Finite Group Databasebeta

← C22×D3×#157 #11724731

Subgroups — C22×D3×#157 #11724731

Order 864.

804 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C22×D3×C3⋊D3C211yesmaximal, characteristic
2C22×C3⋊#157C211yesmaximal, characteristic, complemented
3C22×C3×#157C211yesmaximal, characteristic, complemented
4C2×D3×#157C2112yesmaximal, direct product factor
5C23×#157—31nomaximal
6C22×C3⋊(C3⋊D3)C411yescharacteristic, complemented
7C22×C3×C3⋊D3C2211yescharacteristic
8C22×C32×D3C411yescharacteristic, complemented
9C2×D3×C3⋊D3C416yescomplemented
10C2×D3×C3⋊D3C2216yes—
11C2×C3⋊#157C2216yescomplemented
12C2×C3×#157C2216yescomplemented
13D3×#157C22116yesdirect product factor
14C23×C3⋊D3—31no—
15C22×D32—62no—
16C22×#157—31no—
17C22×#157—312no—
18C22×#157D311yescharacteristic, direct product factor
19C22×C33C2×C411yescharacteristic, complemented
20C2×C3⋊(C3⋊D3)C2×C416yescomplemented
21C2×C3×C3⋊D3C2×C413yescomplemented
22C2×C3×C3⋊D3C2313yes—
23C2×C32×D3C2×C416yescomplemented
24D3×C3⋊D3C2×C4112yescomplemented
25D3×C3⋊D3C2314yes—
26C3⋊#157C2314yescomplemented
27C3×#157C2314yescomplemented
28C22×C4×D3—91nomaximal
29C23×C32—31no—
30C22×C3⋊D3—31no—
31C22×C3⋊D3—36no—
32C22×C3⋊D3—36no—
33C22×C3⋊D3—62no—
34C22×C3⋊D3—122no—
35C22×C3⋊D3C2×D311yescharacteristic
36C22×C3×D3—22no—
37C22×C3×D3—62no—
38C2×D32—612no—
39C2×D32—612no—
40C2×#157—36no—
41C2×#157—316no—
42C2×#157C2×D316yesdirect product factor
43C2×C33C22×C413yescomplemented
44C3⋊(C3⋊D3)C22×C414yescomplemented
45C3×C3⋊D3C22×C413yescomplemented
46C3×C3⋊D3C2411yescharacteristic
47C32×D3C22×C414yescomplemented
48C23×D3—91no—
49C23×D3—182no—
50C22×C3×C4—91no—
51C22×Dic3—91no—
52C2×C4×D3—912no—
53C22×C32—22no—
54C22×C32—36no—
55C22×C32—42no—
56C22×C32C4×D311yescharacteristic, complemented
57C2×C3⋊D3—36no—
58C2×C3⋊D3—312no—
59C2×C3⋊D3—34no—
60C2×C3⋊D3—612no—
61C2×C3⋊D3—1212no—
62C2×C3⋊D3C4×D313yescomplemented
63C2×C3⋊D3C22×D313yes—
64C2×C3×D3—212no—
65C2×C3×D3—66no—
66C2×C3×D3—66no—
67D32—624no—
68D32—68no—
69#157—34no—
70#157C22×D314yesdirect product factor
71C23×C4—271noSylow
72C33C23×C411yescharacteristic, complemented, Sylow
73C23×C3—62no—
74C23×C3—91no—
75C22×D3—62no—
76C22×D3—91no—
77C22×D3—96no—
78C22×D3—96no—
79C22×D3—1812no—
80C22×D3—1812no—
81C22×D3—182no—
82C22×D3—362no—
83C22×D3#15711yescharacteristic, direct product factor
84C2×C3×C4—96no—
85C2×Dic3—96no—
86C4×D3—916no—
87C2×C32—26no—
88C2×C32—34no—
89C2×C32—46no—
90C2×C32C2×C4×D313yescomplemented
91C3⋊D3—34no—
92C3⋊D3—68no—
93C3⋊D3—128no—
94C3⋊D3C2×C4×D313yescomplemented
95C3⋊D3C23×D311yescharacteristic
