Finite Group Databasebeta

← C2×#11722827 #13170521

Subgroups — C2×#11722827 #13170521

Order 1728.

1314 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 252 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
1C2×#89042C211yesmaximal, characteristic
2C2×C3⋊D3×C3⋊C8C213yesmaximal, complemented
3C2×C3×#3712C213yesmaximal, complemented
4#11722827C218yesmaximal, direct product factor
5C2×#63805—33nomaximal
6C2×C3×C4×C3⋊D3C2213yes—
7C2×#6296C411yescharacteristic
8C2×#6295C413yes—
9#89042C414yes—
10#89042C2214yes—
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C2×C32×C3⋊C8C2213yes—
13C3⋊D3×C3⋊C8C22112yescomplemented
14C3×#3712C22112yescomplemented
15C2×#3798—33no—
16C2×C8×C3⋊D3—33no—
17C2×C3×ΓL(1,𝔽25)—36no—
18C2×#3712D313yescomplemented
19C2×D3×C3⋊C8—36no—
20C2×D3×C3⋊C8—63no—
21#63805—324no—
22C22×C3×C3⋊D3C2×C413yes—
23C2×C33×C4C2311yescharacteristic
24C2×C3×C3⋊Dic3C2×C413yes—
25C3×C4×C3⋊D3C2×C416yes—
26C3×C4×C3⋊D3C2316yes—
27#6296C2×C414yes—
28#6295C2×C4112yes—
29C3⋊(C3⋊(C3⋊C8))C2312yes—
30C32×C3⋊C8C2316yes—
31C2×#907—93no—
32C2×C4×C3⋊D3C2×D313yes—
33C2×C3×C4×D3—36no—
34C2×C3×C4×D3—63no—
35C2×#627—33no—
36C2×#626—36no—
37C2×#625—33no—
38#3798—312no—
39#3798—312no—
40C2×C32×C8—33no—
41C2×C3⋊(C3⋊C8)—33no—
42C2×C3⋊(C3⋊C8)—63no—
43C2×C3⋊(C3⋊C8)—121no—
44C8×C3⋊D3—312no—
45C2×C3×C3⋊C8—23no—
46C2×C3×C3⋊C8C2×D316yes—
47C3×ΓL(1,𝔽25)—324no—
48#3712C2×D3112yescomplemented
49D3×C3⋊C8—324no—
50D3×C3⋊C8—612no—
51C22×C33C22×C411yescharacteristic
52C2×C3×C3⋊D3—26no—
53C2×C3×C3⋊D3C22×C416yes—
54C33×C4C22×C411yescharacteristic
55C33×C4C2411yescharacteristic
56C3×C3⋊Dic3C22×C416yes—
57C2×#235—93no—
58C2×C3×M16—93no—
59C22×C3⋊C8—93no—
60#907—924no—
61C2×ΓL(1,𝔽25)—36no—
62C2×D3×C8—96no—
63C2×D3×C8—183no—
64C22×C3⋊D3C4×D313yes—
65C22×C3×D3—36no—
66C22×C3×D3—63no—
67C2×C32×C4—23no—
68C2×C32×C4—41no—
69C2×C32×C4C22×D313yes—
70C2×C3⋊Dic3C4×D313yes—
71C4×C3⋊D3C4×D316yes—
72C4×C3⋊D3C22×D316yes—
73C2×C3×Dic3—36no—
74C2×C3×Dic3—63no—
75C3×C4×D3—312no—
76C3×C4×D3—312no—
77C3×C4×D3—66no—
78C3×C4×D3—66no—
79#627—312no—
80#626—324no—
81#625—312no—
82C32×C8—36no—
83C3⋊(C3⋊C8)—36no—
84C3⋊(C3⋊C8)—66no—
85C3⋊(C3⋊C8)—122no—
86C3×C3⋊C8—26no—
87C3×C3⋊C8C22×D3112yes—
88C2×#131—271noSylow
89C2×C33#13112yescomplemented
90C2×C33C23×C411yescharacteristic
91C3×C3⋊D3—26no—
92C22×C3×C4—93no—
93C2×#67—96no—
94#235—912no—
95#235—912no—
96C2×D12—93no—
97C2×C4×D3—36no—
98C2×C4×D3—63no—
99C2×Dic6—93no—
100C3×M16—912no—
101C2×C3×C8—36no—
102C2×C3×C8—63no—
103C2×C3⋊C8—93no—
104C2×C3⋊C8—912no—
105C2×C3⋊C8—183no—
106C2×C3⋊C8—361no—
107C2×C3⋊C8D3213yes—
108ΓL(1,𝔽25)—324no—
109D3×C8—924no—
110D3×C8—1812no—
111C22×C32—23no—
112C22×C32—41no—
113C22×C32C2×C4×D313yes—
114C2×C3⋊D3—26no—
115C2×C3⋊D3C2×C4×D316yes—
