Finite Group Databasebeta

← #10764034

Subgroups — #10764034

Order 768.

1248 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86195C211yesmaximal, characteristic
2#79293C211yesmaximal, characteristic, complemented
3#79051C211yesmaximal, characteristic, complemented
4C3×#1545C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic, complemented
6#70773C211yesmaximal, characteristic, complemented
7#70505C211yesmaximal, characteristic, complemented
8#68387C212yesmaximal, complemented
9#68385C212yesmaximal, complemented
10#68385C212yesmaximal, complemented
11D3×#331C212yesmaximal, complemented
12#17770—31nomaximal, Sylow
13C2×#1004C412yes—
14C2×C3×#127C2211yescharacteristic
15#5805C412yes—
16D4×#67—22no—
17D4×#67—22no—
18#5718—21no—
19#5718—21no—
20#5717—21no—
21#5717—21no—
22#5716C2212yes—
23#5715C414yes—
24D3×#127C2212yes—
25#5477C2211yescharacteristic
26C4×#235C2211yescharacteristic
27C3×#431C2211yescharacteristic, complemented
28C3×#423C2211yescharacteristic, complemented
29#5127—22no—
30#4925C2211yescharacteristic, complemented
31#4924C2211yescharacteristic, complemented
32#4835C2212yescomplemented
33#4835C2212yescomplemented
34#4834—21no—
35D3×C4⋊C8C2212yescomplemented
36D3×C4⋊C8C2212yescomplemented
37#4831—21no—
38#4728—22no—
39C8×D12—21no—
40#4679—21no—
41C3×#331C2212yescomplemented
42C3×#331C2212yescomplemented
43#4544C2212yescomplemented
44#4544C2212yescomplemented
45#4485C2211yescharacteristic, complemented
46#4485C2212yescomplemented
47#4485C2211yescharacteristic, complemented
48#4481C2212yescomplemented
49#4481C2211yescharacteristic, complemented
50#4481C2211yescharacteristic, complemented
51#3498—31no—
52#2990—31no—
53#2990—31no—
54#1560—31no—
55#1555—32no—
56#1551—32no—
57#1551—32no—
58#1545—31no—
59#1545D311yescharacteristic, complemented
60#1543—31no—
61C2×#331—32no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2×C412yes—
65C22×#67—24no—
66#1004—48no—
67C2×#237—22no—
68C2×#237C2×C414yes—
69C2×D3×D4—22no—
70C2×D3×D4—22no—
71C2×D3×D4—22no—
72C2×D3×D4C2×C414yes—
73C2×#235C2×C411yescharacteristic
74C2×#235C2×C412yes—
75C2×C3×M16—21no—
76C3×#127—21no—
77C3×#127—21no—
78C3×#127C2×C412yes—
79C3×#127C2312yes—
80C2×C3×C42C2311yescharacteristic
81#953—24no—
82#940—22no—
83#940—22no—
84#940C2×C414yes—
85#939—24no—
86#939—24no—
87#938—22no—
88#938—22no—
89#938—22no—
90#938—22no—
91#937—24no—
92#937—24no—
93#936C2×C414yes—
94D4×Dic3—22no—
95D4×Dic3—22no—
96#930—21no—
97#930—21no—
98C4×#67D412yes—
99C4×#67D412yes—
100#926C2×C411yescharacteristic
101C2×#208—21no—
102C2×D3×C8—22no—
103C2×D3×C8—22no—
104#875C2×C412yes—
105#874C2311yescharacteristic
106C4×D12—21no—
107C4×D12—21no—
108C4×D12D411yescharacteristic
109C4×D12D411yescharacteristic
110#872C2×C412yes—
111C42×D3C2312yes—
112#869C2311yescharacteristic
113C4×Dic6D411yescharacteristic
114C4×Dic6D411yescharacteristic
115C3×C4⋊C8—21no—
116C3×C4⋊C8C2312yes—
117C3×C4⋊C8C2311yescharacteristic
118C3×C4⋊C8C2311yescharacteristic
119C3×C22⋊C8—21no—
120C3×C4×C8—21no—
121#830—21no—
122(C2×D3)⋊C8—22no—
123(C2×D3)⋊C8—22no—
124#815—22no—
