Finite Group Databasebeta

← #11784621

Subgroups — #11784621

Order 1152.

1502 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C22×#3832C211yesmaximal, characteristic
2#10593811C212yesmaximal, complemented
3C2×#63774C214yesmaximal, complemented
4C22×C32×Q8C411yescharacteristic, complemented
5C2×#3832—28no—
6C2×#3832—24no—
7C22×C4⋊(C3⋊D3)C411yescharacteristic, complemented
8C22×C4⋊(C3⋊D3)C2212yes—
9C22×C4×C3⋊D3C412yescomplemented
10C22×C4×C3⋊D3C2211yescharacteristic
11C2×C22⋊#157C2214yes—
12#63781—24no—
13#63774—28no—
14C22×#239—62no—
15C23×C3⋊D3C2×C412yes—
16C23×C3⋊D3C2311yescharacteristic
17C22×#157—24no—
18C2×C32×Q8—24no—
19C2×C32×Q8—22no—
20C22×C32×C4C2×C412yes—
21C22×C32×C4C2×C411yescharacteristic
22C22×C3⋊Dic3C2×C411yescharacteristic
23#3832—232no—
24C2×C4⋊(C3⋊D3)—24no—
25C2×C4⋊(C3⋊D3)—24no—
26C2×C4⋊(C3⋊D3)—24no—
27C2×C4⋊(C3⋊D3)—22no—
28C2×C4⋊(C3⋊D3)D418yes—
29C2×C4×C3⋊D3—24no—
30C2×C4×C3⋊D3—24no—
31C2×C4×C3⋊D3—24no—
32C2×C4×C3⋊D3—24no—
33C2×C4×C3⋊D3D414yes—
34C22⋊#157—28no—
35#1782—91nomaximal, Sylow
36C22×C3×Q8—22no—
37C2×#239—616no—
38C2×#239—632no—
39C22×D12—62no—
40C22×D12—64no—
41C22×C4×D3—64no—
42C22×C4×D3—62no—
43C23×C32C22×C411yescharacteristic
44C22×C3⋊D3—22no—
45C22×C3⋊D3—22no—
46C22×C3⋊D3—42no—
47C22×C3⋊D3—44no—
48C22×C3⋊D3C22⋊C414yes—
49C22×C3⋊D3C2×D412yes—
50C22×C3⋊D3C2×D414yes—
51C2×#157—48no—
52C2×#157—44no—
53C32×Q8—28no—
54C2×C32×C4—22no—
55C2×C32×C4—22no—
56C2×C32×C4—22no—
57C2×C32×C4C22⋊C414yes—
58C2×C32×C4C22⋊C412yes—
59C2×C3⋊Dic3—22no—
60C2×C3⋊Dic3C22⋊C412yes—
61C4⋊(C3⋊D3)—28no—
62C4⋊(C3⋊D3)—216no—
63C4×C3⋊D3—216no—
64C4×C3⋊D3—28no—
65C22×#41—91no—
66C2×#99—94no—
67#381—92no—
68C23×D3—64no—
69C23×D3—62no—
70C2×C3×Q8—216no—
71C2×C3×Q8—28no—
72C22×C3×C4—24no—
73C22×C3×C4—22no—
74C22×Dic3—62no—
75#239—6128no—
76C2×D12—616no—
77C2×D12—616no—
78C2×D12—616no—
79C2×D12—616no—
80C2×D12—68no—
81C2×C4×D3—68no—
82C2×C4×D3—616no—
83C2×C4×D3—616no—
84C2×C4×D3—616no—
85C2×C4×D3—616no—
86C22×C32—22no—
87C22×C32C2×C22⋊C412yes—
88C22×C32C2×C22⋊C411yescharacteristic
89C2×C3⋊D3—24no—
90C2×C3⋊D3—24no—
91C2×C3⋊D3—44no—
92C2×C3⋊D3—44no—
93C2×C3⋊D3—44no—
94C2×C3⋊D3—44no—
95C2×C3⋊D3#12014yes—
96#157—84no—
97C32×C4—24no—
98C32×C4—22no—
99C3⋊Dic3—22no—
100C2×#41—184no—
101C2×#41—188no—
102C22×Q8#15711yescharacteristic, complemented
103C22×D4—92no—
104C22×D4—91no—
105C23×C4—91no—
106C23×C4—92no—
107C2×C22⋊C4—94no—
108#99—188no—
