Finite Group Databasebeta

← #10698682

Subgroups — #10698682

Order 768.

386 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 89 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#74217C211yesmaximal, characteristic
2#69827C212yesmaximal
3C2×#4465C212yesmaximal
4#69548C211yesmaximal, characteristic
5C3×#1295C211yesmaximal, characteristic
6#10428—31nomaximal, Sylow
7C2×#870C2212yes—
8C2×C3×#97C2212yes—
9C3×#382C2211yescharacteristic
10#4981—22no—
11#4935C2211yescharacteristic
12C2×#803C2211yescharacteristic
13#4661—22no—
14#4659—22no—
15#4551—22no—
16#4551—22no—
17#4529—22no—
18#4465—24no—
19#2822—31no—
20#1992—32no—
21#1836—31no—
22C2×#330—32no—
23#1295D311yescharacteristic
24C22×C3×C8—22no—
25C2×C3×C4⋊C4D412yes—
26C2×C3×C42C2311yescharacteristic
27C2×C4⋊Dic3—21no—
28C2×C4⋊Dic3—22no—
29C2×C4⋊Dic3—22no—
30C2×C4⋊Dic3D412yes—
31C2×C4⋊Dic3D412yes—
32C2×Dic3⋊C4—22no—
33C2×Dic3⋊C4—22no—
34C22×C3⋊C8—21no—
35#870—22no—
36#870—22no—
37#870—22no—
38#870—22no—
39C3×#97—22no—
40C3×#97—22no—
41C3×#95—22no—
42C3⋊#95—22no—
43C3⋊#95—24no—
44#803—21no—
45#803D412yes—
46C2×#125—32no—
47C2×#97—31no—
48C2×#97C2×D312yes—
49#400—62no—
50#398—62no—
51#398—62no—
52#384—31no—
53#382C2×D311yescharacteristic
54#340—62no—
55#340—62no—
56#340—62no—
57#330—64no—
58C22×C3×C4—22no—
59C22×C3×C4C2×D411yescharacteristic
60C22×C3×C4C2×D412yes—
61C22×Dic3—22no—
62C22×Dic3—22no—
63C22×Dic3—41no—
64C2×C3×C8—24no—
65C2×C3×C8—42no—
66C2×C3×C8—42no—
67C3×C4⋊C4—42no—
68C3×C42C2×D411yescharacteristic
69C3×C42#4112yes—
70C3×C42#4111yescharacteristic
71C4⋊Dic3—22no—
72C4⋊Dic3—22no—
73C4⋊Dic3—22no—
74C4⋊Dic3—22no—
75C4⋊Dic3—42no—
76C4⋊Dic3—44no—
77C4⋊Dic3—82no—
78Dic3⋊C4—42no—
79Dic3⋊C4—42no—
80Dic3⋊C4—42no—
81Dic3⋊C4—42no—
82C2×C3⋊C8—22no—
83C2×C3⋊C8—41no—
84C2×C3⋊C8—41no—
85C22×C8—22no—
86C22×C8—61no—
87#125—62no—
88#125—62no—
89#125—62no—
90#125—62no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—62no—
94C2×C4⋊C4—62no—
95C2×C4⋊C4—61no—
96C2×C4⋊C4—62no—
97C2×C4⋊C4—62no—
98C2×C4⋊C4D1212yes—
99C2×C42C22×D311yescharacteristic
100#97—22no—
101#97—22no—
102#97—32no—
103#97—61no—
104#95—22no—
105#95—64no—
106#95—62no—
107C23×C3#12011yescharacteristic
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—42no—
113C2×C3×C4—42no—
114C2×C3×C4—42no—
115C2×C3×C4#12112yes—
116C2×C3×C4#12111yescharacteristic
117C2×C3×C4#12312yes—
118C2×C3×C4#12411yescharacteristic
119C2×Dic3—42no—
120C2×Dic3—42no—
121C2×Dic3—42no—
122C2×Dic3—42no—
123C2×Dic3—42no—
124C2×Dic3—42no—
125C2×Dic3—81no—
126C2×Dic3—82no—
127C3×C8—44no—
128C3⋊C8—42no—
129C22×C4—22no—
130C22×C4—62no—
131C22×C4—62no—
132C22×C4—121no—
133C22×C4C2×D1211yescharacteristic
134C22×C4D3×D412yes—
135C2×C8—24no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—62no—
139C2×C8—121no—
140C2×C8—121no—
141C4⋊C4—42no—
142C4⋊C4—62no—
143C4⋊C4—62no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—122no—
147C4⋊C4—124no—
148C4⋊C4—122no—
149C4⋊C4—122no—
150C4⋊C4—122no—
151C4⋊C4—122no—
152C4⋊C4—242no—
153C42#23512yes—
154C42D3×D411yescharacteristic
155C42#23911yescharacteristic
156C22×C3#39411yescharacteristic
157C22×C3#45411yescharacteristic
158C22×C3#45412yes—
159C22×C3#45511yescharacteristic
160C22×C3#45612yes—
161C3×C4—22no—
162C3×C4—22no—
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—82no—
166Dic3—82no—
167Dic3—82no—
168Dic3—161no—
169C23#88211yescharacteristic
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—122no—
178C2×C4—122no—
179C2×C4—122no—
180C2×C4—122no—
181C2×C4—122no—
182C2×C4—122no—
183C2×C4—241no—
184C2×C4—242no—
185C2×C4#87511yescharacteristic
186C2×C4#88412yes—
187C2×C4#89412yes—
188C2×C4#89511yescharacteristic
189C8—44no—
190C8—122no—
191C2×C3#165811yescharacteristic
192C2×C3#166311yescharacteristic
193C2×C3#166412yes—
194C2×C3#203812yes—
195C2×C3#204011yescharacteristic
196C22#467211yescharacteristic
197C22#474611yescharacteristic
198C22#475311yescharacteristic
199C22#482612yes—
200C22#482712yes—
201C4—21no—
202C4—21no—
203C4—22no—
204C4—22no—
205C4—82no—
206C4—242no—
207C4—242no—
208C4—481no—
209C3#1042811yescharacteristic, complemented, Sylow
210C2#6835912yes—
211C2#6837511yescharacteristic
212C2#6837711yescharacteristic
213C2#6873412yes—
214C2#6897711yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socle, prime radicalC23×C3#120
hypercenter#1295D3
hypocenterC3#10428
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
209. C3#10428semidirect

Maximal subgroups

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

subgroupindex
#742172
#698272
C2×#44652
#695482
C3×#12952
#104283