Finite Group Databasebeta

← #10707446

Subgroups — #10707446

Order 768.

442 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 59 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#76604C211yesmaximal, characteristic, complemented
2#69694C211yesmaximal, characteristic
3#68133C212yesmaximal
4C3×#1319C211yesmaximal, characteristic, complemented
5#67457C212yesmaximal, complemented
6#12130—31nomaximal, Sylow
7C2×#942—21no—
8#5510C2211yescharacteristic
9C3×#391C2211yescharacteristic
10#5229—21no—
11#4979C2212yes—
12#4929—21no—
13C2×#805—21no—
14#4738—22no—
15#4656—22no—
16#4646—22no—
17C3×#324C2212yes—
18#4530C2211yescharacteristic
19#4470—22no—
20#2811—31no—
21#2007—31no—
22#1604—32no—
23#1354—32no—
24#1319D311yescharacteristic, complemented
25C22×C3×Q8D411yescharacteristic
26C2×#235—21no—
27C22×Dic6—21no—
28C22×Dic6D412yes—
29C2×C3×C4⋊C4C2311yescharacteristic
30C2×C3×C42D411yescharacteristic
31C2×#216—41no—
32#942—42no—
33#942—42no—
34C2×#215—41no—
35#930—22no—
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3—21no—
38C2×Dic3⋊C4—22no—
39C2×C4×Dic3—21no—
40C2×#208—21no—
41C2×#204—42no—
42#897—41no—
43#895—41no—
44C3×C22⋊C8—22no—
45C3×#95—22no—
46C3⋊#95—22no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49C3⋊#95D412yes—
50#816—42no—
51#805—42no—
52C2×#137—61no—
53#537—31no—
54#448—62no—
55C2×#104—61no—
56#395—32no—
57#395—62no—
58#391—62no—
59#391—61no—
60#391—61no—
61#391C2×D311yescharacteristic
62#343—31no—
63#327—62no—
64#324C2×D312yes—
65C2×C3×Q8—21no—
66C2×C3×Q8—41no—
67C2×C3×Q8—41no—
68C2×C3×Q8SD1612yescomplemented
69C22×C3×C4—21no—
70C22×C3×C4—21no—
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4C2×D412yes—
73C2×#67—41no—
74C22×Dic3—22no—
75C22×Dic3—22no—
76#235—41no—
77#235—41no—
78#235—41no—
79#235—41no—
80C2×D12—21no—
81C2×D12—22no—
82C2×C4×D3—41no—
83C2×Dic6—21no—
84C2×Dic6—24no—
85C2×Dic6—42no—
86C2×Dic6—82no—
87C2×Dic6—81no—
88C2×Dic6—81no—
89C2×C3×C8—42no—
90C3×C4⋊C4—41no—
91C3×C42—41no—
92#216—44no—
93#215—44no—
94(C2×D3)⋊C4—42no—
95C4⋊Dic3—22no—
96C4⋊Dic3—81no—
97C4⋊Dic3—81no—
98Dic3⋊C4—82no—
99C4×Dic3—81no—
100#208—22no—
101#208—41no—
102C2×C3⋊C8—41no—
103#204—84no—
104C2×#41—61no—
105C22×Q8—32no—
106C22×Q8—61no—
107C22×Q8D1211yescharacteristic
108#137—122no—
109#137—122no—
110C2×Q16—121no—
111C2×SD16—122no—
112C2×SD16—121no—
113C2×M16—61no—
114C22⋊Q8—121no—
115#121—62no—
116#121—121no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—61no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4C22×D311yescharacteristic
121C2×C42—61no—
122C2×C42D1211yescharacteristic
123#104—122no—
124#103—122no—
125C22⋊C8—22no—
126#95—22no—
127#95—32no—
128#95—62no—
129#95—62no—
130#95—62no—
131C23×C3#12011yescharacteristic
132C22×D3—41no—
133C3×Q8—42no—
134C3×Q8—81no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—41no—
141C2×C3×C4—41no—
142C2×C3×C4—41no—
143C2×C3×C4—41no—
144C2×C3×C4—42no—
