Finite Group Databasebeta

← #10698686

Subgroups — #10698686

Order 768.

402 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 91 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
2C3×#1847C211yesmaximal, characteristic
3C2×#4465C212yesmaximal
4#69583C212yesmaximal
5#67630C211yesmaximal, characteristic
6#10429—31nomaximal, Sylow
7C2×#870C2212yes—
8C2×C3×#107—21no—
9C2×C3×#106—21no—
10C2×C3×#97C2211yescharacteristic
11C3×#384C2211yescharacteristic
12C3×#382C2211yescharacteristic
13#4981—21no—
14#4981—21no—
15C2×#803C2212yes—
16#4661—22no—
17#4659—22no—
18#4529—22no—
19#4529—22no—
20#4465—24no—
21#2822—31no—
22#1992—32no—
23#1847D311yescharacteristic
24C2×#330—32no—
25#1295—31no—
26C22×C3×C8—21no—
27C2×C3×C4⋊C4—22no—
28C2×C3×C4⋊C4—21no—
29C2×C3×C4⋊C4D411yescharacteristic
30C2×C3×C4⋊C4D411yescharacteristic
31C2×C3×C42C2311yescharacteristic
32C2×C4⋊Dic3—21no—
33C2×C4⋊Dic3—21no—
34C2×C4⋊Dic3D412yes—
35C2×C4⋊Dic3D412yes—
36C2×Dic3⋊C4—22no—
37C2×Dic3⋊C4—22no—
38C22×C3⋊C8—22no—
39#870—22no—
40#870—22no—
41#870—22no—
42#870—22no—
43C3×#107—42no—
44C3×#106—42no—
45C3×#97—21no—
46C3×#97D412yes—
47C3×#95—22no—
48C3⋊#95—21no—
49C3⋊#95—21no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52#803—22no—
53#803—22no—
54C2×#125—32no—
55C2×#107—21no—
56C2×#106—21no—
57C2×#97—32no—
58C2×#97C2×D311yescharacteristic
59#400—62no—
60#398—62no—
61#398—61no—
62#398—61no—
63#384C2×D311yescharacteristic
64#382C2×D311yescharacteristic
65#340—62no—
66#340—62no—
67#330—64no—
68C22×C3×C4—21no—
69C22×C3×C4—21no—
70C22×C3×C4—41no—
71C22×C3×C4C2×D412yes—
72C22×C3×C4C2×D411yescharacteristic
73C22×Dic3—22no—
74C22×Dic3—22no—
75C2×C3×C8—22no—
76C2×C3×C8—22no—
77C2×C3×C8—22no—
78C3×C4⋊C4—41no—
79C3×C4⋊C4—41no—
80C3×C4⋊C4—41no—
81C3×C4⋊C4—41no—
82C3×C4⋊C4—44no—
83C3×C42C2×D411yescharacteristic
84C3×C42#4111yescharacteristic
85C3×C42#4112yes—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90C4⋊Dic3—81no—
91C4⋊Dic3—81no—
92Dic3⋊C4—42no—
93Dic3⋊C4—42no—
94Dic3⋊C4—42no—
95Dic3⋊C4—42no—
96C2×C3⋊C8—24no—
97C2×C3⋊C8—42no—
98C2×C3⋊C8—42no—
99C22×C8—21no—
100C22×C8—62no—
101#125—62no—
102#125—62no—
103#125—62no—
104#125—62no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4—61no—
113C2×C4⋊C4#6711yescharacteristic
114C2×C4⋊C4#6711yescharacteristic
115C2×C42C22×D311yescharacteristic
116#107—42no—
117#106—42no—
118#97—21no—
119#97—62no—
120#97—62no—
121#97D1212yes—
122#95—22no—
123#95—62no—
124#95—62no—
125#95—61no—
126#95—61no—
127C23×C3#12011yescharacteristic
128C2×C3×C4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—41no—
133C2×C3×C4—41no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—81no—
137C2×C3×C4—82no—
138C2×C3×C4#12112yes—
139C2×C3×C4#12111yescharacteristic
140C2×C3×C4#12312yes—
141C2×C3×C4#12411yescharacteristic
142C2×Dic3—42no—
143C2×Dic3—42no—
144C2×Dic3—42no—
145C2×Dic3—42no—
146C2×Dic3—42no—
147C2×Dic3—42no—
148C3×C8—24no—
149C3⋊C8—44no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—62no—
154C22×C4—62no—
155C22×C4D3×D412yes—
156C22×C4C2×#6711yescharacteristic
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—64no—
161C2×C8—122no—
162C2×C8—122no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—44no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C4⋊C4—122no—
174C4⋊C4—122no—
175C4⋊C4—122no—
176C4⋊C4—241no—
177C4⋊C4—241no—
178C42C2×D1211yescharacteristic
179C42#23511yescharacteristic
180C42#23912yes—
181C22×C3#39411yescharacteristic
182C22×C3#45311yescharacteristic
183C22×C3#45412yes—
184C22×C3#45612yes—
185C22×C3#45611yescharacteristic
186C3×C4—22no—
187C3×C4—22no—
188C3×C4—21no—
189C3×C4—21no—
190C3×C4—81no—
191C3×C4—81no—
192C3×C4—161no—
193Dic3—82no—
194Dic3—82no—
195C23#93711yescharacteristic
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—82no—
205C2×C4—81no—
206C2×C4—122no—
207C2×C4—122no—
208C2×C4—122no—
209C2×C4—122no—
210C2×C4—122no—
211C2×C4—122no—
212C2×C4#89412yes—
213C2×C4#89512yes—
214C2×C4#93011yescharacteristic
215C2×C4#94711yescharacteristic
216C8—24no—
217C8—124no—
218C2×C3#165711yescharacteristic
219C2×C3#166011yescharacteristic
220C2×C3#166312yes—
221C2×C3#203812yes—
222C2×C3#203911yescharacteristic
223C22#499111yescharacteristic
224C22#507711yescharacteristic
225C22#509511yescharacteristic
226C22#520112yes—
227C22#520512yes—
228C4—22no—
229C4—22no—
230C4—21no—
231C4—21no—
232C4—81no—
233C4—81no—
234C4—161no—
235C4—242no—
236C4—242no—
237C3#1042911yescharacteristic, complemented, Sylow
238C2#6959311yescharacteristic
239C2#7067812yes—
240C2#7132811yescharacteristic
241C2#7133011yescharacteristic
242C2#7139312yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1847C2
socle, prime radicalC23×C3#120
hypercenter#1847D3
hypocenterC3#10429
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
237. C3#10429semidirect

Maximal subgroups

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

subgroupindex
#742172
C3×#18472
C2×#44652
#695832
#676302
#104293