
#214460762
#214459082
#214458492
#214457562
#214456522
#214456382
#214456012
C61×#19262
#2258361