Finite Group Databasebeta

← #10897673

Subgroups — #10897673

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78980C211yesmaximal, characteristic, complemented
2#78489C211yesmaximal, characteristic, complemented
3C3×#1938C211yesmaximal, characteristic, complemented
4#70549C211yesmaximal, characteristic, complemented
5#70546C211yesmaximal, characteristic, complemented
6#70391C211yesmaximal, characteristic, complemented
7#70296C211yesmaximal, characteristic, complemented
8#70293C211yesmaximal, characteristic, complemented
9#70013C211yesmaximal, characteristic
10#70002C211yesmaximal, characteristic, complemented
11#69843C211yesmaximal, characteristic, complemented
12#68947C211yesmaximal, characteristic, complemented
13#68936C211yesmaximal, characteristic, complemented
14#68778C211yesmaximal, characteristic, complemented
15#68749C211yesmaximal, characteristic, complemented
16#29387—31nomaximal, Sylow
17#5822C2211yescharacteristic
18#5822C2211yescharacteristic
19#5819C2211yescharacteristic
20#5748C2211yescharacteristic
21#5745C2211yescharacteristic
22C4×#235C2211yescharacteristic
23C3×#424C2211yescharacteristic
24C2×C3×#104C2211yescharacteristic
25C3×#416C2211yescharacteristic
26C3×#408C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5240C2211yescharacteristic
29#5139C2211yescharacteristic
30#5124C2211yescharacteristic
31#5113C2211yescharacteristic
32#5110C2211yescharacteristic
33#5110C2211yescharacteristic
34#5110C2211yescharacteristic
35#5102C2211yescharacteristic
36#5102C2211yescharacteristic
37#5102C2211yescharacteristic
38#5006C2211yescharacteristic
39#5005C2211yescharacteristic
40C2×#805C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4824—22no—
43#4823—22no—
44#4709—22no—
45#4705—22no—
46C3×#341C2211yescharacteristic
47C3×#339C2211yescharacteristic
48C3⋊#338C2211yescharacteristic
49#4559C2211yescharacteristic
50#4559C2211yescharacteristic
51#4556C2211yescharacteristic
52#4552C2211yescharacteristic
53#4533C2211yescharacteristic
54#4531C2211yescharacteristic
55#4523C2211yescharacteristic
56#2956—31no—
57#2923—31no—
58#2103—31no—
59#2098—31no—
60#2053—31no—
61#2017—31no—
62#1938—31no—
63#1938D311yescharacteristic, complemented
64#1906—31no—
65#1906—31no—
66#1881—31no—
67#1881—31no—
68#1868—31no—
69#1868—31no—
70#1841—31no—
71C2×C3×#41C2311yescharacteristic
72C2×#235C2311yescharacteristic
73C2×#235C2311yescharacteristic
74C2×C3×M16C2311yescharacteristic
75C2×C3×M16C2311yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C3×#117C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80#951—21no—
81#951—21no—
82#950—22no—
83#950—41no—
84#949—21no—
85#949—21no—
86#948—22no—
87#948—41no—
88#942—21no—
89#942—21no—
90#932—21no—
91#932—21no—
92C4×#67—22no—
93#926C2311yescharacteristic
94#926C2311yescharacteristic
95C2×#208—41no—
96C2×#208C2311yescharacteristic
97C2×#208C2311yescharacteristic
98C2×#208C2311yescharacteristic
99C2×#208C2311yescharacteristic
100#909—21no—
101#909—21no—
102#908—21no—
103#908—21no—
104#907—22no—
105D3×M16—41no—
106#904—41no—
107#901—41no—
108C4×D12—21no—
109#872—21no—
110C42×D3—21no—
111C4×Dic6—21no—
112C3×C4⋊C8—41no—
113C3×#104—21no—
114C3×#104—21no—
115C3×#103—21no—
116C3×#102—21no—
117C3×C22⋊C8—41no—
118C3×#97—21no—
119C3×#97—21no—
120#834—22no—
121#825—41no—
122#822—22no—
123#821—21no—
124(C2×D3)⋊C8—22no—
125(C2×D3)⋊C8—22no—
126(C2×D3)⋊C8—41no—
127#816—21no—
128Dic3⋊C8—22no—
129Dic3⋊C8—22no—
130Dic3⋊C8—41no—
131#806—22no—
132#805—21no—
133#805—21no—
134#805—41no—
135#803D412yes—
136C4×C3⋊C8D412yes—
137#575—31no—
138#575—31no—
139#575—31no—
140#568—31no—
141#568—31no—
142C4×#41—31no—
143#444—62no—
144#443—62no—
145#433—62no—
146#432—62no—
147#430—31no—
148#424—31no—
149#424—31no—
150#424—31no—
151#424C2×D311yescharacteristic
152#421—31no—
153#421—31no—
154C2×#104—31no—
155C2×#104C2×D311yescharacteristic
