Finite Group Databasebeta

← C22×#64364 #11919235

Subgroups — C22×#64364 #11919235

Order 1152.

17616 conjugacy classes of subgroups; only the 4761 normal subgroups are shown.

normal subgroups only

This group has 17616 conjugacy classes of subgroups — too many to draw. Only its 4761 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×C32×Q8C211yesmaximal, characteristic, complemented
2C2×#64364C2148yesmaximal, direct product factor
3C22×#3832C214yesmaximal, complemented
4C22×Q8×C3⋊D3C214yesmaximal, complemented
5C22×#3828C216yesmaximal, complemented
6C22×#1007—34nomaximal
7C22×C32×Q8C2214yes—
8C22×C32×Q8C22124yescomplemented
9C23×C32×C4C2213yes—
10C22×#638C2214yes—
11#64364C221256yesdirect product factor
12C2×#3832C22196yescomplemented
13C2×Q8×C3⋊D3C22196yescomplemented
14C2×#3828C221144yescomplemented
15C22×C4⋊(C3⋊D3)C2216yes—
16C22×C4×C3⋊D3C22112yes—
17C22×#634C2216yes—
18C23×C3×Q8D314yescomplemented
19C2×#1007—3192no—
20C22×#239—316no—
21C22×D3×Q8—316no—
22C22×#235—324no—
23C24×C32C2311yescharacteristic
24C23×C3⋊D3C2314yes—
25C2×C32×Q8C23148yes—
26C2×C32×Q8C23164yescomplemented
27C22×C32×C4C2316yes—
28C22×C32×C4C23136yes—
29C2×#638C23148yes—
30C22×C3⋊Dic3C2314yes—
31#3832C231256yescomplemented
32Q8×C3⋊D3C231256yescomplemented
33#3828C231384yescomplemented
34C2×C4⋊(C3⋊D3)C23172yes—
35C2×C4×C3⋊D3C231144yes—
36C2×#634C23172yes—
37C22×2−1+4—91noSylow
38C22×C3×Q8C2×D3116yes—
39C22×C3×Q8C2×D3196yescomplemented
40C23×C3×C4C2×D3112yes—
41C22×#67—316no—
42#1007—31024no—
43C2×#239—3384no—
44C2×D3×Q8—3384no—
45C2×#235—3576no—
46C22×D12—324no—
47C22×C4×D3—348no—
48C22×Dic6—324no—
49C23×C32—24no—
50C23×C32C2416yes—
51C23×C32C2411yescharacteristic
52C22×C3⋊D3—216no—
53C22×C3⋊D3C24124yes—
54C32×Q8C24164yes—
55C2×C32×C4C24136yes—
56C2×C32×C4C24148yes—
57#638C24164yes—
58C2×C3⋊Dic3C24124yes—
59C4⋊(C3⋊D3)C24196yes—
60C4×C3⋊D3C241192yes—
61#634C24196yes—
62C2×2−1+4—948no—
63C22×#41—94no—
64C22×#41—96no—
65C23×Q8—94no—
66C23×Q8C3⋊D311yescharacteristic, complemented
67C24×C3C22×D314yes—
68C23×D3—316no—
69C2×C3×Q8C22×D31192yes—
70C2×C3×Q8C22×D31256yescomplemented
71C22×C3×C4C22×D3124yes—
72C22×C3×C4C22×D31144yes—
73C2×#67—3192no—
74C22×Dic3—316no—
75#239—31024no—
76D3×Q8—31024no—
77#235—31536no—
78C2×D12—3288no—
79C2×C4×D3—3576no—
80C2×Dic6—3288no—
81C22×C32—212no—
82C22×C322−1+414yescomplemented
83C22×C32C2514yes—
84C22×C32C2513yes—
85C2×C3⋊D3—248no—
86C2×C3⋊D3C25116yes—
87C32×C4C25124yes—
88C3⋊Dic3C25116yes—
892−1+4—9256no—
90C2×#41—996no—
91C2×#41—9144no—
92C22×Q8—996no—
93C22×Q8—96no—
94C22×Q8C2×C3⋊D314yes—
95C22×Q8C2×C3⋊D3124yescomplemented
96C22×D4—94no—
97C22×D4—96no—
