Finite Group Databasebeta

← #126×C22⋊C9 #11919707

Subgroups — #126×C22⋊C9 #11919707

Order 1152.

580 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C4⋊C4×C22⋊C9C213yesmaximal, complemented
2C22⋊C4×C22⋊C9C213yesmaximal
3C42×C22⋊C9C211yesmaximal, characteristic, complemented
4C22×C3×#126C311yesmaximal, characteristic
5C23×C22⋊C9C2211yescharacteristic
6C2×C4×C22⋊C9C2213yes—
7C2×C4×C22⋊C9C2213yes—
8C9×#126—41nomaximal
9C2×C3×#126—31no—
10C2×C3×#126—33no—
11C2×C3×#126—33no—
12C2×C3×#126—31no—
13C22×C3×C4⋊C4C2×C313yes—
14C22×C3×C22⋊C4C2×C313yes—
15C22×C3×C42C2×C311yescharacteristic
16C22×C22⋊C9—23no—
17C22×C22⋊C9C2311yescharacteristic
18C4×C22⋊C9—23no—
19C4×C22⋊C9—23no—
20C9×C4⋊C4—43no—
21C9×C22⋊C4—43no—
22C42×C9—41no—
23C22×#126C911yescharacteristic, complemented, Sylow
24C25×C3C22×C311yescharacteristic
25C23×C3×C4C22×C313yes—
26C23×C3×C4C22×C313yes—
27C3×#126—36no—
28C3×#126—32no—
29C3×#126—36no—
30C3×#126—33no—
31C3×#126—33no—
32C3×#126—31no—
33C3×#126A411yescharacteristic
34C2×C3×C4⋊C4—33no—
35C2×C3×C4⋊C4—33no—
36C2×C3×C4⋊C4—36no—
37C2×C3×C22⋊C4—33no—
38C2×C3×C22⋊C4—33no—
39C2×C3×C22⋊C4—33no—
40C2×C3×C22⋊C4—33no—
41C2×C3×C42—31no—
42C2×C3×C42—33no—
43C23×C9—41no—
44C2×C22⋊C9—41no—
45C2×C22⋊C9#4113yes—
46C2×C4×C9—43no—
47C2×C4×C9—43no—
48C2×#126—31no—
49C2×#126—33no—
50C2×#126—33no—
51C2×#126—31no—
52C22×C4⋊C4C2×C913yescomplemented
53C22×C22⋊C4C2×C913yes—
54C22×C42C2×C911yescharacteristic, complemented
55C24×C3—23no—
56C24×C3—31no—
57C24×C3—31no—
58C24×C3—63no—
59C24×C3C23×C311yescharacteristic
60C22×C3×C4—23no—
61C22×C3×C4—23no—
62C22×C3×C4—33no—
63C22×C3×C4—33no—
64C22×C3×C4—33no—
65C22×C3×C4—33no—
66C22×C3×C4—63no—
67C22×C3×C4—63no—
68C3×C4⋊C4—36no—
69C3×C4⋊C4—36no—
70C3×C4⋊C4—33no—
71C3×C4⋊C4C2×A413yes—
72C3×C22⋊C4—36no—
73C3×C22⋊C4—33no—
74C3×C22⋊C4—33no—
75C3×C22⋊C4—33no—
76C3×C22⋊C4C2×A413yes—
77C3×C42—32no—
78C3×C42—33no—
79C3×C42C2×A411yescharacteristic
80C22×C9—41no—
81C22×C9—83no—
82C22⋊C9#12611yescharacteristic, direct product factor
83C4×C9—83no—
84C4×C9—83no—
85C25C22×C911yescharacteristic
86C23×C4C22×C913yes—
87C23×C4C22×C913yes—
88#126—36no—
89#126—32no—
90#126—36no—
91#126—33no—
92#126—33no—
93#126—31no—
94#126C22⋊C911yescharacteristic, direct product factor
95C2×C4⋊C4—33no—
96C2×C4⋊C4—33no—
97C2×C4⋊C4—36no—
98C2×C22⋊C4—33no—
99C2×C22⋊C4—33no—
100C2×C22⋊C4—33no—
101C2×C22⋊C4—33no—
102C2×C42—31no—
103C2×C42—33no—
104C23×C3—33no—
105C23×C3—31no—
106C23×C3—31no—
107C23×C3—41no—
108C23×C3—63no—
109C23×C3—63no—
110C23×C3—63no—
111C23×C3—63no—
112C23×C3—123no—
113C23×C3—122no—
114C23×C3C3×#4113yes—
