Finite Group Databasebeta

← C2×#11723716 #13178783

Subgroups — C2×#11723716 #13178783

Order 1728.

1514 conjugacy classes of subgroups; only the 247 normal subgroups are shown.

normal subgroups only

This group has 1514 conjugacy classes of subgroups — too many to draw. Only its 247 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×C3×#3757C213yesmaximal
2C2×#88815C213yesmaximal, complemented
3#11723716C218yesmaximal, direct product factor
4C2×#88813C211yesmaximal, characteristic, complemented
5C2×#63956—33nomaximal
6C24×C33C2211yescharacteristic
7C22×C3×C3⋊Dic3C2213yes—
8C22×C3×C3⋊Dic3C2213yes—
9C3×#3757C22112yes—
10#88815C22112yescomplemented
11#88813C2214yescomplemented
12C2×#3757D313yes—
13C2×C3×#217—33no—
14C2×C3×#217—33no—
15C2×C3×#217—63no—
16C2×#3724—33no—
17C2×Dic3⋊Dic3—33no—
18C2×Dic3⋊Dic3—33no—
19#63956—324no—
20C2×Dic32—33no—
21C23×C33—26no—
22C23×C33C2312yes—
23C23×C33C2311yescharacteristic
24C2×C3×C3⋊Dic3—26no—
25C2×C3×C3⋊Dic3—26no—
26C2×C3×C3⋊Dic3C2316yes—
27C2×C3×C3⋊Dic3C2316yes—
28C2×#879—93no—
29C24×C32—23no—
30C24×C32—41no—
31C24×C32C2×D313yes—
32C22×C3⋊Dic3C2×D313yes—
33C22×C3⋊Dic3C2×D313yes—
34C22×C3×Dic3—33no—
35C22×C3×Dic3—33no—
36C22×C3×Dic3—33no—
37C22×C3×Dic3—33no—
38C22×C3×Dic3—63no—
39C22×C3×Dic3—63no—
40#3757C2×D3112yes—
41C3×#217—312no—
42C3×#217—312no—
43C3×#217—612no—
44#3724—312no—
45Dic3⋊Dic3—312no—
46Dic3⋊Dic3—312no—
47Dic32—312no—
48C22×C33—26no—
49C22×C33—44no—
50C22×C33#4116yes—
51C22×C33C2411yescharacteristic
52C3×C3⋊Dic3—26no—
53C3×C3⋊Dic3—26no—
54C2×C3×C22⋊C4—93no—
55C2×#217—33no—
56C2×#217—33no—
57C2×#217—63no—
58C2×C4⋊Dic3—93no—
59C2×Dic3⋊C4—93no—
60C2×Dic3⋊C4—93no—
61C2×C4×Dic3—93no—
62#879—924no—
63C23×C32—26no—
64C23×C32—212no—
65C23×C32—26no—
66C23×C32—26no—
67C23×C32—23no—
68C23×C32—26no—
69C23×C32—42no—
70C23×C32—41no—
71C23×C32—46no—
72C23×C32—46no—
73C23×C32—412no—
74C23×C32C22×D316yes—
75C23×C32C22×D313yes—
76C2×C3⋊Dic3—26no—
77C2×C3⋊Dic3—26no—
78C2×C3⋊Dic3C22×D316yes—
79C2×C3⋊Dic3C22×D316yes—
80C2×C3×Dic3—36no—
81C2×C3×Dic3—36no—
82C2×C3×Dic3—36no—
83C2×C3×Dic3—36no—
84C2×C3×Dic3—66no—
85C2×C3×Dic3—66no—
86C2×C3×Dic3—66no—
87C2×C3×Dic3—66no—
88C2×C3×Dic3—66no—
89C2×C3×Dic3—66no—
90C2×C3×Dic3—126no—
91C2×C3×Dic3—126no—
92C2×#126—271noSylow
93C2×C33—42no—
94C2×C33#12614yescomplemented
95C2×C33C2×#4113yes—
96C24×C3—23no—
97C24×C3—41no—
98C24×C3D3213yes—
99C22×C3×C4—93no—
100C22×C3×C4—93no—
101C22×Dic3—33no—
102C22×Dic3—33no—
103C22×Dic3—33no—
104C22×Dic3—33no—
105C22×Dic3—63no—
106C22×Dic3—63no—
107C3×C22⋊C4—912no—
108#217—312no—
109#217—312no—
110#217—612no—
111C4⋊Dic3—912no—
112Dic3⋊C4—912no—
113Dic3⋊C4—912no—
114C4×Dic3—912no—
115C22×C32—26no—
116C22×C32—212no—
117C22×C32—26no—
118C22×C32—26no—
119C22×C32—23no—
120C22×C32—26no—
121C22×C32—26no—
122C22×C32—26no—
123C22×C32—46no—
124C22×C32—41no—
125C22×C32—46no—
126C22×C32—46no—
127C22×C32—412no—
128C22×C32—424no—
129C22×C32—416no—
130C22×C32—412no—
131C22×C32#23516yes—