109#95—184no—
110C23×C3—22no—
111C22×D3—68no—
112C22×D3—64no—
113C22×D3—68no—
114C22×D3—68no—
115C22×D3—68no—
116C22×D3—128no—
117C22×D3—1216no—
118C3×Q8—232no—
119C2×C3×C4—28no—
120C2×C3×C4—28no—
121C2×C3×C4—28no—
122C2×C3×C4—28no—
123C2×C3×C4—24no—
124C2×Dic3—64no—
125C2×Dic3—68no—
126D12—664no—
127D12—632no—
128C4×D3—632no—
129C4×D3—664no—
130C2×C32—22no—
131C2×C32#37911yescharacteristic
132C2×C32#41012yes—
133C3⋊D3—24no—
134C3⋊D3—44no—
135C24—91no—
136C24—92no—
137#41—1832no—
138C2×Q8—24no—
139C2×Q8—22no—
140C2×D4—98no—
141C2×D4—184no—
142C2×D4—184no—
143C2×D4—184no—
144C2×D4—182no—
145C22×C4—91no—
146C22×C4—94no—
147C22×C4—184no—
148C22×C4—184no—
149C22×C4—184no—
150C22×C4—184no—
151C22×C4—184no—
152C22×C4C2×#15712yes—
153C22×C4C2×#15711yescharacteristic
154C22⋊C4—188no—
155C22×C3—28no—
156C22×C3—24no—
157C22×C3—22no—
158C2×D3—68no—
159C2×D3—616no—
160C2×D3—128no—
161C2×D3—1216no—
162C2×D3—1216no—
163C2×D3—1216no—
164C2×D3—1216no—
165C3×C4—216no—
166C3×C4—28no—
167Dic3—68no—
168C32#178211yescharacteristic, complemented, Sylow
169C23—92no—
170C23—94no—
171C23—94no—
172C23—182no—
173C23—182no—
174C23—362no—
175C23—364no—
176C23C22×#15711yescharacteristic
177Q8—28no—
178D4—1816no—
179D4—188no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—92no—
184C2×C4—188no—
185C2×C4—1816no—
186C2×C4—182no—
187C2×C4—364no—
188C2×C4—368no—
189C2×C4C22⋊#15714yes—
190C2×C4C22⋊#15712yes—
191C2×C3—28no—
192C2×C3—24no—
193C2×C3—22no—
194D3—128no—
195D3—1216no—
196C22—22no—
197C22—94no—
198C22—184no—
199C22—184no—
200C22—364no—
201C22—364no—
202C22—364no—
203C22—364no—
204C22C2×C22⋊#15712yes—
205C22C2×C22⋊#15711yescharacteristic
206C4—24no—
207C4—22no—
208C4—182no—
209C4—724no—
210C3—22no—
211C2—22no—
212C2—184no—
213C2—364no—
214C2#1059383212yes—
215C2#1059384711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#157
commutatorC23×C32C22×C4
FrattiniC23C22×#157
FittingC22×C32×Q8C4
socleC22×C32C2×C22⋊C4
prime radicalC22×#3832C2
cosocleC23×C3⋊D3C23
hypercenterC22×Q8#157
hypocenterC32#1782
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #10593811C2semidirect
3. C2×#63774C2semidirect
4. C22×C32×Q8C4semidirect
7. C22×C4⋊(C3⋊D3)C4semidirect
9. C22×C4×C3⋊D3C4semidirect
102. C22×Q8#157semidirect
168. C32#1782semidirect

Maximal subgroups

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

subgroupindex
C22×#38322
#105938112
C2×#637742
#17829