96C3×D3—28no—
97C3×D3—66no—
98C3×D3—62no—
99C24—271no—
100C22×C4—91no—
101C22×C4—2712no—
102C22×C4—271no—
103C22×C3—22no—
104C22×C3—42no—
105C22×C3—612no—
106C22×C3—93no—
107C22×C3—93no—
108C22×C3C2×#15711yescharacteristic, complemented
109C2×D3—66no—
110C2×D3—66no—
111C2×D3—96no—
112C2×D3—94no—
113C2×D3—912no—
114C2×D3—188no—
115C2×D3—1824no—
116C2×D3—1812no—
117C2×D3—3612no—
118C2×D3C2×#15716yesdirect product factor
119C3×C4—94no—
120Dic3—94no—
121C32—22no—
122C32—42no—
123C32C22×C4×D311yescharacteristic, complemented
124C23—31no—
125C23—91no—
126C23—276no—
127C23—276no—
128C23—271no—
129C2×C4—96no—
130C2×C4—2716no—
131C2×C4—276no—
132C2×C3—26no—
133C2×C3—46no—
134C2×C3—68no—
135C2×C3—91no—
136C2×C3—93no—
137C2×C3C22×#15713yescomplemented
138D3—66no—
139D3—62no—
140D3—94no—
141D3—188no—
142D3—368no—
143D3C22×#15714yesdirect product factor
144C22—36no—
145C22—93no—
146C22—93no—
147C22—274no—
148C22—2712no—
149C22—276no—
150C22D3×#15711yescharacteristic, direct product factor
151C4—94no—
152C4—274no—
153C3—22no—
154C3—42no—
155C3C23×#15711yescharacteristic, complemented
156C2—34no—
157C2—91no—
158C2—93no—
159C2—274no—
160C2C2×D3×#15713yesdirect product factor
subgroupG/H
center, hypercenterC22D3×#157
commutator, hypocenterC33C23×C4
FrattiniC1G
Fitting, socleC22×C33C2×C4
prime radicalC22×D3×C3⋊D3C2
cosocleC3×C3⋊D3C24
LCSG ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#157C2semidirect
3. C22×C3×#157C2semidirect
4. C2×D3×#157C2direct (N × H)
6. C22×C3⋊(C3⋊D3)C4semidirect
8. C22×C32×D3C4semidirect
9. C2×D3×C3⋊D3C4semidirect
11. C2×C3⋊#157C22semidirect
12. C2×C3×#157C22semidirect
13. D3×#157C22direct (N × H)
18. C22×#157D3direct (N × H)
19. C22×C33C2×C4semidirect
20. C2×C3⋊(C3⋊D3)C2×C4semidirect
21. C2×C3×C3⋊D3C2×C4semidirect
23. C2×C32×D3C2×C4semidirect
24. D3×C3⋊D3C2×C4semidirect
26. C3⋊#157C23semidirect
27. C3×#157C23semidirect
42. C2×#157C2×D3direct (N × H)
43. C2×C33C22×C4semidirect
44. C3⋊(C3⋊D3)C22×C4semidirect
45. C3×C3⋊D3C22×C4semidirect
47. C32×D3C22×C4semidirect
56. C22×C32C4×D3semidirect
62. C2×C3⋊D3C4×D3semidirect
70. #157C22×D3direct (N × H)
72. C33C23×C4semidirect
83. C22×D3#157direct (N × H)
90. C2×C32C2×C4×D3semidirect
94. C3⋊D3C2×C4×D3semidirect
108. C22×C3C2×#157semidirect
118. C2×D3C2×#157direct (N × H)
123. C32C22×C4×D3semidirect
137. C2×C3C22×#157semidirect
143. D3C22×#157direct (N × H)
150. C22D3×#157direct (N × H)
155. C3C23×#157semidirect
160. C2C2×D3×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C3⋊D32
C22×C3⋊#1572
C22×C3×#1572
C2×D3×#1572
C23×#1573
C22×C4×D39