Finite Group Databasebeta

← C22×#127⋊C9 #11919709

Subgroups — C22×#127⋊C9 #11919709

Order 1152.

1112 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 90 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×#127⋊C9C216yesmaximal, direct product factor
2C22×C42⋊C9C211yesmaximal, characteristic, complemented
3C22×C3×#127C311yesmaximal, characteristic
4C23×C22⋊C9—41nomaximal
5#127⋊C9C2214yesdirect product factor
6C2×C42⋊C9C2213yescomplemented
7C23×C3×D4—31no—
8C23×C3×D4—31no—
9C2×C3×#127—36no—
10C2×C3×#127C2×C316yes—
11C22×C3×C42C2×C311yescharacteristic
12C22×C22⋊C9—41no—
13C22×C22⋊C9—46no—
14C42⋊C9C2311yescharacteristic, complemented
15C22×#127C911yescharacteristic, complemented, Sylow
16C25×C3—31no—
17C25×C3A411yescharacteristic
18C22×C3×D4—36no—
19C22×C3×D4—36no—
20C22×C3×D4—36no—
21C22×C3×D4—36no—
22C22×C3×D4—62no—
23C22×C3×D4—66no—
24C22×C3×D4—66no—
25C22×C3×D4—62no—
26C23×C3×C4—31no—
27C3×#127—312no—
28C3×#127—38no—
29C3×#127C22×C314yes—
30C2×C3×C42—33no—
31C2×C3×C42C22×C313yes—
32C23×C9—161no—
33C2×C22⋊C9—43no—
34C2×C22⋊C9—44no—
35C23×D4—31no—
36C23×D4—31no—
37C2×#127—36no—
38C2×#127C2×C916yescomplemented
39C22×C42C2×C911yescharacteristic, complemented
40C24×C3—36no—
41C24×C3—61no—
42C24×C3—63no—
43C24×C3—63no—
44C24×C3—66no—
45C24×C3—61no—
46C24×C3—62no—
47C24×C3C2×A411yescharacteristic
48C24×C3C2×A416yes—
49C2×C3×D4—34no—
50C2×C3×D4—312no—
51C2×C3×D4—34no—
52C2×C3×D4—312no—
53C2×C3×D4—612no—
54C2×C3×D4—612no—
55C2×C3×D4—612no—
56C2×C3×D4—612no—
57C2×C3×D4—612no—
58C2×C3×D4—612no—
59C2×C3×D4—612no—
60C2×C3×D4—612no—
61C22×C3×C4—36no—
62C22×C3×C4—32no—
63C22×C3×C4—33no—
64C22×C3×C4—33no—
65C3×C42—33no—
66C3×C42—32no—
67C3×C42C23×C311yescharacteristic
68C22×C9—166no—
69C22×C9—161no—
70C22⋊C9—41no—
71C25—31no—
72C25C22⋊C911yescharacteristic
73C22×D4—36no—
74C22×D4—36no—
75C22×D4—36no—
76C22×D4—36no—
77C22×D4—62no—
78C22×D4—66no—
79C22×D4—66no—
80C22×D4—62no—
81C23×C4—31no—
82#127—312no—
83#127—38no—
84#127C22×C914yescomplemented
85C2×C42—33no—
86C2×C42C22×C913yescomplemented
87C23×C3—33no—
88C23×C3—31no—
89C23×C3—34no—
90C23×C3—41no—
91C23×C3—66no—
92C23×C3—66no—
93C23×C3—66no—
94C23×C3—612no—
95C23×C3—66no—
96C23×C3—612no—
97C23×C3—123no—
98C23×C3—122no—
99C23×C3—126no—
100C23×C3—126no—
101C23×C3—123no—
102C23×C3—121no—
103C23×C3C22×A413yes—
104C23×C3C22×A414yes—
105C3×D4—68no—
106C3×D4—624no—
107C3×D4—68no—
108C3×D4—624no—
109C2×C3×C4—36no—
110C2×C3×C4—36no—
111C2×C3×C4—36no—
