Finite Group Databasebeta

← #29670005

Subgroups — #29670005

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758200C211yesmaximal, characteristic, complemented
2#21458631C211yesmaximal, characteristic, complemented
3#21458479C211yesmaximal, characteristic
4#21445476C211yesmaximal, characteristic, complemented
5#21445398C211yesmaximal, characteristic, complemented
6#21445237C211yesmaximal, characteristic, complemented
7#21445228C211yesmaximal, characteristic
8#21445199C211yesmaximal, characteristic, complemented
9#21445003C211yesmaximal, characteristic, complemented
10#21444958C211yesmaximal, characteristic, complemented
11#21444953C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic
13#21444655C211yesmaximal, characteristic, complemented
14#21444638C211yesmaximal, characteristic, complemented
15C61×#3299C211yesmaximal, characteristic, complemented
16#16840497C2211yescharacteristic
17Q8×Dic122C2211yescharacteristic
18#16840158—21no—
19#16840158C2211yescharacteristic
20#16840141C2211yescharacteristic
21#16840123—21no—
22#16840119—21no—
23#16840110C2211yescharacteristic
24#16840110C2211yescharacteristic
25#16840109C2211yescharacteristic
26#16840059—21no—
27#16840057C2211yescharacteristic
28#16840054C2211yescharacteristic
29#16840052—21no—
30#16840052C2211yescharacteristic
31#16840044—21no—
32#16840044C2211yescharacteristic
33#16840041—21no—
34#16839986—21no—
35#16839977—21no—
36#16839962C2211yescharacteristic
37#16839957C2211yescharacteristic
38#16839892C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40Q8×#92418C2211yescharacteristic
41#16839865—21no—
42C4×#11758200—21no—
43C4×#11758200C2211yescharacteristic
44C4×#11758199—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#554C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#484—21no—
60C61×#477C2211yescharacteristic
61C61×#467—21no—
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C4×Q16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#103×C61—21no—
67C2×#11758202—41no—
68C2×#11758200D411yescharacteristic
69C2×#11758201—21no—
70C2×#11758199—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519776—21no—
74#14519776—41no—
75#14519774C2311yescharacteristic
76#14519766—21no—
77#14519765—21no—
78#14519762—41no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122C2311yescharacteristic
82C4×D244—21no—
83#14519739—21no—
84#14519710—21no—
85#14519710—21no—
86#14519708—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708D411yescharacteristic
91C4⋊#92418—21no—
92C4⋊#92418—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—21no—
98#14519700—21no—
99#14519700—21no—
100#14519700D411yescharacteristic
101#14519699—21no—
102#14519699—21no—
103#14519698—21no—
104#14519698D411yescharacteristic
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519687—21no—
111C4×#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C2×Q16×C61—21no—
117C2×Q16×C61D411yescharacteristic
118C22×C8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#125×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C22⋊Q8×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—42no—
148C2×Q8×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154C22×C4×C61—41no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×C4×D61—41no—
159C2×Dic122—21no—
160C2×Dic122—41no—
161C2×Dic122C2×D411yescharacteristic
162Q16×C61—22no—
163Q16×C61—81no—
164C2×C8×C61—21no—
165C2×C8×C61—81no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61C2411yescharacteristic
186#11758202—82no—
187#11758201—42no—
188#11758200—22no—
189#11758199—42no—
190#11758198—41no—
191#11758198—41no—
192#11758198—41no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—21no—
198#11758196—41no—
199C4×Dic61—21no—
200C4×Dic61—41no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C23×C61—21no—
208C22×D61—41no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—82no—
215Q8×C61—81no—
216Q8×C61#13512yes—
217D4×C61—21no—
218D4×C61#13712yes—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—82no—
229C2×C4×C61—161no—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—41no—
236D244—41no—
237D244—41no—
238D244—41no—
239D244—41no—
240D244—81no—
241C4×D61—161no—
242Dic122—21no—
243Dic122—21no—
244Dic122—81no—
245Dic122—161no—
246C8×C61—21no—
247C8×C61—21no—
248C8×C61—41no—
249#92418—21no—
250#92418—21no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#19798—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61D4211yescharacteristic
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C4×C61—22no—
264C4×C61—21no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—81no—
270C4×C61—81no—
271C4×C61C2×#13511yescharacteristic
272C4×C61C2×#13711yescharacteristic
273C4×C61#57711yescharacteristic
274C4×C61#57811yescharacteristic
275Dic61—41no—
276Dic61—41no—
277Dic61—81no—
278Q8×SD16—611no—
279#3375—611no—
280#3361—611no—
281#3299D6111yescharacteristic, complemented
282#1720—611no—
283#1704—611no—
284#1678—611no—
285#1677—611no—
286#1672—611no—
287#1640—611no—
288#1633—611no—
289#1632—611no—
290#1627—611no—
291#1582—611no—
292#1579—611no—
293C2×C61—42no—
294C2×C61#327011yescharacteristic
295C2×C61#328211yescharacteristic
296C2×C61#328811yescharacteristic
297D61—161no—
298#559—611no—
299Q82—611no—
300#554C2×D6111yescharacteristic
301D4×Q8C2×D6111yescharacteristic
