Finite Group Databasebeta

← #76762⋊C3 #11921168

Subgroups — #76762⋊C3 #11921168

Order 1152.

346 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 28 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
1C3×#392⋊C3C211yesmaximal, characteristic, complemented
2D3×#95⋊C3C211yesmaximal, characteristic, complemented
3C3⋊(#399⋊C3)C211yesmaximal, characteristic, complemented
4#76762C311yesmaximal, characteristic, complemented
5#2656⋊C3—31nomaximal
6A4×D12—41nomaximal
7C3×#95⋊C3C2211yescharacteristic
8C2×#938—31no—
9C2×#884—31no—
10C3×#392C2×C311yescharacteristic, complemented
11#5223—31no—
12#4949—31no—
13#4677C2×C311yescharacteristic, complemented
14D3×#95C2×C311yescharacteristic, complemented
15#399⋊C3—31no—
16#392⋊C3—21no—
17#392⋊C3D311yescharacteristic, complemented
18C2×#95⋊C3—31no—
19C2×D3×A4—41no—
20C2×D3×A4—41no—
21C3×C4×A4—41no—
22#2656—31noSylow
23C22×C3×D4—31no—
24C22×#67—31no—
25C22×#67—31no—
26C22×D12A411yescharacteristic
27C22×C4×D3—31no—
28D4×A4—121no—
29C2×C3×C22⋊C4—31no—
30C2×#217—32no—
31#938—62no—
32#938—62no—
33C2×(C2×D3)⋊C4—31no—
34C2×C4⋊Dic3—31no—
35C2×Dic3⋊C4—31no—
36#884—62no—
37#884—62no—
38C3×#95C22×C311yescharacteristic
39C3⋊#95—31no—
40#95⋊C3—21no—
41#95⋊C3C2×D311yescharacteristic
42C2×C3×A4—41no—
43D3×A4—81no—
44D3×A4—81no—
45C3×D12—161no—
46C2×#121—91no—
47C2×#121—91no—
48#399—31no—
49#399—91no—
50#392—91no—
51#392C3×D311yescharacteristic, complemented
52C2×#95—31no—
53C24×C3—31no—
54C23×D3C2×A411yescharacteristic
55C23×D3C2×A411yescharacteristic
56C22×A4—121no—
57C22×A4—121no—
58C2×C3×D4—62no—
59C2×C3×D4—122no—
60C22×C3×C4—31no—
61C22×C3×C4C2×A411yescharacteristic
62C2×#67—62no—
63C2×#67—61no—
64C2×#67—61no—
65C2×#67—62no—
66C2×#67—62no—
67C2×#67—122no—
68C22×Dic3—31no—
69C22×Dic3—31no—
70C2×D12—61no—
71C2×D12—61no—
72C2×C4×D3—32no—
73C2×C4×D3—32no—
74C2×C4×D3—122no—
75C4×A4—41no—
76C4×A4—81no—
77C3×C22⋊C4—62no—
78#217—64no—
79(C2×D3)⋊C4—62no—
80C4⋊Dic3—62no—
81Dic3⋊C4—62no—
82C2×C3×D3—161no—
83C2×C3×D3—161no—
84C3×A4—41no—
85C32×C4—161no—
86C22×D4—31no—
87C22×D4—31no—
88C22×D4—91no—
89C22×D4—91no—
90C23×C4—91no—
91#121—182no—
92#121—182no—
93#121—182no—
94#121—182no—
95C2×C4⋊C4—91no—
96C2×C4⋊C4—91no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—91no—
99C2×C22⋊C4—92no—
100#95—91no—
101#95C2×C3×D311yescharacteristic
102C23×C3—62no—
103C23×C3—61no—
104C23×C3—62no—
105C23×C3—61no—
106C23×C3—61no—
107C23×C3C22×A411yescharacteristic
108C22×D3—21no—
109C22×D3—21no—
110C22×D3—31no—
111C22×D3—31no—
112C22×D3—61no—
113C22×D3—61no—
114C22×D3—61no—
115C2×A4—41no—
116C2×A4—81no—
117C2×A4—241no—
118C2×A4—241no—
119C3×D4—124no—
120C3×D4—481no—
121C2×C3×C4—32no—
122C2×C3×C4—61no—
123C2×C3×C4—62no—
124#67—124no—
125#67—122no—
126#67—121no—
127#67—121no—
