Finite Group Databasebeta

← #10692141

Subgroups — #10692141

Order 768.

438 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82739C211yesmaximal, characteristic
2#78636C211yesmaximal, characteristic
3C3×#1472C211yesmaximal, characteristic, complemented
4#71689C211yesmaximal, characteristic, complemented
5#69943C211yesmaximal, characteristic
6#67686C211yesmaximal, characteristic, complemented
7#67679C211yesmaximal, characteristic, complemented
8#9331—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10#5791C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13#5228C411yescharacteristic, complemented
14#5227C411yescharacteristic, complemented
15#5211C2211yescharacteristic
16#5207C2211yescharacteristic
17#5207C2211yescharacteristic
18#5128—21no—
19#5127—21no—
20#5052—21no—
21#5049—21no—
22#5047—21no—
23#5045—21no—
24#4970C411yescharacteristic, complemented
25#4968C411yescharacteristic, complemented
26#4729—21no—
27#4728—21no—
28#4727—21no—
29#4720—21no—
30C3×#324—21no—
31C3×#324—21no—
32#4538—21no—
33#4538—21no—
34#4470—21no—
35#4467—21no—
36#3196—31no—
37#2978—31no—
38#1516—31no—
39#1511—31no—
40#1472D311yescharacteristic, complemented
41#1306—31no—
42#1293—31no—
43C22×C3×Q8C2×C411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C2×#235D411yescharacteristic
46C2×C3×M16—21no—
47C22×C3×C8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4C2×C411yescharacteristic
54C2×C3×C22⋊C4—21no—
55#953—21no—
56C2×#217—21no—
57C2×#216—21no—
58C2×#215—21no—
59#930C2×C411yescharacteristic
60#930D411yescharacteristic
61#929—21no—
62C4×#67—21no—
63C4×#67—21no—
64#926D411yescharacteristic
65#924C2×C411yescharacteristic
66#924D411yescharacteristic
67C2×#208—21no—
68C2×#208—21no—
69C2×ΓL(1,𝔽25)—21no—
70C2×D3×C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D411yescharacteristic
75C3×C22⋊C8D411yescharacteristic
76#835—21no—
77#835—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#830C2×C411yescharacteristic
83#830C2×C411yescharacteristic
84#830D411yescharacteristic
85#830D411yescharacteristic
86(C2×D3)⋊C8—21no—
87(C2×D3)⋊C8—21no—
88#815—21no—
89Dic3⋊C8—21no—
90Dic3⋊C8—21no—
91C8×Dic3—21no—
92#810—21no—
93#809—21no—
94#808—21no—
95#807—21no—
96#525—31no—
97C2×C22⋊Q8C2×D311yescharacteristic
98#485—61no—
99#484—31no—
100#481—31no—
101#470—61no—
102#468—61no—
103#452—61no—
104#451—31no—
105#448—31no—
106#436—61no—
107#436—61no—
108#435—61no—
109#435—61no—
110#433—61no—
111#408—61no—
112#407—31no—
113#407—31no—
114#407C2×D311yescharacteristic
115C2×C22⋊C8C2×D311yescharacteristic
116#381—31no—
117#328—61no—
118#328—61no—
119#328—61no—
120#327—61no—
121#324—21no—
122#324—21no—
123C24×C3#4111yescharacteristic
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—81no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—81no—
130C22×C3×C4C22×C411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C2×#67—21no—
137C2×#67—21no—
138C2×#67—21no—
139C22×Dic3—41no—
140#235—81no—
141C2×D12C22⋊C411yescharacteristic
142C2×C4×D3—21no—
143C2×Dic6C22⋊C411yescharacteristic
144C3×M16—81no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—41no—
148C2×C3×C8—81no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155#217—21no—
156#217—21no—
157#217—21no—
158#217—81no—
159#216—81no—
160#215—81no—
161(C2×D3)⋊C4—21no—
162(C2×D3)⋊C4—21no—
163C4⋊Dic3C22⋊C411yescharacteristic
164C4⋊Dic3C22⋊C411yescharacteristic
165Dic3⋊C4—21no—
166Dic3⋊C4—21no—
167C4×Dic3—21no—
168#208—81no—
169#208—81no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—21no—
174ΓL(1,𝔽25)—81no—
175D3×C8—81no—
176C2×#41—31no—
177C22×Q8C4×D311yescharacteristic
178C23×C4C22×D311yescharacteristic
179C2×Q16—61no—
180C2×SD16—61no—
181C2×M16—21no—
182C2×M16—61no—
183C2×M16—61no—
184C2×M16—61no—
185C22×C8—21no—
186C22×C8—61no—
187#123—61no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—31no—
193C22⋊Q8—31no—
194#121—31no—
195#121—31no—
196#120—61no—
197C4×D4—61no—
198C4×D4—61no—
