Finite Group Databasebeta

← #29699472

Subgroups — #29699472

Order 15616.

752 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519799C211yesmaximal, characteristic
2#21454487C211yesmaximal, characteristic, complemented
3#21454388C211yesmaximal, characteristic, complemented
4#21446521C212yesmaximal, complemented
5#21446417C212yesmaximal, complemented
6#21446147C212yesmaximal
7#21445836C211yesmaximal, characteristic, complemented
8#21445836C211yesmaximal, characteristic, complemented
9C8×C4⋊Dic61C211yesmaximal, characteristic, complemented
10#21445655C212yesmaximal, complemented
11C61×#1837C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519785C2212yes—
14C4×#11758222C2211yescharacteristic
15C4×#11758222C2211yescharacteristic
16C4×Dic244C411yescharacteristic, complemented
17C4×Dic244C411yescharacteristic, complemented
18C4×#11758190C411yescharacteristic, complemented
19C4×#11758190C411yescharacteristic, complemented
20C4×#11758190C411yescharacteristic, complemented
21C4×#11758190C411yescharacteristic, complemented
22C4×D488C411yescharacteristic, complemented
23C4×D488C411yescharacteristic, complemented
24#16839822—22no—
25#16839818—21no—
26#16839814—21no—
27#16839800C2211yescharacteristic
28#16839790C2212yes—
29#16839787—22no—
30#16839784—21no—
31C2×C4⋊#92418C2211yescharacteristic
32C4⋊C8×D61—21no—
33C2×#14519703C2212yes—
34C2×#14519703C2212yes—
35#16839759C2211yescharacteristic
36#16839759C2211yescharacteristic
37#16839730C2212yes—
38#16839720C2212yes—
39C2×C8×Dic61C2212yes—
40C4×C4⋊Dic61C2211yescharacteristic
41#16839523C2212yes—
42#16839523C2212yes—
43#16839516C2212yes—
44C61×#423C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4×C8×C61C2211yescharacteristic
47C61×#341C2212yes—
48C61×#341C2212yes—
49C2×#11758222C2311yescharacteristic
50C2×#11758222C2311yescharacteristic
51#14519799—22no—
52#14519799—22no—
53C2×Dic244C2×C411yescharacteristic
54C2×#11758190C2×C411yescharacteristic
55C2×#11758190C2×C411yescharacteristic
56C2×D488C2×C411yescharacteristic
57#14519785—24no—
58#14519785—42no—
59C2×#11758189—22no—
60C22×#92418C2312yes—
61C2×C8×D61—22no—
62C4×Dic122C2×C411yescharacteristic
63C4×Dic122C2×C411yescharacteristic
64C4×Dic122C2×C411yescharacteristic
65C4×Dic122C2×C411yescharacteristic
66C4×D244C2×C411yescharacteristic
67C4×D244C2×C411yescharacteristic
68C4×D244C2×C411yescharacteristic
69C4×D244C2×C411yescharacteristic
70C4×C22⋊D61—21no—
71C4×C22⋊D61—21no—
72C4×C22⋊D61—21no—
73C4×C22⋊D61—21no—
74#14519736C2311yescharacteristic
75#14519736C2311yescharacteristic
76#14519734—21no—
77#14519734—21no—
78C2×C4⋊Dic61—21no—
79C2×C4×Dic61C2312yes—
80C42×D61—21no—
81C42×D61—21no—
82#14519711C2×C411yescharacteristic
83#14519711C2×C411yescharacteristic
84#14519709C2×C411yescharacteristic
85#14519709C2×C411yescharacteristic
86C4⋊#92418D411yescharacteristic
87C4⋊#92418D411yescharacteristic
88C4⋊#92418D412yes—
89#14519706—22no—
90#14519706—22no—
91#14519703—24no—
92#14519703—42no—
93#14519702C2×C411yescharacteristic
94#14519702C2×C411yescharacteristic
95#14519702C2×C411yescharacteristic
96#14519702C2×C411yescharacteristic
97#14519699C2×C411yescharacteristic
98#14519699C2×C411yescharacteristic
99#14519699C2×C411yescharacteristic
100#14519699C2×C411yescharacteristic
