Finite Group Databasebeta

← C32×#3523 #11770572

Subgroups — C32×#3523 #11770572

Order 1152.

2976 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#557C213yesmaximal, complemented
2C32×#557C211yesmaximal, characteristic, complemented
3C32×#556C213yesmaximal, complemented
4C32×#552C216yesmaximal, complemented
5C32×D4×Q8C212yesmaximal, complemented
6C32×D4×Q8C211yesmaximal, characteristic, complemented
7C32×#541C216yesmaximal, complemented
8C32×#533C213yesmaximal, complemented
9C32×#529C213yesmaximal
10C32×#519C212yesmaximal, complemented
11C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
12C3×#3523C314yesmaximal, direct product factor
13C22×C32×Q8C2211yescharacteristic
14C22×C32×Q8C2212yes—
15C32×C4⋊Q8C2213yes—
16C32×C4⋊Q8C2213yes—
17C32×C4⋊Q8C2216yes—
18C32×#126C22112yes—
19C32×#126C2216yes—
20C32×#125C22112yes—
21C32×#125C2216yes—
22C32×#125C2213yes—
23C32×#125C2216yes—
24C32×#124C2213yes—
25C32×#123C2216yes—
26C32×C22⋊Q8C2216yes—
27C32×C22⋊Q8C2216yes—
28C32×C22⋊Q8C2216yes—
29C32×C22⋊Q8C2213yes—
30C32×#121C2211yescharacteristic
31C32×C4×Q8C2216yes—
32C32×C4×Q8C2213yes—
33C32×C4×Q8C2216yes—
34C32×C4×Q8C2213yes—
35C32×C4×Q8C2211yescharacteristic
36C32×C4×Q8C2212yes—
37C32×C4×Q8C2211yescharacteristic
38C32×C4×Q8C2213yes—
39C32×C4×Q8C2216yes—
40C32×C4×D4C2216yes—
41C32×C4×D4C2213yes—
42C32×#117C2216yes—
43C32×#117C2216yes—
44C32×#117C2216yes—
45C2×C32×C4⋊C4C2213yes—
46C2×C32×C42C2213yes—
47C3×#557C2×C314yescomplemented
48C3×#557C2×C3112yescomplemented
49C3×#556C2×C3112yescomplemented
50C3×#552C2×C3124yescomplemented
51C3×D4×Q8C2×C318yescomplemented
52C3×D4×Q8C2×C314yescomplemented
53C3×#541C2×C3124yescomplemented
54C3×#533C2×C3112yescomplemented
55C3×#529C2×C3112yes—
56C3×#519C2×C318yescomplemented
57C2×C3×C4×Q8C2×C314yescomplemented
58C2×C32×Q8—24no—
59C2×C32×Q8—28no—
60C2×C32×Q8C2313yes—
61C2×C32×Q8C2316yes—
62C2×C32×Q8C2313yes—
63C2×C32×Q8C2311yescharacteristic
64C2×C32×D4C2312yes—
65C2×C32×D4C2311yescharacteristic
66C22×C32×C4C2313yes—
67C22×C32×C4C2316yes—
68C22×C32×C4C2311yescharacteristic
69C22×C32×C4C2313yes—
70C32×C4⋊C4C2313yes—
71C32×C4⋊C4C2313yes—
72C32×C4⋊C4C2316yes—
73C32×C4⋊C4C2316yes—
74C32×C4⋊C4C2316yes—
75C32×C4⋊C4C2316yes—
76C32×C4⋊C4C2316yes—
77C32×C4⋊C4C2316yes—
78C32×C4⋊C4C2313yes—
79C32×C4⋊C4C2316yes—
80C32×C4⋊C4C2311yescharacteristic
81C32×C4⋊C4C23112yes—
82C32×C4⋊C4C2316yes—
83C32×C22⋊C4C2312yes—
84C32×C22⋊C4C2316yes—
85C32×C22⋊C4C2316yes—
86C32×C22⋊C4C2316yes—
87C32×C22⋊C4C2316yes—
88C32×C42C2313yes—
89C32×C42C2313yes—
90C32×C42C2313yes—
91C32×C42C2313yes—
