Finite Group Databasebeta

← #11802913

Subgroups — #11802913

Order 1152.

1490 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1490 conjugacy classes of subgroups — too many to draw. Only its 199 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×C22⋊#157C211yesmaximal, characteristic
2C2×#63974C212yesmaximal
3#10593516C214yesmaximal, complemented
4#10593485C218yesmaximal, complemented
5C24×C3⋊D3C2211yescharacteristic
6C23×#157C2212yes—
7C22×#624C2214yes—
8C2×C22⋊#157C418yescomplemented
9C2×C22⋊#157C2214yes—
10C2×#3773C2218yescomplemented
11C2×C3×#217—22no—
12#63974—22no—
13#63974—22no—
14#63974C2214yes—
15#63974C2214yes—
16C2×#3722—22no—
17#63737C2218yescomplemented
18C2×D3×C22⋊C4—62no—
19C24×C32D411yescharacteristic
20C23×C3⋊D3—44no—
21C23×C3⋊D3C2×C414yes—
22C23×C3⋊D3D412yes—
23C23×C3⋊D3D411yescharacteristic
24C23×C3⋊D3Q814yes—
25C23×C3⋊D3C2312yes—
26C23×C3⋊D3C2311yescharacteristic
27C22×#157—44no—
28C22×#157C2×C418yescomplemented
29C22×#157C2314yes—
30C22×C3×Dic3—24no—
31C2×#624—24no—
32C2×#624—48no—
33C2×#624C2318yes—
34C22⋊#157C2×C4116yescomplemented
35#3773—48no—
36C3×#217—24no—
37C3×#217—24no—
38#3722—24no—
39#3722—24no—
40#2315—91nomaximal, Sylow
41C24×D3—62no—
42C22×C4×D3—64no—
43C2×C3×C22⋊C4—62no—
44C2×#217—22no—
45C2×(C2×D3)⋊C4—64no—
46D3×C22⋊C4—64no—
47D3×C22⋊C4—64no—
48D3×C22⋊C4—64no—
49D3×C22⋊C4—64no—
50C23×C32—42no—
51C23×C32C4⋊C414yescomplemented
52C23×C32C2×D411yescharacteristic
53C23×C32C2×D412yes—
54C22×C3⋊D3—28no—
55C22×C3⋊D3—24no—
56C22×C3⋊D3—48no—
57C22×C3⋊D3—48no—
58C22×C3⋊D3—42no—
59C22×C3⋊D3—44no—
60C22×C3⋊D3C4⋊C418yescomplemented
61C22×C3⋊D3C4⋊C414yescomplemented
62C22×C3⋊D3C22×C412yes—
63C22×C3⋊D3C22×C414yes—
64C22×C3⋊D3C2×D411yescharacteristic
65C22×C3⋊D3C2×D412yes—
66C22×C3⋊D3C2×Q814yes—
67C22×C3⋊D3C2×Q812yes—
68C22×C3⋊D3C2411yescharacteristic
69C2×#157—28no—
70C2×#157—48no—
71C2×#157C22×C418yes—
72C2×C3×Dic3—28no—
73C2×C3×Dic3—48no—
74#624—216no—
75#624—48no—
76C22×C22⋊C4—92no—
77C22×C22⋊C4—91no—
78#385—98no—
79#380—94no—
80C24×C3—22no—
81C23×D3—62no—
82C23×D3—68no—
83C23×D3—62no—
84C23×D3—62no—
85C23×D3—62no—
86C23×D3—68no—
87C23×D3—62no—
88C23×D3—62no—
89C23×D3—1216no—
90C22×C3×C4—64no—
91C22×Dic3—24no—
92C2×C4×D3—68no—
93C2×C4×D3—68no—
94C2×C4×D3—1216no—
95C3×C22⋊C4—64no—
96C3×C22⋊C4—64no—
97#217—24no—
98#217—24no—
99(C2×D3)⋊C4—68no—
100(C2×D3)⋊C4—68no—
101C22×C32—24no—
102C22×C32—22no—
103C22×C32—44no—
104C22×C32C2×C4⋊C414yes—
105C22×C32C2×C4⋊C412yes—
106C22×C32C22×D411yescharacteristic
107C2×C3⋊D3—24no—
108C2×C3⋊D3—28no—
109C2×C3⋊D3—28no—
110C2×C3⋊D3—22no—
111C2×C3⋊D3—24no—
112C2×C3⋊D3—24no—
113C2×C3⋊D3—44no—
114C2×C3⋊D3—48no—
115C2×C3⋊D3C2×C4⋊C414yes—
116C2×C3⋊D3C2×C4⋊C412yes—
117C2×C3⋊D3C23×C411yescharacteristic
118C2×C3⋊D3C22×Q811yescharacteristic
119C2×C3⋊D3H(5,𝔽2)14yes—
120#157—28no—
121C3×Dic3—48no—
122C25—91no—
123C23×C4—94no—
124C23×C4—92no—
125C2×C4⋊C4—98no—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—94no—
128C2×C22⋊C4—94no—
129C2×C22⋊C4—98no—
130C2×C22⋊C4—94no—
131C2×C22⋊C4—182no—
132C2×C22⋊C4—182no—
133C2×C22⋊C4—182no—
134#95—98no—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—22no—
138C23×C3—28no—
