Finite Group Databasebeta

← C32×#3509 #11770558

Subgroups — C32×#3509 #11770558

Order 1152.

2976 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#557C214yesmaximal, complemented
2C32×#556C214yesmaximal, complemented
3C32×#552C218yesmaximal, complemented
4C32×#541C214yesmaximal, complemented
5C32×#536C212yesmaximal
6C32×#533C211yesmaximal, characteristic, complemented
7C32×#533C214yesmaximal, complemented
8C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
9C32×#519C212yesmaximal, complemented
10C32×C4×#41C211yesmaximal, characteristic
11C3×#3509C314yesmaximal, direct product factor
12C2×C32×#41C2211yescharacteristic
13C22×C32×Q8C2212yes—
14C32×C4⋊Q8C2214yes—
15C32×C4⋊Q8C2214yes—
16C32×C4⋊Q8C2212yes—
17C32×C4⋊Q8C2212yes—
18C32×C4⋊Q8C2212yes—
19C32×#126C22116yes—
20C32×#125C2218yes—
21C32×#125C22116yes—
22C32×#125C2212yes—
23C32×#124C2214yes—
24C32×#123C2218yes—
25C32×C22⋊Q8C2214yes—
26C32×C22⋊Q8C2218yes—
27C32×C22⋊Q8C2218yes—
28C32×C4×Q8C2218yes—
29C32×C4×Q8C2218yes—
30C32×C4×Q8C2214yes—
31C32×C4×Q8C2212yes—
32C32×C4×Q8C2218yes—
33C32×C4×D4C2214yes—
34C32×C4×D4C2212yes—
35C32×#117C2218yes—
36C32×#117C2218yes—
37C32×#117C2211yescharacteristic
38C32×#117C2212yes—
39C32×#117C2212yes—
40C2×C32×C4⋊C4C2214yes—
41C2×C32×C42C2211yescharacteristic
42C2×C32×C42C2212yes—
43C3×#557C2×C3116yescomplemented
44C3×#556C2×C3116yescomplemented
45C3×#552C2×C3132yescomplemented
46C3×#541C2×C3116yescomplemented
47C3×#536C2×C318yes—
48C3×#533C2×C314yescomplemented
49C3×#533C2×C3116yescomplemented
50C2×C3×C4⋊Q8C2×C314yescomplemented
51C3×#519C2×C318yescomplemented
52C3×C4×#41C2×C314yes—
53C32×#41—24no—
54C2×C32×Q8—28no—
55C2×C32×Q8C2314yes—
56C2×C32×Q8C2314yes—
57C2×C32×Q8C2311yescharacteristic
58C2×C32×Q8C2314yes—
59C2×C32×D4C2312yes—
60C2×C32×D4C2311yescharacteristic
61C22×C32×C4C2314yes—
62C22×C32×C4C2312yes—
63C22×C32×C4C2312yes—
64C22×C32×C4C2314yes—
65C22×C32×C4C2311yescharacteristic
66C32×C4⋊C4C2318yes—
67C32×C4⋊C4C2318yes—
68C32×C4⋊C4C2314yes—
69C32×C4⋊C4C2312yes—
70C32×C4⋊C4C2314yes—
71C32×C4⋊C4C2314yes—
72C32×C4⋊C4C23116yes—
73C32×C4⋊C4C23116yes—
74C32×C4⋊C4C2318yes—
75C32×C22⋊C4C2312yes—
76C32×C22⋊C4C2314yes—
77C32×C22⋊C4C2314yes—
78C32×C22⋊C4C23116yes—
79C32×C42C2311yescharacteristic
80C32×C42C2312yes—
81C32×C42C2311yescharacteristic
82C32×C42C2314yes—
83C32×C42C2312yes—
84C32×C42C23116yes—
85C32×C42C2314yes—
86#3509C3211yescharacteristic, direct product factor, Sylow
87C2×C3×#41C22×C314yes—
88C22×C3×Q8C22×C318yes—
89C3×C4⋊Q8C22×C318yes—
90C3×C4⋊Q8C22×C318yes—
91C3×C4⋊Q8C22×C3116yes—
92C3×C4⋊Q8C22×C318yes—
93C3×C4⋊Q8C22×C3116yes—
94C3×#126C22×C3164yes—
95C3×#125C22×C318yes—
96C3×#125C22×C3132yes—
97C3×#125C22×C3164yes—
98C3×#124C22×C3116yes—
99C3×#123C22×C3132yes—
100C3×C22⋊Q8C22×C3132yes—
101C3×C22⋊Q8C22×C3116yes—
102C3×C22⋊Q8C22×C3132yes—
103C3×C4×Q8C22×C318yes—
104C3×C4×Q8C22×C3132yes—
105C3×C4×Q8C22×C3132yes—
106C3×C4×Q8C22×C3132yes—
107C3×C4×Q8C22×C3116yes—
108C3×C4×D4C22×C3116yes—
109C3×C4×D4C22×C318yes—
110C3×#117C22×C314yes—
111C3×#117C22×C318yes—
112C3×#117C22×C318yes—
113C3×#117C22×C3132yes—
