Finite Group Databasebeta

← #29697315

Subgroups — #29697315

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×#16839762C211yesmaximal, characteristic
2#21448687C211yesmaximal, characteristic, complemented
3#21448657C211yesmaximal, characteristic
4#21446515C211yesmaximal, characteristic, complemented
5#21446503C211yesmaximal, characteristic, complemented
6#21446447C211yesmaximal, characteristic, complemented
7#21446438C211yesmaximal, characteristic, complemented
8#21446089C211yesmaximal, characteristic, complemented
9#21445908C211yesmaximal, characteristic, complemented
10#21445853C211yesmaximal, characteristic, complemented
11#21445758C211yesmaximal, characteristic
12#21445649C211yesmaximal, characteristic, complemented
13#103×Dic61C211yesmaximal, characteristic, complemented
14#21445601C211yesmaximal, characteristic, complemented
15C61×#1863C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519735—21no—
20C2×C4⋊C4×D61C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22#16839762C418yescomplemented
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839664—21no—
33#16839664C2211yescharacteristic
34#16839663—21no—
35#16839663C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839649—21no—
38#16839643—21no—
39#16839624—21no—
40#16839619—21no—
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C4×#11758198—21no—
46C4×#11758198C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C2×#11758226—22no—
61C2×#11758226—22no—
62C2×#11758226—22no—
63C2×#11758226—41no—
64C2×#11758226—41no—
65C2×#11758226C2×C414yes—
66C22×D244—21no—
67C22×D244C2311yescharacteristic
68C22×C4×D61—21no—
69C22×C4×D61—21no—
70C22×C4×D61C2311yescharacteristic
71C2×#11758189—41no—
72C2×#11758189—41no—
73C2×#11758189C2×C414yes—
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76Q8×Dic61—22no—
77Q8×Dic61—41no—
78#14519735—44no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C4⋊C4×D61—22no—
86C4⋊C4×D61C2×C414yes—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198D411yescharacteristic
94C2×#11758198D411yescharacteristic
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99#14519709—41no—
100#14519701—22no—
101#14519701C2×C414yes—
102#14519699—41no—
103#14519699C2×C414yes—
104#14519697—41no—
105#14519697C2×C414yes—
106#14519689—22no—
107#14519688—41no—
108C8×Dic61—41no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×Q8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4×Q8×C61—22no—
120C4×Q8×C61—41no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#107×C61—41no—
129#103×C61—22no—
130#103×C61C2×C414yes—
131#95×C61—21no—
132C23×D61—21no—
133C23×D61C2×D411yescharacteristic
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61C22×C412yes—
138C2×Q8×C61#4112yes—
139C2×Q8×C61#4112yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153#11758226—28no—
154#11758226—44no—
155#11758226—44no—
156#11758226—44no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—24no—
162C2×D244—21no—
163C2×D244—42no—
164C2×D244—42no—
165C2×D244C22×C412yes—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—24no—
171C2×C4×D61—24no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61C22×C412yes—
177C2×C4×D61#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×C61—42no—
190C4⋊C4×C61—42no—
191C4⋊C4×C61C22×C412yes—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—42no—
195#11758198—22no—
196#11758198—41no—
197#11758198—42no—
198#11758198—42no—
199#11758198—42no—
200#11758198—81no—
201#11758198—81no—
202#11758198—81no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—42no—
209C4⋊Dic61—42no—
210C4⋊Dic61C22×C412yes—
211#11758196—24no—
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×C61C22×D411yescharacteristic
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—22no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61C4×D412yes—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61#11714yescomplemented
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—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
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×Dic61C4×D412yes—
284C2×Dic61AGL(1,ℤ/8)12yes—
285D244—22no—
286D244—22no—
287D244—48no—
288D244—41no—
289D244—41no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—48no—
295C4×D61—48no—
296C8×C61—22no—
297C8×C61—41no—
298#92418—22no—
299#92418—41no—
300#22583—611nomaximal, Sylow
301C22×C61C2×C4×D411yescharacteristic
302C22×C61#54611yescharacteristic
303C22×C61#54611yescharacteristic
304C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C61#57511yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57811yescharacteristic
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—81no—
319C2×D61—81no—
320C2×D61—81no—
321C2×D61—81no—
322C4×C61—24no—
323C4×C61—41no—
324C4×C61—42no—
325C4×C61—41no—
326C4×C61—81no—
327C4×C61—81no—
328C4×C61C2×#11711yescharacteristic
329C4×C61#51811yescharacteristic
330C4×C61#55111yescharacteristic
331C4×C61#55111yescharacteristic
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#1926—611no—
344#1925—611no—
345#1865—611no—
346#1863D6111yescharacteristic, complemented
347#1824—611no—
348#1809—611no—
349#1788—611no—
350#1763—611no—
351C4×#103—611no—
352#1753—611no—
