Finite Group Databasebeta

← C2×#11724403 #13181258

Subgroups — C2×#11724403 #13181258

Order 1728.

6210 conjugacy classes of subgroups; only the 724 normal subgroups are shown.

normal subgroups only

This group has 6210 conjugacy classes of subgroups — too many to draw. Only its 724 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×D33C212yesmaximal, complemented
2C2×C4×#1113C211yesmaximal, characteristic, complemented
3C2×C3⋊D3×D12C213yesmaximal, complemented
4C2×C3×#3802C213yesmaximal, complemented
5C2×#88952C216yesmaximal, complemented
6#11724403C2116yesmaximal, direct product factor
7C2×D32×D4—33nomaximal
8C22×#1113C2211yescharacteristic
9C22×D3×C3⋊D3C2216yescomplemented
10C22×C3×D32C2216yescomplemented
11C2×D33—216no—
12C2×D33C22116yescomplemented
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
14C2×C3×C4×C3⋊D3C2213yescomplemented
15C2×C32×D12C2213yescomplemented
16C2×#6294C2213yescomplemented
17C4×#1113C2218yescomplemented
18C2×#6291C2216yescomplemented
19C3⋊D3×D12C22124yescomplemented
20C2×C3×#625C2216yescomplemented
21C3×#3802C22124yescomplemented
22#88952C22148yescomplemented
23C23×D32—36no—
24C2×D4×C3⋊D3—33no—
25C2×C3×D3×D4—36no—
26C2×#3811—36no—
27C2×D3×#67—312no—
28D32×D4—348no—
29C2×#3802D313yescomplemented
30C2×D3×D12—36no—
31C2×D3×D12—63no—
32C2×C4×D32—33no—
33C22×C3⋊(C3⋊D3)C2312yescomplemented
34C22×C3×C3⋊D3C2313yes—
35C22×C32×D3C2316yescomplemented
36C2×#1113D418yescomplemented
37C2×#1113C2314yes—
38C2×D3×C3⋊D3—224no—
39C2×D3×C3⋊D3C23124yescomplemented
40C2×C3×D32—224no—
41C2×C3×D32C23124yescomplemented
42D33—264no—
43C2×C33×C4C2311yescharacteristic, complemented
44C4⋊(C3⋊(C3⋊D3))C2314yescomplemented
45C2×C3×C3⋊Dic3C2313yescomplemented
46C3×C4×C3⋊D3C23112yescomplemented
47C32×D12C23112yescomplemented
48#6294C23112yescomplemented
49#6291C23124yescomplemented
50C3×#625C23124yescomplemented
51C22×D3×D4—93no—
52C23×C3⋊D3—36no—
53C23×C3×D3—312no—
54C22×D32—33no—
55C22×D32—312no—
56C22×D32—348no—
57C22×D32—648no—
58C22×D32—66no—
59C22×D32C2×D316yescomplemented
60C2×C32×D4—33no—
61C2×#638—36no—
62D4×C3⋊D3—324no—
63C2×C4⋊(C3⋊D3)—33no—
64C2×C4⋊(C3⋊D3)—63no—
65C2×C4⋊(C3⋊D3)—121no—
66C2×C4×C3⋊D3C2×D313yescomplemented
67C2×C3×#67—312no—
68C3×D3×D4—348no—
69C2×C3×D12—23no—
70C2×C3×D12C2×D316yescomplemented
71C2×C3×C4×D3—36no—
72C2×C3×C4×D3—63no—
73#3811—348no—
74D3×#67—396no—
75C2×#626—312no—
76C2×#626—66no—
77C2×#625C2×D316yescomplemented
78C2×#624—33no—
79C2×D3×Dic3—36no—
80#3802C2×D3124yescomplemented
81D3×D12—348no—
82D3×D12—624no—
83C4×D32—324no—
84C22×C33C2411yescharacteristic
85C2×C3⋊(C3⋊D3)—24no—
86C2×C3⋊(C3⋊D3)C2414yescomplemented
87C2×C3×C3⋊D3C2×D4112yescomplemented
88C2×C3×C3⋊D3C2416yes—