125#815—22no—
126Dic3⋊C8—22no—
127Dic3⋊C8—22no—
128C4⋊(C3⋊C8)—21no—
129C4⋊(C3⋊C8)C2312yes—
130C4⋊(C3⋊C8)C2311yescharacteristic
131C4⋊(C3⋊C8)C2311yescharacteristic
132C4×C3⋊C8—21no—
133C2×H(5,𝔽2)—32no—
134#550—61no—
135#550—61no—
136D42—61no—
137D42—61no—
138D42—62no—
139D42—62no—
140#535—32no—
141#535—34no—
142#532—32no—
143#532—31no—
144C2×#127—32no—
145C2×#127C2×D311yescharacteristic
146C4×#41—31no—
147#446—61no—
148#446—61no—
149#435—62no—
150#435—62no—
151C8×D4—61no—
152C8×D4—61no—
153#432—32no—
154#432—32no—
155#431—31no—
156#431C2×D311yescharacteristic, complemented
157#423—31no—
158#423C2×D311yescharacteristic, complemented
159C2×C4⋊C8—32no—
160C2×C4⋊C8—32no—
161#333—31no—
162#333—31no—
163#333—32no—
164#331—32no—
165#331—32no—
166#331—31no—
167#331—32no—
168#331—31no—
169#331C2×D312yescomplemented
170#331C2×D312yescomplemented
171C24×C3—22no—
172C23×D3—24no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—44no—
179C2×C3×D4—81no—
180C2×C3×D4—81no—
181C2×C3×D4—81no—
182C2×C3×D4—81no—
183C2×C3×D4C22×C414yes—
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C2×#67—22no—
188C2×#67—24no—
189C2×#67—24no—
190C2×#67—24no—
191C2×#67—24no—
192C2×#67—416no—
193C2×#67C22⋊C412yes—
194C22×Dic3—24no—
195#237—48no—
196#237—48no—
197D3×D4—48no—
198D3×D4—48no—
199D3×D4—82no—
200D3×D4—82no—
201D3×D4—82no—
202D3×D4—82no—
203#235—24no—
204#235—44no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12C22⋊C411yescharacteristic
208C2×D12C22×C412yes—
209C2×C4×D3—22no—
210C2×C4×D3C22×C412yes—
211C2×C4×D3C2×D412yes—
212C2×C4×D3C2×D412yes—
213C2×Dic6C22⋊C411yescharacteristic
214C2×Dic6C22×C412yes—
215C3×M16—81no—
216C2×C3×C8—21no—
217C2×C3×C8—22no—
218C2×C3×C8—21no—
219C3×C42C22×C411yescharacteristic
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D411yescharacteristic
222C3×C42C2411yescharacteristic
223#217—24no—
224#217—24no—
225#217C22⋊C412yes—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4C22⋊C412yes—
229(C2×D3)⋊C4C22⋊C412yes—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3C22⋊C411yescharacteristic
233C4⋊Dic3C22⋊C411yescharacteristic
234Dic3⋊C4C22⋊C412yes—
235Dic3⋊C4C22⋊C412yes—
236C4×Dic3C22×C412yes—
237C4×Dic3C2×D412yes—
238C4×Dic3C2×D412yes—
239#208—81no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—22no—
243D3×C8—82no—
244D3×C8—82no—
245H(5,𝔽2)—128no—
246C2×#41—31no—
247C2×#41—34no—
248C2×#41—32no—
249C2×#41—62no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—34no—
253C22×D4—62no—
254C22×D4—62no—
255C22×D4—62no—
256C22×D4—64no—
257C22×D4C4×D312yes—
258C2×M16—21no—
259C2×M16—61no—
260C22×C8—62no—
261C22×C8—62no—
262C4⋊Q8—31no—
263#127—21no—
264#127—21no—
265#127—34no—
266#127—31no—
267#127—62no—
268#127—62no—
269#127C4×D312yes—
270#127C22×D312yes—
271#124—34no—
272#124—32no—
273#121—62no—
274#121—62no—
275#121—64no—
276#121—64no—
277#121—62no—
278#121—62no—
279#121—61no—
280#121—61no—
281#120—64no—
282#120—64no—
283#120—64no—
284C4×Q8—31no—
285C4×Q8—31no—
286C4×D4—32no—
287C4×D4—32no—
288C4×D4—31no—
289C4×D4—31no—
290C4×D4—62no—
291C4×D4—62no—
292C4×D4—61no—
293C4×D4—61no—