156#416—31no—
157#416C2×D311yescharacteristic
158#410—31no—
159#410—31no—
160#408—31no—
161#408—31no—
162#408—31no—
163#408C2×D311yescharacteristic
164C4×M16—31no—
165C2×#97C2×D311yescharacteristic
166#344—31no—
167#344—31no—
168#344—31no—
169#344—31no—
170#341—31no—
171#341C2×D311yescharacteristic
172#339—31no—
173#339C2×D311yescharacteristic
174#338—31no—
175H(3,ℤ/4)—31no—
176C3×#41—21no—
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C2×C3×Q8#4111yescharacteristic
181C2×C3×D4—21no—
182C2×C3×D4#4111yescharacteristic
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2411yescharacteristic
186C2×#67—21no—
187C2×#67—21no—
188#235—21no—
189#235—21no—
190#235—21no—
191#235—21no—
192#235—21no—
193#235—21no—
194#235—21no—
195#235—21no—
196C2×D12#4111yescharacteristic
197C2×D12#4111yescharacteristic
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—22no—
201C2×Dic6#4111yescharacteristic
202C2×Dic6#4111yescharacteristic
203C3×M16—21no—
204C3×M16—21no—
205C3×M16—21no—
206C3×M16—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C3×C4⋊C4—21no—
216C3×C22⋊C4—21no—
217C3×C42—21no—
218C3×C42—21no—
219C3×C42C2×D411yescharacteristic
220C3×C42C2×D411yescharacteristic
221#217—21no—
222#217#4112yes—
223#216—22no—
224#216—41no—
225#215—22no—
226#215—41no—
227#214—22no—
228#214—41no—
229#213—22no—
230#213—41no—
231(C2×D3)⋊C4—22no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234Dic3⋊C4—22no—
235C4×Dic3—22no—
236C4×Dic3—21no—
237C4×Dic3—21no—
238#208—21no—
239#208—21no—
240#208—21no—
241#208—21no—
242#208—21no—
243#208—21no—
244#208—21no—
245#208—21no—
246#208—42no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—42no—
250C2×C3⋊C8C2×D412yes—
251C2×C3⋊C8C2×D412yes—
252Dic12—41no—
253D24—41no—
254#204—41no—
255#204—41no—
256ΓL(1,𝔽25)—42no—
257D3×C8—42no—
258C2×#41—31no—
259C2×#41—31no—
260C2×#41C22×D311yescharacteristic
261#137—61no—
262#137—61no—
263#137—61no—
264#137—61no—
265#137—61no—
266#137—61no—
267AGL(1,ℤ/8)—61no—
268AGL(1,ℤ/8)—61no—
269AGL(1,ℤ/8)—61no—
270AGL(1,ℤ/8)—61no—
271AGL(1,ℤ/8)—61no—
272AGL(1,ℤ/8)—61no—
273#135—62no—
274#135—121no—
275#135—121no—
276#131—62no—
277#131—62no—
278#131—121no—
279#131—121no—
280C2×M16—31no—
281C2×M16—31no—
282C2×M16—31no—
283C2×M16—31no—
284C2×M16—121no—
285C2×M16—121no—
286C2×M16C22×D311yescharacteristic
287C2×M16C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290C4×Q8—61no—
291C4×D4—61no—
292C4×D4—62no—
293#117—31no—
294#117—31no—
295#117—61no—
296#117C22×D311yescharacteristic
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#108—62no—
300C4⋊C8—41no—
301C4⋊C8—62no—
302C4⋊C8—62no—
303C4⋊C8—121no—
304#104—21no—
305#104—21no—
306#104—61no—
307#104—61no—
308#104—121no—
309#104—121no—
310#103—21no—
311#103—61no—
312#102—21no—
313#102—61no—
314#99—62no—
315#99—62no—
316C22⋊C8—41no—
317C22⋊C8—62no—
318C22⋊C8—62no—
319C22⋊C8—121no—
320#97—21no—
321#97—21no—
322#97—32no—
323C4×C8—32no—
324C23×C3—21no—
325C23×C3C2×#4111yescharacteristic
326C22×D3—21no—
327C22×D3—21no—
328C3×Q8—21no—
329C3×Q8—21no—
330C3×D4—21no—
331C3×D4—21no—
332C3×D4—41no—
333C3×D4—41no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—81no—
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4H(5,𝔽2)11yescharacteristic
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347#67—41no—
348#67—41no—
349#67—41no—
350#67—41no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—41no—
355D12—21no—
356D12—21no—
357D12—21no—
358D12—21no—
359C4×D3—41no—
360C4×D3—41no—
361C4×D3—41no—
362C4×D3—41no—
363C4×D3—82no—
364Dic6—21no—
365Dic6—21no—
366Dic6—21no—
367Dic6—21no—
368C3×C8—41no—
369C3×C8—41no—
370C3×C8—41no—
371C3×C8—41no—
372C3⋊C8—22no—
373C3⋊C8—22no—
374C3⋊C8—41no—
375C3⋊C8—41no—
376#41—21no—
377#41—21no—
378#41—21no—
379#41—21no—
380#41—61no—
381#41—61no—
382#41—61no—
383#41—61no—
384#41—61no—
385#41—61no—