92C32×C42C23112yes—
93C32×C42C2316yes—
94#3523C3211yescharacteristic, direct product factor, Sylow
95C22×C3×Q8C22×C314yes—
96C22×C3×Q8C22×C318yes—
97C3×C4⋊Q8C22×C3112yes—
98C3×C4⋊Q8C22×C3112yes—
99C3×C4⋊Q8C22×C3124yes—
100C3×#126C22×C3124yes—
101C3×#126C22×C3148yes—
102C3×#125C22×C3112yes—
103C3×#125C22×C3124yes—
104C3×#125C22×C3148yes—
105C3×#125C22×C3124yes—
106C3×#124C22×C3112yes—
107C3×#123C22×C3124yes—
108C3×C22⋊Q8C22×C3112yes—
109C3×C22⋊Q8C22×C3124yes—
110C3×C22⋊Q8C22×C3124yes—
111C3×C22⋊Q8C22×C3124yes—
112C3×#121C22×C314yes—
113C3×C4×Q8C22×C314yes—
114C3×C4×Q8C22×C3112yes—
115C3×C4×Q8C22×C314yes—
116C3×C4×Q8C22×C3112yes—
117C3×C4×Q8C22×C3124yes—
118C3×C4×Q8C22×C318yes—
119C3×C4×Q8C22×C3124yes—
120C3×C4×Q8C22×C3112yes—
121C3×C4×Q8C22×C3124yes—
122C3×C4×D4C22×C3124yes—
123C3×C4×D4C22×C3112yes—
124C3×#117C22×C3124yes—
125C3×#117C22×C3124yes—
126C3×#117C22×C3124yes—
127C2×C3×C4⋊C4C22×C3112yes—
128C2×C3×C42C22×C3112yes—
129C23×C32C2412yes—
130C23×C32C2411yescharacteristic
131C32×Q8—26no—
132C32×Q8—212no—
133C32×Q8—26no—
134C32×Q8#4114yes—
135C32×D4—24no—
136C32×D4—22no—
137C2×C32×C4—26no—
138C2×C32×C4—26no—
139C2×C32×C4—212no—
140C2×C32×C4—22no—
141C2×C32×C4C2411yescharacteristic
142C2×C32×C4C2413yes—
143C2×C32×C4C2416yes—
144C2×C32×C4C2412yes—
145C2×C32×C4C2413yes—
146C2×C32×C4C2413yes—
147C2×C32×C4C2416yes—
148C2×C32×C4C2413yes—
149C2×C32×C4C2411yescharacteristic
150#557C2×C3211yescharacteristic, complemented
151#557C2×C3213yescomplemented
152#556C2×C3213yescomplemented
153#552C2×C3216yescomplemented
154D4×Q8C2×C3212yescomplemented
155D4×Q8C2×C3211yescharacteristic, complemented
156#541C2×C3216yescomplemented
157#533C2×C3213yescomplemented
158#529C2×C3213yes—
159#519C2×C3212yescomplemented
160C2×C4×Q8C2×C3211yescharacteristic, complemented
161C2×C3×Q8—216no—
162C2×C3×Q8—232no—
163C2×C3×Q8C23×C3112yes—
164C2×C3×Q8C23×C3112yes—
165C2×C3×Q8C23×C3124yes—
166C2×C3×Q8C23×C314yes—
167C2×C3×D4C23×C318yes—
168C2×C3×D4C23×C314yes—
169C22×C3×C4C23×C314yes—
170C22×C3×C4C23×C3112yes—
171C22×C3×C4C23×C3112yes—
172C22×C3×C4C23×C3124yes—
173C3×C4⋊C4C23×C3112yes—
174C3×C4⋊C4C23×C3112yes—
175C3×C4⋊C4C23×C3124yes—
176C3×C4⋊C4C23×C3124yes—
177C3×C4⋊C4C23×C3124yes—
178C3×C4⋊C4C23×C3124yes—
179C3×C4⋊C4C23×C3124yes—
180C3×C4⋊C4C23×C3124yes—
181C3×C4⋊C4C23×C3112yes—
182C3×C4⋊C4C23×C3124yes—
183C3×C4⋊C4C23×C314yes—
184C3×C4⋊C4C23×C3148yes—
185C3×C4⋊C4C23×C3124yes—
186C3×C22⋊C4C23×C318yes—
187C3×C22⋊C4C23×C3124yes—
188C3×C22⋊C4C23×C3124yes—
189C3×C22⋊C4C23×C3124yes—
190C3×C22⋊C4C23×C3124yes—
191C3×C42C23×C3112yes—
192C3×C42C23×C3112yes—
