Finite Group Databasebeta

← #11781153

Subgroups — #11781153

Order 1152.

363 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 96 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
1Q8×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C3×#5031C212yesmaximal
3#10590312C211yesmaximal, characteristic
4#10589100C211yesmaximal, characteristic
5#10589093C212yesmaximal
6#71053—32nomaximal
7C32×C4×Q8C2211yescharacteristic
8C2×C3×#216—22no—
9C3×#869C2212yes—
10C4⋊(C3⋊(C3⋊C8))—21no—
11C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C4×C3⋊(C3⋊C8)—21no—
13C4×C3⋊(C3⋊C8)C2211yescharacteristic
14C3×#810—22no—
15C3×C4⋊(C3⋊C8)C2212yes—
16#63576—21no—
17#63576—21no—
18C3×#462—32no—
19#5078—32no—
20#5031D312yes—
21Q8×C3⋊C8—32no—
22Q8×C3⋊C8—61no—
23#4547—32no—
24#4482—32no—
25#4479—32no—
26C2×C32×Q8D411yescharacteristic
27C2×C3×Dic6—22no—
28C32×C4⋊C4—21no—
29C32×C4⋊C4D411yescharacteristic
30C32×C42—21no—
31C32×C42C2311yescharacteristic
32C2×C3⋊(C3⋊C8)—21no—
33C2×C3⋊(C3⋊C8)D411yescharacteristic
34C2×C3⋊(C3⋊C8)D411yescharacteristic
35C3×#216—44no—
36C3×C4⋊Dic3—22no—
37C3×C4⋊Dic3—22no—
38C2×C3×C3⋊C8—22no—
39#1674—91noSylow
40C2×C3×Q16—62no—
41C3×C4⋊Q8—32no—
42C3×C4×Q8—21no—
43C3×C4×Q8C2×D312yes—
44C2×#216—22no—
45#869C2×D312yes—
46C3×C4⋊C8—32no—
47C3×#103—62no—
48#810—22no—
49#807—62no—
50#807—62no—
51C4⋊(C3⋊C8)—32no—
52C4⋊(C3⋊C8)—61no—
53C4⋊(C3⋊C8)—62no—
54C4⋊(C3⋊C8)—62no—
55C4⋊(C3⋊C8)C2×D312yes—
56C4×C3⋊C8—32no—
57C4×C3⋊C8—61no—
58C4×C3⋊C8—62no—
59C4×C3⋊C8—62no—
60C32×Q8—21no—
61C32×Q8D812yes—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4C2×D411yescharacteristic
65C2×C32×C4C2×D411yescharacteristic
66C2×C32×C4#4111yescharacteristic
67C2×C3×Dic3—42no—
68C3×Dic6—42no—
69C3×Dic6—42no—
70C3⋊(C3⋊C8)—21no—
71C3⋊(C3⋊C8)—41no—
72C3⋊(C3⋊C8)Q1612yescomplemented
73C3×C3⋊C8—42no—
74#500—91no—
75#462—32no—
76C8×Q8—91no—
77#335—91no—
78#332—92no—
79C2×C3×Q8—21no—
80C2×C3×Q8—62no—
81C2×C3×Q8#6712yes—
82C2×Dic6—22no—
83C3×Q16—124no—
84C2×C3×C8—62no—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—22no—
87C3×C4⋊C4—21no—
88C3×C4⋊C4—62no—
89C3×C4⋊C4—62no—
90C3×C4⋊C4#6712yes—
91C3×C42—22no—
92C3×C42—22no—
93C3×C42—21no—
94C3×C42C22×D312yes—
95#216—44no—
96C4⋊Dic3—22no—
97C4⋊Dic3—22no—
98C2×C3⋊C8—22no—
99C2×C3⋊C8—32no—
100C2×C3⋊C8—32no—
101C2×C3⋊C8—61no—
102C2×C3⋊C8—61no—
103C2×C3⋊C8—62no—
104C2×C3⋊C8—62no—
105C22×C32#12111yescharacteristic
106C32×C4—22no—
107C32×C4—21no—
108C32×C4—41no—
109C32×C4C2×D811yescharacteristic
110C32×C4C2×Q1611yescharacteristic
111C32×C4#13511yescharacteristic
112C32×C4#13711yescharacteristic
113C3×Dic3—82no—
114C2×Q16—62no—
115C4⋊Q8—32no—
116C4×Q8D3211yescharacteristic
117#107—181no—
118#107—181no—
119C4⋊C8—32no—
120C4⋊C8—91no—
121C4⋊C8—181no—
122#103—62no—
123C4×C8—91no—
124C4×C8—181no—
125C3×Q8—22no—
126C3×Q8—22no—
127C3×Q8—22no—
128C3×Q8—122no—
129C3×Q8—122no—
130C3×Q8#21314yes—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—122no—
139C2×C3×C4D3×D412yes—
140C2×C3×C4#23712yes—
141C2×C3×C4C2×#6712yes—
142C2×Dic3—42no—
143Dic6—42no—
144Dic6—42no—
145C3×C8—122no—
146C3⋊C8—34no—
147C3⋊C8—42no—
148C3⋊C8—62no—
149C3⋊C8—62no—
150C3⋊C8—121no—
151C3⋊C8—122no—
152C3⋊C8—122no—
153C2×C32#46211yescharacteristic
154C2×C32#46611yescharacteristic
155C2×C32#47311yescharacteristic
156C2×Q8—62no—
157C2×Q8#62511yescharacteristic
158Q16—124no—
159C2×C8—62no—
160C2×C8—91no—
161C2×C8—91no—
162C2×C8—181no—
163C4⋊C4—21no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C4⋊C4#62511yescharacteristic
167C42—21no—
168C42C2×D3211yescharacteristic
169C22×C3—21no—
170C22×C3#93912yes—
171C3×C4—24no—
172C3×C4—22no—
173C3×C4—24no—
174C3×C4—21no—
175C3×C4—21no—
176C3×C4—21no—
177C3×C4—21no—
178C3×C4—41no—
179C3×C4—42no—
180C3×C4—42no—
181C3×C4—242no—
182C3×C4D3×Q1612yes—
183C3×C4#91812yes—
184C3×C4C2×#21312yes—
185C3×C4#95012yes—
186Dic3—82no—
187C32#167411yescharacteristic, complemented, Sylow
188Q8—21no—
189Q8—122no—
190Q8—122no—
191Q8#371312yes—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—122no—
195C2×C4#380511yescharacteristic
196C2×C4C2×#62511yescharacteristic
197C2×C4#381111yescharacteristic
198C8—92no—
199C8—122no—
200C8—181no—
201C8—361no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3#503812yes—
206C2×C3#518212yes—
207C2×C3#520412yes—
208C22#6398011yescharacteristic
209C4—22no—
210C4—21no—
211C4—41no—
212C4—242no—
213C4C2×#371311yescharacteristic
214C4#6382211yescharacteristic
215C4#6394411yescharacteristic
216C4#6394511yescharacteristic
217C3—21no—
218C3#7105312yescomplemented
219C2#1059116611yescharacteristic
220C2#1059163311yescharacteristic
221C2#1059168311yescharacteristic
subgroupG/H
centerC22#63980
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socle, prime radicalC22×C32#121
hypercenterC4×Q8D32
hypocenterC32#1674
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
72. C3⋊(C3⋊C8)Q16semidirect
187. C32#1674semidirect
218. C3#71053semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊(C3⋊C8)2
C3×#50312
#105903122
#105891002
#105890932
#710533