Finite Group Databasebeta

← #10950940

Subgroups — #10950940

Order 768.

590 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 163 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×#5116C212yesmaximal
2C2×C3×C22⋊C16C211yesmaximal, characteristic
3#71416C214yesmaximal, complemented
4#69113C214yesmaximal, complemented
5#69099C214yesmaximal, complemented
6#33755—31nomaximal, Sylow
7C23×C3×C8C2211yescharacteristic
8C2×#924—22no—
9C22×Dic12C2212yes—
10C22×C3×C16C2212yes—
11#5116—22no—
12#5116—22no—
13#5116C2214yes—
14#5116C2214yes—
15C2×#818C2212yes—
16C2×#816—22no—
17C2×Dic24C2214yescomplemented
18C3×C22⋊C16C2214yes—
19#4504C2214yescomplemented
20#4503C2218yescomplemented
21C2×#467—32no—
22#2236—34no—
23#2216—34no—
24#2188—34no—
25C2×C22⋊C16D311yescharacteristic
26C23×C3×C4D411yescharacteristic
27C22×Dic6—22no—
28C22×C3×C8—42no—
29C22×C3×C8D412yes—
30C22×C3×C8D411yescharacteristic
31C22×C3×C8C2311yescharacteristic
32C22×C3×C8C2312yes—
33C2×#217—42no—
34C2×C4⋊Dic3—22no—
35#924—24no—
36#924—24no—
37#924—42no—
38#924—42no—
39C2×Dic3⋊C4—42no—
40C2×Dic12—22no—
41C2×Dic12—44no—
42C2×Dic12C2314yes—
43C2×C3×C16—42no—
44C2×C3×C16C2314yes—
45#818—22no—
46#818C2314yes—
47#816—24no—
48#816—24no—
49Dic24—44no—
50C22×Q16—32no—
51C23×C8C2×D311yescharacteristic
52C2×C22⋊Q8—62no—
53C2×Q32—34no—
54C22×C16C2×D312yes—
55#467—34no—
56#467—34no—
57#467—62no—
58#467—62no—
59C2×#107—32no—
60C2×#103—62no—
61#366—34no—
62#358—38no—
63C22⋊C16C2×D314yes—
64C24×C3D811yescharacteristic
65C22×C3×C4—41no—
66C22×C3×C4—41no—
67C22×C3×C4D812yes—
68C22×C3×C4D811yescharacteristic
69C22×C3×C4C2×D411yescharacteristic
70C22×C3×C4C2×D412yes—
71C22×Dic3—42no—
72C22×Dic3—42no—
73C2×Dic6—24no—
74C2×Dic6—44no—
75C2×Dic6—44no—
76C2×Dic6—42no—
77C2×C3×C8—22no—
78C2×C3×C8—24no—
79C2×C3×C8—41no—
80C2×C3×C8—42no—
81C2×C3×C8C2×D411yescharacteristic
82C2×C3×C8C2×D412yes—
83C2×C3×C8C2411yescharacteristic
84#217—44no—
85#217—44no—
86C4⋊Dic3—24no—
87C4⋊Dic3—42no—
88Dic3⋊C4—44no—
89Dic3⋊C4—44no—
90Dic12—28no—
91Dic12—44no—
92C3×C16—24no—
93C22×Q8—62no—
94C23×C4D1211yescharacteristic
95C2×Q16—34no—
96C2×Q16—62no—
97C2×Q16—124no—
98C22×C8—42no—
99C22×C8D1212yes—
100C22×C8D1211yescharacteristic
101C22×C8C22×D312yes—
102C22×C8C22×D311yescharacteristic
103C22⋊Q8—64no—
104C22⋊Q8—64no—
105C22⋊Q8—122no—
106C22⋊Q8—122no—
107C2×C4⋊C4—62no—
108C2×C4⋊C4—122no—
109C2×C22⋊C4—122no—
110Q32—124no—
111C2×C16—42no—
112C2×C16C22×D314yes—
113#107—34no—
114#107—62no—
115#103—64no—
116#103—64no—
117C23×C3—41no—
118C23×C3—41no—
119C23×C3Q3214yescomplemented
120C23×C3C2×D811yescharacteristic
121C23×C3C2×D812yes—
122C2×C3×C4—22no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—41no—
