Finite Group Databasebeta

← C2×#11724633 #13181695

Subgroups — C2×#11724633 #13181695

Order 1728.

4434 conjugacy classes of subgroups; only the 663 normal subgroups are shown.

normal subgroups only

This group has 4434 conjugacy classes of subgroups — too many to draw. Only its 663 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×C3×#638C213yesmaximal, complemented
2#11724633C2116yesmaximal, direct product factor
3C2×#89049C216yesmaximal, complemented
4C2×#89046C216yesmaximal, complemented
5C2×#64330—33nomaximal
6C24×C33C2211yescharacteristic, complemented
7C23×C3×C3⋊D3C2213yescomplemented
8C22×#1113C2212yescomplemented
9C2×C3×#638C22112yescomplemented
10C2×C3×#638C22124yescomplemented
11C22×C3×C3⋊Dic3C2213yescomplemented
12#89049C22148yescomplemented
13C2×#6296C2212yescomplemented
14C2×#6295C2216yescomplemented
15#89046C22148yescomplemented
16C2×#6294C2216yescomplemented
17C22×#638D313yescomplemented
18C22×C3×#67—36no—
19C22×C3×#67—63no—
20#64330—348no—
21C2×#3811—36no—
22C2×D3×#67—312no—
23C2×#3805—36no—
24C2×#3804—312no—
25C23×C33—42no—
26C23×C33C2313yes—
27C23×C33C2314yescomplemented
28C22×C3×C3⋊D3—46no—
29C22×C3×C3⋊D3C2316yes—
30C22×C3×C3⋊D3C23112yescomplemented
31C2×#1113—44no—
32C2×#1113C2318yescomplemented
33C3×#638C23148yescomplemented
34C2×C3×C3⋊Dic3C2316yes—
35C2×C3×C3⋊Dic3C23112yescomplemented
36#6296C2318yescomplemented
37#6295C23124yescomplemented
38#6294C23124yescomplemented
39C2×#1004—93no—
40C24×C32—23no—
41C24×C32—41no—
42C24×C32C2×D313yescomplemented
43C23×C3⋊D3C2×D313yescomplemented
44C23×C3×D3—36no—
45C23×C3×D3—63no—
46C22×D32—36no—
47C2×#638C2×D3112yescomplemented
48C2×#638C2×D3124yescomplemented
49C22×C3⋊Dic3C2×D313yescomplemented
50C2×C3×#67—312no—
51C2×C3×#67—312no—
52C2×C3×#67—348no—
53C2×C3×#67—612no—
54C2×C3×#67—624no—
55C22×C3×Dic3—36no—
56C22×C3×Dic3—63no—
57#3811—348no—
58D3×#67—396no—
59C2×#627—36no—
60C2×#626—312no—
61C2×#625—36no—
62C2×#624—36no—
63#3805—348no—
64#3804—396no—
65C2×D3×Dic3—312no—
66C22×C33—26no—
67C22×C33—44no—
68C22×C33C2416yes—
69C22×C33C2411yescharacteristic
70C2×C3×C3⋊D3—212no—
71C2×C3×C3⋊D3—412no—
72C2×C3×C3⋊D3C24112yes—
73#1113—48no—
74C3×C3⋊Dic3C24112yes—
75C22×C3×D4—93no—
76C22×#67—36no—
77C22×#67—63no—
78C22×#67—96no—
79#1004—948no—
80C2×#237—912no—
81C2×D3×D4—912no—
82C2×#235—96no—
83C23×C32—23no—
84C23×C32—26no—
85C23×C32—212no—
86C23×C32—43no—
87C23×C32—44no—
88C23×C32—46no—
89C23×C32—412no—
90C23×C32—48no—
91C23×C32C22×D316yes—
92C23×C32C22×D313yes—
93C23×C32C22×D3112yescomplemented
94C22×C3⋊D3—46no—
95C22×C3⋊D3C22×D316yes—
96C22×C3⋊D3C22×D3112yescomplemented
97C22×C3×D3—312no—
98C22×C3×D3—324no—
99C22×C3×D3—66no—
100C22×C3×D3—612no—
101C22×C3×D3—1212no—
102C22×C3×D3—1212no—
103C2×D32—324no—
104C2×D32—1212no—
