Finite Group Databasebeta

← #10764122

Subgroups — #10764122

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86214C211yesmaximal, characteristic
2#79293C211yesmaximal, characteristic, complemented
3#79051C211yesmaximal, characteristic, complemented
4C3×#1548C211yesmaximal, characteristic, complemented
5#71774C211yesmaximal, characteristic, complemented
6#70773C211yesmaximal, characteristic, complemented
7#70505C211yesmaximal, characteristic, complemented
8#68487C212yesmaximal, complemented
9#68485C212yesmaximal, complemented
10#68485C212yesmaximal, complemented
11D3×#333C212yesmaximal, complemented
12#17779—31nomaximal, Sylow
13C2×#1007C412yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15#5816C412yes—
16#5727—21no—
17#5727—21no—
18#5726—21no—
19#5726—21no—
20#5725C2212yes—
21#5724C414yes—
22D3×C4⋊Q8C2212yes—
23#5507—22no—
24#5507—22no—
25#5477C2211yescharacteristic
26C4×#235C2211yescharacteristic
27C3×#431C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5127—22no—
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32#4835C2212yes—
33#4835C2212yes—
34#4834—21no—
35D3×C4⋊C8C2212yes—
36D3×C4⋊C8C2212yes—
37#4831—21no—
38#4728—22no—
39C8×D12—21no—
40#4679—21no—
41C3×#333C2212yes—
42C3×#333C2212yes—
43#4546C2212yes—
44#4546C2212yes—
45#4485C2212yes—
46#4485C2211yescharacteristic
47#4485C2211yescharacteristic
48#4481C2212yes—
49#4481C2211yescharacteristic
50#4481C2211yescharacteristic
51#3499—31no—
52#2990—31no—
53#2990—31no—
54#1560—31no—
55#1558—32no—
56#1551—32no—
57#1551—32no—
58#1548—31no—
59#1548D311yescharacteristic, complemented
60#1543—31no—
61C2×#333—32no—
62C22×C3×Q8C2×C412yes—
63#1007—48no—
64C2×#239—22no—
65C2×#239C2×C414yes—
66C2×D3×Q8—22no—
67C2×D3×Q8C2×C414yes—
68C2×#235—24no—
69C2×#235C2×C411yescharacteristic
70C2×#235C2×C412yes—
71C2×C3×M16—21no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8C2×C412yes—
75C3×C4⋊Q8C2312yes—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C42C2311yescharacteristic
79#947C2×C414yes—
80#946—22no—
81#946—22no—
82#944C2×C414yes—
83#930—21no—
84#930—21no—
85#929—24no—
86C4×#67D412yes—
87C4×#67D412yes—
88#926C2×C411yescharacteristic
89C2×#208—21no—
90C2×D3×C8—22no—
91C2×D3×C8—22no—
92#896—24no—
93#896—24no—
94#895—22no—
95#895—22no—
96#894—24no—
97#894—24no—
98#893—22no—
99#893—22no—
100D3×C4⋊C4—22no—
101D3×C4⋊C4—22no—
102#888—22no—
103#888—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—
133C2×2−1+4—32no—
134D4×Q8—61no—
135D4×Q8—61no—
136#547—62no—
137#547—62no—
138#547—61no—
139#547—61no—
140#536—32no—
141#536—34no—
142#532—32no—
143#532—31no—
144C2×C4⋊Q8—32no—
145C2×C4⋊Q8C2×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
157#423—31no—
158#423C2×D311yescharacteristic
159C2×C4⋊C8—32no—
160C2×C4⋊C8—32no—
161#333—32no—
162#333—32no—
163#333—31no—
164#333—32no—
165#333—31no—
166#333C2×D312yes—
167#333C2×D312yes—
168#331—31no—
169#331—31no—
170#331—32no—
171C2×C3×Q8—22no—
172C2×C3×Q8—44no—
173C2×C3×Q8C22×C414yes—
174C22×C3×C4—22no—
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C2×#67—22no—
179C2×#67C22⋊C412yes—
180#239—48no—
181#239—48no—
182D3×Q8—48no—
183D3×Q8—48no—
184#235—24no—
185#235—416no—
186#235—44no—
187C2×D12—24no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12C22⋊C411yescharacteristic
191C2×D12C22×C412yes—
192C2×C4×D3—24no—
