Finite Group Databasebeta

← #11780550

Subgroups — #11780550

Order 1152.

419 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 76 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×C3⋊#564C211yesmaximal, characteristic, complemented
2#10592766C211yesmaximal, characteristic
3#10590282C212yesmaximal
4C3×#4543C211yesmaximal, characteristic, complemented
5#10589088C212yesmaximal, complemented
6#71960—31nomaximal
7#68375—31nomaximal
8C32×#125C2211yescharacteristic
9#64083C2211yescharacteristic
10C3×#890—22no—
11C3×#889—22no—
12C3×#889—22no—
13C3×#870C2212yes—
14#63636—22no—
15#63630C2211yescharacteristic
16C3×#803C2212yes—
17#63564—22no—
18#63564—22no—
19C3×#564—31no—
20C3⋊#564D311yescharacteristic, complemented
21#5067—31no—
22#5067—31no—
23#5067—32no—
24#5067—61no—
25#4713—32no—
26C3×#330—31no—
27#4543—32no—
28#4543D311yescharacteristic, complemented
29#4465—32no—
30C2×C4⋊(C3⋊D3)—21no—
31C2×C4⋊(C3⋊D3)—22no—
32C32×C4⋊C4—22no—
33C32×C4⋊C4D412yes—
34C32×C42C2311yescharacteristic
35C2×C3⋊(C3⋊C8)—21no—
36C3×C4⋊Dic3—22no—
37C3×C4⋊Dic3D412yes—
38C3×C4⋊Dic3D412yes—
39C3×Dic3⋊C4—22no—
40C3×Dic3⋊C4—22no—
41C3×Dic3⋊C4—22no—
42C3×Dic3⋊C4—22no—
43C3×C4×Dic3—22no—
44C2×C3×C3⋊C8—22no—
45#1658—91noSylow
46C3×#125—21no—
47C3×#125—32no—
48C3×#125—62no—
49C3×#125—62no—
50C3×#125—62no—
51C3×#125C2×D311yescharacteristic
52C3×#125C2×D311yescharacteristic
53#890—22no—
54#889—22no—
55#889—22no—
56#874—31no—
57#874—31no—
58#874—61no—
59#870C2×D312yes—
60C3×#97—32no—
61#821—62no—
62#821—62no—
63#809—62no—
64#809—62no—
65#809—64no—
66#809—62no—
67#809—62no—
68#803—31no—
69#803—31no—
70#803—61no—
71#803C2×D312yes—
72C22×C3⋊D3—41no—
73C2×C32×C4—22no—
74C2×C32×C4C2×D411yescharacteristic
75C2×C32×C4C2×D412yes—
76C4⋊(C3⋊D3)—42no—
77C4⋊(C3⋊D3)—44no—
78C2×C3×Dic3—22no—
79C2×C3×Dic3—22no—
80C3⋊(C3⋊C8)—41no—
81C3×C3⋊C8—42no—
82#564—31no—
83#490—92no—
84#490—91no—
85#330—31no—
86#330—92no—
87C2×D12—62no—
88C2×D12—61no—
89C2×D12—62no—
90C2×D12—61no—
91C2×D12—62no—
92C2×D12—122no—
93C2×C3×C8—62no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—32no—
98C3×C4⋊C4—32no—
99C3×C4⋊C4—42no—
100C3×C4⋊C4—62no—
101C3×C4⋊C4—62no—
102C3×C4⋊C4—62no—
103C3×C4⋊C4—62no—
104C3×C4⋊C4—62no—
105C3×C4⋊C4D1212yes—
106C3×C4⋊C4#6712yes—
107C3×C42—21no—
108C3×C42—62no—
109C3×C42C22×D311yescharacteristic
110C3×C42C22×D311yescharacteristic
111C4⋊Dic3—22no—
112C4⋊Dic3#6712yes—
113C4⋊Dic3#6712yes—
114Dic3⋊C4—22no—
115Dic3⋊C4—22no—
116Dic3⋊C4—22no—
117Dic3⋊C4—22no—
118C4×Dic3—22no—
119C2×C3⋊C8—22no—
120C2×C3⋊C8—62no—
121C2×C3⋊C8—61no—
122C2×C3⋊C8—61no—
123C22×C32#12011yescharacteristic
124C2×C3⋊D3—81no—
125C2×C3⋊D3—82no—
126C32×C4—21no—