105#638C22×D3148yescomplemented
106C2×C3⋊Dic3C22×D316yes—
107C2×C3⋊Dic3C22×D3112yescomplemented
108C3×#67—348no—
109C3×#67—348no—
110C3×#67—648no—
111C2×C3×Dic3—312no—
112C2×C3×Dic3—324no—
113C2×C3×Dic3—66no—
114C2×C3×Dic3—612no—
115#627—324no—
116#626—348no—
117#625—324no—
118#624—324no—
119D3×Dic3—348no—
120C2×H(5,𝔽2)—271noSylow
121C2×C33—26no—
122C2×C33H(5,𝔽2)12yescomplemented
123C2×C33C2511yescharacteristic
124C3×C3⋊D3—212no—
125C24×C3—23no—
126C24×C3—41no—
127C24×C3—93no—
128C24×C3D3213yescomplemented
129C23×D3—36no—
130C23×D3—63no—
131C23×D3—96no—
132C2×C3×D4—912no—
133C2×C3×D4—924no—
134C22×C3×C4—93no—
135C2×#67—312no—
136C2×#67—312no—
137C2×#67—348no—
138C2×#67—612no—
139C2×#67—624no—
140C2×#67—912no—
141C2×#67—948no—
142C22×Dic3—36no—
143C22×Dic3—63no—
144C22×Dic3—96no—
145#237—996no—
146D3×D4—996no—
147#235—948no—
148C2×D12—96no—
149C2×C4×D3—912no—
150C2×Dic6—96no—
151C22×C32—26no—
152C22×C32—212no—
153C22×C32—212no—
154C22×C32—26no—
155C22×C32—23no—
156C22×C32—212no—
157C22×C32—46no—
158C22×C32—41no—
159C22×C32—412no—
160C22×C32—412no—
161C22×C32—412no—
162C22×C32—424no—
163C22×C32—416no—
164C22×C32C23×D3112yes—
165C22×C32C23×D316yes—
166C22×C32C23×D313yes—
167C2×C3⋊D3—212no—
168C2×C3⋊D3—412no—
169C2×C3⋊D3C23×D3112yes—
170C2×C3×D3—324no—
171C2×C3×D3—612no—
172C2×C3×D3—624no—
173C2×C3×D3—1212no—
174C2×C3×D3—1224no—
175C2×C3×D3—1224no—
176D32—1224no—
177C3⋊Dic3C23×D3112yes—
178C3×Dic3—324no—
179C3×Dic3—612no—
180H(5,𝔽2)—2716no—
181C2×#41—276no—
182C22×D4—93no—
183C22×D4—276no—
184C33C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
185C23×C3—23no—
186C23×C3—26no—
187C23×C3—212no—
188C23×C3—46no—
189C23×C3—412no—
190C23×C3—48no—
191C23×C3—43no—
192C23×C3—44no—
193C23×C3—96no—
194C23×C3—912no—
195C23×C3—366no—
196C23×C3C2×D3216yes—
197C23×C3C2×D3213yes—
198C23×C3C2×D32112yescomplemented
199C22×D3—312no—
200C22×D3—324no—
201C22×D3—66no—
202C22×D3—612no—
203C22×D3—924no—
204C22×D3—1212no—
205C22×D3—1212no—
206C22×D3—3612no—
207C3×D4—948no—
208C2×C3×C4—96no—
209C2×C3×C4—912no—
210#67—348no—
211#67—348no—
212#67—648no—
213#67—948no—
214C2×Dic3—312no—
215C2×Dic3—324no—
216C2×Dic3—66no—
217C2×Dic3—612no—
218C2×Dic3—924no—
219D12—924no—
220C4×D3—948no—
221Dic6—924no—
222C2×C32—26no—
223C2×C32—212no—
224C2×C32—212no—
225C2×C32—26no—
226C2×C32—23no—
227C2×C32—42no—
228C2×C32—41no—
229C2×C32—412no—
230C2×C32—412no—
231C2×C32#100416yescomplemented
232C2×C32C24×D313yes—
233C3⋊D3—212no—
234C3×D3—624no—
235C3×D3—1212no—
236C24—93no—
237C24—272no—
238C24#111311yescharacteristic, complemented
239#41—2748no—
240C2×Q8—272no—
241C2×D4—912no—
242C2×D4—924no—
243C2×D4—276no—
244C2×D4—2748no—
245C22×C4—93no—
246C22×C4—276no—
247C22×C3—212no—
248C22×C3—212no—
249C22×C3—26no—
250C22×C3—26no—
251C22×C3—212no—