101#14519689—22no—
102#14519689—22no—
103#14519687—22no—
104C8×Dic61—22no—
105C8×Dic61—42no—
106#14519684—21no—
107C2×M16×C61C2312yes—
108C2×M16×C61C2312yes—
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112C4⋊C8×C61—21no—
113C4⋊C8×C61—21no—
114C4⋊C8×C61—21no—
115C4⋊C8×C61—21no—
116C4×C8×C61C2×C411yescharacteristic
117C4×C8×C61C2×C411yescharacteristic
118C4×C8×C61C2×C411yescharacteristic
119C4×C8×C61C2×C411yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—21no—
124C2×C22⋊D61—21no—
125C22×Dic61—22no—
126#11758222—21no—
127#11758222—21no—
128#11758222—21no—
129#11758222—21no—
130#11758222—21no—
131#11758222—21no—
132#11758222#4112yes—
133#11758222#4112yes—
134C2×D244C22×C411yescharacteristic
135C2×D244C22×C411yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×Dic122C22×C411yescharacteristic
143C2×Dic122C22×C411yescharacteristic
144M16×C61—22no—
145M16×C61—22no—
146M16×C61—22no—
147M16×C61—22no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61C22×C411yescharacteristic
155C2×C8×C61C22×C411yescharacteristic
156C2×C8×C61C22×C411yescharacteristic
157C2×C8×C61C22×C411yescharacteristic
158C42×C61C22×C411yescharacteristic
159C42×C61C22×C411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C22⋊Dic61—21no—
163C22⋊Dic61—21no—
164#11758198—21no—
165#11758198—21no—
166#11758198—21no—
167#11758198—21no—
168C4⋊Dic61—24no—
169C4⋊Dic61C22×C411yescharacteristic
170C4⋊Dic61C22×C411yescharacteristic
171C4⋊Dic61C22×C411yescharacteristic
172C4⋊Dic61C22×C411yescharacteristic
173#11758196—21no—
174#11758196—21no—
175#11758196—21no—
176#11758196—21no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—22no—
184C4×Dic61—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418C2×D412yes—
188C2×#92418C2×D412yes—
189Dic244—22no—
190D488—22no—
191#11758190—22no—
192#11758190—22no—
193#11758189—44no—
194C8×D61—44no—
195C23×C61C2×#4111yescharacteristic
196C22×D61—21no—
197C22×D61—21no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61C4×D411yescharacteristic
201C2×C4×C61C4×D411yescharacteristic
202C2×C4×C61C4×D411yescharacteristic
203C2×C4×C61C4×D411yescharacteristic
204C2×C4×C61C23×C411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61H(5,𝔽2)11yescharacteristic
209C2×C4×C612−1+411yescharacteristic
210C22⋊D61—22no—
211C22⋊D61—22no—
212C22⋊D61—41no—
213C22⋊D61—41no—
214C2×Dic61—21no—
215C2×Dic61—24no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223D244—21no—
224D244—21no—
225D244—21no—
226D244#11712yes—
227C4×D61—22no—
228C4×D61—22no—
229C4×D61—41no—
230C4×D61—41no—
231C4×D61—42no—
232C4×D61—42no—
233C4×D61—81no—
234C4×D61—81no—
235Dic122—21no—
236Dic122—21no—
237Dic122—21no—
238Dic122#11712yes—
239C8×C61—21no—
240C8×C61—21no—
241C8×C61—21no—
242C8×C61—21no—
243C8×C61—42no—
244C8×C61—42no—
245#92418—22no—
246#92418—22no—
247#22783—611nomaximal, Sylow
248C22×C61—21no—
249C22×C61—21no—
250C22×C61C2×C4×D411yescharacteristic
251C22×C61#55211yescharacteristic
252C22×C61#55311yescharacteristic
253C2×D61—22no—
254C2×D61—41no—
255C2×D61—41no—
256C2×D61—41no—
257C2×D61—41no—
258C2×D61—41no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61#44312yes—
263C4×C61C2×#11711yescharacteristic
264C4×C61#52011yescharacteristic
265C4×C61#54611yescharacteristic