112C2×C3×C4—36no—
113C2×C3×C4—31no—
114C2×C3×C4—33no—
115C2×C9—164no—
116C2×C9—163no—
117C24—36no—
118C24—61no—
119C24—63no—
120C24—63no—
121C24—66no—
122C24—61no—
123C24—62no—
124C24C2×C22⋊C911yescharacteristic
125C24C2×C22⋊C916yes—
126C2×D4—34no—
127C2×D4—312no—
128C2×D4—34no—
129C2×D4—312no—
130C2×D4—612no—
131C2×D4—612no—
132C2×D4—612no—
133C2×D4—612no—
134C2×D4—612no—
135C2×D4—612no—
136C2×D4—612no—
137C2×D4—612no—
138C22×C4—36no—
139C22×C4—32no—
140C22×C4—33no—
141C22×C4—33no—
142C42—33no—
143C42—32no—
144C42C23×C911yescharacteristic, complemented
145C22×C3—33no—
146C22×C3—32no—
147C22×C3—33no—
148C22×C3—33no—
149C22×C3—46no—
150C22×C3—64no—
151C22×C3—64no—
152C22×C3—68no—
153C22×C3—126no—
154C22×C3—1212no—
155C22×C3—126no—
156C22×C3—126no—
157C22×C3#127⋊C311yescharacteristic
158C22×C3C23×A411yescharacteristic
159C3×C4—32no—
160C3×C4—36no—
161C9—161noSylow
162C23—33no—
163C23—31no—
164C23—34no—
165C23—41no—
166C23—66no—
167C23—66no—
168C23—66no—
169C23—612no—
170C23—66no—
171C23—612no—
172C23—123no—
173C23—122no—
174C23—126no—
175C23—126no—
176C23—123no—
177C23—121no—
178C23C22×C22⋊C913yes—
179C23C22×C22⋊C914yes—
180D4—68no—
181D4—624no—
182D4—68no—
183D4—624no—
184C2×C4—36no—
185C2×C4—36no—
186C2×C4—36no—
187C2×C4—36no—
188C2×C4—31no—
189C2×C4—33no—
190C2×C3—31no—
191C2×C3—33no—
192C2×C3—44no—
193C2×C3—124no—
194C2×C3C2×#127⋊C313yes—
195C22—33no—
196C22—32no—
197C22—33no—
198C22—33no—
199C22—46no—
200C22—64no—
201C22—64no—
202C22—68no—
203C22—126no—
204C22—1212no—
205C22—126no—
206C22—126no—
207C22#127⋊C911yescharacteristic, direct product factor
208C22C23×C22⋊C911yescharacteristic
209C4—32no—
210C4—36no—
211C3C22×#127⋊C311yescharacteristic
212C2—31no—
213C2—33no—
214C2—44no—
215C2—124no—
216C2C2×#127⋊C913yesdirect product factor
subgroupG/H
center, hypercenterC22×C3#127⋊C3
commutator, hypocenterC42C23×C9
FrattiniC22×C3C23×A4
Fitting, prime radicalC22×C3×#127C3
socleC24×C3C2×A4
cosocleC3×C42C23×C3
LCSG ▷ C42
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×#127⋊C9C2direct (N × H)
2. C22×C42⋊C9C2semidirect
5. #127⋊C9C22direct (N × H)
6. C2×C42⋊C9C22semidirect
14. C42⋊C9C23semidirect
15. C22×#127C9semidirect
38. C2×#127C2×C9semidirect
39. C22×C42C2×C9semidirect
84. #127C22×C9semidirect
86. C2×C42C22×C9semidirect
144. C42C23×C9semidirect
207. C22#127⋊C9direct (N × H)
216. C2C2×#127⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#127⋊C92
C22×C42⋊C92
C22×C3×#1273
C23×C22⋊C94