193C2×C4×D3—22no—
194C2×C4×D3—24no—
195C2×C4×D3C22×C412yes—
196C2×C4×D3C2×D412yes—
197C2×C4×D3C2×D412yes—
198C2×Dic6—24no—
199C2×Dic6C22⋊C411yescharacteristic
200C2×Dic6C22×C412yes—
201C3×M16—81no—
202C2×C3×C8—21no—
203C2×C3×C8—22no—
204C2×C3×C8—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C42C22×C411yescharacteristic
210C3×C42C2×D411yescharacteristic
211C3×C42C2×D411yescharacteristic
212C3×C42C2411yescharacteristic
213#217C22⋊C412yes—
214(C2×D3)⋊C4—22no—
215(C2×D3)⋊C4—22no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4C22⋊C412yes—
219(C2×D3)⋊C4C22⋊C412yes—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3C22⋊C411yescharacteristic
223C4⋊Dic3C22⋊C411yescharacteristic
224Dic3⋊C4—24no—
225Dic3⋊C4—24no—
226Dic3⋊C4C22⋊C412yes—
227Dic3⋊C4C22⋊C412yes—
228C4×Dic3C22×C412yes—
229C4×Dic3C2×D412yes—
230C4×Dic3C2×D412yes—
231#208—81no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—22no—
235D3×C8—82no—
236D3×C8—82no—
2372−1+4—128no—
238C2×#41—31no—
239C2×#41—34no—
240C2×#41—32no—
241C2×#41—64no—
242C2×#41—62no—
243C22×Q8—34no—
244C22×Q8—62no—
245C22×Q8C4×D312yes—
246C2×M16—21no—
247C2×M16—61no—
248C22×C8—62no—
249C22×C8—62no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—34no—
253C4⋊Q8—31no—
254C4⋊Q8—62no—
255C4⋊Q8—62no—
256C4⋊Q8C4×D312yes—
257C4⋊Q8C22×D312yes—
258#127—31no—
259#124—34no—
260#124—32no—
261#123—64no—
262#123—64no—
263#123—64no—
264C22⋊Q8—64no—
265C22⋊Q8—64no—
266C22⋊Q8—62no—
267C22⋊Q8—62no—
268#121—62no—
269#121—62no—
270#121—61no—
271#121—61no—
272C4×Q8—31no—
273C4×Q8—31no—
274C4×D4—32no—
275C4×D4—32no—
276C4×D4—31no—
277C4×D4—31no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4—61no—
281C4×D4—61no—
282#117—31no—
283#117—32no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C42—32no—
289C2×C42C22×D311yescharacteristic
290C4⋊C8—21no—
291C4⋊C8—31no—
292C4⋊C8—32no—
293C4⋊C8—31no—
294C4⋊C8—62no—
295C4⋊C8—62no—
296C4⋊C8—61no—
297C4⋊C8C22×D311yescharacteristic
298C4⋊C8C22×D312yes—
299C4⋊C8C22×D311yescharacteristic
300C22⋊C8—21no—
301C22⋊C8—62no—
302C22⋊C8—62no—
303C22⋊C8—61no—
304#97—62no—
305#97—62no—
306C4×C8—21no—
307C4×C8—61no—
308C23×C3C2×C22⋊C411yescharacteristic
309C22×D3—22no—
310C22×D3C2×C22⋊C412yes—
311C3×Q8—44no—
312C3×Q8—44no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—44no—
320C2×C3×C4C2×C22⋊C411yescharacteristic
321C2×C3×C4C2×C22⋊C411yescharacteristic
322C2×C3×C4C2×C22⋊C411yescharacteristic
323C2×C3×C4C23×C411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326#67—24no—
327#67—44no—
328C2×Dic3—22no—
329C2×Dic3C2×C22⋊C412yes—
330C2×Dic3C2×C22⋊C412yes—
331C2×Dic3C2×C22⋊C412yes—
332D12—42no—
333D12—48no—
334D12—41no—
335D12—41no—
336D12—41no—
337D12—41no—
338D12#10014yes—
339C4×D3—24no—
340C4×D3—22no—
341C4×D3—22no—
342C4×D3—22no—
343C4×D3—22no—
344C4×D3—48no—
345C4×D3—48no—
346C4×D3—44no—
347Dic6—22no—
348Dic6—48no—
349Dic6—42no—
350C3×C8—41no—
351C3×C8—42no—
352C3×C8—41no—
353C3⋊C8—41no—
354C3⋊C8—42no—
355C3⋊C8—41no—
356#41—64no—
357#41—124no—
358#41—1216no—
359#41—128no—
360#41—128no—
361C2×Q8—22no—
362C2×Q8—31no—
363C2×Q8—32no—
364C2×Q8—44no—
365C2×Q8—64no—
366C2×Q8—128no—
367C2×Q8—128no—
368C2×Q8C2×C4×D314yes—
369C2×D4—32no—
370C2×D4—31no—
371C2×D4—32no—
372C2×D4—64no—
373C2×D4—62no—
374C2×D4—61no—
375C2×D4—61no—
376C22×C4—22no—
377C22×C4—32no—
