Finite Group Databasebeta

← #10696903

Subgroups — #10696903

Order 768.

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

normal subgroups only

This group has 472 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#79351C211yesmaximal, characteristic
2#79271C211yesmaximal, characteristic, complemented
3C2×#4714C211yesmaximal, characteristic
4#70824C211yesmaximal, characteristic, complemented
5#70784C211yesmaximal, characteristic
6#69801C211yesmaximal, characteristic, complemented
7C3×#1302C211yesmaximal, characteristic, complemented
8#10141—31nomaximal, Sylow
9C2×#875C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15C2×#821—21no—
16C2×#816—21no—
17#4926C2211yescharacteristic
18#4851—22no—
19#4845—22no—
20#4739—21no—
21#4731—21no—
22#4721—22no—
23#4715—21no—
24#4714—22no—
25#4714—21no—
26#4714—22no—
27#4681—21no—
28#4680—21no—
29#4481—22no—
30#4468—22no—
31#4465—22no—
32#3229—31no—
33#3215—31no—
34C2×#489—31no—
35#1558—31no—
36#1531—31no—
37#1469—31no—
38#1302D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×Dic6—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C2×C3×C42C2311yescharacteristic
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46#926—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49#924—21no—
50#924—21no—
51C2×Dic12—22no—
52C2×#204—22no—
53#875—21no—
54#875—21no—
55#875—21no—
56#875—21no—
57#870—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3×C22⋊C8—21no—
69C3×C22⋊C8D412yes—
70C3×#97—21no—
71C3×#97—21no—
72C3×#97D412yes—
73C3×C4×C8—21no—
74#830—21no—
75#821—24no—
76#821—22no—
77#817—24no—
78#816—24no—
79#816—24no—
80#816—22no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×#124—31no—
86#498—61no—
87#497—61no—
88#489—62no—
89#489—61no—
90#489—62no—
91#487—61no—
92#483—62no—
93#475—62no—
94#464—62no—
95#451—61no—
96#450—61no—
97#432—31no—
98#432C2×D311yescharacteristic
99#431C2×D311yescharacteristic
100C2×#103—61no—
101C2×#102—61no—
102C2×#97C2×D311yescharacteristic
103#333—62no—
104#330—62no—
105#323—62no—
106C23×D3—21no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—41no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—42no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—22no—
120C2×Dic6—42no—
121C2×Dic6—41no—
122C2×Dic6—82no—
123C2×C3×C8—22no—
124C2×C3×C8—22no—
125C2×C3×C8—22no—
126C2×C3×C8—42no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131#217—41no—
132(C2×D3)⋊C4—41no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—22no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—41no—
147C2×C3⋊C8—41no—
148Dic12—82no—
149#204—82no—
150C22×Q8—31no—
151C22×Q8—61no—
152C22×D4—31no—
153C2×Q16—62no—
154C2×SD16—62no—
155C22×C8—21no—
156C4⋊Q8—32no—
157C4⋊Q8—31no—
158C4⋊Q8—31no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162#125—61no—
163#124—61no—
164#124—61no—
165#124—61no—
166#124—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171#117—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C42C22×D311yescharacteristic
177#106—64no—
178C4⋊C8—21no—
179C4⋊C8—61no—
180C4⋊C8D1212yes—
181#103—64no—
182#103—62no—
183#103—64no—
184#102—64no—
185#102—62no—
186C22⋊C8—21no—
187C22⋊C8—61no—
188C22⋊C8D1212yes—
189#97—21no—
190#97—21no—
191#97—61no—
192#97D1212yes—
193C4×C8—21no—
194C23×C3#12111yescharacteristic
195C22×D3—41no—
196C22×D3—42no—
197C22×D3—81no—
198C22×D3—81no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12711yescharacteristic
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—42no—
211C2×Dic3—41no—
212C2×Dic3—82no—
213C2×Dic3—81no—
214D12—42no—
215D12—42no—
216Dic6—42no—
217Dic6—42no—
218Dic6—42no—
219Dic6—42no—
220Dic6—82no—
221Dic6—82no—
222C3×C8—42no—
223C3×C8—42no—
224C3×C8—42no—
225C3⋊C8—81no—
226C24—61no—
227C2×Q8—61no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—62no—
232C2×Q8—122no—
233C2×Q8—121no—
234C2×Q8—242no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—122no—
238C22×C4—61no—
239C22×C4—121no—
240C22×C4C2×D1211yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4#23911yescharacteristic
243Q16—242no—
244SD16—242no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—42no—
249C2×C8—121no—
250C4⋊C4—62no—
251C4⋊C4—61no—
252C4⋊C4—62no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C22⋊C4—121no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C42—121no—
266C42C2×D1211yescharacteristic
267C42C2×D1211yescharacteristic
268C42D3×D411yescharacteristic
269C42D3×D411yescharacteristic
270C22×C3—21no—
271C22×C3—21no—
272C22×C3#39211yescharacteristic
273C22×C3#47112yes—
274C2×D3—81no—
275C2×D3—81no—
276C2×D3—81no—
277C2×D3—162no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4#45512yes—
284Dic3—81no—
285Dic3—81no—
286Dic3—81no—
287Dic3—82no—
288C23—121no—
289C23—122no—
290C23—241no—
291C23—241no—
292C23#89511yescharacteristic
293Q8—122no—
294Q8—122no—
295Q8—122no—
296Q8—122no—
297Q8—242no—
298Q8—242no—
299D4—122no—
300D4—122no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—122no—
308C2×C4—242no—
309C2×C4—241no—
310C2×C4#87411yescharacteristic
311C2×C4#88211yescharacteristic
312C2×C4#88211yescharacteristic
313C2×C4#88211yescharacteristic
314C2×C4#89511yescharacteristic
315C2×C4#89511yescharacteristic
316C8—42no—
317C8—42no—
318C8—42no—
319C8—241no—
320C2×C3—22no—
321C2×C3#200411yescharacteristic
322C2×C3#200911yescharacteristic
323C2×C3#201911yescharacteristic
324D3—161no—
325C22—21no—
326C22—21no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22—482no—
331C22#467111yescharacteristic
332C22#489812yes—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—242no—
339C4—241no—
340C4—241no—
341C4—241no—
342C4#482512yes—
343C3#1014111yescharacteristic, complemented, Sylow
344C2—22no—
345C2—481no—
346C2#6876211yescharacteristic
347C2#6894311yescharacteristic
348C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1302C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1302D3
hypocenterC3#10141
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1302

Complements

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

NH ≅ G/Ntype
2. #79271C2semidirect
4. #70824C2semidirect
6. #69801C2semidirect
7. C3×#1302C2semidirect
38. #1302D3semidirect
343. C3#10141semidirect

Maximal subgroups

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

subgroupindex
#793512
#792712
C2×#47142
#708242
#707842
#698012
C3×#13022
#101413