Finite Group Databasebeta

← C32×#3522 #11770571

Subgroups — C32×#3522 #11770571

Order 1152.

3312 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#559C213yesmaximal, complemented
2C32×Q82C211yesmaximal, characteristic, complemented
3C32×#554C213yesmaximal, complemented
4C32×#550C211yesmaximal, characteristic, complemented
5C32×#548C212yesmaximal, complemented
6C32×#547C216yesmaximal, complemented
7C32×#542C216yesmaximal, complemented
8C32×#532C213yesmaximal, complemented
9C32×#529C213yesmaximal, complemented
10C32×#520C212yesmaximal, complemented
11C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
12C3×#3522C314yesmaximal, direct product factor
13C2×C32×#41C2212yes—
14C2×C32×#41C2216yes—
15C22×C32×Q8C2211yescharacteristic
16C32×C4⋊Q8C2213yes—
17C32×C4⋊Q8C2216yes—
18C32×#127C2213yes—
19C32×#126C2216yes—
20C32×#125C2213yes—
21C32×#124C2213yes—
22C32×#124C2216yes—
23C32×#124C2216yes—
24C32×#123C22112yes—
25C32×#123C2216yes—
26C32×C22⋊Q8C22112yes—
27C32×C22⋊Q8C2216yes—
28C32×C22⋊Q8C2211yescharacteristic
29C32×#121C2213yes—
30C32×#121C2216yes—
31C32×#121C2216yes—
32C32×C4×Q8C2213yes—
33C32×C4×Q8C2213yes—
34C32×C4×Q8C2211yescharacteristic
35C32×C4×Q8C2212yes—
36C32×C4×Q8C2211yescharacteristic
37C32×C4×Q8C2213yes—
38C32×C4×D4C2216yes—
39C32×C4×D4C2213yes—
40C32×C4×D4C2216yes—
41C32×C4×D4C2216yes—
42C32×C4×D4C2216yes—
43C32×#117C2216yes—
44C2×C32×C4⋊C4C2216yes—
45C2×C32×C4⋊C4C2213yes—
46C2×C32×C42C2213yes—
47C3×#559C2×C3112yescomplemented
48C3×Q82C2×C314yescomplemented
49C3×#554C2×C3112yescomplemented
50C3×#550C2×C314yescomplemented
51C3×#548C2×C318yescomplemented
52C3×#547C2×C3124yescomplemented
53C3×#542C2×C3124yescomplemented
54C3×#532C2×C3112yescomplemented
55C3×#529C2×C3112yescomplemented
56C3×#520C2×C318yescomplemented
57C2×C3×C4×Q8C2×C314yescomplemented
58C32×#41—28no—
59C32×#41—412no—
60C2×C32×Q8—24no—
61C2×C32×Q8C2313yes—
62C2×C32×Q8C2311yescharacteristic
63C2×C32×Q8C2316yes—
64C2×C32×Q8C2311yescharacteristic
65C2×C32×D4C2316yes—
66C2×C32×D4C2316yes—
67C2×C32×D4C2313yes—
68C2×C32×D4C2316yes—
69C22×C32×C4C2313yes—
70C22×C32×C4C2313yes—
71C22×C32×C4C2316yes—
72C22×C32×C4C2311yescharacteristic
73C22×C32×C4C23112yes—
74C32×C4⋊C4C2313yes—
75C32×C4⋊C4C2316yes—
76C32×C4⋊C4C2316yes—
77C32×C4⋊C4C2313yes—
78C32×C4⋊C4C2316yes—
79C32×C4⋊C4C2312yes—
80C32×C4⋊C4C2311yescharacteristic
81C32×C4⋊C4C2313yes—
82C32×C4⋊C4C2316yes—
83C32×C4⋊C4C2316yes—
84C32×C22⋊C4C2316yes—
85C32×C22⋊C4C2316yes—
86C32×C22⋊C4C2312yes—
87C32×C22⋊C4C2316yes—
88C32×C22⋊C4C23112yes—
89C32×C22⋊C4C2316yes—
90C32×C42C2313yes—
91C32×C42C2313yes—
92C32×C42C2313yes—
93C32×C42C2313yes—
94C32×C42C2316yes—
