Finite Group Databasebeta

← C32×#3512 #11770561

Subgroups — C32×#3512 #11770561

Order 1152.

3096 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#553C214yesmaximal, complemented
2C32×#553C218yesmaximal, complemented
3C32×#552C214yesmaximal, complemented
4C32×#541C212yesmaximal, complemented
5C32×#540C212yesmaximal, complemented
6C32×#537C212yesmaximal, complemented
7C32×#532C211yesmaximal, characteristic, complemented
8C32×#529C214yesmaximal, complemented
9C2×C32×#125C211yesmaximal, characteristic, complemented
10C32×#520C212yesmaximal, complemented
11C32×C4×#41C211yesmaximal, characteristic, complemented
12C3×#3512C314yesmaximal, direct product factor
13C2×C32×#41C2211yescharacteristic
14C2×C32×#41C2212yes—
15C32×C4⋊Q8C2211yescharacteristic
16C32×#127C2211yescharacteristic
17C32×#126C2218yes—
18C32×#126C2218yes—
19C32×#125C2214yes—
20C32×#125C2218yes—
21C32×#125C2212yes—
22C32×#125C2212yes—
23C32×#124C2212yes—
24C32×#123C2218yes—
25C32×#123C2218yes—
26C32×C22⋊Q8C2214yes—
27C32×C22⋊Q8C2214yes—
28C32×#121C2214yes—
29C32×#121C2218yes—
30C32×#121C2214yes—
31C32×#121C2218yes—
32C32×#121C2214yes—
33C32×C4×Q8C2212yes—
34C32×C4×Q8C2214yes—
35C32×C4×D4C2214yes—
36C32×C4×D4C2214yes—
37C32×C4×D4C2212yes—
38C32×C4×D4C2218yes—
39C32×C4×D4C2218yes—
40C32×C4×D4C2214yes—
41C32×#117C2214yes—
42C32×#117C2218yes—
43C32×#117C2211yescharacteristic
44C32×#117C2212yes—
45C2×C32×C4⋊C4C2214yes—
46C2×C32×C4⋊C4C2214yes—
47C2×C32×C4⋊C4C2212yes—
48C2×C32×C42C2211yescharacteristic
49C2×C32×C42C2212yes—
50C3×#553C2×C3116yescomplemented
51C3×#553C2×C3132yescomplemented
52C3×#552C2×C3116yescomplemented
53C3×#541C2×C318yescomplemented
54C3×#540C2×C318yescomplemented
55C3×#537C2×C318yescomplemented
56C3×#532C2×C314yescomplemented
57C3×#529C2×C3116yescomplemented
58C2×C3×#125C2×C314yescomplemented
59C3×#520C2×C318yescomplemented
60C3×C4×#41C2×C314yescomplemented
61C32×#41—24no—
62C32×#41—28no—
63C2×C32×Q8C2311yescharacteristic
64C2×C32×Q8C2312yes—
65C2×C32×D4C2312yes—
66C2×C32×D4C2314yes—
67C2×C32×D4C2311yescharacteristic
68C2×C32×D4C2312yes—
69C2×C32×D4C2318yes—
70C2×C32×D4C2314yes—
71C22×C32×C4C2312yes—
72C22×C32×C4C2314yes—
73C22×C32×C4C2312yes—
74C22×C32×C4C2314yes—
75C22×C32×C4C2314yes—
76C22×C32×C4C2318yes—
77C22×C32×C4C2311yescharacteristic
78C32×C4⋊C4C2312yes—
79C32×C4⋊C4C2314yes—
80C32×C4⋊C4C2318yes—
81C32×C4⋊C4C2314yes—
82C32×C4⋊C4C2318yes—
83C32×C4⋊C4C2318yes—
84C32×C4⋊C4C2314yes—
85C32×C4⋊C4C2318yes—
86C32×C4⋊C4C2314yes—
87C32×C22⋊C4C2314yes—
88C32×C22⋊C4C2312yes—
89C32×C22⋊C4C2314yes—
90C32×C22⋊C4C2318yes—
91C32×C22⋊C4C2318yes—
92C32×C22⋊C4C2314yes—
93C32×C22⋊C4C2318yes—
94C32×C42C2311yescharacteristic
95C32×C42C2312yes—
96C32×C42C2314yes—
97C32×C42C2311yescharacteristic
98C32×C42C2312yes—