302#498—611no—
303#498—1221no—
304#494—611no—
305#489—1221no—
306#488—1221no—
307#487C2×D6111yescharacteristic
308#486—611no—
309#486—611no—
310#486—611no—
311#484—21no—
312#478—1221no—
313#477—611no—
314#477—611no—
315#477C2×D6111yescharacteristic
316#476—611no—
317#476—1221no—
318#475—611no—
319#475—1221no—
320#474—1221no—
321#467—21no—
322#464—1221no—
323#463—1221no—
324#462C2×D6111yescharacteristic
325#460—611no—
326#459—611no—
327#451—21no—
328#446—611no—
329C8×Q8—611no—
330C8×Q8—611no—
331#440—1221no—
332C4×Q16C2×D6111yescharacteristic
333C4×SD16—611no—
334C4×SD16—1221no—
335C4×D8—1221no—
336C8×D4C2×D6111yescharacteristic
337C2×#103—21no—
338#334—611no—
339#334—611no—
340#333—611no—
341#333—611no—
342#332—611no—
343#331—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346Q8⋊C8—611no—
347D4⋊C8—611no—
348#322—611no—
349C61#1979811yescharacteristic, complemented, Sylow
350C22×Q8—21no—
351C2×Q16—21no—
352C2×Q16—2441no—
353C2×Q16C22⋊D6111yescharacteristic
354C2×SD16—611no—
355C2×SD16—1221no—
356C2×D8—1221no—
357C22×C8—21no—
358C4⋊Q8—21no—
359C4⋊Q8—611no—
360C4⋊Q8—1221no—
361C4⋊Q8—1221no—
362C4⋊Q8—2441no—
363C4⋊Q8C22×D6111yescharacteristic
364C4⋊Q8C22×D6111yescharacteristic
365#127—611no—
366#125—21no—
367#124—1221no—
368#124—1221no—
369#123—2441no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—41no—
373C22⋊Q8—41no—
374C4×Q8—611no—
375C4×Q8—1221no—
376C4×Q8—1221no—
377C4×Q8C22×D6111yescharacteristic
378C4×Q8C22×D6111yescharacteristic
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—1221no—
382C4×D4—1221no—
383C4×D4C22×D6111yescharacteristic
384C2×C4⋊C4—21no—
385#107—21no—
386#107—1221no—
387#107—1221no—
388#107C22⋊D6111yescharacteristic
389#106—611no—
390#106—1221no—
391#106—1221no—
392#106—1221no—
393#106—1221no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8C22×D6111yescharacteristic
401#103—21no—
402#103—21no—
403#103—21no—
404#103—21no—
405#103—611no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103C22⋊D6111yescharacteristic
410#103C22⋊D6111yescharacteristic
411#102—611no—
412#102—1221no—
413#102—1221no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—2441no—
420C22⋊C8—21no—
421#97—1221no—
422C4×C8—611no—
423C4×C8—611no—
424C4×C8—1221no—
425C4×C8—1221no—
426C4×C8C22×D6111yescharacteristic
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—41no—
430C2×Q8—42no—
431C2×Q8—81no—
432C2×Q8—611no—
433C2×Q8—1221no—
434C2×Q8—2441no—
435C2×Q8C2×C22⋊D6111yescharacteristic
436C2×Q8C2×C22⋊D6111yescharacteristic
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4D4×D6111yescharacteristic
441C22×C4—21no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—2441no—
445Q16—22no—
446Q16—81no—
447Q16—4882no—
448SD16—1222no—
449SD16—2442no—
450D8—2442no—
451C2×C8—21no—
452C2×C8—81no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8C2×C22⋊D6111yescharacteristic
460C2×C8C2×C22⋊D6111yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—611no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4D4×D6111yescharacteristic
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C22⋊C4—21no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—2441no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C42—21no—
486C42—21no—
487C42—1221no—
488C42—2441no—
489C42C23×D6111yescharacteristic
490C23—21no—
491C23—2441no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—41no—
496Q8—41no—
497Q8—82no—
498Q8—81no—
499Q8—1221no—
500Q8—1221no—
501Q8—9761no—
502Q8—4881no—
503Q8#1451979712yes—
504D4—21no—
505D4—4881no—
506D4—2441no—
507D4—2441no—
508D4—2441no—
509D4—2441no—
510D4#1451980912yes—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—82no—
521C2×C4—161no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—9761no—
525C2×C4—2441no—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4#1451982211yescharacteristic
529C8—21no—
530C8—21no—
531C8—41no—
532C8—1221no—
533C8—1221no—
534C8—2441no—
535C8—2441no—
536C8—2441no—
537C8—2441no—
538C22—22no—
539C22—41no—
540C22—41no—
541C22—4881no—
542C22—4881no—
543C22—4881no—
544C22D4×C22⋊D6111yescharacteristic
545C4—22no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—81no—
552C4—81no—
553C4—4881no—
554C4—2441no—
555C4—2441no—
556C4C2×#1451979711yescharacteristic
557C4C2×#1451980911yescharacteristic
558C4#1684057311yescharacteristic
559C4#1684057511yescharacteristic
560C2—42no—
561C2—9761no—
562C2#2145738911yescharacteristic
563C2#2145752211yescharacteristic
564C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3299C2
socleC22×C61D42
prime radical#16839957C22
hypercenter#3299D61
hypocenterC61#19798
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3299

Complements

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

NH ≅ G/Ntype
1. Q8×#11758200C2semidirect
2. #21458631C2semidirect
4. #21445476C2semidirect
5. #21445398C2semidirect
6. #21445237C2semidirect
8. #21445199C2semidirect
9. #21445003C2semidirect
10. #21444958C2semidirect
11. #21444953C2semidirect
13. #21444655C2semidirect
14. #21444638C2semidirect
15. C61×#3299C2semidirect
281. #3299D61semidirect
349. C61#19798semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582002
#214586312
#214584792
#214454762
#214453982
#214452372
#214452282
#214451992
#214450032
#214449582
#214449532
#214449222
#214446552
#214446382
C61×#32992
#1979861