128C2×Dic3—32no—
129C2×Dic3—62no—
130C2×Dic3—61no—
131C2×Dic3—62no—
132D12—41no—
133D12—121no—
134C4×D3—122no—
135C4×D3—122no—
136C2×C32—161no—
137C3×D3—321no—
138C3×D3—321no—
139C24—31no—
140C24—31no—
141C24—31no—
142C2×D4—62no—
143C2×D4—122no—
144C2×D4—181no—
145C2×D4—181no—
146C2×D4—182no—
147C2×D4—181no—
148C2×D4—182no—
149C2×D4—181no—
150C2×D4—182no—
151C2×D4—362no—
152C22×C4—31no—
153C22×C4—92no—
154C22×C4—92no—
155C22×C4—91no—
156C22×C4—91no—
157C22×C4—362no—
158C22×C4D3×A411yescharacteristic
159C4⋊C4—182no—
160C4⋊C4—182no—
161C22⋊C4—62no—
162C22⋊C4—182no—
163C22⋊C4—184no—
164C22×C3—31no—
165C22×C3—31no—
166C22×C3—121no—
167C22×C3—122no—
168C22×C3—122no—
169C22×C3—121no—
170C22×C3—121no—
171C22×C3—481no—
172C22×C3—481no—
173C22×C32−1+4⋊C311yescharacteristic
174C2×D3—31no—
175C2×D3—41no—
176C2×D3—61no—
177C2×D3—61no—
178C2×D3—61no—
179C2×D3—61no—
180C2×D3—121no—
181C2×D3—121no—
182C2×D3#127⋊C311yescharacteristic
183A4—41no—
184A4—81no—
185C3×C4—41no—
186C3×C4—62no—
187C3×C4—161no—
188C3×C4—321no—
189Dic3—62no—
190Dic3—121no—
191C32—161noSylow
192C23—61no—
193C23—62no—
194C23—61no—
195C23—62no—
196C23—61no—
197C23—61no—
198C23—61no—
199C23—91no—
200C23—91no—
201C23—181no—
202C23—181no—
203C23—181no—
204C23C2×D3×A411yescharacteristic
205D4—121no—
206D4—124no—
207D4—361no—
208D4—364no—
209D4—361no—
210D4—361no—
211D4—362no—
212C2×C4—32no—
213C2×C4—61no—
214C2×C4—62no—
215C2×C4—92no—
216C2×C4—182no—
217C2×C4—181no—
218C2×C4—182no—
219C2×C4—362no—
220C2×C4—362no—
221C2×C3—31no—
222C2×C3—31no—
223C2×C3—161no—
224C2×C3—241no—
225C2×C3—321no—
226C2×C3—961no—
227C2×C3—961no—
228C2×C3C2×#127⋊C311yescharacteristic
229D3—21no—
230D3—61no—
231D3—81no—
232C22—31no—
233C22—31no—
234C22—31no—
235C22—91no—
236C22—122no—
237C22—121no—
238C22—121no—
239C22—122no—
240C22—121no—
241C22—121no—
242C22—181no—
243C22—181no—
244C22—181no—
245C22—181no—
246C22—361no—
247C22—361no—
248C22#1010⋊C311yescharacteristic
249C4—41no—
250C4—62no—
251C4—182no—
252C4—361no—
253C3—161no—
254C3—321no—
255C3#2656⋊C311yescharacteristic, complemented
256C2—31no—
257C2—31no—
258C2—61no—
259C2—181no—
260C2—241no—
261C2—241no—
262C2D3×#127⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2D3×#127⋊C3
commutator, cosocle, hypocenterC3×#95C22×C3
FrattiniC23C2×D3×A4
FittingC3×#392C2×C3
socleC23×C3C22×A4
LCSG ▷ C3×#95
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C3×#392⋊C3C2semidirect
2. D3×#95⋊C3C2semidirect
3. C3⋊(#399⋊C3)C2semidirect
4. #76762C3semidirect
10. C3×#392C2×C3semidirect
13. #4677C2×C3semidirect
14. D3×#95C2×C3semidirect
17. #392⋊C3D3semidirect
51. #392C3×D3semidirect
255. C3#2656⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#392⋊C32
D3×#95⋊C32
C3⋊(#399⋊C3)2
#767623
#2656⋊C33
A4×D124