266C4×C61#54811yescharacteristic
267Dic61—22no—
268Dic61—44no—
269Dic61—41no—
270Dic61—41no—
271Dic61—41no—
272Dic61—41no—
273Dic61—41no—
274C4×#135—611no—
275#2925—611no—
276#2906—611no—
277#1937—612no—
278#1922—612no—
279#1876—612no—
280#1837D6111yescharacteristic, complemented
281#1806—611no—
282#1806—611no—
283C8×C4⋊C4—611no—
284#1764—612no—
285C2×C61—21no—
286C2×C61—21no—
287C2×C61#235211yescharacteristic
288C2×C61C2×#44311yescharacteristic
289C2×C61#295711yescharacteristic
290D61—42no—
291D61—81no—
292C2×#135—611no—
293C2×#131—612no—
294C4×#41—611no—
295C4×#41—611no—
296C4×Q16—611no—
297C4×Q16—611no—
298C4×SD16—611no—
299C4×SD16—611no—
300C4×SD16—611no—
301C4×SD16—611no—
302C4×D8—611no—
303C4×D8—611no—
304#433—1222no—
305#432—1221no—
306#431—1221no—
307#427—611no—
308#424—612no—
309#424—1222no—
310#423—1221no—
311#423C2×D6111yescharacteristic
312#423C2×D6111yescharacteristic
313C2×C4⋊C8—611no—
314C2×C4⋊C8—1221no—
315C2×#104—612no—
316C2×#104—612no—
317#416—611no—
318#416—611no—
319#408—612no—
320#405—612no—
321C2×C4×C8—612no—
322C2×C4×C8C2×D6111yescharacteristic
323C4×C4⋊C4—611no—
324#341C2×D6112yes—
325#341C2×D6112yes—
326#339—612no—
327#339—612no—
328#336—612no—
329C61#2278311yescharacteristic, complemented, Sylow
330C2×#41—611no—
331C2×#41—611no—
332#135—1222no—
333#135—1222no—
334C2×Q16—611no—
335C2×SD16—611no—
336C2×SD16—611no—
337C2×D8—611no—
338#131—1224no—
339#131—2442no—
340C2×M16—1222no—
341C2×M16C22×D6112yes—
342C2×M16C22×D6112yes—
343C22×C8—612no—
344C22×C8—1222no—
345C22×C8C22×D6111yescharacteristic
346C22×C8C22×D6111yescharacteristic
347C4×Q8—611no—
348C4×Q8—611no—
349C4×Q8—611no—
350C4×Q8—611no—
351C4×D4—611no—
352C4×D4—611no—
353C4×D4—611no—
354C4×D4—611no—
355C4×D4—1221no—
356C4×D4—1221no—
357C4×D4—1221no—
358C4×D4—1221no—
359#117—611no—
360#117—611no—
361#117—1221no—
362#117—1221no—
363C2×C4⋊C4—1221no—
364C2×C42—612no—
365C2×C42—1221no—
366C2×C42—1221no—
367C2×C42C22×D6111yescharacteristic
368#107—611no—
369#107—611no—
370#106—611no—
371#106—611no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—21no—
375C4⋊C8—21no—
376C4⋊C8—611no—
377C4⋊C8—611no—
378C4⋊C8—612no—
379C4⋊C8—1222no—
380C4⋊C8—1222no—
381#104—1224no—
382#104—2442no—
383#103—611no—
384#103—611no—
385#103—611no—
386#103—611no—
387#102—611no—
388#102—611no—
389#102—611no—
390#102—611no—
391C22⋊C8—1222no—
392C22⋊C8—1222no—
393#97—1222no—
394C4×C8—1222no—
395C4×C8—2442no—
396C4×C8C4×D6111yescharacteristic
397C4×C8C4×D6111yescharacteristic
398C4×C8C4×D6111yescharacteristic
399C4×C8C4×D6111yescharacteristic
400#95—1221no—
401#41—612no—
402#41—612no—
403#41—1221no—
404#41—1221no—
405#41—1221no—
406#41—1221no—
407#41—1221no—
408#41—1221no—
409C2×Q8—611no—
410C2×Q8—611no—
411C2×D4—611no—
412C2×D4—611no—
413C2×D4—1221no—
414C2×D4—1221no—
415C22×C4—1221no—
416C22×C4—1222no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4#1175822611yescharacteristic
423C22×C4#1175822611yescharacteristic
424C22×C4C23×D6111yescharacteristic
425Q16—1222no—
426SD16—1222no—
427SD16—1222no—
428D8—1222no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—2444no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—612no—
441C2×C8—612no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—2444no—
445C2×C8C2×C4×D6111yescharacteristic
