Finite Group Databasebeta

← C22×#64231 #11900707

Subgroups — C22×#64231 #11900707

Order 1152.

4362 conjugacy classes of subgroups; only the 583 normal subgroups are shown.

normal subgroups only

This group has 4362 conjugacy classes of subgroups — too many to draw. Only its 583 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
1C23×GO+(4,𝔽2)C214yesmaximal, complemented
2C22×#3802C212yesmaximal, complemented
3C22×C4×#157C211yesmaximal, characteristic, complemented
4C2×#64231C2124yesmaximal, direct product factor
5C23×D32C2214yes—
6C23×#157C2212yes—
7C22×GO+(4,𝔽2)—232no—
8C22×GO+(4,𝔽2)C22148yescomplemented
9C22×C4×C3⋊D3C2211yescharacteristic
10C22×C3×D12—22no—
11C22×#625—22no—
12C2×#3802—212no—
13C2×#3802C22124yescomplemented
14C2×C4×#157C22112yescomplemented
15#64231C22164yesdirect product factor
16C22×D3×D4—62no—
17C23×C3⋊D3C2311yescharacteristic
18C23×C3×D3—24no—
19C22×D32—232no—
20C22×D32—212no—
21C22×D32C23124yes—
22C22×#157D4116yescomplemented
23C22×#157C23112yes—
24C2×GO+(4,𝔽2)—2192no—
25C2×GO+(4,𝔽2)C23164yescomplemented
26C22×C32×C4D411yescharacteristic, complemented
27C22×C3⋊Dic3D411yescharacteristic, complemented
28C2×C4×C3⋊D3D416yescomplemented
29C2×C4×C3⋊D3C2316yes—
30C2×C3×D12—224no—
31C2×#625—224no—
32#3802—248no—
33#3802C23132yescomplemented
34C4×#157C23116yescomplemented
35C22×#127—91nomaximal, Sylow
36C24×D3—64no—
37C22×C3×D4—62no—
38C22×#67—64no—
39C2×D3×D4—624no—
40C2×D3×D4—624no—
41C22×D12—22no—
42C22×C4×D3—62no—
43C23×C32C2×D411yescharacteristic
44C22×C3⋊D3C2×D413yes—
45C22×C3⋊D3C2×D418yes—
46C22×C3⋊D3C2413yes—
47C22×C3×D3—224no—
48C22×C3×D3—416no—
49C2×D32—224no—
50C2×D32—296no—
51C2×D32—448no—
52C2×D32C24116yes—
53C2×#157C2×D4148yescomplemented
54C2×#157C2418yes—
55GO+(4,𝔽2)—2128no—
56C2×C32×C4C2×D416yescomplemented
57C2×C3⋊Dic3C2×D416yescomplemented
58C4×C3⋊D3C2×D4112yescomplemented
59C4×C3⋊D3C2414yes—
60C3×D12—232no—
61#625—232no—
62C23×D4—92no—
63C23×D4—94no—
64C2×#127—924no—
65C22×C42—91no—
66C24×C3—64no—
67C23×D3—24no—
68C23×D3—624no—
69C23×D3—62no—
70C23×D3—624no—
71C23×D3—1232no—
72C2×C3×D4—624no—
73C22×C3×C4—22no—
74C2×#67—648no—
75C22×Dic3—62no—
76D3×D4—632no—
77D3×D4—696no—
78C2×D12—224no—
79C2×C4×D3—612no—
80C2×C4×D3—612no—
81C22×C32#12714yescomplemented
82C22×C32C22×D413yes—
83C2×C3⋊D3#127112yescomplemented
84C2×C3⋊D3C22×D413yes—
85C2×C3⋊D3C22×D4112yes—
86C2×C3⋊D3C2511yescharacteristic
87C2×C3×D3—216no—
88C2×C3×D3—448no—
89D32—232no—
90D32—448no—
91#157C22×D4116yescomplemented
92C32×C4C22×D414yescomplemented
93C3⋊Dic3C22×D414yescomplemented
94C25—94no—
95C22×D4—62no—
96C22×D4—924no—
97C22×D4—948no—
98C22×D4—1812no—
99C22×D4—182no—
100C22×D4—1832no—
101C23×C4—92no—
102C23×C4—91no—
103#127—964no—
104C2×C42—912no—
105C23×C3—22no—
106C23×C3—624no—
107C23×C3—1216no—
108C22×D3—224no—
109C22×D3—416no—
110C22×D3—616no—
111C22×D3—66no—
112C22×D3—616no—
113C22×D3—66no—
114C22×D3—648no—
115C22×D3—1296no—
116C22×D3—1296no—
