Finite Group Databasebeta

← C22×D3×#239 #11919200

Subgroups — C22×D3×#239 #11919200

Order 1152.

13485 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3832C211yesmaximal, characteristic, complemented
2C22×C3×#239C211yesmaximal, characteristic, complemented
3C22×C3×D3×Q8C211yesmaximal, characteristic, complemented
4C2×D3×#239C2148yesmaximal, direct product factor
5C22×D3×D12C213yesmaximal, complemented
6C22×C4×D32C213yesmaximal, complemented
7C22×#3799C213yesmaximal, complemented
8C22×#3796C213yesmaximal, complemented
9C22×D3×#41—31nomaximal
10C23×#239—31nomaximal
11C23×D32C2213yes—
12C22×C32×Q8C2211yescharacteristic, complemented
13C2×#3832C22124yescomplemented
14C22×C4⋊(C3⋊D3)C2213yescomplemented
15C22×C4×C3⋊D3C2213yescomplemented
16C2×C3×#239C22124yescomplemented
17C2×C3×D3×Q8C22124yescomplemented
18C22×C3×D12C2213yescomplemented
19C22×C3×C4×D3C2213yes—
20C22×C3×C4×D3C2213yescomplemented
21C22×C3×Dic6C2213yescomplemented
22C22×#626C2216yescomplemented
23C22×#624C2213yescomplemented
24C22×D3×Dic3C2211yescharacteristic
25C22×D3×Dic3C2213yescomplemented
26D3×#239C221256yesdirect product factor
27C2×D3×D12C22172yescomplemented
28C2×C4×D32C22172yescomplemented
29C2×#3799C22172yescomplemented
30C2×#3796C22172yescomplemented
31C22×C3×#41—31no—
32C23×C3×Q8—31no—
33C2×D3×#41—348no—
34C22×#239—31no—
35C22×#239—31no—
36C22×#239—348no—
37C22×#239—33no—
38C22×#239—33no—
39C22×#239—61no—
40C22×#239D311yescharacteristic, direct product factor
41C22×D3×Q8D311yescharacteristic, complemented
42C22×#237—33no—
43C22×D3×D4—33no—
44C22×#235—33no—
45C23×D12—33no—
46C23×C4×D3—33no—
47C23×C4×D3—33no—
48C23×C3⋊D3C2313yes—
49C23×C3×D3C2313yes—
50C23×C3×D3C2311yescharacteristic
51C22×D32—224no—
52C22×D32C23136yes—
53C2×C32×Q8C23112yescomplemented
54C22×C32×C4C2313yes—
55C22×C3⋊Dic3C2311yescharacteristic
56#3832C23164yescomplemented
57C2×C4⋊(C3⋊D3)C23136yescomplemented
58C2×C4×C3⋊D3C23136yescomplemented
59C22×C3×Dic3C2311yescharacteristic
60C22×C3×Dic3C2313yes—
61C3×#239C23164yescomplemented
62C3×D3×Q8C23164yescomplemented
63C2×C3×D12C23136yescomplemented
64C2×C3×C4×D3C23136yes—
65C2×C3×C4×D3C23136yescomplemented
66C2×C3×Dic6C23136yescomplemented
67C2×#626C23172yescomplemented
68C2×#624C23136yescomplemented
69C2×D3×Dic3C23112yes—
70C2×D3×Dic3C23136yescomplemented
71D3×D12C231192yescomplemented
72C4×D32C231192yescomplemented
73#3799C231192yescomplemented
74#3796C231192yescomplemented
75C23×#41—91noSylow
76C24×D3—33no—
77C24×D3—33no—
78C2×C3×#41—324no—