353C2×C61#234211yescharacteristic
354C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
355C2×C61#294711yescharacteristic
356C2×C61#330811yescharacteristic
357C2×C61#331711yescharacteristic
358C2×C61#332711yescharacteristic
359C2×C61#333211yescharacteristic
360D61—42no—
361D61—82no—
362C22×#41—611no—
363C22×M16—611no—
364C2×C4×Q8—611no—
365C2×C4×Q8C2×D6111yescharacteristic
366C2×#117—1221no—
367C22×C4⋊C4—611no—
368C2×#107C2×D6111yescharacteristic
369C2×#106—611no—
370#417—618no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103C2×D6111yescharacteristic
374C2×#103C2×D6111yescharacteristic
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×C22⋊C8—611no—
380C2×#97—611no—
381C2×C4×C8—611no—
382#390—611no—
383#390—611no—
384#390—1221no—
385#390—1221no—
386#389—611no—
387#389C2×D6111yescharacteristic
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—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×Q8C22×D6111yescharacteristic
414C22×D4—611no—
415C22×D4—1221no—
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—22no—
427C4×Q8—41no—
428C4×Q8—1222no—
429C4×Q8—2441no—
430#117—2444no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—614no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—1222no—
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—611no—
445C2×C22⋊C4—611no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
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—41no—
459#106—2441no—
460#103—22no—
461#103—614no—
462#103—1222no—
463#103C4×D6114yes—
464#102—614no—
465#102—614no—
466#102—2441no—
467#102—2441no—
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—611no—
480C24—1221no—
481#41—1228no—
482#41—2444no—
483#41—2444no—
484#41—2444no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8C2×C4×D6112yes—
489C2×Q8#1175822412yes—
490C2×Q8#1175822612yes—
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—611no—
505C22×C4—611no—
506C22×C4—612no—
507C22×C4—612no—
508C22×C4—611no—
509C22×C4—1224no—
510C22×C4—1224no—
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—42no—
544C4⋊C4—42no—
545C4⋊C4—612no—
546C4⋊C4—1224no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—2442no—
553C4⋊C4—2442no—
554C4⋊C4C2×C4×D6112yes—
555C22⋊C4—1222no—
556C22⋊C4—4881no—
557C22⋊C4—4881no—
558C22⋊C4—4881no—
559C22⋊C4—2441no—
560C22⋊C4—2442no—
561C22⋊C4—2442no—
562C22⋊C4—2442no—
563C42—21no—
564C42—21no—
565C42—42no—
566C42—1221no—
567C42—1221no—
568C42—2442no—
569C42—2442no—
570C42—2442no—
571C23—612no—
572C23—1221no—
573C23—1221no—
574C23—1221no—
575C23—1221no—
576C23—1221no—
577C23—1221no—
578C23—1222no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23—2441no—
583C23—2441no—
584C23—2441no—
585C23C2×D4×D6111yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—22no—
589Q8#1451973514yescomplemented
590D4—1222no—
591D4—1222no—
592D4—2448no—
593D4—2441no—
594D4—2441no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—612no—
614C2×C4—612no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
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×C4C2×#1175822411yescharacteristic
640C2×C4C2×#1175822611yescharacteristic
641C2×C4#1451982211yescharacteristic
642C2×C4#1451982311yescharacteristic
643C8—22no—
644C8—41no—
645C8—1222no—
646C8—2441no—
647C22—1222no—
648C22—1222no—
649C22—1222no—
650C22—4881no—
651C22—4881no—
652C22—4881no—
653C22—4881no—
654C22—2442no—
655C22—2442no—
656C22—2442no—
657C22—2441no—
658C22—2441no—
659C22—2441no—
660C22—2441no—
661C22C4×D4×D6111yescharacteristic
662C22#1684039911yescharacteristic
663C22#1684040311yescharacteristic
664C22AGL(1,ℤ/8)×D6111yescharacteristic
665C22#1684056111yescharacteristic
666C22#1684056911yescharacteristic
667C22#1684057511yescharacteristic
668C4—24no—
669C4—42no—
670C4—41no—
671C4—41no—
672C4—81no—
673C4—81no—
674C4—1222no—
675C4—1222no—
676C4—4881no—
677C4—4881no—
678C4—2441no—
679C4—2442no—
680C4C2×#1451973511yescharacteristic
681C4#1684023411yescharacteristic
682C4#1684043511yescharacteristic
683C4#1684043911yescharacteristic
684C2—4882no—
685C2—2442no—
686C2#2144866611yescharacteristic
687C2#2145462411yescharacteristic
688C2#2145465711yescharacteristic
689C2#2145780711yescharacteristic
690C2#2145794211yescharacteristic
691C2#2145809111yescharacteristic
692C2#2145813911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1863C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
cosocleC22×C4×C61C24
hypercenter#1863D61
hypocenterC61#22583
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
2. #21448687C2semidirect
4. #21446515C2semidirect
5. #21446503C2semidirect
6. #21446447C2semidirect
7. #21446438C2semidirect
8. #21446089C2semidirect
9. #21445908C2semidirect
10. #21445853C2semidirect
12. #21445649C2semidirect
13. #103×Dic61C2semidirect
14. #21445601C2semidirect
15. C61×#1863C2semidirect
22. #16839762C4semidirect
242. Q8×C61#117semidirect
346. #1863D61semidirect
406. C61#22583semidirect
589. Q8#14519735semidirect

Maximal subgroups

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

subgroupindex
C2×#168397622
#214486872
#214486572
#214465152
#214465032
#214464472
#214464382
#214460892
#214459082
#214458532
#214457582
#214456492
#103×Dic612
#214456012
C61×#18632
#2258361