Finite Group Databasebeta

← C22×C3×#127⋊C3 #11919708

Subgroups — C22×C3×#127⋊C3 #11919708

Order 1152.

1256 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×C3×#127⋊C3C216yesmaximal, direct product factor
2C22×C3×C42⋊C3C211yesmaximal, characteristic, complemented
3C22×C3×#127C311yesmaximal, characteristic, complemented
4C22×#127⋊C3C313yesmaximal, direct product factor
5C23×C3×A4—41nomaximal
6C3×#127⋊C3C2214yesdirect product factor
7C2×C3×C42⋊C3C2213yescomplemented
8C23×C3×D4—31no—
9C23×C3×D4—31no—
10C2×C3×#127—36no—
11C2×C3×#127C2×C316yescomplemented
12C22×C3×C42C2×C311yescharacteristic, complemented
13C2×#127⋊C3C2×C3118yesdirect product factor
14C22×C42⋊C3C2×C313yescomplemented
15C22×C3×A4—41no—
16C22×C3×A4—46no—
17C3×C42⋊C3C2311yescharacteristic, complemented
18C22×#127C3211yescharacteristic, complemented, Sylow
19C25×C3—31no—
20C25×C3A411yescharacteristic
21C23×A4—43no—
22C22×C3×D4—36no—
23C22×C3×D4—36no—
24C22×C3×D4—36no—
25C22×C3×D4—36no—
26C22×C3×D4—62no—
27C22×C3×D4—66no—
28C22×C3×D4—66no—
29C22×C3×D4—62no—
30C23×C3×C4—31no—
31C3×#127—312no—
32C3×#127—38no—
33C3×#127C22×C314yescomplemented
34C2×C3×C42—33no—
35C2×C3×C42C22×C313yescomplemented
36#127⋊C3C22×C3112yesdirect product factor
37C2×C42⋊C3C22×C319yescomplemented
38C23×C32—161no—
39C2×C3×A4—43no—
40C2×C3×A4—44no—
41C23×D4—31no—
42C23×D4—31no—
43C2×#127—36no—
44C2×#127C2×C3216yescomplemented
45C22×C42C2×C3211yescharacteristic, complemented
46C24×C3—36no—
47C24×C3—61no—
48C24×C3—63no—
49C24×C3—63no—
50C24×C3—66no—
51C24×C3—61no—
52C24×C3—62no—
53C24×C3C2×A411yescharacteristic
54C24×C3C2×A416yes—
55C22×A4—43no—
56C22×A4—418no—
57C2×C3×D4—34no—
58C2×C3×D4—312no—
59C2×C3×D4—34no—
60C2×C3×D4—312no—
61C2×C3×D4—612no—
62C2×C3×D4—612no—
63C2×C3×D4—612no—
64C2×C3×D4—612no—
65C2×C3×D4—612no—
66C2×C3×D4—612no—
67C2×C3×D4—612no—
68C2×C3×D4—612no—
69C22×C3×C4—36no—
70C22×C3×C4—32no—
71C22×C3×C4—33no—
72C22×C3×C4—33no—
73C3×C42—33no—
74C3×C42—32no—
75C3×C42C23×C311yescharacteristic, complemented
76C42⋊C3C23×C313yescomplemented
77C22×C32—166no—
78C22×C32—161no—
79C3×A4—41no—
80C25—31no—
81C25C3×A411yescharacteristic
82C22×D4—36no—
83C22×D4—36no—
84C22×D4—36no—
85C22×D4—36no—
86C22×D4—62no—
87C22×D4—66no—
88C22×D4—66no—
89C22×D4—62no—
90C23×C4—31no—
91#127—312no—
92#127—38no—
93#127C22×C3214yescomplemented
94C2×C42—33no—
95C2×C42C22×C3213yescomplemented
96C23×C3—33no—
97C23×C3—31no—
98C23×C3—34no—
99C23×C3—41no—
100C23×C3—66no—
101C23×C3—66no—
102C23×C3—66no—
103C23×C3—612no—
104C23×C3—66no—
105C23×C3—612no—
106C23×C3—123no—
107C23×C3—122no—
108C23×C3—126no—
109C23×C3—126no—
110C23×C3—123no—
111C23×C3—121no—
112C23×C3—163no—
113C23×C3C22×A413yes—
114C23×C3C22×A414yes—
115C2×A4—49no—
116C2×A4—412no—
117C3×D4—68no—
118C3×D4—624no—
119C3×D4—68no—
120C3×D4—624no—
121C2×C3×C4—36no—
122C2×C3×C4—36no—
123C2×C3×C4—36no—
124C2×C3×C4—36no—
125C2×C3×C4—31no—
126C2×C3×C4—33no—
127C2×C32—164no—
128C2×C32—163no—
129C24—36no—
130C24—61no—
131C24—63no—