79C22×C3×Q8—21no—
80C22×C3×Q8—324no—
81C22×C3×Q8—33no—
82C22×C3×Q8C2×D311yescharacteristic, complemented
83C22×C3×Q8C2×D311yescharacteristic, complemented
84C22×C3×D4—33no—
85C23×C3×C4—33no—
86C23×C3×C4—33no—
87C22×#67—36no—
88C23×Dic3—31no—
89C23×Dic3—33no—
90D3×#41—3256no—
91C2×#239—324no—
92C2×#239—324no—
93C2×#239—3256no—
94C2×#239—372no—
95C2×#239—372no—
96C2×#239—624no—
97C2×#239C2×D3124yesdirect product factor
98C2×D3×Q8C2×D3124yescomplemented
99C2×#237—372no—
100C2×D3×D4—372no—
101C2×#235—372no—
102C22×D12—36no—
103C22×D12—33no—
104C22×D12—33no—
105C22×D12—372no—
106C22×D12—63no—
107C22×D12C2×D313yescomplemented
108C22×C4×D3—31no—
109C22×C4×D3—33no—
110C22×C4×D3—33no—
111C22×C4×D3—33no—
112C22×C4×D3—33no—
113C22×C4×D3—33no—
114C22×C4×D3—372no—
115C22×C4×D3—372no—
116C22×C4×D3—63no—
117C22×C4×D3C2×D313yes—
118C22×C4×D3C2×D313yescomplemented
119C22×Dic6C2×D313yescomplemented
120C23×C32C2411yescharacteristic
121C22×C3⋊D3—212no—
122C22×C3⋊D3C24118yes—
123C22×C3×D3—212no—
124C22×C3×D3#4118yescomplemented
125C22×C3×D3C24118yes—
126C22×C3×D3C2416yes—
127C2×D32—2144no—
128C2×D32C24148yes—
129C32×Q8C24116yescomplemented
130C2×C32×C4C24118yes—
131C2×C3⋊Dic3C2416yes—
132C4⋊(C3⋊D3)C24148yescomplemented
133C4×C3⋊D3C24148yescomplemented
134C2×C3×Dic3C2416yes—
135C2×C3×Dic3C24118yes—
136C3×D12C24148yescomplemented
137C3×C4×D3C24148yes—
138C3×C4×D3C24148yescomplemented
139C3×Dic6C24148yescomplemented
140#626C24196yescomplemented
141#624C24148yescomplemented
142D3×Dic3C24116yes—
143D3×Dic3C24148yescomplemented
144C22×#41—31no—
145C22×#41—93no—
146C22×#41—93no—
147C22×#41—948no—
148C22×#41—91no—
149C23×Q8—31no—
150C23×D4—93no—
151C24×C4—93no—
152C24×C3—31no—
153C24×C3—33no—
154C23×D3—33no—
155C23×D3—33no—
156C23×D3—336no—
157C23×D3—336no—
158C23×D3—63no—
159C23×D3—624no—
160C23×D3—624no—
161C23×D3C22×D313yes—
162C23×D3C22×D311yescharacteristic
163C3×#41—364no—
164C2×C3×Q8—212no—
165C2×C3×Q8—364no—
166C2×C3×Q8—336no—
167C2×C3×Q8C22×D3112yescomplemented
168C2×C3×Q8C22×D3112yescomplemented
169C2×C3×D4—336no—
170C22×C3×C4—23no—
171C22×C3×C4—31no—
172C22×C3×C4—33no—
173C22×C3×C4—336no—
174C22×C3×C4—336no—
175C22×C3×C4C22×D313yes—
176C22×C3×C4C22×D313yes—
177C2×#67—372no—
178C22×Dic3—31no—
179C22×Dic3—31no—
180C22×Dic3—312no—
181C22×Dic3—336no—
182C22×Dic3—61no—
183C22×Dic3C22×D313yes—
184C22×Dic3C22×D311yescharacteristic
185#239—364no—
186#239—364no—
187#239—3192no—
188#239—3192no—
189#239—664no—
190#239C22×D3164yesdirect product factor