114C3×#117C22×C3132yes—
115C2×C3×C4⋊C4C22×C3116yes—
116C2×C3×C42C22×C314yes—
117C2×C3×C42C22×C318yes—
118C23×C32C2412yes—
119C23×C32C2411yescharacteristic
120C32×Q8—28no—
121C32×Q8—28no—
122C32×Q8—22no—
123C32×Q8—28no—
124C32×D4—24no—
125C32×D4—22no—
126C2×C32×C4—28no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—28no—
130C2×C32×C4#4114yes—
131C2×C32×C4C2414yes—
132C2×C32×C4C2418yes—
133C2×C32×C4C2412yes—
134C2×C32×C4C2418yes—
135C2×C32×C4C2412yes—
136C2×C32×C4C2411yescharacteristic
137C2×C32×C4C2411yescharacteristic
138C2×C32×C4C2412yes—
139#557C2×C3214yescomplemented
140#556C2×C3214yescomplemented
141#552C2×C3218yescomplemented
142#541C2×C3214yescomplemented
143#536C2×C3212yes—
144#533C2×C3211yescharacteristic, complemented
145#533C2×C3214yescomplemented
146C2×C4⋊Q8C2×C3211yescharacteristic, complemented
147#519C2×C3212yescomplemented
148C4×#41C2×C3211yescharacteristic
149C3×#41—216no—
150C2×C3×Q8—232no—
151C2×C3×Q8C23×C3116yes—
152C2×C3×Q8C23×C3116yes—
153C2×C3×Q8C23×C3116yes—
154C2×C3×Q8C23×C314yes—
155C2×C3×D4C23×C318yes—
156C2×C3×D4C23×C314yes—
157C22×C3×C4C23×C318yes—
158C22×C3×C4C23×C3116yes—
159C22×C3×C4C23×C318yes—
160C22×C3×C4C23×C3116yes—
161C22×C3×C4C23×C314yes—
162C3×C4⋊C4C23×C3132yes—
163C3×C4⋊C4C23×C3132yes—
164C3×C4⋊C4C23×C3116yes—
165C3×C4⋊C4C23×C318yes—
166C3×C4⋊C4C23×C3116yes—
167C3×C4⋊C4C23×C3116yes—
168C3×C4⋊C4C23×C3164yes—
169C3×C4⋊C4C23×C3164yes—
170C3×C4⋊C4C23×C3132yes—
171C3×C22⋊C4C23×C318yes—
172C3×C22⋊C4C23×C3116yes—
173C3×C22⋊C4C23×C3116yes—
174C3×C22⋊C4C23×C3164yes—
175C3×C42C23×C314yes—
176C3×C42C23×C318yes—
177C3×C42C23×C314yes—
178C3×C42C23×C3116yes—
179C3×C42C23×C318yes—
180C3×C42C23×C3164yes—
181C3×C42C23×C3116yes—
182C22×C32—22no—
183C22×C32—22no—
184C22×C32—24no—
185C22×C32C2×#4112yes—
186C22×C32C2511yescharacteristic
187C32×C4—22no—
188C32×C4—28no—
189C32×C4—24no—
190C32×C4—22no—
191C32×C4—28no—
192C32×C4C2×#4112yes—
193C32×C4C2×#4112yes—
194C32×C42−1+414yes—
195C2×#41C22×C3211yescharacteristic
196C22×Q8C22×C3212yes—
197C4⋊Q8C22×C3212yes—
198C4⋊Q8C22×C3212yes—
199C4⋊Q8C22×C3214yes—
200C4⋊Q8C22×C3212yes—
201C4⋊Q8C22×C3214yes—
202#126C22×C32116yes—
203#125C22×C3212yes—
204#125C22×C3218yes—
205#125C22×C32116yes—
206#124C22×C3214yes—
207#123C22×C3218yes—
208C22⋊Q8C22×C3218yes—
209C22⋊Q8C22×C3214yes—
210C22⋊Q8C22×C3218yes—
211C4×Q8C22×C3212yes—
212C4×Q8C22×C3218yes—
213C4×Q8C22×C3218yes—
214C4×Q8C22×C3218yes—
215C4×Q8C22×C3214yes—
216C4×D4C22×C3214yes—
217C4×D4C22×C3212yes—
218#117C22×C3211yescharacteristic
219#117C22×C3212yes—
220#117C22×C3212yes—
221#117C22×C3218yes—
222#117C22×C3218yes—
223C2×C4⋊C4C22×C3214yes—
224C2×C42C22×C3211yescharacteristic
225C2×C42C22×C3212yes—
226C23×C3C24×C318yes—
227C23×C3C24×C314yes—
228C3×Q8—232no—
229C3×Q8—232no—
230C3×Q8—28no—
231C3×Q8—232no—
232C3×D4—216no—
233C3×D4—28no—
234C2×C3×C4—232no—
235C2×C3×C4—216no—
236C2×C3×C4—216no—
237C2×C3×C4—232no—
238C2×C3×C4C3×#41116yes—
239C2×C3×C4C24×C3116yes—
240C2×C3×C4C24×C3132yes—
241C2×C3×C4C24×C318yes—
242C2×C3×C4C24×C3132yes—
243C2×C3×C4C24×C318yes—
244C2×C3×C4C24×C314yes—