99C32×C42C2318yes—
100#3512C3211yescharacteristic, direct product factor, Sylow
101C2×C3×#41C22×C314yes—
102C2×C3×#41C22×C318yes—
103C3×C4⋊Q8C22×C314yes—
104C3×#127C22×C314yes—
105C3×#126C22×C3132yes—
106C3×#126C22×C3132yes—
107C3×#125C22×C318yes—
108C3×#125C22×C318yes—
109C3×#125C22×C3116yes—
110C3×#125C22×C3132yes—
111C3×#124C22×C318yes—
112C3×#123C22×C3132yes—
113C3×#123C22×C3132yes—
114C3×C22⋊Q8C22×C3116yes—
115C3×C22⋊Q8C22×C3116yes—
116C3×#121C22×C3116yes—
117C3×#121C22×C3116yes—
118C3×#121C22×C3132yes—
119C3×#121C22×C3116yes—
120C3×#121C22×C3132yes—
121C3×C4×Q8C22×C318yes—
122C3×C4×Q8C22×C3116yes—
123C3×C4×D4C22×C3116yes—
124C3×C4×D4C22×C318yes—
125C3×C4×D4C22×C3116yes—
126C3×C4×D4C22×C3132yes—
127C3×C4×D4C22×C3132yes—
128C3×C4×D4C22×C3116yes—
129C3×#117C22×C314yes—
130C3×#117C22×C318yes—
131C3×#117C22×C3116yes—
132C3×#117C22×C3132yes—
133C2×C3×C4⋊C4C22×C318yes—
134C2×C3×C4⋊C4C22×C3116yes—
135C2×C3×C4⋊C4C22×C3116yes—
136C2×C3×C42C22×C314yes—
137C2×C3×C42C22×C318yes—
138C23×C32C2414yes—
139C23×C32C2412yes—
140C23×C32C2411yescharacteristic
141C32×Q8—22no—
142C32×Q8—24no—
143C32×D4—24no—
144C32×D4—28no—
145C32×D4—22no—
146C32×D4—24no—
147C32×D4—48no—
148C32×D4—44no—
149C2×C32×C4—28no—
150C2×C32×C4—24no—
151C2×C32×C4—44no—
152C2×C32×C4—48no—
153C2×C32×C4—42no—
154C2×C32×C4—44no—
155C2×C32×C4#4114yes—
156C2×C32×C4C2412yes—
157C2×C32×C4C2414yes—
158C2×C32×C4C2412yes—
159C2×C32×C4C2414yes—
160C2×C32×C4C2412yes—
161C2×C32×C4C2418yes—
162C2×C32×C4C2411yescharacteristic
163C2×C32×C4C2411yescharacteristic
164#553C2×C3214yescomplemented
165#553C2×C3218yescomplemented
166#552C2×C3214yescomplemented
167#541C2×C3212yescomplemented
168#540C2×C3212yescomplemented
169#537C2×C3212yescomplemented
170#532C2×C3211yescharacteristic, complemented
171#529C2×C3214yescomplemented
172C2×#125C2×C3211yescharacteristic, complemented
173#520C2×C3212yescomplemented
174C4×#41C2×C3211yescharacteristic, complemented
175C3×#41—216no—
176C3×#41—232no—
177C2×C3×Q8C23×C314yes—
178C2×C3×Q8C23×C318yes—
179C2×C3×D4C23×C318yes—
180C2×C3×D4C23×C3116yes—
181C2×C3×D4C23×C314yes—
182C2×C3×D4C23×C318yes—
183C2×C3×D4C23×C3132yes—
184C2×C3×D4C23×C3116yes—
185C22×C3×C4C23×C318yes—
186C22×C3×C4C23×C3116yes—
187C22×C3×C4C23×C318yes—
188C22×C3×C4C23×C3116yes—
189C22×C3×C4C23×C3116yes—
190C22×C3×C4C23×C3132yes—
191C22×C3×C4C23×C314yes—
192C3×C4⋊C4C23×C318yes—
193C3×C4⋊C4C23×C3116yes—
194C3×C4⋊C4C23×C3132yes—
195C3×C4⋊C4C23×C3116yes—
196C3×C4⋊C4C23×C3132yes—
197C3×C4⋊C4C23×C3132yes—
198C3×C4⋊C4C23×C3116yes—
199C3×C4⋊C4C23×C3132yes—
200C3×C4⋊C4C23×C3116yes—
201C3×C22⋊C4C23×C3116yes—
202C3×C22⋊C4C23×C318yes—
203C3×C22⋊C4C23×C3116yes—
204C3×C22⋊C4C23×C3132yes—
205C3×C22⋊C4C23×C3132yes—
206C3×C22⋊C4C23×C3116yes—
