Finite Group Databasebeta

← #10714873

Subgroups — #10714873

Order 768.

416 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 81 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
2#70102C212yesmaximal
3#69672C211yesmaximal, characteristic
4C3×#1394C211yesmaximal, characteristic
5#67699C212yesmaximal
6#13464—31nomaximal, Sylow
7#5812C2211yescharacteristic
8C2×C3×#101C2212yes—
9C3×#391C2211yescharacteristic
10#5229C2212yes—
11C2×#836C2211yescharacteristic
12#5121—22no—
13#4979—21no—
14#4964—21no—
15#4929—21no—
16#4656—22no—
17#4646—22no—
18#4555—22no—
19#4523—21no—
20#2811—31no—
21#1857—32no—
22#1857—31no—
23#1394—32no—
24#1394D311yescharacteristic
25C22×C3×Q8C2311yescharacteristic
26C22×Dic6—21no—
27C22×Dic6—21no—
28C2×C3×M16—22no—
29C2×C3×C4⋊C4D411yescharacteristic
30C2×C3×C42D411yescharacteristic
31Q8×Dic3—41no—
32Q8×Dic3—41no—
33#926—22no—
34#926—21no—
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3D412yes—
37C2×Dic3⋊C4—22no—
38C2×C4×Dic3D412yes—
39C2×#208—21no—
40C3×#101—22no—
41C3×#101Q814yescomplemented
42C3×#95—22no—
43#836—21no—
44#836Q812yescomplemented
45C3⋊#95—21no—
46C3⋊#95—22no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49#818—42no—
50#817—42no—
51#808—41no—
52#807—41no—
53#519—31no—
54#428—62no—
55#428—61no—
56C2×#101—31no—
57C2×#101C2×D312yes—
58#395—62no—
59#395—61no—
60#391—32no—
61#391—62no—
62#391—61no—
63#391C2×D311yescharacteristic
64#339—62no—
65#339—61no—
66C2×C3×Q8—41no—
67C2×C3×Q8—41no—
68C2×C3×Q8C2×Q812yes—
69C2×C3×Q8C2×Q811yescharacteristic
70C2×C3×Q8#4111yescharacteristic
71C22×C3×C4—21no—
72C22×C3×C4—21no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D412yes—
75C22×Dic3—22no—
76C22×Dic3—22no—
77C2×Dic6—24no—
78C2×Dic6—81no—
79C2×Dic6—81no—
80C2×Dic6—82no—
81C2×Dic6—81no—
82C3×M16—24no—
83C3×M16—42no—
84C2×C3×C8—42no—
85C3×C4⋊C4—41no—
86C3×C42—41no—
87#217—41no—
88C4⋊Dic3—21no—
89C4⋊Dic3—21no—
90C4⋊Dic3—22no—
91C4⋊Dic3—22no—
92C4⋊Dic3—42no—
93C4⋊Dic3—81no—
94Dic3⋊C4—82no—
95C4×Dic3—41no—
96C4×Dic3—42no—
97C4×Dic3—42no—
98#208—22no—
99#208—41no—
100C2×C3⋊C8—41no—
101C22×Q8—61no—
102C22×Q8—61no—
103C22×Q8C22×D311yescharacteristic
104C2×M16—22no—
105C2×M16—61no—
106C4×Q8—121no—
107C4×Q8—121no—
108#117—61no—
109#117—62no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—61no—
113C2×C4⋊C4D1211yescharacteristic
114C2×C42—32no—
115C2×C42D1211yescharacteristic
116#107—121no—
117#107—122no—
118#106—121no—
119#106—122no—
120#101—22no—
121#101—32no—
122#101—61no—
123#101Dic614yescomplemented
124#95—22no—
125#95—62no—
126#95—61no—
127#95—62no—
128#95—62no—
129C23×C3#12011yescharacteristic
130C3×Q8—42no—
131C3×Q8—41no—
132C3×Q8—41no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—41no—
139C2×C3×C4—41no—
140C2×C3×C4—42no—
141C2×C3×C4—41no—
142C2×C3×C4C22⋊Q812yes—
143C2×C3×C4C22⋊Q811yescharacteristic
144C2×C3×C4C4⋊Q811yescharacteristic
145C2×C3×C4C4⋊Q812yes—
146C2×Dic3—24no—
147C2×Dic3—42no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—42no—
151C2×Dic3—41no—
152C2×Dic3—41no—
153C2×Dic3—42no—
154Dic6—84no—
155Dic6—84no—
156C3×C8—44no—
157C3⋊C8—42no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—64no—
161C2×Q8—241no—
162C2×Q8—241no—
163C2×Q8—242no—
164C2×Q8—241no—
165C2×Q8C2×Dic612yes—
166C2×Q8#23711yescharacteristic
167C2×Q8D3×Q811yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4C2×D1211yescharacteristic
173C22×C4D3×D412yes—
174M16—24no—
175M16—42no—
176M16—62no—
177M16—121no—
178C2×C8—42no—
179C2×C8—121no—
180C4⋊C4—41no—
181C4⋊C4—61no—
182C4⋊C4—62no—
183C4⋊C4—61no—
184C4⋊C4—62no—
185C4⋊C4—122no—
186C4⋊C4—242no—
187C4⋊C4—241no—
188C22⋊C4—121no—
189C42—41no—
190C42—122no—
191C42—121no—
192C42—122no—
193C22×C3—21no—
194C22×C3—21no—
195C22×C3#39511yescharacteristic
196C22×C3B(4,𝔽2)11yescharacteristic
197C22×C3#45811yescharacteristic
198C3×C4—21no—
199C3×C4—22no—
200C3×C4—21no—
201C3×C4—22no—
202C3×C4—42no—
203C3×C4—81no—
204Dic3—44no—
205Dic3—82no—
206Dic3—81no—
207Dic3—81no—
208C23#88211yescharacteristic
209Q8—42no—
210Q8—41no—
211Q8—41no—
212Q8—244no—
213Q8—244no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—64no—
224C2×C4—122no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4—122no—
228C2×C4—121no—
229C2×C4—121no—
230C2×C4—122no—
231C2×C4#86911yescharacteristic
232C2×C4#87812yes—
233C2×C4#88812yes—
234C2×C4#89711yescharacteristic
235C8—44no—
236C8—122no—
237C2×C3—21no—
238C2×C3—21no—
239C2×C3#206211yescharacteristic
240C2×C3#220211yescharacteristic
241C2×C3#220511yescharacteristic
242C22—21no—
243C22—21no—
244C22#465611yescharacteristic
245C22#474111yescharacteristic
246C22#474211yescharacteristic
247C4—22no—
248C4—21no—
249C4—21no—
250C4—22no—
251C4—42no—
252C4—81no—
253C4—124no—
254C4—242no—
255C4—241no—
256C4—241no—
257C3#1346411yescharacteristic, complemented, Sylow
258C2—21no—
259C2—21no—
260C2#6895811yescharacteristic
261C2#6916011yescharacteristic
262C2#6916911yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC22×C3×Q8C23
FrattiniC22×Q8C22×D3
FittingC3×#1394C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1394D3
hypocenterC3#13464
LCSG ▷ C22×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
41. C3×#101Q8semidirect
44. #836Q8semidirect
123. #101Dic6semidirect
257. C3#13464semidirect

Maximal subgroups

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

subgroupindex
#766042
#701022
#696722
C3×#13942
#676992
#134643