Finite Group Databasebeta

← #29669999

Subgroups — #29669999

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×#11758201C211yesmaximal, characteristic, complemented
2#21458475C211yesmaximal, characteristic
3#21457697C211yesmaximal, characteristic, complemented
4#21445480C211yesmaximal, characteristic, complemented
5#21445401C211yesmaximal, characteristic, complemented
6#21445233C211yesmaximal, characteristic, complemented
7#21445225C211yesmaximal, characteristic
8#21445203C211yesmaximal, characteristic, complemented
9#21445005C211yesmaximal, characteristic, complemented
10#21444959C211yesmaximal, characteristic, complemented
11#21444952C211yesmaximal, characteristic, complemented
12#21444920C211yesmaximal, characteristic
13#21444653C211yesmaximal, characteristic, complemented
14#21444638C211yesmaximal, characteristic, complemented
15C61×#3375C211yesmaximal, characteristic, complemented
16#16840472C2211yescharacteristic
17Q8×D244C2211yescharacteristic
18#16840158—21no—
19#16840158C2211yescharacteristic
20#16840141C2211yescharacteristic
21#16840116—21no—
22#16840113C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840099—21no—
26#16840062—21no—
27#16840057C2211yescharacteristic
28#16840057C2211yescharacteristic
29#16840049—21no—
30#16840049C2211yescharacteristic
31#16840047—21no—
32#16840047C2211yescharacteristic
33#16840002—21no—
34#16839974—21no—
35#16839972C2211yescharacteristic
36#16839954C2211yescharacteristic
37#16839916—21no—
38Q8×#92418C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C4×#11758201—21no—
41C4×#11758201C2211yescharacteristic
42C4×#11758199—21no—
43C8×D244C2211yescharacteristic
44#103×D61—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#559C2211yescharacteristic
57Q82×C61C2211yescharacteristic
58C61×#489—21no—
59C61×#486C2211yescharacteristic
60C61×#477C2211yescharacteristic
61C61×#474—21no—
62C61×#464—21no—
63C61×#460C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#440—21no—
66C4×Q16×C61C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×#11758202—21no—
69C2×#11758202—41no—
70C2×#11758201D411yescharacteristic
71C2×#11758199—21no—
72C2×C8×D61—21no—
73#14519777—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519769—21no—
77#14519757—21no—
78#14519757—41no—
79#14519755—41no—
80#14519753—21no—
81C4×D244—21no—
82C4×D244—21no—
83C4×D244C2311yescharacteristic
84C4⋊C4×D61—21no—
85#14519710—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708D411yescharacteristic
92C4⋊#92418—21no—
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519702—21no—
98#14519702—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—41no—
108#14519697—21no—
109#14519697D411yescharacteristic
110#14519689—21no—
111C4×#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×Q16×C61D411yescharacteristic
116C2×SD16×C61—21no—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—41no—
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61C2311yescharacteristic
122#124×C61—21no—
123#124×C61—21no—
124#123×C61—41no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132#107×C61—21no—
133#107×C61D411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61D411yescharacteristic
138#103×C61D411yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—41no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D411yescharacteristic
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C22×C4×C61—41no—
153Q8×D61—42no—
154Q8×D61—81no—
155C2×D244C2×D411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—41no—
162Q16×C61—22no—
163SD16×C61—42no—
164C2×C8×C61—21no—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C22⋊C4×C61—41no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—41no—
184C42×C61C2411yescharacteristic
185#11758202—81no—
186#11758202—82no—
187#11758201—22no—
188#11758199—42no—
189#11758198—21no—
190#11758198—41no—
191#11758198—41no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—41no—
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—41no—
199#11758196—41no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C8×D61—81no—
207C23×C61—41no—
208C22×D61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—81no—
215Q8×C61—161no—
216Q8×C61#13512yes—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—81no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—161no—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236D244—21no—
237D244#13712yes—
238C4×D61—41no—
239C4×D61—41no—
240C4×D61—82no—
241C4×D61—161no—
242Dic122—41no—
243Dic122—41no—
244Dic122—82no—
245Dic122—81no—
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—81no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61D4211yescharacteristic
260C2×D61—22no—
261C2×D61—41no—
262C2×D61—41no—
263C4×C61—22no—
264C4×C61—21no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—81no—
271C4×C61C2×#13511yescharacteristic
272C4×C61C2×#13711yescharacteristic
273C4×C61#57711yescharacteristic
274C4×C61#57811yescharacteristic
275Dic61—41no—
276Dic61—81no—
277Dic61—81no—
