Finite Group Databasebeta

← #10696525

Subgroups — #10696525

Order 768.

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

normal subgroups only

This group has 500 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
1C2×#4702C212yesmaximal
2#74217C211yesmaximal, characteristic
3C2×#4465C212yesmaximal
4#69552C211yesmaximal, characteristic
5C3×#1295C211yesmaximal, characteristic
6#10087—31nomaximal, Sylow
7C2×#870C2212yes—
8C2×#869C2211yescharacteristic
9C2×C3×#97C2212yes—
10C3×#382C2211yescharacteristic
11C2×#818—22no—
12C2×#817—22no—
13#4981—22no—
14C2×#810—22no—
15C2×#803C2211yescharacteristic
16#4702—24no—
17#4702—24no—
18#4661—22no—
19#4659—22no—
20#4465—24no—
21C2×#501—32no—
22#2822—31no—
23#1960—31no—
24C2×#330—32no—
25#1295D311yescharacteristic
26C22×Dic6—21no—
27C22×C3×C8—22no—
28C2×C3×C4⋊C4D412yes—
29C2×C3×C42C2311yescharacteristic
30C2×C4⋊Dic3—22no—
31C2×C4⋊Dic3—22no—
32C2×C4⋊Dic3D412yes—
33C2×C4⋊Dic3D412yes—
34C2×Dic3⋊C4—22no—
35C2×Dic3⋊C4—22no—
36C22×C3⋊C8—21no—
37#870—22no—
38#870—22no—
39#870—22no—
40#870—22no—
41#869—22no—
42#869—21no—
43#869D412yes—
44C3×#97—22no—
45C3×#97D414yes—
46C3×#95—22no—
47C3⋊#95—22no—
48C3⋊#95—24no—
49#818—28no—
50#817—28no—
51#810—44no—
52#803—22no—
53C2×C4⋊Q8—31no—
54C2×#125—32no—
55#501—64no—
56#501—64no—
57C2×#107—62no—
58C2×#106—62no—
59C2×#103—62no—
60C2×#97—31no—
61C2×#97C2×D312yes—
62#400—62no—
63#398—62no—
64#398—62no—
65#382C2×D311yescharacteristic
66#330—64no—
67C22×C3×C4—22no—
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4C2×D412yes—
70C22×Dic3—22no—
71C22×Dic3—22no—
72C22×Dic3—41no—
73C2×Dic6—22no—
74C2×Dic6—42no—
75C2×Dic6—42no—
76C2×Dic6—41no—
77C2×C3×C8—28no—
78C2×C3×C8—24no—
79C3×C4⋊C4—42no—
80C3×C42C2×D412yes—
81C3×C42C2×D411yescharacteristic
82C3×C42#4111yescharacteristic
83C4⋊Dic3—22no—
84C4⋊Dic3—24no—
85C4⋊Dic3—22no—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88C4⋊Dic3—42no—
89C4⋊Dic3—82no—
90Dic3⋊C4—42no—
91Dic3⋊C4—42no—
92Dic3⋊C4—42no—
93Dic3⋊C4—42no—
94C2×C3⋊C8—22no—
95C2×C3⋊C8—41no—
96C2×C3⋊C8—41no—
97C22×Q8—61no—
98C22×C8—22no—
99C22×C8—61no—
100C4⋊Q8—32no—
101C4⋊Q8—62no—
102C4⋊Q8—61no—
103#125—62no—
104#125—62no—
105#125—62no—
106#125—62no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—62no—
113C2×C4⋊C4D1212yes—
114C2×C42C22×D311yescharacteristic
115#107—68no—
116#106—68no—
117#103—124no—
118#97—22no—
119#97—62no—
120#97D1214yes—
121#95—22no—
122#95—64no—
123#95—62no—
124C23×C3#12011yescharacteristic
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—24no—
128C2×C3×C4—42no—
129C2×C3×C4—44no—
130C2×C3×C4#12012yes—
131C2×C3×C4#12111yescharacteristic
132C2×C3×C4#12112yes—
133C2×C3×C4#12711yescharacteristic
134C2×Dic3—42no—
135C2×Dic3—42no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—42no—
139C2×Dic3—44no—
140C2×Dic3—81no—
141C2×Dic3—81no—
142Dic6—44no—
143Dic6—82no—
144C3×C8—28no—
145C3⋊C8—42no—
146C2×Q8—62no—
147C2×Q8—122no—
148C2×Q8—122no—
149C2×Q8—121no—
150C22×C4—22no—
151C22×C4—62no—
152C22×C4—62no—
153C22×C4—121no—
154C22×C4C2×D1211yescharacteristic
155C22×C4D3×D412yes—
156C2×C8—28no—
157C2×C8—24no—
158C2×C8—62no—
159C2×C8—121no—
160C2×C8—121no—
161C4⋊C4—42no—
162C4⋊C4—62no—
163C4⋊C4—64no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C4⋊C4—62no—
167C4⋊C4—122no—
168C4⋊C4—122no—
169C4⋊C4—122no—
170C4⋊C4—122no—
171C4⋊C4—122no—
172C4⋊C4—242no—
173C42C2×D1212yes—
174C42D3×D411yescharacteristic
175C42#23911yescharacteristic
176C22×C3#39211yescharacteristic
177C22×C3#45311yescharacteristic
178C22×C3#45411yescharacteristic
179C22×C3#45614yes—
180C3×C4—22no—
181C3×C4—22no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—82no—
185Dic3—82no—
186Dic3—82no—
187Dic3—82no—
188C23#88211yescharacteristic
189Q8—124no—
190Q8—242no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—24no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—122no—
197C2×C4—122no—
198C2×C4—122no—
199C2×C4—122no—
200C2×C4—122no—
201C2×C4—124no—
202C2×C4—241no—
203C2×C4—241no—
204C2×C4#87411yescharacteristic
205C2×C4#88212yes—
206C2×C4#89512yes—
207C2×C4#89511yescharacteristic
208C8—28no—
209C8—122no—
210C2×C3#166012yes—
211C2×C3#166312yes—
212C2×C3#200512yes—
213C2×C3#200611yescharacteristic
214C22#467111yescharacteristic
215C22#474711yescharacteristic
216C22#475311yescharacteristic
217C22#482614yes—
218C4—21no—
219C4—21no—
220C4—22no—
221C4—22no—
222C4—82no—
223C4—242no—
224C4—242no—
225C4—242no—
226C3#1008711yescharacteristic, complemented, Sylow
227C2#6835512yes—
228C2#6837712yes—
229C2#6876812yes—
230C2#6900811yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socle, prime radicalC23×C3#120
hypercenter#1295D3
hypocenterC3#10087
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
226. C3#10087semidirect

Maximal subgroups

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

subgroupindex
C2×#47022
#742172
C2×#44652
#695522
C3×#12952
#100873