116C2×C3×D3—312no—
117C2×C3×D3—66no—
118C2×C3×D3—612no—
119C2×C3×D3—612no—
120C32×C4—23no—
121C32×C4—23no—
122C32×C4—41no—
123C32×C4—41no—
124C32×C4C2×C4×D313yes—
125C32×C4C23×D313yes—
126C3⋊Dic3C2×C4×D316yes—
127C3×Dic3—312no—
128C3×Dic3—66no—
129C2×#41—271no—
130#131—278no—
131C2×M16—93no—
132C22×C8—273no—
133C33C2×#13111yescharacteristic, complemented, Sylow
134C23×C3—93no—
135C22×D3—36no—
136C22×D3—63no—
137C2×C3×C4—23no—
138C2×C3×C4—41no—
139C2×C3×C4—93no—
140C2×C3×C4—96no—
141C2×C3×C4—96no—
142C2×C3×C4C2×D3213yes—
143#67—924no—
144C2×Dic3—36no—
145C2×Dic3—63no—
146D12—912no—
147C4×D3—312no—
148C4×D3—312no—
149C4×D3—66no—
150C4×D3—66no—
151Dic6—912no—
152C3×C8—312no—
153C3×C8—66no—
154C3⋊C8—96no—
155C3⋊C8—186no—
156C3⋊C8—362no—
157C3⋊C8C2×D3216yes—
158C2×C32—23no—
159C2×C32—26no—
160C2×C32—42no—
161C2×C32—41no—
162C2×C32#90716yescomplemented
163C2×C32C22×C4×D313yes—
164C3⋊D3—26no—
165C3×D3—612no—
166C3×D3—612no—
167#41—274no—
168#41—274no—
169C2×Q8—271no—
170C2×D4—273no—
171C22×C4—93no—
172M16—912no—
173C2×C8—33no—
174C2×C8—2712no—
175C2×C8—271no—
176C22×C3—23no—
177C22×C3—41no—
178C22×C3—96no—
179C22×C3—186no—
180C22×C3C4×D3213yes—
181C2×D3—312no—
182C2×D3—66no—
183C2×D3—612no—
184C2×D3—612no—
185C3×C4—23no—
186C3×C4—23no—
187C3×C4—41no—
188C3×C4—41no—
189C3×C4—96no—
190C3×C4C4×D3213yes—
191C3×C4C22×D3213yes—
192Dic3—312no—
193Dic3—66no—
194C32—23no—
195C32—41no—
196C32C2×#90713yescomplemented
197C23—93no—
198Q8—274no—
199D4—2712no—
200C2×C4—93no—
201C2×C4—96no—
202C2×C4—96no—
203C2×C4D3311yescharacteristic
204C8—36no—
205C8—272no—
206C2×C3—26no—
207C2×C3—23no—
208C2×C3—42no—
209C2×C3—41no—
210C2×C3—186no—
211C2×C3#6380516yescomplemented
212C2×C3C2×C4×D3213yes—
213D3—612no—
214D3—612no—
215C22—96no—
216C22—186no—
217C22#8894911yescharacteristic
218C4—96no—
219C4#8894911yescharacteristic
220C4C2×D3311yescharacteristic
221C3—23no—
222C3—41no—
223C3C2×#6380513yescomplemented
224C2—186no—
225C2#1172282712yesdirect product factor
226C2C2×#8894911yescharacteristic
subgroupG/H
center, hypercenterC2×C4D33
commutatorC2×C33C23×C4
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC2×#89042C2
cosocleC33×C4C24
hypocenterC33C2×#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊D3×C3⋊C8C2semidirect
3. C2×C3×#3712C2semidirect
4. #11722827C2direct (N × H)
13. C3⋊D3×C3⋊C8C22semidirect
14. C3×#3712C22semidirect
18. C2×#3712D3semidirect
48. #3712C2×D3semidirect
89. C2×C33#131semidirect
133. C33C2×#131semidirect
162. C2×C32#907semidirect
196. C32C2×#907semidirect
211. C2×C3#63805semidirect
223. C3C2×#63805semidirect
225. C2#11722827direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890422
C2×C3⋊D3×C3⋊C82
C2×C3×#37122
#117228272
C2×#638053