95#3522C3211yescharacteristic, direct product factor, Sylow
96C2×C3×#41C22×C318yes—
97C2×C3×#41C22×C3124yes—
98C22×C3×Q8C22×C314yes—
99C3×C4⋊Q8C22×C3112yes—
100C3×C4⋊Q8C22×C3124yes—
101C3×#127C22×C3112yes—
102C3×#126C22×C3124yes—
103C3×#125C22×C3112yes—
104C3×#124C22×C3112yes—
105C3×#124C22×C3124yes—
106C3×#124C22×C3124yes—
107C3×#123C22×C3148yes—
108C3×#123C22×C3124yes—
109C3×C22⋊Q8C22×C314yes—
110C3×C22⋊Q8C22×C3148yes—
111C3×C22⋊Q8C22×C3124yes—
112C3×#121C22×C3112yes—
113C3×#121C22×C3124yes—
114C3×#121C22×C3124yes—
115C3×C4×Q8C22×C3112yes—
116C3×C4×Q8C22×C314yes—
117C3×C4×Q8C22×C318yes—
118C3×C4×Q8C22×C3112yes—
119C3×C4×Q8C22×C314yes—
120C3×C4×Q8C22×C3112yes—
121C3×C4×D4C22×C3124yes—
122C3×C4×D4C22×C3124yes—
123C3×C4×D4C22×C3124yes—
124C3×C4×D4C22×C3124yes—
125C3×C4×D4C22×C3112yes—
126C3×#117C22×C3124yes—
127C2×C3×C4⋊C4C22×C3112yes—
128C2×C3×C4⋊C4C22×C3124yes—
129C2×C3×C42C22×C3112yes—
130C23×C32C2416yes—
131C23×C32C2411yescharacteristic
132C32×Q8—26no—
133C32×Q8—22no—
134C32×Q8—46no—
135C32×Q8#4114yescomplemented
136C32×D4—212no—
137C32×D4—212no—
138C32×D4—26no—
139C32×D4—46no—
140C2×C32×C4—26no—
141C2×C32×C4—26no—
142C2×C32×C4—212no—
143C2×C32×C4—22no—
144C2×C32×C4—412no—
145C2×C32×C4C2411yescharacteristic
146C2×C32×C4C2412yes—
147C2×C32×C4C2413yes—
148C2×C32×C4C2413yes—
149C2×C32×C4C2413yes—
150C2×C32×C4C2416yes—
151C2×C32×C4C2412yes—
152C2×C32×C4C2413yes—
153C2×C32×C4C2411yescharacteristic
154#559C2×C3213yescomplemented
155Q82C2×C3211yescharacteristic, complemented
156#554C2×C3213yescomplemented
157#550C2×C3211yescharacteristic, complemented
158#548C2×C3212yescomplemented
159#547C2×C3216yescomplemented
160#542C2×C3216yescomplemented
161#532C2×C3213yescomplemented
162#529C2×C3213yescomplemented
163#520C2×C3212yescomplemented
164C2×C4×Q8C2×C3211yescharacteristic, complemented
165C3×#41—232no—
166C3×#41—448no—
167C2×C3×Q8—216no—
168C2×C3×Q8C23×C314yes—
169C2×C3×Q8C23×C3112yes—
170C2×C3×Q8C23×C3124yes—
171C2×C3×Q8C23×C314yes—
172C2×C3×D4C23×C3112yes—
173C2×C3×D4C23×C3124yes—
174C2×C3×D4C23×C3124yes—
175C2×C3×D4C23×C3124yes—
176C22×C3×C4C23×C314yes—
177C22×C3×C4C23×C3112yes—
178C22×C3×C4C23×C3112yes—
179C22×C3×C4C23×C3124yes—
180C22×C3×C4C23×C3148yes—
181C3×C4⋊C4C23×C3112yes—
182C3×C4⋊C4C23×C3124yes—
183C3×C4⋊C4C23×C3124yes—
184C3×C4⋊C4C23×C3112yes—
185C3×C4⋊C4C23×C3124yes—
186C3×C4⋊C4C23×C318yes—
187C3×C4⋊C4C23×C314yes—
188C3×C4⋊C4C23×C3112yes—
189C3×C4⋊C4C23×C3124yes—
190C3×C4⋊C4C23×C3124yes—
191C3×C22⋊C4C23×C3124yes—
192C3×C22⋊C4C23×C3124yes—
193C3×C22⋊C4C23×C318yes—
194C3×C22⋊C4C23×C3124yes—
195C3×C22⋊C4C23×C3148yes—
196C3×C22⋊C4C23×C3124yes—
197C3×C42C23×C3112yes—
198C3×C42C23×C3112yes—
199C3×C42C23×C3112yes—