191D3×Q8C22×D3164yescomplemented
192#237—3192no—
193D3×D4—3192no—
194#235—3192no—
195C2×D12—372no—
196C2×D12—336no—
197C2×D12—336no—
198C2×D12—3192no—
199C2×D12—636no—
200C2×D12C22×D3136yescomplemented
201C2×C4×D3—312no—
202C2×C4×D3—336no—
203C2×C4×D3—336no—
204C2×C4×D3—336no—
205C2×C4×D3—336no—
206C2×C4×D3—336no—
207C2×C4×D3—3192no—
208C2×C4×D3—3192no—
209C2×C4×D3—636no—
210C2×C4×D3C22×D3136yes—
211C2×C4×D3C22×D3136yescomplemented
212C2×Dic6C22×D3136yescomplemented
213C22×C32C2×#4114yescomplemented
214C22×C32C2513yes—
215C2×C3⋊D3—236no—
216C2×C3⋊D3C25112yes—
217C2×C3×D3—236no—
218C2×C3×D3C2×#41124yescomplemented
219C2×C3×D3C25112yes—
220C2×C3×D3C2514yes—
221D32—296no—
222C32×C4C25112yes—
223C3⋊Dic3C2514yes—
224C3×Dic3C2514yes—
225C3×Dic3C25112yes—
226C25—93no—
227C2×#41—324no—
228C2×#41—972no—
229C2×#41—972no—
230C2×#41—9256no—
231C2×#41—924no—
232C22×Q8—324no—
233C22×Q8—33no—
234C22×Q8D3211yescharacteristic, complemented
235C22×D4—33no—
236C22×D4—972no—
237C22×D4—96no—
238C22×D4—93no—
239C23×C4—33no—
240C23×C4—33no—
241C23×C4—93no—
242C23×C4—972no—
243C23×C4—91no—
244C23×C4—93no—
245C23×C4—93no—
246C23×C3—21no—
247C23×C3—38no—
248C23×C3—36no—
249C23×C3—318no—
250C23×C3—612no—
251C23×C3C23×D311yescharacteristic
252C23×C3C23×D311yescharacteristic
253C22×D3—212no—
254C22×D3—318no—
255C22×D3—318no—
256C22×D3—348no—
257C22×D3—348no—
258C22×D3—618no—
259C22×D3—6144no—
260C22×D3—612no—
261C22×D3—612no—
262C22×D3—6144no—
263C22×D3—1212no—
264C22×D3#23918yesdirect product factor
265C22×D3C23×D3118yes—
266C22×D3C23×D316yes—
267C3×Q8—216no—
268C3×Q8—348no—
269C3×Q8C23×D3116yescomplemented
270C3×Q8C23×D3116yescomplemented
271C3×D4—348no—
272C2×C3×C4—218no—
273C2×C3×C4—36no—
274C2×C3×C4—318no—
275C2×C3×C4—348no—
276C2×C3×C4—348no—
277C2×C3×C4C23×D3118yes—
278C2×C3×C4C23×D3118yes—
279#67—396no—
280C2×Dic3—36no—
281C2×Dic3—36no—
282C2×Dic3—316no—
283C2×Dic3—348no—
284C2×Dic3—66no—
285C2×Dic3C23×D3118yes—
286C2×Dic3C23×D316yes—
287D12—396no—
288D12—348no—
289D12—348no—
290D12—648no—
291D12C23×D3148yescomplemented
292C4×D3—316no—
293C4×D3—348no—
294C4×D3—348no—
295C4×D3—348no—
296C4×D3—348no—
297C4×D3—348no—
298C4×D3—648no—
299C4×D3C23×D3148yes—
300C4×D3C23×D3148yescomplemented
301Dic6C23×D3148yescomplemented
302C2×C32C22×#4116yescomplemented
303C2×C32C2611yescharacteristic
304C3⋊D3—212no—
305C3×D3—212no—
306C3×D3C22×#4118yescomplemented
307C24—31no—
308C24—33no—
309C24—936no—
310C24—93no—
311C24—1824no—
312#41—364no—
313#41—9192no—
314#41—9192no—
315#41—964no—
