Finite Group Databasebeta

← #11784600

Subgroups — #11784600

Order 1152.

2242 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 2242 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C23×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2#10593847C212yesmaximal, complemented
3C2×#63774C214yesmaximal, complemented
4C24×C3⋊D3C411yescharacteristic, complemented
5C24×C3⋊D3C2211yescharacteristic
6C23×C32×C4C411yescharacteristic, complemented
7C22×C4⋊(C3⋊D3)—28no—
8C22×C4⋊(C3⋊D3)—24no—
9C22×C4⋊(C3⋊D3)C412yescomplemented
10C22×C4⋊(C3⋊D3)C2212yes—
11C2×C22⋊#157—24no—
12C2×C22⋊#157C2214yescomplemented
13#63774—28no—
14C23×D12—62no—
15C24×C32C2×C411yescharacteristic
16C23×C3⋊D3—22no—
17C23×C3⋊D3—24no—
18C23×C3⋊D3—24no—
19C23×C3⋊D3—44no—
20C23×C3⋊D3—44no—
21C23×C3⋊D3C2×C411yescharacteristic
22C23×C3⋊D3C2×C412yes—
23C23×C3⋊D3D414yes—
24C23×C3⋊D3C2311yescharacteristic
25C22×#157—24no—
26C22×C32×C4—22no—
27C22×C32×C4—24no—
28C22×C32×C4C2×C412yes—
29C2×C4⋊(C3⋊D3)—232no—
30C2×C4⋊(C3⋊D3)—24no—
31C2×C4⋊(C3⋊D3)—24no—
32C2×C4⋊(C3⋊D3)—24no—
33C2×C4⋊(C3⋊D3)—24no—
34C2×C4⋊(C3⋊D3)—24no—
35C2×C4⋊(C3⋊D3)D418yes—
36C22⋊#157—28no—
37C22⋊#157—48no—
38#1781—91nomaximal, Sylow
39C24×D3—62no—
40C24×D3—62no—
41C23×C3×C4—22no—
42C22×D12—64no—
43C22×D12—64no—
44C22×D12—632no—
45C22×D12—616no—
46C23×C32—24no—
47C23×C32—22no—
48C23×C32C22⋊C412yes—
49C23×C32C22×C411yescharacteristic
50C22×C3⋊D3—28no—
51C22×C3⋊D3—22no—
52C22×C3⋊D3—22no—
53C22×C3⋊D3—22no—
54C22×C3⋊D3—28no—
55C22×C3⋊D3—28no—
56C22×C3⋊D3—42no—
57C22×C3⋊D3—44no—
58C22×C3⋊D3—42no—
59C22×C3⋊D3—416no—
60C22×C3⋊D3—416no—
61C22×C3⋊D3—44no—
62C22×C3⋊D3—44no—
63C22×C3⋊D3—44no—
64C22×C3⋊D3C22⋊C412yes—
65C22×C3⋊D3C22⋊C414yescomplemented
66C22×C3⋊D3C2×D412yes—
67C22×C3⋊D3C2×D414yes—
68C2×#157—48no—
69C2×#157—44no—
70C2×C32×C4—28no—
71C2×C32×C4—22no—
72C2×C32×C4—22no—
73C2×C32×C4C22⋊C414yescomplemented
74C4⋊(C3⋊D3)—216no—
75C4⋊(C3⋊D3)—216no—
76C23×D4—91no—
77C2×#99—94no—
78#379—92no—
79C24×C3—22no—
80C23×D3—62no—
81C23×D3—64no—
82C23×D3—68no—
83C23×D3—62no—
84C23×D3—616no—
85C23×D3—616no—
86C23×D3—68no—
87C23×D3—1216no—
88C23×D3—1216no—
89C22×C3×C4—28no—
90C22×C3×C4—216no—
91C22×C3×C4—24no—
92C2×D12—616no—
93C2×D12—616no—
94C2×D12—616no—
95C2×D12—616no—
96C2×D12—616no—
97C2×D12—6128no—
98C2×D12—616no—
99C22×C32—28no—
100C22×C32—22no—
101C22×C32—22no—
102C22×C32—22no—
103C22×C32#9914yescomplemented
104C22×C32C2×C22⋊C412yes—
105C22×C32C2×C22⋊C411yescharacteristic
106C2×C3⋊D3—22no—
107C2×C3⋊D3—24no—
108C2×C3⋊D3—24no—
109C2×C3⋊D3—24no—
110C2×C3⋊D3—44no—
111C2×C3⋊D3—416no—
112C2×C3⋊D3—44no—
113C2×C3⋊D3—416no—
114C2×C3⋊D3—44no—
115C2×C3⋊D3—44no—
116C2×C3⋊D3—44no—
117C2×C3⋊D3#12014yes—
118#157—84no—
119C32×C4—24no—
120C25—91no—
121C25—91no—
122C22×D4—92no—
123C22×D4—92no—
124C22×D4—188no—
125C22×D4—184no—
126C23×C4#15711yescharacteristic, complemented
127C2×C22⋊C4—94no—
128C2×C22⋊C4—184no—
129#99—188no—
130C23×C3—28no—
131C23×C3—216no—
132C23×C3—22no—
133C23×C3—24no—
134C22×D3—64no—
135C22×D3—68no—
136C22×D3—64no—