200C3×C42C23×C3112yes—
201C3×C42C23×C3124yes—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—26no—
205C22×C32—26no—
206C22×C32—26no—
207C22×C32H(5,𝔽2)12yes—
208C22×C32C2511yescharacteristic
209C32×C4—22no—
210C32×C4—23no—
211C32×C4—23no—
212C32×C4—23no—
213C32×C4—26no—
214C32×C4—21no—
215C32×C4—22no—
216C32×C4C2×#4116yes—
217C32×C42−1+412yes—
218C2×#41C22×C3212yes—
219C2×#41C22×C3216yes—
220C22×Q8C22×C3211yescharacteristic
221C4⋊Q8C22×C3213yes—
222C4⋊Q8C22×C3216yes—
223#127C22×C3213yes—
224#126C22×C3216yes—
225#125C22×C3213yes—
226#124C22×C3213yes—
227#124C22×C3216yes—
228#124C22×C3216yes—
229#123C22×C3216yes—
230#123C22×C32112yes—
231C22⋊Q8C22×C3211yescharacteristic
232C22⋊Q8C22×C32112yes—
233C22⋊Q8C22×C3216yes—
234#121C22×C3213yes—
235#121C22×C3216yes—
236#121C22×C3216yes—
237C4×Q8C22×C3211yescharacteristic
238C4×Q8C22×C3213yes—
239C4×Q8C22×C3211yescharacteristic
240C4×Q8C22×C3213yes—
241C4×Q8C22×C3212yes—
242C4×Q8C22×C3213yes—
243C4×D4C22×C3216yes—
244C4×D4C22×C3216yes—
245C4×D4C22×C3216yes—
246C4×D4C22×C3213yes—
247C4×D4C22×C3216yes—
248#117C22×C3216yes—
249C2×C4⋊C4C22×C3213yes—
250C2×C4⋊C4C22×C3216yes—
251C2×C42C22×C3213yes—
252C23×C3C24×C3124yes—
253C23×C3C24×C314yes—
254C3×Q8—224no—
255C3×Q8—28no—
256C3×Q8—424no—
257C3×Q8C3×#41116yescomplemented
258C3×D4—248no—
259C3×D4—248no—
260C3×D4—224no—
261C3×D4—424no—
262C2×C3×C4—224no—
263C2×C3×C4—224no—
264C2×C3×C4—248no—
265C2×C3×C4—28no—
266C2×C3×C4—448no—
267C2×C3×C4C24×C314yes—
268C2×C3×C4C24×C318yes—
269C2×C3×C4C24×C3112yes—
270C2×C3×C4C24×C3112yes—
271C2×C3×C4C24×C3112yes—
272C2×C3×C4C24×C3124yes—
273C2×C3×C4C24×C318yes—
274C2×C3×C4C24×C3112yes—
275C2×C3×C4C24×C314yes—
276C2×C32—22no—
277C2×C32—46no—
278C2×C32C22×#4111yescharacteristic
279C2×C32C2×H(5,𝔽2)11yescharacteristic
280C2×C32C2×2−1+411yescharacteristic
281#41—28no—
282#41—412no—
283C2×Q8—24no—
284C2×Q8C23×C3211yescharacteristic
285C2×Q8C23×C3213yes—
286C2×Q8C23×C3216yes—
287C2×Q8C23×C3211yescharacteristic
288C2×D4C23×C3213yes—
289C2×D4C23×C3216yes—
290C2×D4C23×C3216yes—
291C2×D4C23×C3216yes—
292C22×C4C23×C3211yescharacteristic
293C22×C4C23×C3213yes—
294C22×C4C23×C3213yes—
295C22×C4C23×C3216yes—
296C22×C4C23×C32112yes—
297C4⋊C4C23×C3213yes—
298C4⋊C4C23×C3216yes—
299C4⋊C4C23×C3216yes—
300C4⋊C4C23×C3213yes—
301C4⋊C4C23×C3216yes—
302C4⋊C4C23×C3212yes—
303C4⋊C4C23×C3211yescharacteristic
304C4⋊C4C23×C3213yes—
305C4⋊C4C23×C3216yes—
306C4⋊C4C23×C3216yes—
307C22⋊C4C23×C3216yes—
308C22⋊C4C23×C3216yes—
309C22⋊C4C23×C3212yes—
310C22⋊C4C23×C3216yes—
311C22⋊C4C23×C32112yes—
312C22⋊C4C23×C3216yes—
313C42C23×C3213yes—
314C42C23×C3213yes—
315C42C23×C3213yes—
316C42C23×C3213yes—