127C2×C3×C4—42no—
128C2×C3×C4C2×D811yescharacteristic
129C2×C3×C4C2×D812yes—
130C2×C3×C4C22×D411yescharacteristic
131C2×Dic3—44no—
132C2×Dic3—44no—
133C2×Dic3—84no—
134C2×Dic3—84no—
135Dic6—44no—
136Dic6—44no—
137Dic6—84no—
138C3×C8—22no—
139C3×C8—21no—
140C3×C8H(5,𝔽2)12yes—
141C24D2411yescharacteristic
142C2×Q8—64no—
143C2×Q8—124no—
144C2×Q8—124no—
145C2×Q8—122no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—122no—
149C22×C4—122no—
150C22×C4D2412yes—
151C22×C4D2411yescharacteristic
152C22×C4C2×D1212yes—
153C22×C4C2×D1211yescharacteristic
154Q16—68no—
155Q16—124no—
156C2×C8—22no—
157C2×C8—24no—
158C2×C8—41no—
159C2×C8—42no—
160C2×C8C2×D1211yescharacteristic
161C2×C8C2×D1212yes—
162C2×C8C23×D311yescharacteristic
163C4⋊C4—64no—
164C4⋊C4—122no—
165C4⋊C4—124no—
166C4⋊C4—124no—
167C22⋊C4—124no—
168C22⋊C4—124no—
169C16—24no—
170C22×C3—22no—
171C22×C3—21no—
172C22×C3—22no—
173C22×C3—21no—
174C22×C3—44no—
175C22×C3C2×Q3212yes—
176C22×C3C2×Q3214yes—
177C22×C3C22×D811yescharacteristic
178C3×C4—22no—
179C3×C4—21no—
180C3×C4#53411yescharacteristic
181C3×C4#57511yescharacteristic
182Dic3—84no—
183Dic3—84no—
184C23—41no—
185C23—41no—
186C23Dic2414yescomplemented
187C23C2×D2411yescharacteristic
188C23C2×D2412yes—
189Q8—124no—
190Q8—124no—
191Q8—244no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—124no—
199C2×C4—124no—
200C2×C4—244no—
201C2×C4—244no—
202C2×C4C2×D2411yescharacteristic
203C2×C4C2×D2412yes—
204C2×C4C22×D1211yescharacteristic
205C8—22no—
206C8—21no—
207C8#100412yes—
208C2×C3—24no—
209C2×C3—22no—
210C2×C3#318611yescharacteristic
211C2×C3C22×Q3211yescharacteristic
212C2×C3#341411yescharacteristic
213C22—22no—
214C22—21no—
215C22—22no—
216C22—21no—
217C22—44no—
218C22C2×Dic2412yes—
219C22C2×Dic2414yes—
220C22C22×D2411yescharacteristic
221C4—22no—
222C4—21no—
223C4—244no—
224C4—244no—
225C4#549311yescharacteristic
226C4#574811yescharacteristic
227C3#3375511yescharacteristic, complemented, Sylow
228C2—24no—
229C2—22no—
230C2#7861111yescharacteristic
231C2C22×Dic2411yescharacteristic
232C2#8200911yescharacteristic
subgroupG/H
centerC22C22×D24
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×C22⋊C16C2
socleC22×C3C22×D8
prime radicalC24×C3D8
hypercenterC2×C22⋊C16D3
hypocenterC3#33755
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
3. #71416C2semidirect
4. #69113C2semidirect
5. #69099C2semidirect
17. C2×Dic24C22semidirect
19. #4504C22semidirect
20. #4503C22semidirect
119. C23×C3Q32semidirect
186. C23Dic24semidirect
227. C3#33755semidirect

Maximal subgroups

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

subgroupindex
C2×#51162
C2×C3×C22⋊C162
#714162
#691132
#690992
#337553