316C2×Q8—364no—
317C2×Q8—336no—
318C2×Q8C2×D32112yescomplemented
319C2×D4—336no—
320C2×D4—9192no—
321C2×D4—972no—
322C2×D4—936no—
323C22×C4—31no—
324C22×C4—33no—
325C22×C4—336no—
326C22×C4—336no—
327C22×C4—936no—
328C22×C4—9192no—
329C22×C4—912no—
330C22×C4—936no—
331C22×C4—91no—
332C22×C4—936no—
333C22×C4C2×D3213yes—
334C22×C3—24no—
335C22×C3—23no—
336C22×C3—324no—
337C22×C3—34no—
338C22×C3—312no—
339C22×C3—636no—
340C22×C3C2×#23914yescomplemented
341C22×C3D3×#4114yescomplemented
342C22×C3C24×D313yes—
343C22×C3C24×D313yes—
344C2×D3—236no—
345C2×D3—312no—
346C2×D3—312no—
347C2×D3—612no—
348C2×D3—696no—
349C2×D3—636no—
350C2×D3—636no—
351C2×D3—696no—
352C2×D3—1236no—
353C2×D3C2×#239124yesdirect product factor
354C2×D3C24×D3112yes—
355C2×D3C24×D314yes—
356C3×C4—212no—
357C3×C4—34no—
358C3×C4—312no—
359C3×C4C24×D3112yes—
360C3×C4C24×D3112yes—
361Dic3—34no—
362Dic3—34no—
363Dic3—64no—
364Dic3C24×D3112yes—
365Dic3C24×D314yes—
366C32C23×#4111yescharacteristic, complemented, Sylow
367C23—38no—
368C23—36no—
369C23—318no—
370C23—612no—
371C23—948no—
372C23—918no—
373C23—18144no—
374C23—1812no—
375C23C22×D3211yescharacteristic
376Q8—348no—
377Q8C22×D32116yescomplemented
378D4—348no—
379D4—996no—
380D4—948no—
381C2×C4—36no—
382C2×C4—318no—
383C2×C4—348no—
384C2×C4—348no—
385C2×C4—948no—
386C2×C4—916no—
387C2×C4—948no—
388C2×C4—96no—
389C2×C4—948no—
390C2×C4C22×D32118yes—
391C2×C3—26no—
392C2×C3—21no—
393C2×C3—38no—
394C2×C3—612no—
395C2×C3C22×#23916yescomplemented
396C2×C3C2×D3×#4116yescomplemented
397C2×C3C25×D311yescharacteristic
398C2×C3C25×D311yescharacteristic
399D3—212no—
400D3—612no—
401D3—612no—
402D3—1212no—
403D3C22×#23918yesdirect product factor
404C22—324no—
405C22—34no—
406C22—312no—
407C22—636no—
408C22—912no—
409C22—1896no—
410C22—1836no—
411C22D3×#23914yesdirect product factor
412C22C23×D3213yes—
413C4—34no—
414C4—312no—
415C4—94no—
416C4C23×D32112yes—
417C3—21no—
418C3C23×#23911yescharacteristic, complemented
419C3C22×D3×#4111yescharacteristic, complemented
420C2—38no—
421C2—612no—
422C2—1812no—
423C2C2×D3×#23916yesdirect product factor
424C2C24×D3211yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC2×C32C26
FrattiniC2C24×D32
FittingC22×C32×Q8C22
socleC23×C32C24
hypercenterC22×Q8D32
hypocenterC32C23×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#3832C2semidirect
2. C22×C3×#239C2semidirect
3. C22×C3×D3×Q8C2semidirect
4. C2×D3×#239C2direct (N × H)
5. C22×D3×D12C2semidirect
6. C22×C4×D32C2semidirect
7. C22×#3799C2semidirect