89C2×C32×D3—212no—
90C2×C32×D3C24112yescomplemented
91#1113C2×D4116yescomplemented
92D3×C3⋊D3—248no—
93C3×D32—248no—
94C33×C4C2412yescomplemented
95C3×C3⋊Dic3C2416yescomplemented
96C24×D3—96no—
97C22×C3×D4—93no—
98C22×#67—96no—
99C2×D3×D4—36no—
100C2×D3×D4—96no—
101C2×D3×D4—948no—
102C2×D3×D4—912no—
103C2×D3×D4—183no—
104C22×D12—93no—
105C22×C4×D3—93no—
106C23×C32—36no—
107C22×C3⋊D3—36no—
108C22×C3⋊D3—324no—
109C22×C3⋊D3—66no—
110C22×C3⋊D3—624no—
111C22×C3⋊D3—122no—
112C22×C3⋊D3C22×D313yes—
113C22×C3×D3—26no—
114C22×C3×D3—36no—
115C22×C3×D3—348no—
116C22×C3×D3—63no—
117C22×C3×D3—648no—
118C22×C3×D3C22×D3112yescomplemented
119C2×D32—224no—
120C2×D32—324no—
121C2×D32—312no—
122C2×D32—348no—
123C2×D32—624no—
124C2×D32—6192no—
125C2×D32—648no—
126C2×D32—1224no—
127C2×D32C22×D3124yescomplemented
128C32×D4—312no—
129C2×C32×C4—23no—
130C2×C32×C4—41no—
131C2×C32×C4C22×D313yescomplemented
132#638—324no—
133C2×C3⋊Dic3C22×D313yescomplemented
134C4⋊(C3⋊D3)—312no—
135C4⋊(C3⋊D3)—612no—
136C4⋊(C3⋊D3)—124no—
137C4×C3⋊D3C22×D3112yescomplemented
138C3×#67—348no—
139C2×C3×Dic3—36no—
140C2×C3×Dic3—63no—
141C3×D12—212no—
142C3×D12C22×D3124yescomplemented
143C3×C4×D3—324no—
144C3×C4×D3—612no—
145#626—348no—
146#626—624no—
147#625C22×D3124yescomplemented
148#624—312no—
149D3×Dic3—324no—
150C23×D4—271noSylow
151C2×C33C22×D412yescomplemented
152C2×C33C2511yescharacteristic
153C3⋊(C3⋊D3)—24no—
154C3×C3⋊D3C22×D4112yescomplemented
155C32×D3—212no—
156C24×C3—96no—
157C23×D3—312no—
158C23×D3—93no—
159C23×D3—948no—
160C23×D3—912no—
161C23×D3—96no—
162C23×D3—186no—
163C23×D3—1848no—
164C2×C3×D4—36no—
165C2×C3×D4—63no—
166C2×C3×D4—924no—
167C2×C3×D4—96no—
168C22×C3×C4—93no—
169C2×#67—312no—
170C2×#67—912no—
171C2×#67—948no—
172C2×#67—186no—
173C22×Dic3—93no—
174D3×D4—348no—
175D3×D4—948no—
176D3×D4—996no—
177D3×D4—1824no—
178C2×D12—96no—
179C2×D12—93no—
180C2×D12—924no—
181C2×D12—183no—
182C2×D12—361no—
183C2×D12D3213yescomplemented
184C2×C4×D3—36no—
185C2×C4×D3—63no—
186C2×C4×D3—924no—
187C2×C4×D3—96no—
188C22×C32—23no—
189C22×C32—312no—
190C22×C32—41no—
191C22×C32—612no—
192C22×C32C23×D313yes—
193C2×C3⋊D3—312no—
194C2×C3⋊D3—612no—
195C2×C3⋊D3—612no—
196C2×C3⋊D3—648no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—1212no—
199C2×C3⋊D3—244no—
200C2×C3⋊D3D3×D4112yescomplemented
201C2×C3⋊D3C23×D316yes—
202C2×C3×D3—224no—
203C2×C3×D3—212no—
204C2×C3×D3—324no—
205C2×C3×D3—312no—
206C2×C3×D3—412no—
207C2×C3×D3—612no—
208C2×C3×D3—66no—
209C2×C3×D3—696no—
210C2×C3×D3C23×D3124yescomplemented
211D32—248no—
212D32—348no—
213D32—696no—
214D32—1248no—
215C32×C4—26no—
216C32×C4—42no—
217C32×C4C23×D316yescomplemented
218C3⋊Dic3C23×D316yescomplemented
219C3×Dic3—312no—
