Finite Group Databasebeta

← C2×#11723729 #13178805

Subgroups — C2×#11723729 #13178805

Order 1728.

3490 conjugacy classes of subgroups; only the 295 normal subgroups are shown.

normal subgroups only

This group has 3490 conjugacy classes of subgroups — too many to draw. Only its 295 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×#1113C211yesmaximal, characteristic, complemented
2C22×C3×#638C213yesmaximal, complemented
3#11723729C218yesmaximal, direct product factor
4C2×#88814C213yesmaximal, complemented
5C2×#63983—33nomaximal
6C24×C33C2211yescharacteristic, complemented
7C23×C3×C3⋊D3C2213yes—
8C22×#1113—212no—
9C22×#1113C2214yescomplemented
10C2×C3×#638—212no—
11C2×C3×#638C22112yescomplemented
12C22×C3×C3⋊Dic3C2213yescomplemented
13#88814C22112yescomplemented
14C23×D32—33no—
15C22×#638D313yescomplemented
16C22×C3×#67—36no—
17C22×C3×#67—63no—
18#63983—324no—
19C2×#3722—33no—
20C2×(C2×D3)⋊Dic3—36no—
21C23×C33—26no—
22C23×C33C2311yescharacteristic
23C23×C33C2312yescomplemented
24C22×C3×C3⋊D3—212no—
25C22×C3×C3⋊D3D4112yescomplemented
26C22×C3×C3⋊D3C2316yes—
27C2×#1113—232no—
28C2×#1113—224no—
29C3×#638—224no—
30C2×C3×C3⋊Dic3—26no—
31C2×C3×C3⋊Dic3C2316yescomplemented
32C2×#937—93no—
33C24×C32—23no—
34C24×C32—41no—
35C24×C32C2×D313yescomplemented
36C23×C3⋊D3C2×D313yes—
37C23×C3×D3—36no—
38C23×C3×D3—63no—
39C22×D32—312no—
40C22×D32—624no—
41C22×D32—612no—
42C2×#638—212no—
43C2×#638C2×D3112yescomplemented
44C22×C3⋊Dic3C2×D313yescomplemented
45C2×C3×#67—324no—
46C2×C3×#67—624no—
47C2×C3×#67—612no—
48C2×C3×#67—624no—
49C22×C3×Dic3—36no—
50C22×C3×Dic3—63no—
51#3722—312no—
52(C2×D3)⋊Dic3—324no—
53C22×C33—26no—
54C22×C33—44no—
55C22×C33C2×D416yes—
56C22×C33C2411yescharacteristic
57C2×C3×C3⋊D3—224no—
58C2×C3×C3⋊D3—212no—
59C2×C3×C3⋊D3C2×D4112yescomplemented
60#1113—232no—
61C3×C3⋊Dic3—26no—
62C24×D3—93no—
63C22×C3×D4—93no—
64C22×#67—36no—
65C22×#67—63no—
66C2×#217—93no—
67#937—924no—
68C2×(C2×D3)⋊C4—96no—
69C23×C32—26no—
70C23×C32—212no—
71C23×C32—212no—
72C23×C32—23no—
73C23×C32—26no—
74C23×C32—41no—
75C23×C32—42no—
76C23×C32—412no—
77C23×C32—412no—
78C23×C32C22×D313yes—
79C23×C32C22×D316yescomplemented
80C22×C3⋊D3—212no—
81C22×C3⋊D3#67112yescomplemented
82C22×C3⋊D3C22×D316yes—
83C22×C3×D3—324no—
84C22×C3×D3—312no—
85C22×C3×D3—612no—
86C22×C3×D3—66no—
87C22×C3×D3—612no—
88C22×C3×D3—612no—
89C22×C3×D3—1212no—
90C2×D32—648no—
91C2×D32—696no—
92C2×D32—624no—
93#638—224no—
94C2×C3⋊Dic3—26no—
95C2×C3⋊Dic3C22×D316yescomplemented
96C3×#67—648no—
97C3×#67—648no—
98C2×C3×Dic3—312no—
99C2×C3×Dic3—66no—
100C2×C3×Dic3—612no—
101C2×C3×Dic3—612no—
102C2×#120—271noSylow
103C2×C33—42no—
104C2×C33#12014yescomplemented
105C2×C33C22×D413yes—
106C3×C3⋊D3—212no—
107C24×C3—23no—
108C24×C3—41no—
109C24×C3—93no—
110C24×C3D3213yescomplemented
111C23×D3—36no—
112C23×D3—63no—
113C23×D3—912no—
114C23×D3—1824no—
115C23×D3—1812no—
116C2×C3×D4—912no—
117C2×C3×D4—1812no—
118C22×C3×C4—93no—
119C2×#67—324no—
120C2×#67—612no—
121C2×#67—624no—
122C2×#67—624no—
123C22×Dic3—36no—
124C22×Dic3—63no—
125#217—912no—
126(C2×D3)⋊C4—924no—
127C22×C32—26no—
128C22×C32—212no—
129C22×C32—212no—
130C22×C32—23no—
131C22×C32—212no—
132C22×C32—26no—
133C22×C32—46no—
134C22×C32—41no—
135C22×C32—412no—
136C22×C32—412no—
137C22×C32—424no—
138C22×C32—416no—
139C22×C32—412no—
140C22×C32D3×D4112yes—
141C22×C32C2×#6716yes—
142C22×C32C23×D313yes—
143C2×C3⋊D3—224no—
144C2×C3⋊D3—212no—
145C2×C3⋊D3C2×#67112yescomplemented
146C2×C3×D3—324no—
147C2×C3×D3—612no—
148C2×C3×D3—648no—
149C2×C3×D3—612no—
150C2×C3×D3—612no—
151C2×C3×D3—1212no—
152C2×C3×D3—1224no—
153D32—696no—
154C3⋊Dic3—26no—
155C3×Dic3—612no—