132C24—63no—
133C24—66no—
134C24—61no—
135C24—62no—
136C24C2×C3×A411yescharacteristic
137C24C2×C3×A416yes—
138C2×D4—34no—
139C2×D4—312no—
140C2×D4—34no—
141C2×D4—312no—
142C2×D4—612no—
143C2×D4—612no—
144C2×D4—612no—
145C2×D4—612no—
146C2×D4—612no—
147C2×D4—612no—
148C2×D4—612no—
149C2×D4—612no—
150C22×C4—36no—
151C22×C4—32no—
152C22×C4—33no—
153C22×C4—33no—
154C42—33no—
155C42—32no—
156C42C23×C3211yescharacteristic, complemented
157C22×C3—33no—
158C22×C3—32no—
159C22×C3—33no—
160C22×C3—33no—
161C22×C3—46no—
162C22×C3—64no—
163C22×C3—64no—
164C22×C3—68no—
165C22×C3—126no—
166C22×C3—1212no—
167C22×C3—126no—
168C22×C3—126no—
169C22×C3—1618no—
170C22×C3—163no—
171C22×C3#127⋊C311yescharacteristic, direct product factor
172C22×C3C23×A411yescharacteristic
173A4—43no—
174C3×C4—32no—
175C3×C4—36no—
176C32—161noSylow
177C23—33no—
178C23—31no—
179C23—34no—
180C23—41no—
181C23—66no—
182C23—66no—
183C23—66no—
184C23—612no—
185C23—66no—
186C23—612no—
187C23—123no—
188C23—122no—
189C23—126no—
190C23—126no—
191C23—123no—
192C23—121no—
193C23C22×C3×A413yes—
194C23C22×C3×A414yes—
195D4—68no—
196D4—624no—
197D4—68no—
198D4—624no—
199C2×C4—36no—
200C2×C4—36no—
201C2×C4—36no—
202C2×C4—36no—
203C2×C4—31no—
204C2×C4—33no—
205C2×C3—31no—
206C2×C3—33no—
207C2×C3—44no—
208C2×C3—124no—
209C2×C3—1612no—
210C2×C3—169no—
211C2×C3C2×#127⋊C313yesdirect product factor
212C22—33no—
213C22—32no—
214C22—33no—
215C22—33no—
216C22—46no—
217C22—64no—
218C22—64no—
219C22—68no—
220C22—126no—
221C22—1212no—
222C22—126no—
223C22—126no—
224C22C3×#127⋊C311yescharacteristic, direct product factor
225C22C23×C3×A411yescharacteristic
226C4—32no—
227C4—36no—
228C3—163no—
229C3C22×#127⋊C311yescharacteristic, direct product factor
230C2—31no—
231C2—33no—
232C2—44no—
233C2—124no—
234C2C2×C3×#127⋊C313yesdirect product factor
subgroupG/H
center, hypercenterC22×C3#127⋊C3
commutator, cosocle, hypocenterC42C23×C32
FrattiniC22C23×C3×A4
FittingC22×C3×#127C3
socleC24×C3C2×A4
LCSG ▷ C42
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#127⋊C3C2direct (N × H)
2. C22×C3×C42⋊C3C2semidirect
3. C22×C3×#127C3semidirect
4. C22×#127⋊C3C3direct (N × H)
6. C3×#127⋊C3C22direct (N × H)
7. C2×C3×C42⋊C3C22semidirect
11. C2×C3×#127C2×C3semidirect
12. C22×C3×C42C2×C3semidirect
13. C2×#127⋊C3C2×C3direct (N × H)
14. C22×C42⋊C3C2×C3semidirect
17. C3×C42⋊C3C23semidirect
18. C22×#127C32semidirect
33. C3×#127C22×C3semidirect
35. C2×C3×C42C22×C3semidirect
36. #127⋊C3C22×C3direct (N × H)
37. C2×C42⋊C3C22×C3semidirect
44. C2×#127C2×C32semidirect
45. C22×C42C2×C32semidirect
75. C3×C42C23×C3semidirect
76. C42⋊C3C23×C3semidirect
93. #127C22×C32semidirect
95. C2×C42C22×C32semidirect
156. C42C23×C32semidirect
171. C22×C3#127⋊C3direct (N × H)
211. C2×C3C2×#127⋊C3direct (N × H)
224. C22C3×#127⋊C3direct (N × H)
229. C3C22×#127⋊C3direct (N × H)
234. C2C2×C3×#127⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#127⋊C32
C22×C3×C42⋊C32
C22×C3×#1273
C22×#127⋊C33
C23×C3×A44