245C2×C3×C4C24×C314yes—
246C2×C3×C4C24×C318yes—
247C2×C32—22no—
248C2×C32—42no—
249C2×C32C22×#4111yescharacteristic
250C2×C32C2×2−1+412yes—
251#41—24no—
252C2×Q8—28no—
253C2×Q8C23×C3214yes—
254C2×Q8C23×C3214yes—
255C2×Q8C23×C3214yes—
256C2×Q8C23×C3211yescharacteristic
257C2×D4C23×C3212yes—
258C2×D4C23×C3211yescharacteristic
259C22×C4C23×C3212yes—
260C22×C4C23×C3214yes—
261C22×C4C23×C3212yes—
262C22×C4C23×C3214yes—
263C22×C4C23×C3211yescharacteristic
264C4⋊C4C23×C3218yes—
265C4⋊C4C23×C3218yes—
266C4⋊C4C23×C3214yes—
267C4⋊C4C23×C3212yes—
268C4⋊C4C23×C3214yes—
269C4⋊C4C23×C3214yes—
270C4⋊C4C23×C32116yes—
271C4⋊C4C23×C32116yes—
272C4⋊C4C23×C3218yes—
273C22⋊C4C23×C3212yes—
274C22⋊C4C23×C3214yes—
275C22⋊C4C23×C3214yes—
276C22⋊C4C23×C32116yes—
277C42C23×C3211yescharacteristic
278C42C23×C3212yes—
279C42C23×C3211yescharacteristic
280C42C23×C3214yes—
281C42C23×C3212yes—
282C42C23×C32116yes—
283C42C23×C3214yes—
284C22×C3—28no—
285C22×C3—28no—
286C22×C3—216no—
287C22×C3C2×C3×#4118yes—
288C22×C3C25×C314yes—
289C3×C4—28no—
290C3×C4—232no—
291C3×C4—216no—
292C3×C4—28no—
293C3×C4—232no—
294C3×C4C2×C3×#4118yes—
295C3×C4C2×C3×#4118yes—
296C3×C4C3×2−1+4116yes—
297C32#350911yescharacteristic, direct product factor, Sylow
298C23C24×C3212yes—
299C23C24×C3211yescharacteristic
300Q8—28no—
301Q8—28no—
302Q8—22no—
303Q8—28no—
304D4—24no—
305D4—22no—
306C2×C4—28no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—28no—
310C2×C4C32×#4114yes—
311C2×C4C24×C3214yes—
312C2×C4C24×C3218yes—
313C2×C4C24×C3212yes—
314C2×C4C24×C3218yes—
315C2×C4C24×C3212yes—
316C2×C4C24×C3211yescharacteristic
317C2×C4C24×C3211yescharacteristic
318C2×C4C24×C3212yes—
319C2×C3—28no—
320C2×C3—48no—
321C2×C3C22×C3×#4114yes—
322C2×C3C2×C3×2−1+418yes—
323C22—22no—
324C22—22no—
325C22—24no—
326C22C2×C32×#4112yes—
327C22C25×C3211yescharacteristic
328C4—22no—
329C4—28no—
330C4—24no—
331C4—22no—
332C4—28no—
333C4C2×C32×#4112yes—
334C4C2×C32×#4112yes—
335C4C32×2−1+414yes—
336C3C3×#350914yesdirect product factor
337C2—22no—
338C2—42no—
339C2C22×C32×#4111yescharacteristic
340C2C2×C32×2−1+412yes—
subgroupG/H
center, socleC22×C32C25
commutator, Frattini, cosocleC22C25×C32
prime radicalC2×C32×#41C22
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×#556C2semidirect
3. C32×#552C2semidirect
4. C32×#541C2semidirect
6. C32×#533C2semidirect
7. C32×#533C2semidirect
8. C2×C32×C4⋊Q8C2semidirect
9. C32×#519C2semidirect
11. C3×#3509C3direct (N × H)
43. C3×#557C2×C3semidirect
44. C3×#556C2×C3semidirect
45. C3×#552C2×C3semidirect
46. C3×#541C2×C3semidirect
48. C3×#533C2×C3semidirect
49. C3×#533C2×C3semidirect
50. C2×C3×C4⋊Q8C2×C3semidirect
51. C3×#519C2×C3semidirect
86. #3509C32direct (N × H)
139. #557C2×C32semidirect
140. #556C2×C32semidirect
141. #552C2×C32semidirect
142. #541C2×C32semidirect
144. #533C2×C32semidirect
145. #533C2×C32semidirect
146. C2×C4⋊Q8C2×C32semidirect
147. #519C2×C32semidirect
297. C32#3509direct (N × H)
336. C3C3×#3509direct (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×#5412
C32×#5362
C32×#5332
C2×C32×C4⋊Q82
C32×#5192
C32×C4×#412
C3×#35093