278Q8×SD16—611no—
279#3375D6111yescharacteristic, complemented
280#3361—611no—
281#3299—611no—
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—161no—
294C2×C61#327011yescharacteristic
295C2×C61#328211yescharacteristic
296C2×C61#328811yescharacteristic
297D61—42no—
298#559C2×D6111yescharacteristic
299Q82C2×D6111yescharacteristic
300#554—611no—
301D4×Q8—611no—
302#498—611no—
303#498—1221no—
304#494—611no—
305#489—21no—
306#488—1221no—
307#487—611no—
308#486—611no—
309#486—611no—
310#486C2×D6111yescharacteristic
311#484—1221no—
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—21no—
321#467—1221no—
322#464—21no—
323#463—1221no—
324#462—611no—
325#460C2×D6111yescharacteristic
326#459—611no—
327#451—1221no—
328#446C2×D6111yescharacteristic
329C8×Q8—611no—
330C8×Q8—611no—
331#440—21no—
332C4×Q16C2×D6111yescharacteristic
333C4×SD16—611no—
334C4×SD16—1221no—
335C4×D8—1221no—
336C8×D4—611no—
337C2×#103—1221no—
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—1221no—
351C2×Q16—1221no—
352C2×Q16—2441no—
353C2×Q16C22⋊D6111yescharacteristic
354C2×SD16—21no—
355C2×SD16—611no—
356C2×D8—1221no—
357C22×C8—1221no—
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—41no—
361C4⋊Q8—611no—
362C4⋊Q8—611no—
363C4⋊Q8—1221no—
364C4⋊Q8C22×D6111yescharacteristic
365#127C22×D6111yescharacteristic
366#125—1221no—
367#124—21no—
368#124—21no—
369#123—41no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372C22⋊Q8—2441no—
373C22⋊Q8—2441no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8C22×D6111yescharacteristic
377C4×Q8C22×D6111yescharacteristic
378C4×Q8C22×D6111yescharacteristic
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—611no—
382C4×D4—1221no—
383C4×D4—1221no—
384C2×C4⋊C4—1221no—
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—21no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8C22×D6111yescharacteristic
401#103—21no—
402#103—611no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103C22⋊D6111yescharacteristic
410#103C22⋊D6111yescharacteristic
411#102—21no—
412#102—21no—
413#102—21no—
414#102—611no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—2441no—
420C22⋊C8—1221no—
421#97—21no—
422C4×C8—611no—
423C4×C8—611no—
424C4×C8—1221no—
425C4×C8—1221no—
426C4×C8C22×D6111yescharacteristic
427C2×Q8—21no—
428C2×Q8—41no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—4881no—
432C2×Q8—2441no—
433C2×Q8—2442no—
434C2×Q8D4×D6111yescharacteristic
435C2×Q8C2×C22⋊D6111yescharacteristic
436C2×Q8C2×C22⋊D6111yescharacteristic
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—611no—
441C22×C4—41no—
442C22×C4—1221no—
443C22×C4—2441no—
444C22×C4—2441no—
445Q16—22no—
446Q16—4881no—
447Q16—4882no—
448SD16—42no—
449SD16—1222no—
450D8—2442no—
451C2×C8—21no—
452C2×C8—611no—
453C2×C8—611no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—4881no—
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—41no—
467C4⋊C4—41no—
468C4⋊C4—611no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4D4×D6111yescharacteristic
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—1221no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—41no—
489C42C23×D6111yescharacteristic
490C23—41no—
491C23—1221no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—81no—
498Q8—161no—
499Q8—4881no—
500Q8—4882no—
501Q8—2441no—
502Q8—2441no—
503Q8#1451979712yes—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—81no—
509D4—612no—
510D4—1221no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—161no—
519C2×C4—1221no—
520C2×C4—9761no—
521C2×C4—4882no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
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—81no—
539C22—81no—
540C22—81no—
541C22—1222no—
542C22—2441no—
543C22—2441no—
544C22D4×C22⋊D6111yescharacteristic
545C4—22no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—81no—
553C4—4881no—
554C4—4881no—
555C4—2441no—
556C4C2×#1451979711yescharacteristic
557C4#137×D6111yescharacteristic
558C4#1684057011yescharacteristic
559C4#1684057811yescharacteristic
560C2—161no—
561C2—2442no—
562C2#2145738311yescharacteristic
563C2#2145752711yescharacteristic
564C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3375C2
socleC22×C61D42
prime radical#16839954C22
hypercenter#3375D61
hypocenterC61#19798
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3375

Complements

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

NH ≅ G/Ntype
1. Q8×#11758201C2semidirect
3. #21457697C2semidirect
4. #21445480C2semidirect
5. #21445401C2semidirect
6. #21445233C2semidirect
8. #21445203C2semidirect
9. #21445005C2semidirect
10. #21444959C2semidirect
11. #21444952C2semidirect
13. #21444653C2semidirect
14. #21444638C2semidirect
15. C61×#3375C2semidirect
279. #3375D61semidirect
349. C61#19798semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582012
#214584752
#214576972
#214454802
#214454012
#214452332
#214452252
#214452032
#214450052
#214449592
#214449522
#214449202
#214446532
#214446382
C61×#33752
#1979861