137C22×D3—68no—
138C22×D3—632no—
139C22×D3—632no—
140C22×D3—68no—
141C22×D3—68no—
142C22×D3—68no—
143C22×D3—1216no—
144C22×D3—128no—
145C22×D3—1216no—
146C22×D3—128no—
147C22×D3—1216no—
148C22×D3—1264no—
149C22×D3—1264no—
150C22×D3—1216no—
151C22×D3—1216no—
152C2×C3×C4—232no—
153C2×C3×C4—28no—
154C2×C3×C4—28no—
155C2×C3×C4—28no—
156D12—664no—
157D12—664no—
158C2×C32—24no—
159C2×C32—22no—
160C2×C32#37911yescharacteristic
161C2×C32C2×#9912yes—
162C3⋊D3—24no—
163C3⋊D3—42no—
164C3⋊D3—44no—
165C24—91no—
166C24—94no—
167C24—91no—
168C24—92no—
169C24—184no—
170C24—184no—
171C24—182no—
172C24—364no—
173C24—364no—
174C24C2×#15711yescharacteristic
175C2×D4—98no—
176C2×D4—184no—
177C2×D4—184no—
178C2×D4—184no—
179C2×D4—184no—
180C2×D4—1832no—
181C2×D4—184no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—184no—
185C22×C4C2×#15712yes—
186C22⋊C4—188no—
187C22⋊C4—368no—
188C22×C3—28no—
189C22×C3—232no—
190C22×C3—28no—
191C22×C3—28no—
192C22×C3—28no—
193C22×C3—24no—
194C22×C3—22no—
195C2×D3—68no—
196C2×D3—68no—
197C2×D3—616no—
198C2×D3—128no—
199C2×D3—128no—
200C2×D3—1216no—
201C2×D3—1264no—
202C2×D3—1216no—
203C2×D3—1264no—
204C2×D3—1216no—
205C2×D3—1216no—
206C2×D3—1216no—
207C3×C4—216no—
208C32#178111yescharacteristic, complemented, Sylow
209C23—24no—
210C23—22no—
211C23—92no—
212C23—94no—
213C23—92no—
214C23—94no—
215C23—188no—
216C23—188no—
217C23—188no—
218C23—182no—
219C23—182no—
220C23—182no—
221C23—362no—
222C23—362no—
223C23—364no—
224C23—364no—
225C23—3616no—
226C23—3616no—
227C23—364no—
228C23—364no—
229C23C22⋊#15712yes—
230C23C22×#15711yescharacteristic
231D4—1816no—
232D4—1816no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—364no—
237C2×C4—368no—
238C2×C4C22⋊#15714yescomplemented
239C2×C3—216no—
240C2×C3—28no—
241C2×C3—22no—
242C2×C3—24no—
243D3—128no—
244D3—128no—
245D3—1216no—
246C22—28no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—94no—
251C22—184no—
252C22—182no—
253C22—184no—
254C22—184no—
255C22—364no—
256C22—3616no—
257C22—364no—
258C22—3616no—
259C22—364no—
260C22—364no—
261C22—364no—
262C22#6377414yescomplemented
263C22C2×C22⋊#15712yes—
264C22C2×C22⋊#15711yescharacteristic
265C4—24no—
266C4—724no—
267C3—22no—
268C2—24no—
269C2—22no—
270C2—184no—
271C2—362no—
272C2—364no—
273C2C2×#6377412yes—
274C2#1059384711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#157
commutatorC23×C32C22×C4
FrattiniC23C22×#157
FittingC23×C32×C4C4
socleC22×C32C2×C22⋊C4
prime radicalC23×C4⋊(C3⋊D3)C2
cosocleC23×C3⋊D3C23
hypercenterC23×C4#157
hypocenterC32#1781
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. #10593847C2semidirect
3. C2×#63774C2semidirect
4. C24×C3⋊D3C4semidirect
6. C23×C32×C4C4semidirect
9. C22×C4⋊(C3⋊D3)C4semidirect
12. C2×C22⋊#157C22semidirect
65. C22×C3⋊D3C22⋊C4semidirect
73. C2×C32×C4C22⋊C4semidirect
103. C22×C32#99semidirect
126. C23×C4#157semidirect
208. C32#1781semidirect
238. C2×C4C22⋊#157semidirect
262. C22#63774semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊(C3⋊D3)2
#105938472
C2×#637742
#17819