139C23×C3—48no—
140C22×D3—64no—
141C22×D3—62no—
142C22×D3—64no—
143C22×D3—62no—
144C22×D3—68no—
145C22×D3—68no—
146C22×D3—68no—
147C22×D3—64no—
148C22×D3—64no—
149C22×D3—62no—
150C22×D3—62no—
151C22×D3—64no—
152C22×D3—64no—
153C22×D3—616no—
154C22×D3—128no—
155C22×D3—128no—
156C22×D3—128no—
157C22×D3—128no—
158C22×D3—1232no—
159C22×D3—1232no—
160C22×D3—128no—
161C2×C3×C4—68no—
162C2×C3×C4—128no—
163C2×Dic3—28no—
164C2×Dic3—48no—
165C4×D3—1216no—
166C4×D3—1216no—
167C2×C32—24no—
168C2×C32—22no—
169C2×C32C22×C4⋊C411yescharacteristic
170C2×C32#53412yes—
171C3⋊D3—24no—
172C3⋊D3—22no—
173C3⋊D3#51812yes—
174C3⋊D3#54312yes—
175C24—91no—
176C24—92no—
177C24—92no—
178C24—91no—
179C24—94no—
180C24—94no—
181C24—364no—
182C24GO+(4,𝔽2)11yescharacteristic
183C22×C4—64no—
184C22×C4—98no—
185C22×C4—94no—
186C22×C4—98no—
187C22×C4—184no—
188C22×C4—368no—
189C22×C4—364no—
190C4⋊C4—368no—
191C22⋊C4—64no—
192C22⋊C4—64no—
193C22⋊C4—916no—
194C22⋊C4—184no—
195C22⋊C4—184no—
196C22×C3—28no—
197C22×C3—22no—
198C22×C3—24no—
199C22×C3—24no—
200C22×C3—24no—
201C22×C3—44no—
202C22×C3—44no—
203C22×C3—416no—
204C2×D3—68no—
205C2×D3—64no—
206C2×D3—62no—
207C2×D3—62no—
208C2×D3—64no—
209C2×D3—64no—
210C2×D3—68no—
211C2×D3—616no—
212C2×D3—616no—
213C2×D3—128no—
214C2×D3—128no—
215C2×D3—128no—
216C2×D3—128no—
217C2×D3—124no—
218C2×D3—124no—
219C2×D3—1216no—
220C2×D3—1216no—
221C2×D3—1216no—
222C3×C4—128no—
223Dic3—48no—
224C32#231511yescharacteristic, complemented, Sylow
225C23—42no—
226C23—91no—
227C23—91no—
228C23—92no—
229C23—94no—
230C23—92no—
231C23—94no—
232C23—92no—
233C23—94no—
234C23—98no—
235C23—188no—
236C23—184no—
237C23—364no—
238C23—362no—
239C23—368no—
240C23—368no—
241C23#377314yescomplemented
242C23C2×GO+(4,𝔽2)11yescharacteristic
243C23C2×GO+(4,𝔽2)12yes—
244C2×C4—68no—
245C2×C4—98no—
246C2×C4—128no—
247C2×C4—1816no—
248C2×C4—188no—
249C2×C4—368no—
250C2×C4—368no—
251C2×C3—28no—
252C2×C3—22no—
253C2×C3—24no—
254C2×C3—44no—
255C2×C3—44no—
256D3—64no—
257D3—64no—
258D3—68no—
259D3—124no—
260D3—124no—
261C22—24no—
262C22—22no—
263C22—44no—
264C22—91no—
265C22—91no—
266C22—94no—
267C22—94no—
268C22—92no—
269C22—184no—
270C22—188no—
271C22—188no—
272C22—184no—
273C22—184no—
274C22—182no—
275C22—368no—
276C22—364no—
277C22C2×#377314yes—
278C22C2×#377312yes—
279C22C22×GO+(4,𝔽2)11yescharacteristic
280C4—128no—
281C4—188no—
282C3—22no—
283C2—24no—
284C2—22no—
285C2—92no—
286C2—92no—
287C2—184no—
288C2—182no—
289C2C22×#377311yescharacteristic
290C2#1059653812yes—
subgroupG/H
center, FrattiniC22C22×GO+(4,𝔽2)
commutatorC2×C3⋊D3C23×C4
FittingC24×C32D4
socleC22×C32C22×D4
prime radicalC24×C3⋊D3C22
cosocleC22×C3⋊D3C24
hypercenterC24GO+(4,𝔽2)
hypocenterC32#2315
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
3. #10593516C2semidirect
4. #10593485C2semidirect
8. C2×C22⋊#157C4semidirect
10. C2×#3773C22semidirect
17. #63737C22semidirect
28. C22×#157C2×C4semidirect
34. C22⋊#157C2×C4semidirect
51. C23×C32C4⋊C4semidirect
60. C22×C3⋊D3C4⋊C4semidirect
61. C22×C3⋊D3C4⋊C4semidirect
224. C32#2315semidirect
241. C23#3773semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#1572
C2×#639742
#105935162
#105934852
#23159