199#117—31no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C4×D311yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—61no—
204#107—61no—
205#106—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#103—61no—
209#103—61no—
210#103—61no—
211#102—61no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—31no—
216C22⋊C8—31no—
217C22⋊C8—31no—
218C22⋊C8—31no—
219C22⋊C8—61no—
220C22⋊C8—61no—
221C22⋊C8—61no—
222C22⋊C8—61no—
223C22⋊C8#6711yescharacteristic
224C22⋊C8#6711yescharacteristic
225#97—61no—
226C4×C8—61no—
227#95—61no—
228#95—61no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3C4×D411yescharacteristic
233C23×C3#12111yescharacteristic
234C23×C3#12311yescharacteristic
235C22×D3—21no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×Q8—81no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—81no—
249C2×C3×C4C2×C22⋊C411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253#67—81no—
254#67—81no—
255#67—81no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—81no—
260C2×Dic3—81no—
261C2×Dic3—81no—
262D12—41no—
263C4×D3—41no—
264C4×D3—41no—
265Dic6—41no—
266C3×C8—41no—
267C3×C8—41no—
268C3×C8—81no—
269C3⋊C8—41no—
270C3⋊C8—41no—
271C3⋊C8—41no—
272C3⋊C8—41no—
273C24#23511yescharacteristic
274#41—241no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—31no—
278C2×Q8—81no—
279C2×D4—31no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—61no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—61no—
286C22×C4—81no—
287C22×C4—121no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4#23711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294Q16—241no—
295SD16—241no—
296M16—81no—
297M16—241no—
298M16—241no—
299M16—241no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—41no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—81no—
308C2×C8—241no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—61no—
320C22⋊C4—61no—
321C22⋊C4—61no—
322C22⋊C4—61no—
323C22⋊C4—61no—
324C22⋊C4—241no—
325C42—61no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—82no—
336C22×C3#38611yescharacteristic
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—81no—
345C3×C4—81no—
346Dic3—41no—
347Dic3—41no—
348Dic3—41no—
349Dic3—161no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—61no—
354C23#88311yescharacteristic
355C23#88411yescharacteristic
356C23C4×#6711yescharacteristic
357Q8—41no—
358Q8—41no—
359Q8—81no—
360Q8—121no—
361D4—121no—
362D4—241no—
363D4—241no—
364D4—241no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—81no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—241no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4D3×C22⋊C411yescharacteristic
384C2×C4#67⋊C411yescharacteristic
385C2×C4#93711yescharacteristic
386C2×C4#93811yescharacteristic
387C8—41no—
388C8—41no—
389C8—81no—
390C8—121no—
391C8—121no—
392C8—121no—
393C8—121no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3#188111yescharacteristic
398C2×C3#190211yescharacteristic
399C2×C3#191311yescharacteristic
400D3—81no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—82no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22#495611yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—81no—
421C4—121no—
422C4—121no—
423C4—121no—
424C4—481no—
425C3#933111yescharacteristic, complemented, Sylow
426C2—41no—
427C2—41no—
428C2—41no—
429C2—241no—
430C2#7001211yescharacteristic
431C2#7003411yescharacteristic
432C2#7012211yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1472C2
socleC22×C3#386
prime radical#5791C22
cosocleC23×C3×C4C23
hypercenter#1472D3
hypocenterC3#9331
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
3. C3×#1472C2semidirect
4. #71689C2semidirect
6. #67686C2semidirect
7. #67679C2semidirect
13. #5228C4semidirect
14. #5227C4semidirect
24. #4970C4semidirect
25. #4968C4semidirect
40. #1472D3semidirect
425. C3#9331semidirect

Maximal subgroups

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

subgroupindex
#827392
#786362
C3×#14722
#716892
#699432
#676862
#676792
#93313