117C3×D4—632no—
118C2×C3×C4—212no—
119#67—664no—
120C2×Dic3—612no—
121D12—232no—
122C4×D3—624no—
123C4×D3—68no—
124C2×C32C2×#12716yescomplemented
125C2×C32C23×D411yescharacteristic
126C3⋊D3C2×#12716yescomplemented
127C3⋊D3C23×D412yes—
128C3×D3—416no—
129C24—64no—
130C24—924no—
131C24—91no—
132C24—1812no—
133C24—1832no—
134C2×D4—624no—
135C2×D4—932no—
136C2×D4—964no—
137C2×D4—1848no—
138C2×D4—1824no—
139C2×D4—18192no—
140C22×C4—916no—
141C22×C4—912no—
142C22×C4—91no—
143C22×C4—96no—
144C22×C4—96no—
145C22×C4GO+(4,𝔽2)11yescharacteristic, complemented
146C42—916no—
147C22×C3—26no—
148C22×C3—28no—
149C22×C3—616no—
150C22×C3—1248no—
151C2×D3—216no—
152C2×D3—448no—
153C2×D3—66no—
154C2×D3—62no—
155C2×D3—624no—
156C2×D3—624no—
157C2×D3—1232no—
158C2×D3—1296no—
159C3×C4—28no—
160Dic3—68no—
161C32C22×#12711yescharacteristic, complemented, Sylow
162C23—624no—
163C23—916no—
164C23—93no—
165C23—93no—
166C23—98no—
167C23—1216no—
168C23—1824no—
169C23—1896no—
170C23—3648no—
171C23C2×GO+(4,𝔽2)11yescharacteristic
172D4—632no—
173D4—1832no—
174D4—18128no—
175C2×C4—98no—
176C2×C4—948no—
177C2×C4—96no—
178C2×C4—94no—
179C2×C4—912no—
180C2×C4C2×GO+(4,𝔽2)16yescomplemented
181C2×C3—212no—
182C2×C3—22no—
183C2×C3—1216no—
184D3—416no—
185D3—612no—
186D3—64no—
187C22—616no—
188C22—912no—
189C22—912no—
190C22—91no—
191C22—93no—
192C22—1248no—
193C22—1832no—
194C22—3648no—
195C22#6423114yesdirect product factor
196C22C22×GO+(4,𝔽2)13yes—
197C4—916no—
198C4—94no—
199C4C22×GO+(4,𝔽2)14yescomplemented
200C3—22no—
201C2—92no—
202C2—96no—
203C2—1216no—
204C2C2×#6423116yesdirect product factor
205C2C23×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC23C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C25
FrattiniC2C23×GO+(4,𝔽2)
FittingC22×C32×C4D4
socleC23×C32C2×D4
hypercenterC22×C4GO+(4,𝔽2)
hypocenterC32C22×#127
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C23×GO+(4,𝔽2)C2semidirect
2. C22×#3802C2semidirect
3. C22×C4×#157C2semidirect
4. C2×#64231C2direct (N × H)
8. C22×GO+(4,𝔽2)C22semidirect
13. C2×#3802C22semidirect
14. C2×C4×#157C22semidirect
15. #64231C22direct (N × H)
22. C22×#157D4semidirect
25. C2×GO+(4,𝔽2)C23semidirect
26. C22×C32×C4D4semidirect
27. C22×C3⋊Dic3D4semidirect
28. C2×C4×C3⋊D3D4semidirect
33. #3802C23semidirect
34. C4×#157C23semidirect
53. C2×#157C2×D4semidirect
56. C2×C32×C4C2×D4semidirect
57. C2×C3⋊Dic3C2×D4semidirect
58. C4×C3⋊D3C2×D4semidirect
81. C22×C32#127semidirect
83. C2×C3⋊D3#127semidirect
91. #157C22×D4semidirect
92. C32×C4C22×D4semidirect
93. C3⋊Dic3C22×D4semidirect
124. C2×C32C2×#127semidirect
126. C3⋊D3C2×#127semidirect
145. C22×C4GO+(4,𝔽2)semidirect
161. C32C22×#127semidirect
180. C2×C4C2×GO+(4,𝔽2)semidirect
195. C22#64231direct (N × H)
199. C4C22×GO+(4,𝔽2)semidirect
204. C2C2×#64231direct (N × H)

Maximal subgroups

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

subgroupindex
C23×GO+(4,𝔽2)2
C22×#38022
C22×C4×#1572
C2×#642312
C22×#1279