378C22×C4—32no—
379C22×C4—32no—
380C22×C4—64no—
381C22×C4—64no—
382C22×C4—62no—
383C22×C4C2×C4×D311yescharacteristic
384C22×C4D3×D411yescharacteristic
385C22×C4D3×D411yescharacteristic
386M16—81no—
387M16—241no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—61no—
392C2×C8—61no—
393C2×C8—62no—
394C2×C8—242no—
395C2×C8—242no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—32no—
401C4⋊C4—32no—
402C4⋊C4—31no—
403C4⋊C4—31no—
404C4⋊C4—64no—
405C4⋊C4—64no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C22⋊C4—32no—
409C22⋊C4—32no—
410C22⋊C4—32no—
411C22⋊C4—64no—
412C22⋊C4—64no—
413C22⋊C4—62no—
414C22⋊C4—62no—
415C42—32no—
416C42—32no—
417C42—32no—
418C42C2×C4×D311yescharacteristic
419C42D3×D411yescharacteristic
420C42D3×D411yescharacteristic
421C42C23×D311yescharacteristic
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3C22×C22⋊C411yescharacteristic
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3C2×#10014yes—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—44no—
436C3×C4C2×#10012yes—
437C3×C4#57711yescharacteristic
438C3×C4#57711yescharacteristic
439C3×C4#57811yescharacteristic
440C3×C4#57811yescharacteristic
441Dic3—22no—
442Dic3—22no—
443Dic3—22no—
444Dic3—42no—
445C23—32no—
446C23—62no—
447C23D3×C22⋊C411yescharacteristic
448Q8—44no—
449Q8—44no—
450Q8—62no—
451Q8—122no—
452Q8—128no—
453D4—34no—
454D4—64no—
455D4—122no—
456D4—128no—
457D4—124no—
458D4—121no—
459D4—121no—
460D4—121no—
461D4—121no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—32no—
471C2×C4—44no—
472C2×C4—64no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—128no—
479C2×C4—128no—
480C2×C4—124no—
481C2×C4D3×C22⋊C411yescharacteristic
482C2×C4D3×C22⋊C411yescharacteristic
483C2×C4D3×C22⋊C411yescharacteristic
484C2×C4C22×C4×D311yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×D3×D411yescharacteristic
487C8—42no—
488C8—41no—
489C8—41no—
490C8—121no—
491C8—121no—
492C8—122no—
493C2×C3—41no—
494C2×C3C22×#10011yescharacteristic
495C2×C3#289811yescharacteristic
496C2×C3#289811yescharacteristic
497D3—24no—
498D3—82no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—34no—
503C22—62no—
504C22—62no—
505C22—122no—
506C22—122no—
507C22—122no—
508C22C2×D3×C22⋊C411yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—44no—
513C4—62no—
514C4—62no—
515C4—62no—
516C4—122no—
517C4D3×#10012yes—
518C4#576211yescharacteristic
519C4#576611yescharacteristic
520C4#577811yescharacteristic
521C4#577911yescharacteristic
522C3#1777911yescharacteristic, complemented, Sylow
523C2—41no—
524C2—64no—
525C2—242no—
526C2C2×D3×#10011yescharacteristic
527C2#7879011yescharacteristic
528C2#7895711yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1548C2
socleC22×C3C22×C22⋊C4
prime radical#86214C2
cosocleC3×C42C24
hypercenter#1548D3
hypocenterC3#17779
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
2. #79293C2semidirect
3. #79051C2semidirect
4. C3×#1548C2semidirect
5. #71774C2semidirect
6. #70773C2semidirect
7. #70505C2semidirect
8. #68487C2semidirect
9. #68485C2semidirect
10. #68485C2semidirect
11. D3×#333C2semidirect
59. #1548D3semidirect
522. C3#17779semidirect

Maximal subgroups

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

subgroupindex
#862142
#792932
#790512
C3×#15482
#717742
#707732
#705052
#684872
#684852
D3×#3332
#177793