Finite Group Databasebeta

← #10698321

Subgroups — #10698321

Order 768.

416 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 97 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#73397C211yesmaximal, characteristic
2#70792C212yesmaximal
3#69441C211yesmaximal, characteristic
4C3×#1274C211yesmaximal, characteristic
5#67746C212yesmaximal
6#10371—31nomaximal, Sylow
7#5470C2211yescharacteristic
8C2×C4×Dic6—21no—
9C3×#431C2212yes—
10C3×#376C2211yescharacteristic
11C4×C4⋊Dic3—21no—
12C4×C4⋊Dic3C2212yes—
13#4924C2211yescharacteristic
14#4661—22no—
15#4656—22no—
16#4523—21no—
17#4523—21no—
18#4467—22no—
19#4467—22no—
20#2399—31no—
21#1995—31no—
22#1503—32no—
23#1303—32no—
24#1274D311yescharacteristic
25C22×Dic6—21no—
26C2×C3×C42D411yescharacteristic
27C2×C3×C42D411yescharacteristic
28C2×C3×C42C2311yescharacteristic
29#926—21no—
30C2×C4⋊Dic3—22no—
31C2×C4⋊Dic3—21no—
32C2×C4⋊Dic3—22no—
33#924—22no—
34C2×Dic3⋊C4—22no—
35C2×C4×Dic3—22no—
36C2×#208—21no—
37#870—21no—
38#869Q812yescomplemented
39C4×Dic6—22no—
40C4×Dic6—42no—
41C4×Dic6—42no—
42C3×C4⋊C8—22no—
43C3×C22⋊C8—22no—
44C3×C4×C8—22no—
45C3×#95—22no—
46C3⋊#95—21no—
47C3⋊#95—21no—
48C3⋊#95D412yes—
49C3⋊#95D412yes—
50C4⋊(C3⋊C8)—22no—
51#533—31no—
52C2×C4×Q8—61no—
53#431C2×D312yes—
54#423—31no—
55#398—62no—
56#395—62no—
57C4×C4⋊C4—32no—
58C4×C4⋊C4—61no—
59#376C2×D311yescharacteristic
60#339—61no—
61#339—61no—
62#328—62no—
63#328—62no—
64C22×C3×C4—21no—
65C22×C3×C4—21no—
66C22×C3×C4C2×D411yescharacteristic
67C22×C3×C4C2×D412yes—
68C22×Dic3—22no—
69C22×Dic3—22no—
70C2×Dic6—22no—
71C2×Dic6—44no—
72C2×Dic6—42no—
73C2×C3×C8—42no—
74C3×C42—21no—
75C3×C42—21no—
76C3×C42SD1612yes—
77C3×C42Q1612yes—
78C3×C42C2×Q811yescharacteristic
79C3×C42#4112yes—
80C3×C42#4111yescharacteristic
81#217—41no—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—44no—
85C4⋊Dic3—44no—
86C4⋊Dic3—42no—
87Dic3⋊C4—44no—
88Dic3⋊C4—42no—
89C4×Dic3—41no—
90C4×Dic3—42no—
91C4×Dic3—42no—
92#208—41no—
93#208—41no—
94C2×C3⋊C8—41no—
95C22×Q8—61no—
96C2×M16—61no—
97C4⋊Q8—32no—
98#125—61no—
99C22⋊Q8—62no—
100C4×Q8—62no—
101C4×Q8—122no—
102C4×Q8—122no—
103#117—61no—
104C2×C4⋊C4—62no—
105C2×C4⋊C4—61no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—62no—
108C2×C42—62no—
109C2×C42D1211yescharacteristic
110C2×C42D1211yescharacteristic
111C2×C42C22×D311yescharacteristic
112C4⋊C8—22no—
113C4⋊C8—62no—
114C22⋊C8—22no—
115C4×C8—22no—
116#95—22no—
117#95—32no—
118#95—32no—
119#95—61no—
120#95—61no—
121C23×C3#12011yescharacteristic
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×C3×C4C22⋊Q812yes—
128C2×C3×C4C22⋊Q811yescharacteristic
129C2×C3×C4#12411yescharacteristic
130C2×C3×C4#12412yes—
131C2×C3×C4C2×SD1611yescharacteristic
132C2×C3×C4C2×Q1611yescharacteristic
133C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
134C2×C3×C4#13711yescharacteristic
135C2×Dic3—24no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—42no—
139C2×Dic3—42no—
140C2×Dic3—42no—
141C2×Dic3—42no—
142Dic6—48no—
143Dic6—82no—
144C3×C8—82no—
145C3⋊C8—81no—
146C2×Q8—62no—
147C2×Q8—122no—
148C2×Q8—124no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—62no—
152C22×C4—62no—
153C22×C4C2×D1211yescharacteristic
154C22×C4D3×D412yes—
155M16—121no—
156M16—121no—
157C2×C8—42no—
158C2×C8—121no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—124no—
162C4⋊C4—122no—
163C4⋊C4—124no—
164C4⋊C4—124no—
165C4⋊C4—122no—
166C22⋊C4—121no—
167C42—21no—
168C42—21no—
169C42—121no—
170C42—122no—
171C42—122no—
172C42#20412yes—
173C42Dic1212yes—
174C42#23512yes—
175C42#23711yescharacteristic
176C42D3×Q811yescharacteristic
177C22×C3—22no—
178C22×C3#39311yescharacteristic
179C22×C3#45011yescharacteristic
180C22×C3#45111yescharacteristic
181C3×C4—22no—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4—42no—
185C3×C4#47511yescharacteristic
186C3×C4#47711yescharacteristic
187C3×C4#48711yescharacteristic
188C3×C4#48911yescharacteristic
189Dic3—44no—
190Dic3—82no—
191Dic3—82no—
192C23#88211yescharacteristic
193Q8—128no—
194Q8—242no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—64no—
201C2×C4—122no—
202C2×C4—122no—
203C2×C4—122no—
204C2×C4—122no—
205C2×C4—122no—
206C2×C4—122no—
207C2×C4#87511yescharacteristic
208C2×C4#88512yes—
209C2×C4#89612yes—
210C2×C4#89711yescharacteristic
211C2×C4C2×#20411yescharacteristic
212C2×C4C2×Dic1211yescharacteristic
213C2×C4#90811yescharacteristic
214C2×C4#90911yescharacteristic
215C8—82no—
216C8—241no—
217C2×C3—21no—
218C2×C3—21no—
219C2×C3#202411yescharacteristic
220C2×C3#203112yes—
221C22—22no—
222C22#467311yescharacteristic
223C22#473111yescharacteristic
224C22#473911yescharacteristic
225C4—22no—
226C4—22no—
227C4—42no—
228C4—42no—
229C4—124no—
230C4—242no—
231C4—242no—
232C4#468111yescharacteristic
233C4#471411yescharacteristic
234C4#485011yescharacteristic
235C4#485111yescharacteristic
236C3#1037111yescharacteristic, complemented, Sylow
237C2—21no—
238C2—21no—
239C2#6868911yescharacteristic
240C2#6871412yes—
subgroupG/H
centerC2×C4#875
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1274D3
hypocenterC3#10371
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
38. #869Q8semidirect
236. C3#10371semidirect

Maximal subgroups

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

subgroupindex
#733972
#707922
#694412
C3×#12742
#677462
#103713