294#117—31no—
295#117—32no—
296C2×C42—32no—
297C2×C42C22×D311yescharacteristic
298C4⋊C8—21no—
299C4⋊C8—31no—
300C4⋊C8—32no—
301C4⋊C8—31no—
302C4⋊C8—62no—
303C4⋊C8—62no—
304C4⋊C8—61no—
305C4⋊C8C22×D311yescharacteristic
306C4⋊C8C22×D312yes—
307C4⋊C8C22×D311yescharacteristic
308C22⋊C8—21no—
309C22⋊C8—62no—
310C22⋊C8—62no—
311C22⋊C8—61no—
312#97—62no—
313#97—62no—
314C4×C8—21no—
315C4×C8—61no—
316C23×C3—24no—
317C23×C3—44no—
318C23×C3—82no—
319C23×C3—82no—
320C23×C3C2×C22⋊C411yescharacteristic
321C22×D3—22no—
322C22×D3—48no—
323C22×D3—84no—
324C22×D3—84no—
325C22×D3C2×C22⋊C412yes—
326C3×D4—44no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—44no—
332C3×D4—82no—
333C3×D4—82no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4C2×C22⋊C411yescharacteristic
340C2×C3×C4C2×C22⋊C411yescharacteristic
341C2×C3×C4C2×C22⋊C411yescharacteristic
342C2×C3×C4C23×C411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345#67—24no—
346#67—48no—
347#67—44no—
348#67—48no—
349#67—84no—
350#67—84no—
351C2×Dic3—22no—
352C2×Dic3—48no—
353C2×Dic3C2×C22⋊C412yes—
354C2×Dic3C2×C22⋊C412yes—
355C2×Dic3C2×C22⋊C412yes—
356D12—42no—
357D12—41no—
358D12—41no—
359D12—41no—
360D12—41no—
361D12#10014yes—
362C4×D3—24no—
363C4×D3—22no—
364C4×D3—22no—
365C4×D3—22no—
366C4×D3—22no—
367C4×D3—44no—
368Dic6—22no—
369Dic6—42no—
370C3×C8—41no—
371C3×C8—42no—
372C3×C8—41no—
373C3⋊C8—41no—
374C3⋊C8—42no—
375C3⋊C8—41no—
376C24—22no—
377C24—64no—
378#41—64no—
379#41—124no—
380#41—128no—
381#41—128no—
382C2×Q8—31no—
383C2×Q8—32no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—32no—
390C2×D4—31no—
391C2×D4—32no—
392C2×D4—44no—
393C2×D4—64no—
394C2×D4—64no—
395C2×D4—64no—
396C2×D4—64no—
397C2×D4—62no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—81no—
401C2×D4—81no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4—128no—
405C2×D4—128no—
406C2×D4—1216no—
407C2×D4—242no—
408C2×D4—242no—
409C2×D4—242no—
410C2×D4—242no—
411C2×D4C2×C4×D314yes—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—32no—
415C22×C4—64no—
416C22×C4—62no—
417C22×C4C2×C4×D311yescharacteristic
418C22×C4D3×D411yescharacteristic
419C22×C4D3×D411yescharacteristic
420M16—81no—
421M16—241no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—22no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—62no—
428C2×C8—242no—
429C2×C8—242no—
430C4⋊C4—32no—
431C4⋊C4—32no—
432C4⋊C4—31no—
433C4⋊C4—31no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C22⋊C4—32no—
437C22⋊C4—32no—
438C22⋊C4—32no—
439C22⋊C4—64no—
440C22⋊C4—64no—
441C22⋊C4—62no—
442C22⋊C4—62no—
443C42—32no—
444C42—32no—
445C42—32no—
446C42C2×C4×D311yescharacteristic
447C42D3×D411yescharacteristic
448C42D3×D411yescharacteristic
449C42C23×D311yescharacteristic
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—44no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—164no—
457C22×C3C22×C22⋊C411yescharacteristic
458C2×D3—22no—
459C2×D3—22no—
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—168no—
464C2×D3C2×#10014yes—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4C2×#10012yes—