145C2×C3×C4C2×SD1611yescharacteristic
146C2×C3×C4#13711yescharacteristic
147#67—81no—
148#67—81no—
149C2×Dic3—24no—
150C2×Dic3—42no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153C2×Dic3—42no—
154C2×Dic3—41no—
155C2×Dic3—42no—
156D12—41no—
157D12—41no—
158D12—82no—
159C4×D3—81no—
160C4×D3—81no—
161Dic6—41no—
162Dic6—41no—
163Dic6—44no—
164Dic6—84no—
165Dic6—82no—
166C3×C8—82no—
167C3⋊C8—42no—
168#41—121no—
169#41—121no—
170#41—121no—
171#41—121no—
172C2×Q8—21no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—64no—
176C2×Q8—61no—
177C2×Q8—122no—
178C2×Q8—242no—
179C2×Q8—241no—
180C2×Q8—241no—
181C2×Q8#20412yescomplemented
182C2×D4—61no—
183C2×D4—62no—
184C2×D4—121no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—62no—
188C22×C4—62no—
189C22×C4—121no—
190C22×C4C2×D1211yescharacteristic
191C22×C4D3×D412yes—
192Q16—124no—
193SD16—124no—
194SD16—244no—
195M16—62no—
196M16—121no—
197C2×C8—42no—
198C2×C8—121no—
199C4⋊C4—41no—
200C4⋊C4—62no—
201C4⋊C4—242no—
202C4⋊C4—241no—
203C4⋊C4—241no—
204C22⋊C4—122no—
205C42—41no—
206C42—241no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3#44811yescharacteristic
210C22×C3#45311yescharacteristic
211C22×C3B(4,𝔽2)11yescharacteristic
212C2×D3—81no—
213C2×D3—81no—
214C2×D3—81no—
215C3×C4—21no—
216C3×C4—21no—
217C3×C4—42no—
218C3×C4—42no—
219C3×C4—42no—
220Dic3—44no—
221Dic3—81no—
222Dic3—82no—
223C23—121no—
224C23#88211yescharacteristic
225Q8—42no—
226Q8—81no—
227Q8—121no—
228Q8—121no—
229Q8—124no—
230Q8—244no—
231Q8—242no—
232D4—121no—
233D4—121no—
234D4—242no—
235D4—241no—
236D4—241no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—64no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4—121no—
253C2×C4—122no—
254C2×C4—241no—
255C2×C4—241no—
256C2×C4C2×#20411yescharacteristic
257C2×C4#90911yescharacteristic
258C8—82no—
259C8—122no—
260C2×C3—21no—
261C2×C3—21no—
262C2×C3#159611yescharacteristic
263C2×C3#219911yescharacteristic
264C2×C3#220511yescharacteristic
265D3—161no—
266C22—21no—
267C22—21no—
268C22—241no—
269C22—241no—
270C22—241no—
271C22#473811yescharacteristic
272C22#474111yescharacteristic
273C22#474711yescharacteristic
274C4—21no—
275C4—21no—
276C4—42no—
277C4—42no—
278C4—42no—
279C4—124no—
280C4—241no—
281C4—242no—
282C3#1213011yescharacteristic, complemented, Sylow
283C2—21no—
284C2—21no—
285C2—481no—
286C2#6811411yescharacteristic
287C2#6915411yescharacteristic
288C2#6916911yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1319C2
socleC22×C3B(4,𝔽2)
prime radical#5510C22
hypercenter#1319D3
hypocenterC3#12130
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
1. #76604C2semidirect
4. C3×#1319C2semidirect
5. #67457C2semidirect
24. #1319D3semidirect
68. C2×C3×Q8SD16semidirect
181. C2×Q8#204semidirect
282. C3#12130semidirect

Maximal subgroups

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

subgroupindex
#766042
#696942
#681332
C3×#13192
#674572
#121303