156C3×Dic3—612no—
157C25—271no—
158C22×D4—93no—
159#120—278no—
160C2×C22⋊C4—273no—
161C33C2×#12011yescharacteristic, complemented, Sylow
162C23×C3—212no—
163C23×C3—212no—
164C23×C3—26no—
165C23×C3—23no—
166C23×C3—26no—
167C23×C3—412no—
168C23×C3—412no—
169C23×C3—41no—
170C23×C3—42no—
171C23×C3—96no—
172C23×C3—912no—
173C23×C3—1812no—
174C23×C3C2×D3213yes—
175C23×C3C2×D3216yescomplemented
176C22×D3—312no—
177C22×D3—324no—
178C22×D3—612no—
179C22×D3—66no—
180C22×D3—612no—
181C22×D3—612no—
182C22×D3—1212no—
183C22×D3—1848no—
184C22×D3—1824no—
185C22×D3—1896no—
186C3×D4—1824no—
187C2×C3×C4—96no—
188C2×C3×C4—186no—
189#67—648no—
190#67—648no—
191C2×Dic3—312no—
192C2×Dic3—66no—
193C2×Dic3—612no—
194C2×Dic3—612no—
195C2×C32—26no—
196C2×C32—23no—
197C2×C32—212no—
198C2×C32—44no—
199C2×C32—43no—
200C2×C32—412no—
201C2×C32—48no—
202C2×C32—46no—
203C2×C32#937112yescomplemented
204C2×C32C2×D3×D416yes—
205C2×C32C22×#6713yes—
206C3⋊D3—212no—
207C3×D3—624no—
208C3×D3—1212no—
209C24—93no—
210C24—274no—
211C24—5412no—
212C24#111311yescharacteristic, complemented
213C2×D4—912no—
214C2×D4—1812no—
215C22×C4—93no—
216C22⋊C4—2712no—
217C22×C3—212no—
218C22×C3—212no—
219C22×C3—26no—
220C22×C3—23no—
221C22×C3—26no—
222C22×C3—212no—
223C22×C3—412no—
224C22×C3—412no—
225C22×C3—424no—
226C22×C3—416no—
227C22×C3—412no—
228C22×C3—41no—
229C22×C3—46no—
230C22×C3—912no—
231C22×C3—1812no—
232C22×C3—1824no—
233C22×C3D3×#67112yes—
234C22×C3#381116yes—
235C22×C3C22×D3213yes—
236C2×D3—324no—
237C2×D3—612no—
238C2×D3—612no—
239C2×D3—612no—
240C2×D3—648no—
241C2×D3—1212no—
242C2×D3—1224no—
243C2×D3—1896no—
244C3×C4—186no—
245Dic3—612no—
246Dic3—612no—
247C32—23no—
248C32—41no—
249C32C2×#93713yescomplemented
250C23—26no—
251C23—96no—
252C23—912no—
253C23—1812no—
254C23—5424no—
255C23—5432no—
256C23C2×#111311yescharacteristic
257C23C2×#111312yescomplemented
258D4—1824no—
259C2×C4—96no—
260C2×C4—186no—
261C2×C3—23no—
262C2×C3—26no—
263C2×C3—212no—
264C2×C3—412no—
265C2×C3—48no—
266C2×C3—46no—
267C2×C3—43no—
268C2×C3—44no—
269C2×C3—1812no—
270C2×C3#63983112yescomplemented
271C2×C3C2×D3×#6716yes—
272C2×C3C2×#381113yes—
273D3—624no—
274D3—1212no—
275C22—26no—
276C22—44no—
277C22—912no—
278C22—1812no—
279C22—1824no—
280C22—5432no—
281C22#8904916yes—
282C22C22×#111311yescharacteristic
283C4—186no—
284C3—23no—
285C3—41no—
286C3C2×#6398313yescomplemented
287C2—42no—
288C2—1812no—
289C2#1172372914yesdirect product factor
290C2C2×#8904913yes—
subgroupG/H
centerC23C2×#1113
commutator, cosocleC22×C33C24
FrattiniC22C22×#1113
FittingC24×C33C22
socleC23×C33C23
hypercenterC24#1113
hypocenterC33C2×#120
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×#1113C2semidirect
2. C22×C3×#638C2semidirect
3. #11723729C2direct (N × H)
4. C2×#88814C2semidirect
6. C24×C33C22semidirect
9. C22×#1113C22semidirect
11. C2×C3×#638C22semidirect
12. C22×C3×C3⋊Dic3C22semidirect
13. #88814C22semidirect
15. C22×#638D3semidirect
23. C23×C33C23semidirect
25. C22×C3×C3⋊D3D4semidirect
31. C2×C3×C3⋊Dic3C23semidirect
35. C24×C32C2×D3semidirect
43. C2×#638C2×D3semidirect
44. C22×C3⋊Dic3C2×D3semidirect
59. C2×C3×C3⋊D3C2×D4semidirect
79. C23×C32C22×D3semidirect
81. C22×C3⋊D3#67semidirect
95. C2×C3⋊Dic3C22×D3semidirect
104. C2×C33#120semidirect
110. C24×C3D32semidirect
145. C2×C3⋊D3C2×#67semidirect
161. C33C2×#120semidirect
175. C23×C3C2×D32semidirect
203. C2×C32#937semidirect
212. C24#1113semidirect
249. C32C2×#937semidirect
257. C23C2×#1113semidirect
270. C2×C3#63983semidirect
286. C3C2×#63983semidirect
289. C2#11723729direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#11132
C22×C3×#6382
#117237292
C2×#888142
C2×#639833