Finite Group Databasebeta

← #10697908

Subgroups — #10697908

Order 768.

640 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 113 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#79305C211yesmaximal, characteristic
2#79271C211yesmaximal, characteristic, complemented
3#73313C211yesmaximal, characteristic
4#70778C211yesmaximal, characteristic, complemented
5#70774C211yesmaximal, characteristic
6#69396C211yesmaximal, characteristic, complemented
7C3×#1276C211yesmaximal, characteristic, complemented
8#10294—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#4851—22no—
20#4848—22no—
21#4731—22no—
22#4721—22no—
23#4694—21no—
24#4693—21no—
25#4681—21no—
26#4680—21no—
27#4523—22no—
28#4486—22no—
29#4485—22no—
30#3229—31no—
31#3208—31no—
32#2858—31no—
33#1994—31no—
34#1546—31no—
35#1545—31no—
36#1276D311yescharacteristic, complemented
37C22×D12—21no—
38C22×D12—21no—
39C22×D12D412yes—
40C22×Dic6—21no—
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⋊Dic3D412yes—
49#924—21no—
50#924—21no—
51C2×#208—21no—
52C2×#204—24no—
53#875—44no—
54#875—44no—
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#817—24no—
76#816—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79#533—31no—
80C2×C4⋊Q8—31no—
81C2×#127—31no—
82C2×#124—62no—
83#498—61no—
84#492—61no—
85#487—61no—
86#486—61no—
87#483—62no—
88#475—62no—
89#460—62no—
90#450—62no—
91#431C2×D311yescharacteristic
92#431C2×D311yescharacteristic
93#423—31no—
94#390—62no—
95#390—62no—
96C43C2×D311yescharacteristic
97#339—62no—
98#332—62no—
99#331—62no—
100C23×D3—22no—
101C22×C3×C4—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—22no—
106C2×D12—22no—
107C2×D12—22no—
108C2×D12—44no—
109C2×D12—44no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—42no—
115C2×D12—42no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C2×Dic6—42no—
121C2×Dic6—42no—
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C3×C42—22no—
125C3×C42—42no—
126C3×C42—42no—
127C3×C42SD1614yescomplemented
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132#217—41no—
133(C2×D3)⋊C4—416no—
134C4⋊Dic3—22no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—22no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144C4×Dic3—41no—
145#208—42no—
146C2×C3⋊C8—41no—
147#204—84no—
148C22×Q8—61no—
149C22×D4—32no—
150C22×D4—61no—
151C22×D4—61no—
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—124no—
166#124—124no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171#117—61no—
172C2×C4⋊C4—32no—
173C2×C4⋊C4—61no—
174C2×C22⋊C4—68no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42D1212yes—
178C2×C42C22×D311yescharacteristic
179#106—64no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8D1212yes—
183C4⋊C8D1212yes—
184#103—64no—
185#102—64no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C4×C8—21no—
189C4×C8—21no—
190C23×C3#12011yescharacteristic
191C22×D3—44no—
192C22×D3—44no—
193C22×D3—42no—
194C22×D3—44no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12711yescharacteristic
206C2×C3×C4C2×SD1612yes—
207C2×C3×C4AGL(1,ℤ/8)12yes—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—42no—
211C2×Dic3—44no—
212D12—44no—
213D12—82no—
214D12—82no—
215D12—82no—
216D12—82no—
217D12—81no—
218D12—81no—
219Dic6—42no—
220Dic6—42no—
221Dic6—82no—
222Dic6—81no—
223Dic6—81no—
224C3×C8—42no—
225C3×C8—42no—
226C3⋊C8—81no—
227C24—62no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—121no—
231C2×Q8—121no—
232C2×Q8—122no—
233C2×Q8—122no—
234C2×D4—62no—
235C2×D4—62no—
236C2×D4—124no—
237C2×D4—124no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—122no—
243C2×D4—122no—
244C22×C4—22no—
245C22×C4—62no—
246C22×C4C2×D1211yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4D3×D411yescharacteristic
249SD16—244no—
250M16—122no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—121no—
254C4⋊C4—61no—
255C4⋊C4—62no—
256C4⋊C4—61no—
257C4⋊C4—62no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—1216no—
265C22⋊C4—121no—
266C42—22no—
267C42—42no—
268C42—42no—
269C42—121no—
270C42#20414yescomplemented
271C42C2×D1211yescharacteristic
272C42C2×D1211yescharacteristic
273C42D3×D411yescharacteristic
274C42#23911yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3#39211yescharacteristic
278C22×C3#45012yes—
279C2×D3—88no—
280C2×D3—82no—
281C2×D3—82no—
282C2×D3—82no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—44no—
286C3×C4#45412yes—
287C3×C4#45512yes—
288C3×C4#46012yes—
289C3×C4#49211yescharacteristic
290C3×C4#49611yescharacteristic
291Dic3—81no—
292Dic3—81no—
293Dic3—82no—
294C23—124no—
295C23—124no—
296C23—122no—
297C23—124no—
298C23#88211yescharacteristic
299Q8—122no—
300Q8—122no—
301Q8—241no—
302Q8—241no—
303Q8—242no—
304D4—124no—
305D4—242no—
306D4—242no—
307D4—241no—
308D4—241no—
309D4—242no—
310D4—242no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—122no—
319C2×C4—124no—
320C2×C4#87411yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#88211yescharacteristic
323C2×C4#89511yescharacteristic
324C2×C4#89511yescharacteristic
325C2×C4#89511yescharacteristic
326C2×C4C2×#20412yes—
327C2×C4#90812yes—
328C8—42no—
329C8—42no—
330C8—241no—
331C2×C3—22no—
332C2×C3#200311yescharacteristic
333C2×C3#200411yescharacteristic
334C2×C3#201411yescharacteristic
335D3—162no—
336C22—21no—
337C22—21no—
338C22—248no—
339C22—242no—
340C22—242no—
341C22—242no—
342C22#467111yescharacteristic
343C22#473112yes—
344C4—21no—
345C4—21no—
346C4—44no—
347C4—242no—
348C4—241no—
349C4—241no—
350C4#469311yescharacteristic
351C4#471211yescharacteristic
352C4#482512yes—
353C4#482712yes—
354C4#484812yes—
355C3#1029411yescharacteristic, complemented, Sylow
356C2—22no—
357C2—482no—
358C2#6867611yescharacteristic
359C2#6876211yescharacteristic
360C2#6876711yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#1276D3
hypocenterC3#10294
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
2. #79271C2semidirect
4. #70778C2semidirect
6. #69396C2semidirect
7. C3×#1276C2semidirect
36. #1276D3semidirect
127. C3×C42SD16semidirect
270. C42#204semidirect
355. C3#10294semidirect

Maximal subgroups

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

subgroupindex
#793052
#792712
#733132
#707782
#707742
#693962
C3×#12762
#102943