193C3×C42C23×C3112yes—
194C3×C42C23×C3112yes—
195C3×C42C23×C3148yes—
196C3×C42C23×C3124yes—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C322−1+412yes—
203C22×C32C2511yescharacteristic
204C32×C4—23no—
205C32×C4—26no—
206C32×C4—23no—
207C32×C4—23no—
208C32×C4—21no—
209C32×C4—22no—
210C32×C4—26no—
211C32×C4C2×#4116yes—
212C32×C42−1+412yes—
213C22×Q8C22×C3211yescharacteristic
214C22×Q8C22×C3212yes—
215C4⋊Q8C22×C3213yes—
216C4⋊Q8C22×C3213yes—
217C4⋊Q8C22×C3216yes—
218#126C22×C3216yes—
219#126C22×C32112yes—
220#125C22×C3213yes—
221#125C22×C3216yes—
222#125C22×C32112yes—
223#125C22×C3216yes—
224#124C22×C3213yes—
225#123C22×C3216yes—
226C22⋊Q8C22×C3213yes—
227C22⋊Q8C22×C3216yes—
228C22⋊Q8C22×C3216yes—
229C22⋊Q8C22×C3216yes—
230#121C22×C3211yescharacteristic
231C4×Q8C22×C3211yescharacteristic
232C4×Q8C22×C3213yes—
233C4×Q8C22×C3211yescharacteristic
234C4×Q8C22×C3213yes—
235C4×Q8C22×C3216yes—
236C4×Q8C22×C3212yes—
237C4×Q8C22×C3216yes—
238C4×Q8C22×C3213yes—
239C4×Q8C22×C3216yes—
240C4×D4C22×C3216yes—
241C4×D4C22×C3213yes—
242#117C22×C3216yes—
243#117C22×C3216yes—
244#117C22×C3216yes—
245C2×C4⋊C4C22×C3213yes—
246C2×C42C22×C3213yes—
247C23×C3C24×C318yes—
248C23×C3C24×C314yes—
249C3×Q8—224no—
250C3×Q8—248no—
251C3×Q8—224no—
252C3×Q8C3×#41116yes—
253C3×D4—216no—
254C3×D4—28no—
255C2×C3×C4—224no—
256C2×C3×C4—224no—
257C2×C3×C4—248no—
258C2×C3×C4—28no—
259C2×C3×C4C24×C314yes—
260C2×C3×C4C24×C3112yes—
261C2×C3×C4C24×C3124yes—
262C2×C3×C4C24×C318yes—
263C2×C3×C4C24×C3112yes—
264C2×C3×C4C24×C3112yes—
265C2×C3×C4C24×C3124yes—
266C2×C3×C4C24×C3112yes—
267C2×C3×C4C24×C314yes—
268C2×C32—22no—
269C2×C32—42no—
270C2×C32C22×#4111yescharacteristic
271C2×C32C2×2−1+411yescharacteristic
272C2×C32C2×2−1+411yescharacteristic
273C2×Q8—24no—
274C2×Q8—28no—
275C2×Q8C23×C3213yes—
276C2×Q8C23×C3213yes—
277C2×Q8C23×C3216yes—
278C2×Q8C23×C3211yescharacteristic
279C2×D4C23×C3212yes—
280C2×D4C23×C3211yescharacteristic
281C22×C4C23×C3211yescharacteristic
282C22×C4C23×C3213yes—
283C22×C4C23×C3213yes—
284C22×C4C23×C3216yes—
285C4⋊C4C23×C3213yes—
286C4⋊C4C23×C3213yes—
287C4⋊C4C23×C3216yes—
288C4⋊C4C23×C3216yes—
289C4⋊C4C23×C3216yes—
290C4⋊C4C23×C3216yes—
291C4⋊C4C23×C3216yes—
292C4⋊C4C23×C3216yes—
293C4⋊C4C23×C3213yes—
294C4⋊C4C23×C3216yes—
295C4⋊C4C23×C3211yescharacteristic
296C4⋊C4C23×C32112yes—
297C4⋊C4C23×C3216yes—
298C22⋊C4C23×C3212yes—
299C22⋊C4C23×C3216yes—
300C22⋊C4C23×C3216yes—
301C22⋊C4C23×C3216yes—
302C22⋊C4C23×C3216yes—
303C42C23×C3213yes—
304C42C23×C3213yes—
305C42C23×C3213yes—
306C42C23×C3213yes—
307C42C23×C32112yes—