8. C22×#3796C2semidirect
12. C22×C32×Q8C22semidirect
13. C2×#3832C22semidirect
14. C22×C4⋊(C3⋊D3)C22semidirect
15. C22×C4×C3⋊D3C22semidirect
16. C2×C3×#239C22semidirect
17. C2×C3×D3×Q8C22semidirect
18. C22×C3×D12C22semidirect
20. C22×C3×C4×D3C22semidirect
21. C22×C3×Dic6C22semidirect
22. C22×#626C22semidirect
23. C22×#624C22semidirect
25. C22×D3×Dic3C22semidirect
26. D3×#239C22direct (N × H)
27. C2×D3×D12C22semidirect
28. C2×C4×D32C22semidirect
29. C2×#3799C22semidirect
30. C2×#3796C22semidirect
40. C22×#239D3direct (N × H)
41. C22×D3×Q8D3semidirect
53. C2×C32×Q8C23semidirect
56. #3832C23semidirect
57. C2×C4⋊(C3⋊D3)C23semidirect
58. C2×C4×C3⋊D3C23semidirect
61. C3×#239C23semidirect
62. C3×D3×Q8C23semidirect
63. C2×C3×D12C23semidirect
65. C2×C3×C4×D3C23semidirect
66. C2×C3×Dic6C23semidirect
67. C2×#626C23semidirect
68. C2×#624C23semidirect
70. C2×D3×Dic3C23semidirect
71. D3×D12C23semidirect
72. C4×D32C23semidirect
73. #3799C23semidirect
74. #3796C23semidirect
82. C22×C3×Q8C2×D3semidirect
83. C22×C3×Q8C2×D3semidirect
97. C2×#239C2×D3direct (N × H)
98. C2×D3×Q8C2×D3semidirect
107. C22×D12C2×D3semidirect
118. C22×C4×D3C2×D3semidirect
119. C22×Dic6C2×D3semidirect
124. C22×C3×D3#41semidirect
129. C32×Q8C24semidirect
132. C4⋊(C3⋊D3)C24semidirect
133. C4×C3⋊D3C24semidirect
136. C3×D12C24semidirect
138. C3×C4×D3C24semidirect
139. C3×Dic6C24semidirect
140. #626C24semidirect
141. #624C24semidirect
143. D3×Dic3C24semidirect
167. C2×C3×Q8C22×D3semidirect
168. C2×C3×Q8C22×D3semidirect
190. #239C22×D3direct (N × H)
191. D3×Q8C22×D3semidirect
200. C2×D12C22×D3semidirect
211. C2×C4×D3C22×D3semidirect
212. C2×Dic6C22×D3semidirect
213. C22×C32C2×#41semidirect
218. C2×C3×D3C2×#41semidirect
234. C22×Q8D32semidirect
264. C22×D3#239direct (N × H)
269. C3×Q8C23×D3semidirect
270. C3×Q8C23×D3semidirect
291. D12C23×D3semidirect
300. C4×D3C23×D3semidirect
301. Dic6C23×D3semidirect
302. C2×C32C22×#41semidirect
306. C3×D3C22×#41semidirect
318. C2×Q8C2×D32semidirect
340. C22×C3C2×#239semidirect
341. C22×C3D3×#41semidirect
353. C2×D3C2×#239direct (N × H)
366. C32C23×#41semidirect
377. Q8C22×D32semidirect
395. C2×C3C22×#239semidirect
396. C2×C3C2×D3×#41semidirect
403. D3C22×#239direct (N × H)
411. C22D3×#239direct (N × H)
418. C3C23×#239semidirect
419. C3C22×D3×#41semidirect
423. C2C2×D3×#239direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#38322
C22×C3×#2392
C22×C3×D3×Q82
C2×D3×#2392
C22×D3×D122
C22×C4×D322
C22×#37992
C22×#37962
C22×D3×#413
C23×#2393