Finite Group Databasebeta

← C22×D3×#622 #13180998

Subgroups — C22×D3×#622 #13180998

Order 1728.

1256 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 290 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⋊Dic3C211yesmaximal, characteristic
2C22×C3⋊#622C211yesmaximal, characteristic, complemented
3C22×C3×#622C211yesmaximal, characteristic, complemented
4C2×D3×#622C2112yesmaximal, direct product factor
5C23×#622—31nomaximal
6C23×C32×D3C411yescharacteristic
7C22×C3⋊(C3⋊Dic3)C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×D3×C3⋊Dic3C416yes—
10C2×D3×C3⋊Dic3C2216yes—
11C2×C3⋊#622C2216yescomplemented
12C2×C3×#622C2216yescomplemented
13D3×#622C22116yesdirect product factor
14C23×C3⋊Dic3—31no—
15C22×D3×Dic3—62no—
16C22×#622—31no—
17C22×#622—312no—
18C22×#622D311yescharacteristic, direct product factor
19C23×C33C2×C411yescharacteristic
20C22×C32×D3C818yescomplemented
21C22×C32×D3C2×C416yes—
22C2×C3⋊(C3⋊Dic3)C2×C416yes—
23C2×C3×C3⋊Dic3C2×C413yes—
24C2×C3×C3⋊Dic3C2313yes—
25D3×C3⋊Dic3C2×C4112yes—
26D3×C3⋊Dic3C2314yes—
27C3⋊#622C2314yescomplemented
28C3×#622C2314yescomplemented
29C22×D3×C8—91nomaximal
30C24×C32—31no—
31C23×C3×D3—22no—
32C22×C3⋊Dic3—31no—
33C22×C3⋊Dic3—36no—
34C22×C3⋊Dic3—36no—
35C22×C3⋊Dic3—62no—
36C22×C3⋊Dic3—122no—
37C22×C3⋊Dic3C2×D311yescharacteristic
38C22×C3×Dic3—62no—
39C2×D3×Dic3—612no—
40C2×D3×Dic3—612no—
41C2×#622—36no—
42C2×#622—316no—
43C2×#622C2×D316yesdirect product factor
44C22×C33C2×C814yescomplemented
45C22×C33C22×C413yes—
46C2×C32×D3C2×C8124yescomplemented
47C2×C32×D3C22×C414yes—
48C3⋊(C3⋊Dic3)C22×C414yes—
49C3×C3⋊Dic3C22×C413yes—
50C3×C3⋊Dic3C2411yescharacteristic
51C23×Dic3—182no—
52C22×C4×D3—91no—
53C22×C3×C8—91no—
54C22×C3⋊C8—91no—
55C2×D3×C8—912no—
56C23×C32—22no—
57C23×C32—38no—
58C23×C32—36no—
59C23×C32—42no—
60C23×C32C4×D311yescharacteristic
61C22×C3×D3—216no—
62C22×C3×D3—212no—
63C2×C3⋊Dic3—36no—
64C2×C3⋊Dic3—312no—
65C2×C3⋊Dic3—34no—
66C2×C3⋊Dic3—612no—
67C2×C3⋊Dic3—1212no—
68C2×C3⋊Dic3C4×D313yes—
69C2×C3⋊Dic3C22×D313yes—
70C2×C3×Dic3—66no—
71C2×C3×Dic3—66no—
72D3×Dic3—624no—
73D3×Dic3—68no—
74#622—34no—
75#622C22×D314yesdirect product factor
76C23×C8—271noSylow
77C2×C33C22×C816yescomplemented
78C2×C33C23×C411yescharacteristic
79C32×D3C22×C818yescomplemented
80C24×C3—62no—
81C23×D3#15711yescharacteristic
82C22×C3×C4—91no—
83C22×Dic3—62no—
84C22×Dic3—91no—
85C22×Dic3—1812no—
86C22×Dic3—1812no—
87C22×Dic3—182no—
88C22×Dic3—362no—
89C2×C4×D3—96no—
90C2×C4×D3—96no—
91C2×C3×C8—96no—
92C2×C3⋊C8—96no—
93D3×C8—916no—
94C22×C32—26no—
95C22×C32—28no—
96C22×C32—324no—
97C22×C32—34no—
98C22×C32—48no—
99C22×C32—46no—
100C22×C32D3×C814yescomplemented
101C22×C32C2×C4×D313yes—
102C2×C3×D3—28no—
103C2×C3×D3—248no—
104C3⋊Dic3—34no—
105C3⋊Dic3—68no—
106C3⋊Dic3—128no—
107C3⋊Dic3C2×C4×D313yes—
108C3⋊Dic3C23×D311yescharacteristic
109C3×Dic3—66no—
110C3×Dic3—62no—
111C23×C4—271no—
112C22×C8—91no—
113C22×C8—2712no—
114C22×C8—271no—