115C23×C3C22×A411yescharacteristic
116C2×C3×C4—33no—
117C2×C3×C4—33no—
118C2×C3×C4—63no—
119C2×C3×C4—63no—
120C2×C3×C4—63no—
121C2×C3×C4—63no—
122C2×C3×C4—63no—
123C2×C3×C4—63no—
124C2×C3×C4—63no—
125C2×C3×C4—63no—
126C2×C3×C4C22×A413yes—
127C2×C3×C4C22×A413yes—
128C2×C9—43no—
129C2×C9—161no—
130C24—23no—
131C24—31no—
132C24—31no—
133C24—63no—
134C24C23×C911yescharacteristic
135C22×C4—23no—
136C22×C4—23no—
137C22×C4—33no—
138C22×C4—33no—
139C22×C4—33no—
140C22×C4—33no—
141C22×C4—63no—
142C22×C4—63no—
143C4⋊C4—36no—
144C4⋊C4—36no—
145C4⋊C4—33no—
146C4⋊C4C2×C22⋊C913yescomplemented
147C22⋊C4—36no—
148C22⋊C4—33no—
149C22⋊C4—33no—
150C22⋊C4—33no—
151C22⋊C4C2×C22⋊C913yes—
152C42—32no—
153C42—33no—
154C42C2×C22⋊C911yescharacteristic, complemented
155C22×C3—23no—
156C22×C3—33no—
157C22×C3—32no—
158C22×C3—33no—
159C22×C3—33no—
160C22×C3—63no—
161C22×C3—121no—
162C22×C3—123no—
163C22×C3—121no—
164C22×C3—123no—
165C22×C3C3×#12611yescharacteristic
166C22×C3C23×A411yescharacteristic
167C3×C4—23no—
168C3×C4—23no—
169C3×C4—63no—
170C3×C4—63no—
171C9—41noSylow
172C23—33no—
173C23—31no—
174C23—31no—
175C23—41no—
176C23—63no—
177C23—63no—
178C23—63no—
179C23—63no—
180C23—123no—
181C23—122no—
182C23C9×#4113yes—
183C23C22×C22⋊C911yescharacteristic
184C2×C4—33no—
185C2×C4—33no—
186C2×C4—63no—
187C2×C4—63no—
188C2×C4—63no—
189C2×C4—63no—
190C2×C4—63no—
191C2×C4—63no—
192C2×C4—63no—
193C2×C4—63no—
194C2×C4C22×C22⋊C913yes—
195C2×C4C22×C22⋊C913yes—
196C2×C3—33no—
197C2×C3—31no—
198C2×C3—41no—
199C2×C3—121no—
200C2×C3A4×#4113yes—
201C22—23no—
202C22—33no—
203C22—32no—
204C22—33no—
205C22—33no—
206C22—63no—
207C22—121no—
208C22—123no—
209C22—121no—
210C22—123no—
211C22C9×#12611yescharacteristic, complemented
212C22C23×C22⋊C911yescharacteristic
213C4—23no—
214C4—23no—
215C4—63no—
216C4—63no—
217C3A4×#12611yescharacteristic
218C2—31no—
219C2—33no—
220C2—41no—
221C2—121no—
222C2#41×C22⋊C913yes—
subgroupG/H
center, FrattiniC22×C3C23×A4
commutatorC24C23×C9
FittingC22×C3×#126C3
socle, cosocleC24×C3C23×C3
prime radicalC25×C3C22×C3
hypercenterC3×#126A4
hypocenterC22C9×#126
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22×C3 ◁ C3×#126

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×C22⋊C9C2semidirect
3. C42×C22⋊C9C2semidirect
23. C22×#126C9semidirect
52. C22×C4⋊C4C2×C9semidirect
54. C22×C42C2×C9semidirect
82. C22⋊C9#126direct (N × H)
94. #126C22⋊C9direct (N × H)
146. C4⋊C4C2×C22⋊C9semidirect
154. C42C2×C22⋊C9semidirect
211. C22C9×#126semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C22⋊C92
C22⋊C4×C22⋊C92
C42×C22⋊C92
C22×C3×#1263
C9×#1264