446C2×C8C2×C4×D6111yescharacteristic
447C2×C8C2×C4×D6111yescharacteristic
448C2×C8C2×C4×D6111yescharacteristic
449C4⋊C4—611no—
450C4⋊C4—611no—
451C4⋊C4—611no—
452C4⋊C4—611no—
453C4⋊C4—1224no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C22⋊C4—1221no—
459C22⋊C4—1221no—
460C22⋊C4—1221no—
461C22⋊C4—1221no—
462C22⋊C4—1221no—
463C22⋊C4—1221no—
464C42—1221no—
465C42—1221no—
466C42—1221no—
467C42—1221no—
468C42—1221no—
469C42—1221no—
470C42—1222no—
471C42—1222no—
472C42C2×C4×D6111yescharacteristic
473C42C2×C4×D6111yescharacteristic
474C42D4×D6111yescharacteristic
475C42D4×D6111yescharacteristic
476C23—1221no—
477C23—1221no—
478C23C2×#1175822611yescharacteristic
479Q8—612no—
480Q8—1221no—
481Q8—1221no—
482Q8—1221no—
483D4—612no—
484D4—1222no—
485D4—1222no—
486D4—1221no—
487D4—1221no—
488D4—1221no—
489D4—2441no—
490D4—2441no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—1221no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1224no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—4881no—
503C2×C4—4881no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4#1451973911yescharacteristic
511C2×C4#1451973911yescharacteristic
512C2×C4#1451973911yescharacteristic
513C2×C4#1451973911yescharacteristic
514C2×C4C22×C4×D6111yescharacteristic
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4#41×D6111yescharacteristic
517C2×C4#41×D6111yescharacteristic
518C2×C4#1451982311yescharacteristic
519C2×C4#1451982411yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—42no—
525C8—42no—
526C8—1222no—
527C8—1222no—
528C22—21no—
529C22—21no—
530C22—1222no—
531C22—2441no—
532C22—2441no—
533C22—2441no—
534C22—2441no—
535C22—2441no—
536C22C2×#1451973911yescharacteristic
537C22#1684045011yescharacteristic
538C22#1684046511yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—1222no—
543C4—2444no—
544C4—2441no—
545C4—2441no—
546C4—2441no—
547C4—2441no—
548C4—2441no—
549C4#1683987812yes—
550C4#117×D6111yescharacteristic
551C4#1684024211yescharacteristic
552C4#1684040511yescharacteristic
553C4#1684042011yescharacteristic
554C2—21no—
555C2—21no—
556C2—4881no—
557C2—2442no—
558C2#2144874511yescharacteristic
559C2C2×#1683987811yescharacteristic
560C2#2145473511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1837C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519799C22
cosocleC22×C4×C61C24
hypercenter#1837D61
hypocenterC61#22783
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1837

Complements

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

NH ≅ G/Ntype
2. #21454487C2semidirect
3. #21454388C2semidirect
4. #21446521C2semidirect
5. #21446417C2semidirect
7. #21445836C2semidirect
8. #21445836C2semidirect
9. C8×C4⋊Dic61C2semidirect
10. #21445655C2semidirect
11. C61×#1837C2semidirect
16. C4×Dic244C4semidirect
17. C4×Dic244C4semidirect
18. C4×#11758190C4semidirect
19. C4×#11758190C4semidirect
20. C4×#11758190C4semidirect
21. C4×#11758190C4semidirect
22. C4×D488C4semidirect
23. C4×D488C4semidirect
280. #1837D61semidirect
329. C61#22783semidirect

Maximal subgroups

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

subgroupindex
C4×#145197992
#214544872
#214543882
#214465212
#214464172
#214461472
#214458362
C8×C4⋊Dic612
#214456552
C61×#18372
#2278361