207C3×C22⋊C4C23×C3132yes—
208C3×C42C23×C314yes—
209C3×C42C23×C318yes—
210C3×C42C23×C3116yes—
211C3×C42C23×C314yes—
212C3×C42C23×C318yes—
213C3×C42C23×C3132yes—
214C22×C32—22no—
215C22×C32—24no—
216C22×C32—22no—
217C22×C32—24no—
218C22×C32—24no—
219C22×C32—24no—
220C22×C32C2×#4112yes—
221C22×C32C2511yescharacteristic
222C32×C4—24no—
223C32×C4—22no—
224C32×C4—22no—
225C32×C4—24no—
226C32×C4—28no—
227C32×C4—22no—
228C32×C4C2×#4112yes—
229C32×C4C2×#4112yes—
230C2×#41C22×C3211yescharacteristic
231C2×#41C22×C3212yes—
232C4⋊Q8C22×C3211yescharacteristic
233#127C22×C3211yescharacteristic
234#126C22×C3218yes—
235#126C22×C3218yes—
236#125C22×C3212yes—
237#125C22×C3212yes—
238#125C22×C3214yes—
239#125C22×C3218yes—
240#124C22×C3212yes—
241#123C22×C3218yes—
242#123C22×C3218yes—
243C22⋊Q8C22×C3214yes—
244C22⋊Q8C22×C3214yes—
245#121C22×C3214yes—
246#121C22×C3214yes—
247#121C22×C3218yes—
248#121C22×C3214yes—
249#121C22×C3218yes—
250C4×Q8C22×C3212yes—
251C4×Q8C22×C3214yes—
252C4×D4C22×C3214yes—
253C4×D4C22×C3212yes—
254C4×D4C22×C3214yes—
255C4×D4C22×C3218yes—
256C4×D4C22×C3218yes—
257C4×D4C22×C3214yes—
258#117C22×C3211yescharacteristic
259#117C22×C3212yes—
260#117C22×C3214yes—
261#117C22×C3218yes—
262C2×C4⋊C4C22×C3212yes—
263C2×C4⋊C4C22×C3214yes—
264C2×C4⋊C4C22×C3214yes—
265C2×C42C22×C3211yescharacteristic
266C2×C42C22×C3212yes—
267C23×C3C24×C3116yes—
268C23×C3C24×C318yes—
269C23×C3C24×C314yes—
270C3×Q8—28no—
271C3×Q8—216no—
272C3×D4—216no—
273C3×D4—232no—
274C3×D4—28no—
275C3×D4—216no—
276C3×D4—432no—
277C3×D4—416no—
278C2×C3×C4—232no—
279C2×C3×C4—216no—
280C2×C3×C4—416no—
281C2×C3×C4—432no—
282C2×C3×C4—48no—
283C2×C3×C4—416no—
284C2×C3×C4C3×#41116yes—
285C2×C3×C4C24×C318yes—
286C2×C3×C4C24×C3116yes—
287C2×C3×C4C24×C318yes—
288C2×C3×C4C24×C3116yes—
289C2×C3×C4C24×C318yes—
290C2×C3×C4C24×C3132yes—
291C2×C3×C4C24×C314yes—
292C2×C3×C4C24×C314yes—
293C2×C32—22no—
294C2×C32—44no—
295C2×C32—42no—
296C2×C32C22×#4111yescharacteristic
297C2×C32#58512yes—
298#41—24no—
299#41—28no—
300C2×Q8C23×C3211yescharacteristic
301C2×Q8C23×C3212yes—
302C2×D4C23×C3212yes—
303C2×D4C23×C3214yes—
304C2×D4C23×C3211yescharacteristic
305C2×D4C23×C3212yes—
306C2×D4C23×C3218yes—
307C2×D4C23×C3214yes—
308C22×C4C23×C3212yes—
309C22×C4C23×C3214yes—
310C22×C4C23×C3212yes—
311C22×C4C23×C3214yes—
312C22×C4C23×C3214yes—
313C22×C4C23×C3218yes—
314C22×C4C23×C3211yescharacteristic
315C4⋊C4C23×C3212yes—
316C4⋊C4C23×C3214yes—
317C4⋊C4C23×C3218yes—
318C4⋊C4C23×C3214yes—
319C4⋊C4C23×C3218yes—
320C4⋊C4C23×C3218yes—
321C4⋊C4C23×C3214yes—
322C4⋊C4C23×C3218yes—
323C4⋊C4C23×C3214yes—
324C22⋊C4C23×C3214yes—
325C22⋊C4C23×C3212yes—
326C22⋊C4C23×C3214yes—
327C22⋊C4C23×C3218yes—
328C22⋊C4C23×C3218yes—
329C22⋊C4C23×C3214yes—
330C22⋊C4C23×C3218yes—