132C22×C32#23716yes—
133C22×C32#23716yes—
134C22×C32C23×D313yes—
135C3⋊Dic3—26no—
136C3⋊Dic3—26no—
137C3×Dic3—66no—
138C3×Dic3—66no—
139C3×Dic3—66no—
140C3×Dic3—66no—
141C3×Dic3—126no—
142C3×Dic3—126no—
143#126—278no—
144C2×C4⋊C4—273no—
145C2×C22⋊C4—93no—
146C2×C42—271no—
147C33C2×#12611yescharacteristic, complemented, Sylow
148C23×C3—212no—
149C23×C3—26no—
150C23×C3—26no—
151C23×C3—26no—
152C23×C3—26no—
153C23×C3—23no—
154C23×C3—46no—
155C23×C3—46no—
156C23×C3—412no—
157C23×C3—42no—
158C23×C3—41no—
159C23×C3C2×D3216yes—
160C23×C3C2×D3213yes—
161C2×C3×C4—96no—
162C2×C3×C4—96no—
163C2×C3×C4—186no—
164C2×C3×C4—186no—
165C2×Dic3—36no—
166C2×Dic3—36no—
167C2×Dic3—36no—
168C2×Dic3—36no—
169C2×Dic3—66no—
170C2×Dic3—66no—
171C2×Dic3—66no—
172C2×Dic3—66no—
173C2×Dic3—66no—
174C2×Dic3—66no—
175C2×Dic3—126no—
176C2×Dic3—126no—
177C2×C32—23no—
178C2×C32—23no—
179C2×C32—23no—
180C2×C32—212no—
181C2×C32—44no—
182C2×C32—43no—
183C2×C32—412no—
184C2×C32—48no—
185C2×C32—46no—
186C2×C32#879112yescomplemented
187C2×C32C2×#23513yes—
188C2×C32C2×#23713yes—
189C2×C32C2×#23713yes—
190C24#111311yescharacteristic
191C22×C4—93no—
192C22×C4—93no—
193C4⋊C4—2712no—
194C22⋊C4—912no—
195C42—274no—
196C22×C3—212no—
197C22×C3—26no—
198C22×C3—26no—
199C22×C3—26no—
200C22×C3—23no—
201C22×C3—26no—
202C22×C3—26no—
203C22×C3—26no—
204C22×C3—46no—
205C22×C3—46no—
206C22×C3—412no—
207C22×C3—424no—
208C22×C3—416no—
209C22×C3—412no—
210C22×C3—41no—
211C22×C3—46no—
212C22×C3#380416yes—
213C22×C3#380416yes—
214C22×C3#380516yes—
215C22×C3C22×D3213yes—
216C3×C4—186no—
217C3×C4—186no—
218Dic3—66no—
219Dic3—66no—
220Dic3—66no—
221Dic3—66no—
222Dic3—126no—
223Dic3—126no—
224C32—23no—
225C32—41no—
226C32C2×#87913yescomplemented
227C23—26no—
228C23C2×#111312yes—
229C23C2×#111311yescharacteristic
230C2×C4—96no—
231C2×C4—96no—
232C2×C4—186no—
233C2×C4—186no—
234C2×C3—23no—
235C2×C3—23no—
236C2×C3—23no—
237C2×C3—212no—
238C2×C3—412no—
239C2×C3—48no—
240C2×C3—46no—
241C2×C3—43no—
242C2×C3—44no—
243C2×C3#63956112yescomplemented
244C2×C3C2×#380413yes—
245C2×C3C2×#380413yes—
246C2×C3C2×#380513yes—
247C22—26no—
248C22—44no—
249C22#8904616yes—
250C22C22×#111311yescharacteristic
251C4—186no—
252C4—186no—
253C3—23no—
254C3—41no—
255C3C2×#6395613yescomplemented
256C2—42no—
257C2#1172371614yesdirect product factor
258C2C2×#8904613yes—
subgroupG/H
centerC23C2×#1113
commutator, cosocleC22×C33C24
FrattiniC22C22×#1113
Fitting, prime radicalC24×C33C22
socleC23×C33C23
hypercenterC24#1113
hypocenterC33C2×#126
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#88815C2semidirect
3. #11723716C2direct (N × H)
4. C2×#88813C2semidirect
10. #88815C22semidirect
11. #88813C22semidirect
94. C2×C33#126semidirect
147. C33C2×#126semidirect
186. C2×C32#879semidirect
226. C32C2×#879semidirect
243. C2×C3#63956semidirect
255. C3C2×#63956semidirect
257. C2#11723716direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37572
C2×#888152
#117237162
C2×#888132
C2×#639563