308C42C23×C3216yes—
309C22×C3—24no—
310C22×C3—24no—
311C22×C3—28no—
312C22×C3—28no—
313C22×C3—28no—
314C22×C3C3×2−1+418yes—
315C22×C3C25×C314yes—
316C3×C4—212no—
317C3×C4—224no—
318C3×C4—212no—
319C3×C4—212no—
320C3×C4—24no—
321C3×C4—28no—
322C3×C4—224no—
323C3×C4C2×C3×#41124yes—
324C3×C4C3×2−1+418yes—
325C32#352311yescharacteristic, direct product factor, Sylow
326C23C24×C3212yes—
327C23C24×C3211yescharacteristic
328Q8—26no—
329Q8—212no—
330Q8—26no—
331Q8C32×#4114yes—
332D4—24no—
333D4—22no—
334C2×C4—26no—
335C2×C4—26no—
336C2×C4—212no—
337C2×C4—22no—
338C2×C4C24×C3211yescharacteristic
339C2×C4C24×C3213yes—
340C2×C4C24×C3216yes—
341C2×C4C24×C3212yes—
342C2×C4C24×C3213yes—
343C2×C4C24×C3213yes—
344C2×C4C24×C3216yes—
345C2×C4C24×C3213yes—
346C2×C4C24×C3211yescharacteristic
347C2×C3—28no—
348C2×C3—48no—
349C2×C3C22×C3×#4114yes—
350C2×C3C2×C3×2−1+414yes—
351C2×C3C2×C3×2−1+414yes—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22C32×2−1+412yes—
358C22C25×C3211yescharacteristic
359C4—23no—
360C4—26no—
361C4—23no—
362C4—23no—
363C4—21no—
364C4—22no—
365C4—26no—
366C4C2×C32×#4116yes—
367C4C32×2−1+412yes—
368C3C3×#352314yesdirect product factor
369C2—22no—
370C2—42no—
371C2C22×C32×#4111yescharacteristic
372C2C2×C32×2−1+411yescharacteristic
373C2C2×C32×2−1+411yescharacteristic
subgroupG/H
center, socleC22×C32C25
commutator, Frattini, cosocleC22C25×C32
prime radicalC32×#121C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C32 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×#557C2semidirect
2. C32×#557C2semidirect
3. C32×#556C2semidirect
4. C32×#552C2semidirect
5. C32×D4×Q8C2semidirect
6. C32×D4×Q8C2semidirect
7. C32×#541C2semidirect
8. C32×#533C2semidirect
10. C32×#519C2semidirect
11. C2×C32×C4×Q8C2semidirect
12. C3×#3523C3direct (N × H)
47. C3×#557C2×C3semidirect
48. C3×#557C2×C3semidirect
49. C3×#556C2×C3semidirect
50. C3×#552C2×C3semidirect
51. C3×D4×Q8C2×C3semidirect
52. C3×D4×Q8C2×C3semidirect
53. C3×#541C2×C3semidirect
54. C3×#533C2×C3semidirect
56. C3×#519C2×C3semidirect
57. C2×C3×C4×Q8C2×C3semidirect
94. #3523C32direct (N × H)
150. #557C2×C32semidirect
151. #557C2×C32semidirect
152. #556C2×C32semidirect
153. #552C2×C32semidirect
154. D4×Q8C2×C32semidirect
155. D4×Q8C2×C32semidirect
156. #541C2×C32semidirect
157. #533C2×C32semidirect
159. #519C2×C32semidirect
160. C2×C4×Q8C2×C32semidirect
325. C32#3523direct (N × H)
368. C3C3×#3523direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#5572
C32×#5562
C32×#5522
C32×D4×Q82
C32×#5412
C32×#5332
C32×#5292
C32×#5192
C2×C32×C4×Q82
C3×#35233