Finite Group Databasebeta

← #10697887

Subgroups — #10697887

Order 768.

652 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 105 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#79313C211yesmaximal, characteristic, complemented
2#79305C211yesmaximal, characteristic
3#73313C211yesmaximal, characteristic
4#70779C211yesmaximal, characteristic, complemented
5#70774C211yesmaximal, characteristic
6#69396C211yesmaximal, characteristic, complemented
7C3×#1276C211yesmaximal, characteristic, complemented
8#10291—31nomaximal, Sylow
9C2×#875—22no—
10C2×#874C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×C43C2211yescharacteristic
16#4939—22no—
17#4939—22no—
18#4924C2211yescharacteristic
19#4849—22no—
20#4848—22no—
21#4739—22no—
22#4731—22no—
23#4699—21no—
24#4694—21no—
25#4693—21no—
26#4681—21no—
27#4523—22no—
28#4485—22no—
29#4481—22no—
30#3209—31no—
31#3208—31no—
32#2858—31no—
33#1994—31no—
34#1548—31no—
35#1545—31no—
36#1276D311yescharacteristic, complemented
37C22×D12—21no—
38C22×D12—21no—
39C22×D12D412yes—
40C22×Dic6D412yes—
41C2×C3×C42—21no—
42C2×C3×C42—21no—
43C2×C3×C42D412yes—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—28no—
46#926—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49#924—21no—
50#924—21no—
51C2×#208—21no—
52C2×Dic12—24no—
53C2×#204—24no—
54#875—48no—
55#874—21no—
56#874—21no—
57#874—21no—
58#874—21no—
59#870—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8D412yes—
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74#821—24no—
75#816—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78#533—31no—
79C2×C4⋊Q8—31no—
80C2×#127—31no—
81C2×#124—62no—
82#494—61no—
83#492—61no—
84#487—61no—
85#486—61no—
86#462—62no—
87#460—62no—
88#451—62no—
89#450—62no—
90#431C2×D311yescharacteristic
91#431C2×D311yescharacteristic
92#423—31no—
93#390—62no—
94#390—62no—
95C43C2×D311yescharacteristic
96#339—62no—
97#333—62no—
98#331—62no—
99C23×D3—22no—
100C22×C3×C4—22no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×Dic3—22no—
105C2×D12—22no—
106C2×D12—22no—
107C2×D12—44no—
108C2×D12—44no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—42no—
114C2×D12—42no—
115C2×Dic6—22no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—22no—
119C2×Dic6—44no—
120C2×C3×C8—22no—
121C2×C3×C8—22no—
122C3×C42—24no—
123C3×C42—22no—
124C3×C42—22no—
125C3×C42—42no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130#217—41no—
131(C2×D3)⋊C4—416no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142C4×Dic3—41no—
143#208—42no—
144C2×C3⋊C8—41no—
145Dic12—84no—
146#204—84no—
147C22×Q8—32no—
148C22×D4—32no—
149C22×D4—61no—
150C22×D4—61no—
151C2×Q16—64no—
152C2×SD16—64no—
153C2×M16—61no—
154C4⋊Q8—31no—
155C4⋊Q8—31no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160#127—61no—
161#127—61no—
162#127—61no—
163#127—61no—
164#125—61no—
165#124—128no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170#117—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C22⋊C4—68no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42D1212yes—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—61no—
179C4⋊C8—61no—
180C4⋊C8D1212yes—
181C4⋊C8D1212yes—
182#103—64no—
183#102—64no—
184C22⋊C8—21no—
185C22⋊C8—21no—
186C4×C8—21no—
187C4×C8—21no—
188C23×C3#12011yescharacteristic
189C22×D3—44no—
190C22×D3—44no—
191C22×D3—42no—
192C22×D3—44no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—24no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12711yescharacteristic
204C2×C3×C4#13512yes—
205C2×C3×C4#13712yes—
206C2×Dic3—44no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—42no—
210D12—44no—
211D12—82no—
212D12—82no—
213D12—82no—
214D12—82no—
215D12—81no—
216D12—81no—
217Dic6—42no—
218Dic6—44no—
219Dic6—42no—
220Dic6—82no—
221C3×C8—42no—
222C3×C8—42no—
223C3⋊C8—81no—
224C24—62no—
225C2×Q8—62no—
226C2×Q8—61no—
227C2×Q8—61no—
228C2×Q8—62no—
229C2×Q8—124no—
230C2×D4—62no—
231C2×D4—62no—
232C2×D4—124no—
233C2×D4—124no—
234C2×D4—121no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4—122no—
239C2×D4—122no—
240C22×C4—22no—
241C22×C4—62no—
242C22×C4C2×D1211yescharacteristic
243C22×C4D3×D411yescharacteristic
244C22×C4D3×D411yescharacteristic
245Q16—244no—
246SD16—244no—
247M16—122no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—121no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—1216no—
262C22⋊C4—121no—
263C42—24no—
264C42—22no—
265C42—22no—
266C42—42no—
267C42—121no—
268C42C2×D1211yescharacteristic
269C42C2×D1211yescharacteristic
270C42D3×D411yescharacteristic
271C42#23911yescharacteristic
272C22×C3—21no—
273C22×C3—21no—
274C22×C3#39211yescharacteristic
275C22×C3#45212yes—
276C2×D3—88no—
277C2×D3—82no—
278C2×D3—82no—
279C2×D3—82no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—44no—
283C3×C4#45312yes—
284C3×C4#45512yes—
285C3×C4#46412yes—
286C3×C4#49511yescharacteristic
287C3×C4#49711yescharacteristic
288Dic3—82no—
289Dic3—81no—
290Dic3—81no—
291C23—124no—
292C23—124no—
293C23—122no—
294C23—124no—
295C23#88211yescharacteristic
296Q8—122no—
297Q8—124no—
298Q8—122no—
299Q8—242no—
300D4—124no—
301D4—242no—
302D4—242no—
303D4—241no—
304D4—241no—
305D4—242no—
306D4—242no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—124no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—122no—
316C2×C4#87411yescharacteristic
317C2×C4#88211yescharacteristic
318C2×C4#88211yescharacteristic
319C2×C4#89511yescharacteristic
320C2×C4#89511yescharacteristic
321C2×C4#89511yescharacteristic
322C2×C4#90412yes—
323C2×C4#90912yes—
324C8—42no—
325C8—42no—
326C8—241no—
327C2×C3—22no—
328C2×C3#200211yescharacteristic
329C2×C3#200411yescharacteristic
330C2×C3#201511yescharacteristic
331D3—162no—
332C22—21no—
333C22—21no—
334C22—248no—
335C22—242no—
336C22—242no—
337C22—242no—
338C22#467111yescharacteristic
339C22#473312yes—
340C4—21no—
341C4—21no—
342C4—44no—
343C4—242no—
344C4—241no—
345C4—241no—
346C4#469611yescharacteristic
347C4#471511yescharacteristic
348C4#482212yes—
349C4#482512yes—
350C4#484512yes—
351C3#1029111yescharacteristic, complemented, Sylow
352C2—22no—
353C2—482no—
354C2#6867711yescharacteristic
355C2#6876111yescharacteristic
356C2#6876211yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#1276D3
hypocenterC3#10291
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #79313C2semidirect
4. #70779C2semidirect
6. #69396C2semidirect
7. C3×#1276C2semidirect
36. #1276D3semidirect
351. C3#10291semidirect

Maximal subgroups

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

subgroupindex
#793132
#793052
#733132
#707792
#707742
#693962
C3×#12762
#102913