386#41—61no—
387#41—61no—
388C2×Q8—31no—
389C2×Q8—31no—
390C2×Q8#23511yescharacteristic
391C2×D4—21no—
392C2×D4—31no—
393C2×D4—31no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4#23511yescharacteristic
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4C23×D311yescharacteristic
403Q16—62no—
404Q16—121no—
405Q16—121no—
406SD16—62no—
407SD16—62no—
408SD16—121no—
409SD16—121no—
410SD16—121no—
411SD16—121no—
412D8—62no—
413D8—121no—
414D8—121no—
415M16—21no—
416M16—21no—
417M16—21no—
418M16—21no—
419M16—61no—
420M16—61no—
421M16—61no—
422M16—61no—
423M16—61no—
424M16—61no—
425M16—61no—
426M16—61no—
427M16—122no—
428M16—122no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—32no—
438C2×C8—32no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—122no—
442C2×C8—122no—
443C4⋊C4—21no—
444C4⋊C4—62no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C22⋊C4—21no—
448C22⋊C4—32no—
449C22⋊C4—62no—
450C22⋊C4—61no—
451C42—21no—
452C42—21no—
453C42—62no—
454C42—61no—
455C42—61no—
456C42D3×D411yescharacteristic
457C42D3×D411yescharacteristic
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—81no—
461C22×C3#52511yescharacteristic
462C22×C3#55311yescharacteristic
463C22×C3#55311yescharacteristic
464C2×D3—42no—
465C2×D3—41no—
466C2×D3—41no—
467C2×D3—81no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—81no—
472C3×C4#52511yescharacteristic
473C3×C4#53511yescharacteristic
474C3×C4#55111yescharacteristic
475C3×C4#55111yescharacteristic
476Dic3—42no—
477Dic3—41no—
478Dic3—41no—
479Dic3—81no—
480C23—21no—
481C23—61no—
482C23—61no—
483C23D3×#4111yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—61no—
487Q8—61no—
488Q8—61no—
489Q8—61no—
490D4—21no—
491D4—21no—
492D4—41no—
493D4—41no—
494D4—61no—
495D4—61no—
496D4—61no—
497D4—61no—
498D4—121no—
499D4—121no—
500D4—121no—
501D4—121no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—62no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—81no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—242no—
518C2×C4C2×#23511yescharacteristic
519C2×C4C2×D3×D411yescharacteristic
520C2×C4#100411yescharacteristic
521C2×C4D3×#4111yescharacteristic
522C2×C4D3×#4111yescharacteristic
523C2×C4#100911yescharacteristic
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—62no—
529C8—62no—
530C8—121no—
531C8—121no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—81no—
536C2×C3#259811yescharacteristic
537D3—81no—
538D3—81no—
539C22—41no—
540C22—41no—
541C22—81no—
542C22—122no—
543C22—121no—
544C22—121no—
545C22—241no—
546C22#554011yescharacteristic
547C22#565611yescharacteristic
548C22#570911yescharacteristic
549C4—41no—
550C4—41no—
551C4—41no—
552C4—81no—
553C4—122no—
554C4—121no—
555C4—121no—
556C4—241no—
557C4#552711yescharacteristic
558C4#554711yescharacteristic
559C4#560711yescharacteristic
560C4#567911yescharacteristic
561C3#2938711yescharacteristic, complemented, Sylow
562C2—21no—
563C2—21no—
564C2—21no—
565C2—81no—
566C2—241no—
567C2—241no—
568C2#7589511yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1938C2
socleC2×C3#2598
prime radical#70013C2
hypercenter#1938D3
hypocenterC3#29387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1938

Complements

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

NH ≅ G/Ntype
1. #78980C2semidirect
2. #78489C2semidirect
3. C3×#1938C2semidirect
4. #70549C2semidirect
5. #70546C2semidirect
6. #70391C2semidirect
7. #70296C2semidirect
8. #70293C2semidirect
10. #70002C2semidirect
11. #69843C2semidirect
12. #68947C2semidirect
13. #68936C2semidirect
14. #68778C2semidirect
15. #68749C2semidirect
63. #1938D3semidirect
561. C3#29387semidirect

Maximal subgroups

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

subgroupindex
#789802
#784892
C3×#19382
#705492
#705462
#703912
#702962
#702932
#700132
#700022
#698432
#689472
#689362
#687782
#687492
#293873