252C22×C3—23no—
253C22×C3—412no—
254C22×C3—412no—
255C22×C3—412no—
256C22×C3—424no—
257C22×C3—416no—
258C22×C3—46no—
259C22×C3—41no—
260C22×C3—912no—
261C22×C3—1812no—
262C22×C3—3612no—
263C22×C3C22×D32112yes—
264C22×C3C22×D3216yes—
265C22×C3C22×D3213yes—
266C2×D3—324no—
267C2×D3—612no—
268C2×D3—624no—
269C2×D3—1212no—
270C2×D3—1224no—
271C2×D3—1224no—
272C2×D3—3624no—
273C3×C4—912no—
274Dic3—324no—
275Dic3—612no—
276C32—23no—
277C32—41no—
278C32C2×#100413yescomplemented
279C23—42no—
280C23—96no—
281C23—912no—
282C23—278no—
283C23—366no—
284C23—1084no—
285C23C2×#111313yes—
286C23C2×#111314yescomplemented
287Q8—278no—
288D4—948no—
289D4—2724no—
290C2×C4—912no—
291C2×C4—96no—
292C2×C4—2724no—
293C2×C3—212no—
294C2×C3—212no—
295C2×C3—26no—
296C2×C3—26no—
297C2×C3—23no—
298C2×C3—412no—
299C2×C3—412no—
300C2×C3—42no—
301C2×C3—41no—
302C2×C3—1812no—
303C2×C3#6433016yescomplemented
304C2×C3C23×D3213yes—
305D3—624no—
306D3—1212no—
307C22—26no—
308C22—44no—
309C22—912no—
310C22—1812no—
311C22—3612no—
312C22—1088no—
313C22C22×#111316yes—
314C22C22×#111311yescharacteristic
315C4—912no—
316C3—23no—
317C3—41no—
318C3C2×#6433013yescomplemented
319C2—26no—
320C2—1812no—
321C2#1172463312yesdirect product factor
322C2C23×#111311yescharacteristic
subgroupG/H
centerC22C22×#1113
commutator, cosocleC2×C33C25
FrattiniC2C23×#1113
FittingC24×C33C22
socleC22×C33C24
hypercenterC24#1113
hypocenterC33C2×H(5,𝔽2)
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C3×#638C2semidirect
2. #11724633C2direct (N × H)
3. C2×#89049C2semidirect
4. C2×#89046C2semidirect
6. C24×C33C22semidirect
7. C23×C3×C3⋊D3C22semidirect
8. C22×#1113C22semidirect
9. C2×C3×#638C22semidirect
10. C2×C3×#638C22semidirect
11. C22×C3×C3⋊Dic3C22semidirect
12. #89049C22semidirect
13. C2×#6296C22semidirect
14. C2×#6295C22semidirect
15. #89046C22semidirect
16. C2×#6294C22semidirect
17. C22×#638D3semidirect
27. C23×C33C23semidirect
30. C22×C3×C3⋊D3C23semidirect
32. C2×#1113C23semidirect
33. C3×#638C23semidirect
35. C2×C3×C3⋊Dic3C23semidirect
36. #6296C23semidirect
37. #6295C23semidirect
38. #6294C23semidirect
42. C24×C32C2×D3semidirect
43. C23×C3⋊D3C2×D3semidirect
47. C2×#638C2×D3semidirect
48. C2×#638C2×D3semidirect
49. C22×C3⋊Dic3C2×D3semidirect
93. C23×C32C22×D3semidirect
96. C22×C3⋊D3C22×D3semidirect
105. #638C22×D3semidirect
107. C2×C3⋊Dic3C22×D3semidirect
122. C2×C33H(5,𝔽2)semidirect
128. C24×C3D32semidirect
184. C33C2×H(5,𝔽2)semidirect
198. C23×C3C2×D32semidirect
231. C2×C32#1004semidirect
238. C24#1113semidirect
278. C32C2×#1004semidirect
286. C23C2×#1113semidirect
303. C2×C3#64330semidirect
318. C3C2×#64330semidirect
321. C2#11724633direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
#117246332
C2×#890492
C2×#890462
C2×#643303