220C3×Dic3—66no—
221C25—272no—
222C22×D4—93no—
223C22×D4—273no—
224C22×D4—2716no—
225C22×D4—276no—
226C23×C4—271no—
227C33C23×D411yescharacteristic, complemented, Sylow
228C23×C3—312no—
229C23×C3—66no—
230C23×C3—924no—
231C23×C3—93no—
232C23×C3—1824no—
233C22×D3—36no—
234C22×D3—348no—
235C22×D3—63no—
236C22×D3—648no—
237C22×D3—912no—
238C22×D3—924no—
239C22×D3—948no—
240C22×D3—96no—
241C22×D3—924no—
242C22×D3—1824no—
243C22×D3—186no—
244C22×D3—1824no—
245C22×D3—1848no—
246C22×D3—18192no—
247C22×D3—362no—
248C22×D3—3624no—
249C22×D3C2×D3216yescomplemented
250C3×D4—324no—
251C3×D4—612no—
252C3×D4—924no—
253C2×C3×C4—23no—
254C2×C3×C4—41no—
255C2×C3×C4—93no—
256C2×C3×C4—912no—
257C2×C3×C4C2×D3213yescomplemented
258#67—348no—
259#67—948no—
260#67—1824no—
261C2×Dic3—36no—
262C2×Dic3—63no—
263C2×Dic3—912no—
264D12—924no—
265D12—912no—
266D12—1812no—
267D12—364no—
268D12C2×D32112yescomplemented
269C4×D3—324no—
270C4×D3—612no—
271C4×D3—924no—
272C2×C32—26no—
273C2×C32—23no—
274C2×C32—42no—
275C2×C32—41no—
276C2×C32—612no—
277C2×C32C2×D3×D416yescomplemented
278C2×C32C24×D313yes—
279C3⋊D3—612no—
280C3⋊D3—1212no—
281C3⋊D3—244no—
282C3⋊D3C2×D3×D4112yescomplemented
283C3×D3—224no—
284C3×D3—324no—
285C3×D3—412no—
286C3×D3—612no—
287C24—96no—
288C24—276no—
289C24—271no—
290C24—2716no—
291C24—5416no—
292C2×D4—33no—
293C2×D4—924no—
294C2×D4—96no—
295C2×D4—2724no—
296C2×D4—276no—
297C2×D4—2748no—
298C2×D4—271no—
299C22×C4—93no—
300C22×C4—278no—
301C22×C4—273no—
302C22×C3—23no—
303C22×C3—324no—
304C22×C3—41no—
305C22×C3—612no—
306C22×C3—624no—
307C22×C3—912no—
308C22×C3—96no—
309C22×C3—1212no—
310C22×C3—1848no—
311C22×C3C22×D3213yes—
312C2×D3—212no—
313C2×D3—324no—
314C2×D3—312no—
315C2×D3—66no—
316C2×D3—612no—
317C2×D3—696no—
318C2×D3—948no—
319C2×D3—912no—
320C2×D3—1812no—
321C2×D3—1848no—
322C2×D3—1896no—
323C2×D3—1812no—
324C2×D3—364no—
325C2×D3—3648no—
326C2×D3—3612no—
327C2×D3—724no—
328C2×D3C22×D32112yescomplemented
329C3×C4—26no—
330C3×C4—42no—
331C3×C4—96no—
332C3×C4C22×D3216yescomplemented
333Dic3—312no—
334Dic3—66no—
335C32—23no—
336C32—41no—
337C32C22×D3×D413yescomplemented
338C23—36no—
339C23—924no—
340C23—93no—
341C23—1824no—
342C23—2724no—
343C23—274no—
344C23—278no—
345C23—272no—
346C23—5424no—
347C23—5464no—
348D4—312no—
349D4—924no—
350D4—2724no—
351D4—274no—
352C2×C4—93no—
353C2×C4—912no—
354C2×C4—2712no—
355C2×C4D3311yescharacteristic, complemented
356C2×C3—26no—
357C2×C3—23no—
358C2×C3—42no—
359C2×C3—41no—
360C2×C3—624no—
361C2×C3—912no—
362C2×C3—1212no—
363C2×C3D32×D416yescomplemented
364C2×C3C23×D3213yes—
365D3—212no—
366D3—324no—
367D3—612no—
368D3—1812no—
369D3—3612no—
370D3—724no—
371C22—312no—
372C22—612no—