115C33C23×C811yescharacteristic, complemented, Sylow
116C23×C3—22no—
117C23×C3—42no—
118C23×C3—612no—
119C23×C3—616no—
120C23×C3C2×#15711yescharacteristic
121C22×D3#62218yesdirect product factor
122C22×D3C2×#15716yes—
123C2×C3×C4—93no—
124C2×C3×C4—93no—
125C2×Dic3—66no—
126C2×Dic3—66no—
127C2×Dic3—96no—
128C2×Dic3—188no—
129C2×Dic3—1824no—
130C2×Dic3—1812no—
131C2×Dic3—3612no—
132C4×D3—94no—
133C4×D3—912no—
134C3×C8—94no—
135C3⋊C8—94no—
136C2×C32—212no—
137C2×C32—22no—
138C2×C32—38no—
139C2×C32—412no—
140C2×C32—42no—
141C2×C32C2×D3×C816yescomplemented
142C2×C32C22×C4×D311yescharacteristic
143C3×D3—216no—
144C24—31no—
145C22×C4—91no—
146C22×C4—276no—
147C22×C4—276no—
148C22×C4—271no—
149C2×C8—96no—
150C2×C8—2716no—
151C2×C8—276no—
152C22×C3—26no—
153C22×C3—28no—
154C22×C3—46no—
155C22×C3—48no—
156C22×C3—68no—
157C22×C3—648no—
158C22×C3C2×#62214yescomplemented
159C22×C3C22×#15713yes—
160C2×D3C2×#622124yesdirect product factor
161C2×D3C22×#15714yes—
162C3×C4—91no—
163C3×C4—93no—
164Dic3—66no—
165Dic3—62no—
166Dic3—94no—
167Dic3—188no—
168Dic3—368no—
169C32—22no—
170C32—42no—
171C32C22×D3×C811yescharacteristic, complemented
172C23—36no—
173C23—38no—
174C23D3×#15711yescharacteristic
175C2×C4—93no—
176C2×C4—93no—
177C2×C4—274no—
178C2×C4—2712no—
179C2×C4—276no—
180C8—94no—
181C8—274no—
182C2×C3—22no—
183C2×C3—212no—
184C2×C3—42no—
185C2×C3—412no—
186C2×C3—616no—
187C2×C3C22×#62216yescomplemented
188C2×C3C23×#15711yescharacteristic
189D3C22×#62218yesdirect product factor
190C22—34no—
191C22—324no—
192C22D3×#62214yesdirect product factor
193C22C2×D3×#15713yes—
194C4—91no—
195C4—93no—
196C4—274no—
197C3—22no—
198C3—42no—
199C3C23×#62211yescharacteristic, complemented
200C2—38no—
201C2C2×D3×#62216yesdirect product factor
202C2C22×D3×#15711yescharacteristic
subgroupG/H
center, hypercenterC23D3×#157
commutator, hypocenterC33C23×C8
FrattiniC2C22×D3×#157
Fitting, socleC23×C33C2×C4
prime radicalC23×C32×D3C4
cosocleC3×C3⋊Dic3C24
LCSG ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#622C2semidirect
3. C22×C3×#622C2semidirect
4. C2×D3×#622C2direct (N × H)
11. C2×C3⋊#622C22semidirect
12. C2×C3×#622C22semidirect
13. D3×#622C22direct (N × H)
18. C22×#622D3direct (N × H)
20. C22×C32×D3C8semidirect
27. C3⋊#622C23semidirect
28. C3×#622C23semidirect
43. C2×#622C2×D3direct (N × H)
44. C22×C33C2×C8semidirect
46. C2×C32×D3C2×C8semidirect
75. #622C22×D3direct (N × H)
77. C2×C33C22×C8semidirect
79. C32×D3C22×C8semidirect
100. C22×C32D3×C8semidirect
115. C33C23×C8semidirect
121. C22×D3#622direct (N × H)
141. C2×C32C2×D3×C8semidirect
158. C22×C3C2×#622semidirect
160. C2×D3C2×#622direct (N × H)
171. C32C22×D3×C8semidirect
187. C2×C3C22×#622semidirect
189. D3C22×#622direct (N × H)
192. C22D3×#622direct (N × H)
199. C3C23×#622semidirect
201. C2C2×D3×#622direct (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⋊Dic32
C22×C3⋊#6222
C22×C3×#6222
C2×D3×#6222
C23×#6223
C22×D3×C89