317C42C23×C3216yes—
318C22×C3—24no—
319C22×C3—24no—
320C22×C3—224no—
321C22×C3—224no—
322C22×C3—224no—
323C22×C3C3×H(5,𝔽2)18yes—
324C22×C3C25×C314yes—
325C3×C4—28no—
326C3×C4—212no—
327C3×C4—212no—
328C3×C4—212no—
329C3×C4—224no—
330C3×C4—24no—
331C3×C4—28no—
332C3×C4C2×C3×#41124yes—
333C3×C4C3×2−1+418yes—
334C32#352211yescharacteristic, direct product factor, Sylow
335C23C24×C3216yes—
336C23C24×C3211yescharacteristic
337Q8—26no—
338Q8—22no—
339Q8—46no—
340Q8C32×#4114yescomplemented
341D4—212no—
342D4—212no—
343D4—26no—
344D4—46no—
345C2×C4—26no—
346C2×C4—26no—
347C2×C4—212no—
348C2×C4—22no—
349C2×C4—412no—
350C2×C4C24×C3211yescharacteristic
351C2×C4C24×C3212yes—
352C2×C4C24×C3213yes—
353C2×C4C24×C3213yes—
354C2×C4C24×C3213yes—
355C2×C4C24×C3216yes—
356C2×C4C24×C3212yes—
357C2×C4C24×C3213yes—
358C2×C4C24×C3211yescharacteristic
359C2×C3—28no—
360C2×C3—424no—
361C2×C3C22×C3×#4114yes—
362C2×C3C2×C3×H(5,𝔽2)14yes—
363C2×C3C2×C3×2−1+414yes—
364C22—21no—
365C22—21no—
366C22—26no—
367C22—26no—
368C22—26no—
369C22C32×H(5,𝔽2)12yes—
370C22C25×C3211yescharacteristic
371C4—22no—
372C4—23no—
373C4—23no—
374C4—23no—
375C4—26no—
376C4—21no—
377C4—22no—
378C4C2×C32×#4116yes—
379C4C32×2−1+412yes—
380C3C3×#352214yesdirect product factor
381C2—22no—
382C2—46no—
383C2C22×C32×#4111yescharacteristic
384C2C2×C32×H(5,𝔽2)11yescharacteristic
385C2C2×C32×2−1+411yescharacteristic
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×#559C2semidirect
2. C32×Q82C2semidirect
3. C32×#554C2semidirect
4. C32×#550C2semidirect
5. C32×#548C2semidirect
6. C32×#547C2semidirect
7. C32×#542C2semidirect
8. C32×#532C2semidirect
9. C32×#529C2semidirect
10. C32×#520C2semidirect
11. C2×C32×C4×Q8C2semidirect
12. C3×#3522C3direct (N × H)
47. C3×#559C2×C3semidirect
48. C3×Q82C2×C3semidirect
49. C3×#554C2×C3semidirect
50. C3×#550C2×C3semidirect
51. C3×#548C2×C3semidirect
52. C3×#547C2×C3semidirect
53. C3×#542C2×C3semidirect
54. C3×#532C2×C3semidirect
55. C3×#529C2×C3semidirect
56. C3×#520C2×C3semidirect
57. C2×C3×C4×Q8C2×C3semidirect
95. #3522C32direct (N × H)
135. C32×Q8#41semidirect
154. #559C2×C32semidirect
155. Q82C2×C32semidirect
156. #554C2×C32semidirect
157. #550C2×C32semidirect
158. #548C2×C32semidirect
159. #547C2×C32semidirect
160. #542C2×C32semidirect
161. #532C2×C32semidirect
162. #529C2×C32semidirect
163. #520C2×C32semidirect
164. C2×C4×Q8C2×C32semidirect
257. C3×Q8C3×#41semidirect
334. C32#3522direct (N × H)
340. Q8C32×#41semidirect
380. C3C3×#3522direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#5592
C32×Q822
C32×#5542
C32×#5502
C32×#5482
C32×#5472
C32×#5422
C32×#5322
C32×#5292
C32×#5202
C2×C32×C4×Q82
C3×#35223