Finite Group Databasebeta

← #72238836

Subgroups — #72238836

Order 1536.

1151 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 1151 conjugacy classes of subgroups — too many to draw. Only its 25 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
1#154048C311yesmaximal, characteristic, complemented, Sylow
2C22⋊(C22⋊SL(2,𝔽3))—41nomaximal
3C2×C22⋊(C42⋊C3)—43nomaximal
4C22⋊(#95⋊C3)—41nomaximal
5C2×#1446—31no—
6#7556—31no—
7#7554—33no—
8C22⋊(C42⋊C3)—43no—
9C24×Q8A411yescharacteristic
10C23×C42A413yes—
11C22×#95—33no—
12C22×#95—33no—
13C22×#95—31no—
14C22×#95—33no—
15C22×#95A411yescharacteristic
16#1446—38no—
17C2×C22⋊A4—161no—
18C22⋊SL(2,𝔽3)—164no—
19C22⋊SL(2,𝔽3)—1612no—
20C22⋊SL(2,𝔽3)—164no—
21C2×C42⋊C3—1612no—
22#95⋊C3—164no—
23C23×Q8—34no—
24C23×Q8—123no—
25C23×Q8—121no—
26C24×C4—31no—
27C24×C4—33no—
28C24×C4—31no—
29C22×C42—43no—
30C22×C42—66no—
31C22×C42—123no—
32C2×#95—312no—
33C2×#95—312no—
34C2×#95—34no—
35C2×#95—612no—
36C2×#95—121no—
37C2×#95—126no—
38C2×#95—123no—
39C2×#95—123no—
40C2×#95—121no—
41C2×#95—121no—
42C2×#95—123no—
43C22⋊A4—161no—
44C42⋊C3—1612no—
45C25C22⋊A411yescharacteristic
46C22×Q8—34no—
47C22×Q8—412no—
48C22×Q8—44no—
49C22×Q8—1212no—
50C22×Q8—1212no—
51C22×Q8—124no—
52C22×Q8—1212no—
53C22×Q8C42⋊C314yescomplemented
54C23×C4—36no—
55C23×C4—36no—
56C23×C4—36no—
57C23×C4—36no—
58C23×C4—32no—
59C23×C4—63no—
60C23×C4—66no—
61C23×C4—63no—
62C23×C4—63no—
63C23×C4—66no—
64C23×C4—66no—
65C23×C4—63no—
66C23×C4—66no—
67C23×C4—63no—
68C23×C4—66no—
69C23×C4—63no—
70C23×C4—62no—
71C23×C4—66no—
72C23×C4—63no—
73C23×C4—63no—
74C2×C42—412no—
75C2×C42—126no—
76C2×C42—1212no—
77C2×C42—126no—
78#95—44no—
79#95—124no—
80#95—1212no—
81#95—1212no—
82#95—1212no—
83C2×A4—641no—
84C2×A4—641no—
85C2×A4—643no—
86SL(2,𝔽3)—6416no—
87C24—31no—
88C24—33no—
89C24—31no—
90C24—31no—
91C24—33no—
92C24—31no—
93C24#99711yescharacteristic
94C2×Q8—124no—
95C2×Q8—1212no—
96C2×Q8—1248no—
97C2×Q8—1216no—
98C22×C4—312no—
99C22×C4—34no—
100C22×C4—612no—
101C22×C4—66no—
102C22×C4—612no—
103C22×C4—66no—
104C22×C4—66no—
105C22×C4—66no—
106C22×C4—612no—
107C22×C4—66no—
108C22×C4—612no—
109C22×C4—612no—
110C22×C4—612no—
111C22×C4—66no—
112C22×C4—66no—
113C22×C4—66no—
114C22×C4—612no—
115C22×C4—612no—
116C22×C4—66no—
117C22×C4—66no—
118C22×C4—62no—
119C22×C4—123no—
120C22×C4—126no—
121C22×C4—123no—
122C22×C4—123no—
123C22×C4—126no—
124C22×C4—126no—
125C22×C4—126no—
126C22×C4—126no—
127C22×C4—126no—
128C22×C4—122no—
129C22×C4—121no—
130C22×C4—123no—
131C22×C4—126no—
132C22×C4—123no—
133C22×C4—123no—
134C22×C4—121no—
135C22×C4—123no—
136C22×C4—126no—
137C22×C4—123no—
138C22×C4—126no—
139C22×C4—126no—
140C22×C4—123no—
141C22×C4—121no—
142C42—412no—
143C42—1212no—
144A4—641no—
145A4—643no—
146A4—641no—
147C23—33no—
148C23—33no—
149C23—33no—
150C23—33no—
151C23—33no—
152C23—36no—
153C23—33no—
154C23—33no—
155C23—31no—
156C23—33no—
157C23—33no—
158C23—36no—
159C23—31no—
160C23—31no—
161C23—33no—
162C23—33no—
163C23—31no—
164C23—31no—
165C23C22⋊(C42⋊C3)11yescharacteristic
166C23#546413yes—
167C23#546411yescharacteristic
168Q8—416no—
169Q8—1216no—
170C2×C4—624no—
171C2×C4—612no—
172C2×C4—612no—
173C2×C4—126no—
174C2×C4—126no—
175C2×C4—1212no—
176C2×C4—1212no—
177C2×C4—126no—
178C2×C4—1212no—
179C2×C4—126no—
180C2×C4—126no—
181C2×C4—1212no—
182C2×C4—126no—
183C2×C4—126no—
184C2×C4—126no—
185C2×C4—126no—
186C2×C4—126no—
187C2×C4—122no—
188C2×C4—242no—
189C2×C4—246no—
190C2×C3—2561no—
191C22—33no—
192C22—31no—
193C22—31no—
194C22—36no—
195C22—33no—
196C22—31no—
197C22—33no—
198C22—33no—
199C22—33no—
200C22—36no—
201C22—33no—
202C22—33no—
203C22—33no—
204C22—33no—
205C22—33no—
206C22—31no—
207C22—33no—
208C22—31no—
209C22#7328611yescharacteristic
210C22#2736⋊C313yes—
211C22#7329011yescharacteristic
212C4—1212no—
213C4—124no—
214C4—242no—
215C3—2561noSylow
216C2—33no—
217C2—31no—
218C2—33no—
219C2—31no—
220C2—31no—
221C2—31no—
222C2#1170921911yescharacteristic
subgroupG/H
center, hypercenterC2#11709219
commutator, Fitting, cosocle, hypocenter#154048C3
Frattini, socleC25C22⋊A4
LCSG ▷ #154048
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #154048C3semidirect
53. C22×Q8C42⋊C3semidirect

Maximal subgroups

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

subgroupindex
#1540483
C22⋊(C22⋊SL(2,𝔽3))4
C2×C22⋊(C42⋊C3)4
C22⋊(#95⋊C3)4