373C22—912no—
374C22—96no—
375C22—1848no—
376C22—2716no—
377C22—274no—
378C22—5448no—
379C22—544no—
380C22C2×D3311yescharacteristic
381C4—96no—
382C4C2×D3312yescomplemented
383C3—23no—
384C3—41no—
385C3C2×D32×D413yescomplemented
386C2—612no—
387C2—912no—
388C2—544no—
389C2#1172440312yesdirect product factor
390C2C22×D3311yescharacteristic
subgroupG/H
centerC22C2×D33
commutator, cosocleC2×C33C25
FrattiniC2C22×D33
FittingC2×C33×C4C23
socleC22×C33C24
hypercenterC2×C4D33
hypocenterC33C23×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×D33C2semidirect
2. C2×C4×#1113C2semidirect
3. C2×C3⋊D3×D12C2semidirect
4. C2×C3×#3802C2semidirect
5. C2×#88952C2semidirect
6. #11724403C2direct (N × H)
9. C22×D3×C3⋊D3C22semidirect
10. C22×C3×D32C22semidirect
12. C2×D33C22semidirect
13. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
14. C2×C3×C4×C3⋊D3C22semidirect
15. C2×C32×D12C22semidirect
16. C2×#6294C22semidirect
17. C4×#1113C22semidirect
18. C2×#6291C22semidirect
19. C3⋊D3×D12C22semidirect
20. C2×C3×#625C22semidirect
21. C3×#3802C22semidirect
22. #88952C22semidirect
29. C2×#3802D3semidirect
33. C22×C3⋊(C3⋊D3)C23semidirect
35. C22×C32×D3C23semidirect
36. C2×#1113D4semidirect
39. C2×D3×C3⋊D3C23semidirect
41. C2×C3×D32C23semidirect
43. C2×C33×C4C23semidirect
44. C4⋊(C3⋊(C3⋊D3))C23semidirect
45. C2×C3×C3⋊Dic3C23semidirect
46. C3×C4×C3⋊D3C23semidirect
47. C32×D12C23semidirect
48. #6294C23semidirect
49. #6291C23semidirect
50. C3×#625C23semidirect
59. C22×D32C2×D3semidirect
66. C2×C4×C3⋊D3C2×D3semidirect
70. C2×C3×D12C2×D3semidirect
77. C2×#625C2×D3semidirect
80. #3802C2×D3semidirect
86. C2×C3⋊(C3⋊D3)C24semidirect
87. C2×C3×C3⋊D3C2×D4semidirect
90. C2×C32×D3C24semidirect
91. #1113C2×D4semidirect
94. C33×C4C24semidirect
95. C3×C3⋊Dic3C24semidirect
118. C22×C3×D3C22×D3semidirect
127. C2×D32C22×D3semidirect
131. C2×C32×C4C22×D3semidirect
133. C2×C3⋊Dic3C22×D3semidirect
137. C4×C3⋊D3C22×D3semidirect
142. C3×D12C22×D3semidirect
147. #625C22×D3semidirect
151. C2×C33C22×D4semidirect
154. C3×C3⋊D3C22×D4semidirect
183. C2×D12D32semidirect
200. C2×C3⋊D3D3×D4semidirect
210. C2×C3×D3C23×D3semidirect
217. C32×C4C23×D3semidirect
218. C3⋊Dic3C23×D3semidirect
227. C33C23×D4semidirect
249. C22×D3C2×D32semidirect
257. C2×C3×C4C2×D32semidirect
268. D12C2×D32semidirect
277. C2×C32C2×D3×D4semidirect
282. C3⋊D3C2×D3×D4semidirect
328. C2×D3C22×D32semidirect
332. C3×C4C22×D32semidirect
337. C32C22×D3×D4semidirect
355. C2×C4D33semidirect
363. C2×C3D32×D4semidirect
382. C4C2×D33semidirect
385. C3C2×D32×D4semidirect
389. C2#11724403direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D332
C2×C4×#11132
C2×C3⋊D3×D122
C2×C3×#38022
C2×#889522
#117244032
C2×D32×D43