127C32×C4—22no—
128C32×C4—42no—
129C3×Dic3—42no—
130C3×Dic3—42no—
131#127—91no—
132#125—32no—
133#125—62no—
134#125—62no—
135#125—62no—
136#125D3211yescharacteristic
137#102—182no—
138#102—182no—
139#102—182no—
140#97—32no—
141#97—91no—
142C22×D3—122no—
143C22×D3—121no—
144C22×D3—121no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—22no—
149C2×C3×C4—42no—
150C2×C3×C4—62no—
151C2×C3×C4—62no—
152C2×C3×C4C2×D1211yescharacteristic
153C2×C3×C4D3×D412yes—
154C2×C3×C4C2×#6711yescharacteristic
155C2×C3×C4C2×#6712yes—
156C2×Dic3—22no—
157C2×Dic3—22no—
158D12—124no—
159D12—122no—
160D12—122no—
161D12—124no—
162D12—124no—
163D12—244no—
164C3×C8—122no—
165C3⋊C8—42no—
166C3⋊C8—122no—
167C3⋊C8—121no—
168C3⋊C8—121no—
169C2×C32#45411yescharacteristic
170C2×C32#45412yes—
171C3⋊D3—161no—
172C2×D4—181no—
173C2×D4—182no—
174C2×C8—62no—
175C2×C8—181no—
176C4⋊C4—22no—
177C4⋊C4—32no—
178C4⋊C4—32no—
179C4⋊C4—62no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C4⋊C4#62612yes—
185C42—62no—
186C42C2×D3211yescharacteristic
187C22×C3—21no—
188C22×C3#88211yescharacteristic
189C22×C3#95311yescharacteristic
190C2×D3—242no—
191C2×D3—244no—
192C2×D3—241no—
193C2×D3—242no—
194C2×D3—241no—
195C2×D3—242no—
196C3×C4—22no—
197C3×C4—21no—
198C3×C4—22no—
199C3×C4—21no—
200C3×C4—22no—
201C3×C4—42no—
202C3×C4—42no—
203C3×C4—42no—
204C3×C4—82no—
205C3×C4—122no—
206C3×C4—122no—
207Dic3—42no—
208Dic3—42no—
209C32#165811yescharacteristic, complemented, Sylow
210C23—361no—
211D4—362no—
212D4—364no—
213C2×C4—22no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4C2×#62611yescharacteristic
217C2×C4D3×#6712yes—
218C8—122no—
219C8—361no—
220C2×C3—21no—
221C2×C3—22no—
222C2×C3#475311yescharacteristic
223C2×C3#482712yes—
224C2×C3#524511yescharacteristic
225C2×C3#524512yes—
226D3—482no—
227D3—481no—
228D3—481no—
229C22—721no—
230C22—722no—
231C22#6397711yescharacteristic
232C4—21no—
233C4—22no—
234C4—42no—
235C4—122no—
236C4—122no—
237C3—21no—
238C3#6837511yescharacteristic, complemented
239C3#7196011yescharacteristic, complemented
240C2—1441no—
241C2#1059164611yescharacteristic
242C2#1059168512yes—
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×#125C22
socleC22×C32#120
prime radical#64083C22
hypercenter#125D32
hypocenterC32#1658
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C3×C3⋊#564C2semidirect
4. C3×#4543C2semidirect
5. #10589088C2semidirect
20. C3⋊#564D3semidirect
28. #4543D3semidirect
209. C32#1658semidirect
238. C3#68375semidirect
239. C3#71960semidirect

Maximal subgroups

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

subgroupindex
C3×C3⋊#5642
#105927662
#105902822
C3×#45432
#105890882
#719603
#683753