Finite Group Databasebeta

← #10698317

Subgroups — #10698317

Order 768.

416 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C3×#2399C211yesmaximal, characteristic
2#69471C212yesmaximal
3#69441C211yesmaximal, characteristic
4#67620C212yesmaximal
5#67614C211yesmaximal, characteristic
6#10371—31nomaximal, Sylow
7C2×C3×C4×Q8—21no—
8#5470C2211yescharacteristic
9C3×#398—22no—
10C3×#395—22no—
11C3×C4×C4⋊C4—21no—
12C3×C4×C4⋊C4C2212yes—
13C3×#376C2211yescharacteristic
14#4925C2212yes—
15#4924C2211yescharacteristic
16#4523—21no—
17#4523—21no—
18#4452—22no—
19#4452—22no—
20#2399D311yescharacteristic
21#1995—31no—
22#1503—32no—
23#1303—32no—
24#1274—31no—
25C22×C3×Q8—21no—
26C3×C4×Q8—42no—
27C3×C4×Q8—42no—
28C2×C3×C4⋊C4—22no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4—21no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C42—22no—
33C2×C3×C42D411yescharacteristic
34C2×C3×C42D411yescharacteristic
35C2×C3×C42C2311yescharacteristic
36#926—21no—
37#924—22no—
38C2×#208—21no—
39#870—21no—
40#869Q812yescomplemented
41C4×Dic6—22no—
42C3×#95—21no—
43C3×#95—21no—
44C3×#95—22no—
45C3×#95D412yes—
46C3×#95D412yes—
47#830—22no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)—22no—
50C4×C3⋊C8—22no—
51#533—31no—
52C2×C4×Q8—21no—
53#431—32no—
54#423—31no—
55#398—22no—
56#395—22no—
57C4×C4⋊C4—21no—
58C4×C4⋊C4C2×D312yes—
59#376C2×D311yescharacteristic
60#339—61no—
61#339—61no—
62#328—62no—
63#328—62no—
64C2×C3×Q8—42no—
65C2×C3×Q8—44no—
66C22×C3×C4—21no—
67C22×C3×C4—21no—
68C22×C3×C4—22no—
69C22×C3×C4—22no—
70C22×C3×C4C2×D412yes—
71C22×C3×C4C2×D411yescharacteristic
72C2×Dic6—22no—
73C3×C4⋊C4—44no—
74C3×C4⋊C4—44no—
75C3×C4⋊C4—44no—
76C3×C4⋊C4—42no—
77C3×C42—21no—
78C3×C42—21no—
79C3×C42—42no—
80C3×C42—42no—
81C3×C42SD1612yes—
82C3×C42Q1612yes—
83C3×C42C2×Q811yescharacteristic
84C3×C42#4112yes—
85C3×C42#4111yescharacteristic
86#217—41no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89Dic3⋊C4—42no—
90C4×Dic3—41no—
91#208—41no—
92#208—41no—
93C2×C3⋊C8—42no—
94C2×C3⋊C8—41no—
95C22×Q8—21no—
96C2×M16—61no—
97C4⋊Q8—32no—
98#125—61no—
99C22⋊Q8—62no—
100C4×Q8—42no—
101C4×Q8—42no—
102C4×Q8—62no—
103#117—61no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C42—22no—
109C2×C42#6711yescharacteristic
110C2×C42#6711yescharacteristic
111C2×C42C22×D311yescharacteristic
112C4⋊C8—62no—
113C4⋊C8—62no—
114C22⋊C8—62no—
115C4×C8—62no—
116#95—21no—
117#95—21no—
118#95—22no—
119#95#6712yes—
120#95#6712yes—
121C23×C3#12011yescharacteristic
122C3×Q8—48no—
123C2×C3×C4—24no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4—42no—
129C2×C3×C4—42no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4C22⋊Q812yes—
135C2×C3×C4C22⋊Q811yescharacteristic
136C2×C3×C4#12412yes—
137C2×C3×C4#12411yescharacteristic
138C2×C3×C4C2×SD1611yescharacteristic
139C2×C3×C4C2×Q1611yescharacteristic
140C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
141C2×C3×C4#13711yescharacteristic
142C2×Dic3—42no—
143Dic6—82no—
144C3⋊C8—82no—
145C3⋊C8—81no—
146C2×Q8—42no—
147C2×Q8—44no—
148C2×Q8—62no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4C2×#6712yes—
154C22×C4C2×#6711yescharacteristic
155M16—121no—
156M16—121no—
157C2×C8—122no—
158C2×C8—121no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4—122no—
166C22⋊C4—121no—
167C42—21no—
168C42—21no—
169C42—42no—
170C42—42no—
171C42—121no—
172C42#21412yes—
173C42#21612yes—
174C42D3×Q811yescharacteristic
175C42#23911yescharacteristic
176C42#23912yes—
177C22×C3—22no—
178C22×C3#39311yescharacteristic
179C22×C3#45011yescharacteristic
180C22×C3#45111yescharacteristic
181C3×C4—22no—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4—44no—
185C3×C4—42no—
186C3×C4—82no—
187C3×C4#47511yescharacteristic
188C3×C4#47711yescharacteristic
189C3×C4#48711yescharacteristic
190C3×C4#48911yescharacteristic
191Dic3—82no—
192C23#95311yescharacteristic
193Q8—48no—
194Q8—242no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—122no—
207C2×C4#93211yescharacteristic
208C2×C4C2×#21411yescharacteristic
209C2×C4#94211yescharacteristic
210C2×C4C2×#21611yescharacteristic
211C2×C4#94612yes—
212C2×C4#94611yescharacteristic
213C2×C4#94712yes—
214C2×C4#94711yescharacteristic
215C8—242no—
216C8—241no—
217C2×C3—21no—
218C2×C3—21no—
219C2×C3#202411yescharacteristic
220C2×C3#203112yes—
221C22—22no—
222C22#521811yescharacteristic
223C22#522811yescharacteristic
224C22#523111yescharacteristic
225C4—22no—
226C4—22no—
227C4—44no—
228C4—42no—
229C4—42no—
230C4—82no—
231C4—242no—
232C4#508911yescharacteristic
233C4#509111yescharacteristic
234C4#509311yescharacteristic
235C4#509411yescharacteristic
236C3#1037111yescharacteristic, complemented, Sylow
237C2—21no—
238C2—21no—
239C2#7179411yescharacteristic
240C2#7184212yes—
subgroupG/H
centerC2×C4#947
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2399C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#2399D3
hypocenterC3#10371
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
40. #869Q8semidirect
236. C3#10371semidirect

Maximal subgroups

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

subgroupindex
C3×#23992
#694712
#694412
#676202
#676142
#103713