98C23×C4—912no—
99C23×C4C2×C3⋊D313yes—
100C23×C3—216no—
101C23×C3C23×D3124yes—
102C23×C3C23×D314yes—
103C22×D3—396no—
104C22×D3—664no—
105C3×Q8C23×D31256yes—
106C2×C3×C4C23×D31144yes—
107C2×C3×C4C23×D31192yes—
108#67—3256no—
109C2×Dic3—396no—
110D12—3384no—
111C4×D3—3768no—
112Dic6—3384no—
113C2×C32—24no—
114C2×C32C2×2−1+416yescomplemented
115C2×C32C2611yescharacteristic
116C3⋊D3—216no—
117C24—94no—
118C24C22×C3⋊D311yescharacteristic
119#41—9256no—
120#41—9384no—
121C2×Q8—9256no—
122C2×Q8—972no—
123C2×Q8C22×C3⋊D3148yes—
124C2×Q8C22×C3⋊D3164yescomplemented
125C2×D4—948no—
126C2×D4—972no—
127C22×C4—94no—
128C22×C4—9144no—
129C22×C4C22×C3⋊D3136yes—
130C22×C4C22×C3⋊D316yes—
131C22×C3—248no—
132C22×C3#1007116yescomplemented
133C22×C3C24×D3116yes—
134C22×C3C24×D3112yes—
135C2×D3—364no—
136C2×D3—6192no—
137C3×C4C24×D3196yes—
138Dic3—364no—
139C32C22×2−1+411yescharacteristic, complemented, Sylow
140C23—24no—
141C23—924no—
142C23—1816no—
143C23C23×C3⋊D316yes—
144C23C23×C3⋊D311yescharacteristic
145Q8—996no—
146Q8C23×C3⋊D3164yes—
147D4—964no—
148D4—996no—
149C2×C4—924no—
150C2×C4—9192no—
151C2×C4C23×C3⋊D3136yes—
152C2×C4C23×C3⋊D3148yes—
153C2×C3—216no—
154C2×C3C2×#1007124yescomplemented
155C2×C3C25×D314yes—
156D3—664no—
157C22—212no—
158C22—916no—
159C22—1848no—
160C22#6436414yesdirect product factor
161C22C24×C3⋊D314yes—
162C22C24×C3⋊D313yes—
163C4—916no—
164C4C24×C3⋊D3124yes—
165C3C22×#100714yescomplemented
166C2—24no—
167C2—1816no—
168C2C2×#6436416yesdirect product factor
169C2C25×C3⋊D311yescharacteristic
subgroupG/H
centerC23C23×C3⋊D3
commutator, cosocleC2×C32C26
FrattiniC2C25×C3⋊D3
FittingC23×C32×Q8C2
socleC23×C32C24
hypercenterC23×Q8C3⋊D3
hypocenterC32C22×2−1+4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C23×C32×Q8C2semidirect
2. C2×#64364C2direct (N × H)
3. C22×#3832C2semidirect
4. C22×Q8×C3⋊D3C2semidirect
5. C22×#3828C2semidirect
8. C22×C32×Q8C22semidirect
11. #64364C22direct (N × H)
12. C2×#3832C22semidirect
13. C2×Q8×C3⋊D3C22semidirect
14. C2×#3828C22semidirect
18. C23×C3×Q8D3semidirect
26. C2×C32×Q8C23semidirect
31. #3832C23semidirect
32. Q8×C3⋊D3C23semidirect
33. #3828C23semidirect
39. C22×C3×Q8C2×D3semidirect
66. C23×Q8C3⋊D3semidirect
70. C2×C3×Q8C22×D3semidirect
82. C22×C322−1+4semidirect
95. C22×Q8C2×C3⋊D3semidirect
114. C2×C32C2×2−1+4semidirect
124. C2×Q8C22×C3⋊D3semidirect
132. C22×C3#1007semidirect
139. C32C22×2−1+4semidirect
154. C2×C3C2×#1007semidirect
160. C22#64364direct (N × H)
165. C3C22×#1007semidirect
168. C2C2×#64364direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×Q82
C2×#643642
C22×#38322
C22×Q8×C3⋊D32
C22×#38282
C22×#10073