331C42C23×C3211yescharacteristic
332C42C23×C3212yes—
333C42C23×C3214yes—
334C42C23×C3211yescharacteristic
335C42C23×C3212yes—
336C42C23×C3218yes—
337C22×C3—28no—
338C22×C3—216no—
339C22×C3—28no—
340C22×C3—216no—
341C22×C3—216no—
342C22×C3—216no—
343C22×C3C2×C3×#4118yes—
344C22×C3C25×C314yes—
345C3×C4—216no—
346C3×C4—28no—
347C3×C4—28no—
348C3×C4—216no—
349C3×C4—232no—
350C3×C4—28no—
351C3×C4C2×C3×#4118yes—
352C3×C4C2×C3×#4118yes—
353C32#351211yescharacteristic, direct product factor, Sylow
354C23C24×C3214yes—
355C23C24×C3212yes—
356C23C24×C3211yescharacteristic
357Q8—22no—
358Q8—24no—
359D4—24no—
360D4—28no—
361D4—22no—
362D4—24no—
363D4—48no—
364D4—44no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4C32×#4114yes—
372C2×C4C24×C3212yes—
373C2×C4C24×C3214yes—
374C2×C4C24×C3212yes—
375C2×C4C24×C3214yes—
376C2×C4C24×C3212yes—
377C2×C4C24×C3218yes—
378C2×C4C24×C3211yescharacteristic
379C2×C4C24×C3211yescharacteristic
380C2×C3—28no—
381C2×C3—416no—
382C2×C3—48no—
383C2×C3C22×C3×#4114yes—
384C2×C3C3×#58518yes—
385C22—22no—
386C22—24no—
387C22—22no—
388C22—24no—
389C22—24no—
390C22—24no—
391C22C2×C32×#4112yes—
392C22C25×C3211yescharacteristic
393C4—24no—
394C4—22no—
395C4—22no—
396C4—24no—
397C4—28no—
398C4—22no—
399C4C2×C32×#4112yes—
400C4C2×C32×#4112yes—
401C3C3×#351214yesdirect product factor
402C2—22no—
403C2—44no—
404C2—42no—
405C2C22×C32×#4111yescharacteristic
406C2C32×#58512yes—
subgroupG/H
center, socleC22×C32C25
commutator, Frattini, cosocleC22C25×C32
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C32 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×#553C2semidirect
2. C32×#553C2semidirect
3. C32×#552C2semidirect
4. C32×#541C2semidirect
5. C32×#540C2semidirect
6. C32×#537C2semidirect
7. C32×#532C2semidirect
8. C32×#529C2semidirect
9. C2×C32×#125C2semidirect
10. C32×#520C2semidirect
11. C32×C4×#41C2semidirect
12. C3×#3512C3direct (N × H)
50. C3×#553C2×C3semidirect
51. C3×#553C2×C3semidirect
52. C3×#552C2×C3semidirect
53. C3×#541C2×C3semidirect
54. C3×#540C2×C3semidirect
55. C3×#537C2×C3semidirect
56. C3×#532C2×C3semidirect
57. C3×#529C2×C3semidirect
58. C2×C3×#125C2×C3semidirect
59. C3×#520C2×C3semidirect
60. C3×C4×#41C2×C3semidirect
100. #3512C32direct (N × H)
164. #553C2×C32semidirect
165. #553C2×C32semidirect
166. #552C2×C32semidirect
167. #541C2×C32semidirect
168. #540C2×C32semidirect
169. #537C2×C32semidirect
170. #532C2×C32semidirect
171. #529C2×C32semidirect
172. C2×#125C2×C32semidirect
173. #520C2×C32semidirect
174. C4×#41C2×C32semidirect
353. C32#3512direct (N × H)
401. C3C3×#3512direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#5532
C32×#5522
C32×#5412
C32×#5402
C32×#5372
C32×#5322
C32×#5292
C2×C32×#1252
C32×#5202
C32×C4×#412
C3×#35123