469C3×C4#57611yescharacteristic
470C3×C4#57611yescharacteristic
471C3×C4#57711yescharacteristic
472C3×C4#57711yescharacteristic
473Dic3—22no—
474Dic3—22no—
475Dic3—22no—
476Dic3—42no—
477C23—24no—
478C23—32no—
479C23—44no—
480C23—62no—
481C23—82no—
482C23—82no—
483C23—128no—
484C23—244no—
485C23—244no—
486C23D3×C22⋊C411yescharacteristic
487Q8—62no—
488Q8—122no—
489D4—34no—
490D4—44no—
491D4—42no—
492D4—42no—
493D4—44no—
494D4—42no—
495D4—42no—
496D4—64no—
497D4—82no—
498D4—82no—
499D4—122no—
500D4—128no—
501D4—124no—
502D4—128no—
503D4—121no—
504D4—121no—
505D4—121no—
506D4—121no—
507D4—244no—
508D4—244no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—32no—
515C2×C4—32no—
516C2×C4—32no—
517C2×C4—64no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—128no—
524C2×C4—124no—
525C2×C4D3×C22⋊C411yescharacteristic
526C2×C4D3×C22⋊C411yescharacteristic
527C2×C4D3×C22⋊C411yescharacteristic
528C2×C4C22×C4×D311yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×D3×D411yescharacteristic
531C8—42no—
532C8—41no—
533C8—41no—
534C8—121no—
535C8—121no—
536C8—122no—
537C2×C3—41no—
538C2×C3—84no—
539C2×C3C22×#10011yescharacteristic
540C2×C3#289711yescharacteristic
541C2×C3#289711yescharacteristic
542D3—24no—
543D3—82no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—34no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—62no—
552C22—62no—
553C22—122no—
554C22—122no—
555C22—122no—
556C22—164no—
557C22—488no—
558C22C2×D3×C22⋊C411yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—62no—
563C4—62no—
564C4—62no—
565C4—122no—
566C4D3×#10012yes—
567C4#575711yescharacteristic
568C4#576211yescharacteristic
569C4#577411yescharacteristic
570C4#577511yescharacteristic
571C3#1777011yescharacteristic, complemented, Sylow
572C2—41no—
573C2—64no—
574C2—84no—
575C2—242no—
576C2C2×D3×#10011yescharacteristic
577C2#7874211yescharacteristic
578C2#7892511yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1545C2
socleC22×C3C22×C22⋊C4
prime radical#86195C2
cosocleC3×C42C24
hypercenter#1545D3
hypocenterC3#17770
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
2. #79293C2semidirect
3. #79051C2semidirect
4. C3×#1545C2semidirect
5. #71759C2semidirect
6. #70773C2semidirect
7. #70505C2semidirect
8. #68387C2semidirect
9. #68385C2semidirect
10. #68385C2semidirect
11. D3×#331C2semidirect
27. C3×#431C22semidirect
28. C3×#423C22semidirect
30. #4925C22semidirect
31. #4924C22semidirect
32. #4835C22semidirect
33. #4835C22semidirect
35. D3×C4⋊C8C22semidirect
36. D3×C4⋊C8C22semidirect
41. C3×#331C22semidirect
42. C3×#331C22semidirect
43. #4544C22semidirect
44. #4544C22semidirect
45. #4485C22semidirect
46. #4485C22semidirect
47. #4485C22semidirect
48. #4481C22semidirect
49. #4481C22semidirect
50. #4481C22semidirect
59. #1545D3semidirect
156. #431C2×D3semidirect
158. #423C2×D3semidirect
169. #331C2×D3semidirect
170. #331C2×D3semidirect
571. C3#17770semidirect

Maximal subgroups

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

subgroupindex
#861952
#792932
#790512
C3×#15452
#717592
#707732
#705052
#683872
#683852
D3×#3312
#177703