Finite Group Databasebeta

← C32×#2523 #11769572

Subgroups — C32×#2523 #11769572

Order 1152.

1260 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#397C212yesmaximal
2C32×#397C211yesmaximal, characteristic
3C32×#393C211yesmaximal, characteristic
4C32×#391C212yesmaximal, complemented
5C32×#387C212yesmaximal, complemented
6C32×#387C212yesmaximal, complemented
7C32×#385C211yesmaximal, characteristic
8C32×#383C212yesmaximal, complemented
9C32×C4×C22⋊C4C212yesmaximal
10C3×#2523C314yesmaximal, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C23×C32×C4C2211yescharacteristic
13C2×C32×C4⋊C4C2212yes—
14C2×C32×C4⋊C4C2212yes—
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C22⋊C4C2212yes—
17C2×C32×C22⋊C4C2212yes—
18C2×C32×C22⋊C4C2211yescharacteristic
19C2×C32×C22⋊C4C2211yescharacteristic
20C2×C32×C42C2212yes—
21C2×C32×C42C2212yes—
22C32×#95C2212yes—
23C32×#95C2212yes—
24C32×#95C2211yescharacteristic
25C32×#95C2212yes—
26C32×#95C2212yes—
27C32×#95C2212yes—
28C32×#95C2212yes—
29C32×#95C2212yes—
30C32×#95C2211yescharacteristic
31C32×#95C2212yes—
32C3×#397C2×C318yes—
33C3×#397C2×C314yes—
34C3×#393C2×C314yes—
35C3×#391C2×C318yescomplemented
36C3×#387C2×C318yescomplemented
37C3×#387C2×C318yescomplemented
38C3×#385C2×C314yes—
39C3×#383C2×C318yescomplemented
40C3×C4×C22⋊C4C2×C318yes—
41C24×C32C2311yescharacteristic
42C2×C32×Q8—24no—
43C2×C32×Q8—41no—
44C22×C32×C4—24no—
45C22×C32×C4—41no—
46C22×C32×C4C2311yescharacteristic
47C22×C32×C4C2312yes—
48C22×C32×C4C2312yes—
49C22×C32×C4C2312yes—
50C22×C32×C4C2311yescharacteristic
51C22×C32×C4C2312yes—
52C22×C32×C4C2312yes—
53C22×C32×C4C2312yes—
54C32×C4⋊C4—24no—
55C32×C4⋊C4—24no—
56C32×C4⋊C4—41no—
57C32×C22⋊C4—24no—
58C32×C22⋊C4—22no—
59C32×C22⋊C4—24no—
60C32×C22⋊C4—41no—
61C32×C42—24no—
62C32×C42—42no—
63#2523C3211yescharacteristic, direct product factor, Sylow
64C22×C3×Q8C22×C314yes—
65C23×C3×C4C22×C314yes—
66C2×C3×C4⋊C4C22×C318yes—
67C2×C3×C4⋊C4C22×C318yes—
68C2×C3×C4⋊C4C22×C314yes—
69C2×C3×C22⋊C4C22×C318yes—
70C2×C3×C22⋊C4C22×C318yes—
71C2×C3×C22⋊C4C22×C314yes—
72C2×C3×C22⋊C4C22×C314yes—
73C2×C3×C42C22×C318yes—
74C2×C3×C42C22×C318yes—
75C3×#95C22×C318yes—
76C3×#95C22×C318yes—
77C3×#95C22×C314yes—
78C3×#95C22×C318yes—
79C3×#95C22×C318yes—
80C3×#95C22×C318yes—
81C3×#95C22×C318yes—
82C3×#95C22×C318yes—
83C3×#95C22×C314yes—
84C3×#95C22×C318yes—
85C23×C32—22no—
86C23×C32—22no—
87C23×C32—21no—
88C23×C32—21no—
89C23×C32#4112yes—
90C23×C32C2411yescharacteristic
91C32×Q8—44no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—44no—
113C2×C32×C4#4114yes—
114C2×C32×C4#4114yes—
115#397C2×C3212yes—
116#397C2×C3211yescharacteristic
117#393C2×C3211yescharacteristic
118#391C2×C3212yescomplemented
119#387C2×C3212yescomplemented
120#387C2×C3212yescomplemented
121#385C2×C3211yescharacteristic
122#383C2×C3212yescomplemented
123C4×C22⋊C4C2×C3212yes—
124C24×C3C23×C314yes—
125C2×C3×Q8—216no—
126C2×C3×Q8—44no—
127C22×C3×C4—216no—
128C22×C3×C4—44no—
129C22×C3×C4C23×C318yes—
130C22×C3×C4C23×C314yes—
131C22×C3×C4C23×C318yes—
132C22×C3×C4C23×C318yes—
133C22×C3×C4C23×C318yes—
134C22×C3×C4C23×C314yes—
135C22×C3×C4C23×C318yes—
136C22×C3×C4C23×C318yes—
137C3×C4⋊C4—216no—
138C3×C4⋊C4—216no—
139C3×C4⋊C4—44no—
140C3×C22⋊C4—216no—
141C3×C22⋊C4—28no—
142C3×C22⋊C4—216no—
143C3×C22⋊C4—44no—
144C3×C42—216no—
145C3×C42—48no—
146C22×C32—24no—
147C22×C32—22no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—42no—
151C22×C32C2×#4111yescharacteristic
152C22×C32C2×#4112yes—
153C22×C32C2×#4112yes—
154C22×C32H(5,𝔽2)11yescharacteristic
155C22×C322−1+411yescharacteristic
156C32×C4—24no—
157C32×C4—24no—
158C32×C4—42no—
159C32×C4—42no—
160C32×C4—42no—
161C32×C4—41no—
162C32×C4—41no—
163C32×C4—42no—
164C22×Q8C22×C3211yescharacteristic
165C23×C4C22×C3211yescharacteristic
166C2×C4⋊C4C22×C3212yes—
167C2×C4⋊C4C22×C3212yes—
168C2×C4⋊C4C22×C3211yescharacteristic
169C2×C22⋊C4C22×C3212yes—
170C2×C22⋊C4C22×C3212yes—
171C2×C22⋊C4C22×C3211yescharacteristic
172C2×C22⋊C4C22×C3211yescharacteristic
173C2×C42C22×C3212yes—
174C2×C42C22×C3212yes—
175#95C22×C3212yes—
176#95C22×C3212yes—
177#95C22×C3211yescharacteristic
178#95C22×C3212yes—
179#95C22×C3212yes—
180#95C22×C3212yes—
181#95C22×C3212yes—
182#95C22×C3212yes—
183#95C22×C3211yescharacteristic
184#95C22×C3212yes—
185C23×C3—28no—
186C23×C3—28no—
187C23×C3—24no—
188C23×C3—24no—
189C23×C3C3×#4118yes—
190C23×C3C24×C314yes—
191C3×Q8—416no—
192C2×C3×C4—28no—
193C2×C3×C4—28no—
194C2×C3×C4—28no—
195C2×C3×C4—28no—
196C2×C3×C4—28no—
197C2×C3×C4—28no—
198C2×C3×C4—28no—
199C2×C3×C4—28no—
200C2×C3×C4—28no—
201C2×C3×C4—28no—
202C2×C3×C4—28no—
203C2×C3×C4—28no—
204C2×C3×C4—28no—
205C2×C3×C4—28no—
206C2×C3×C4—28no—
207C2×C3×C4—28no—
208C2×C3×C4—24no—
209C2×C3×C4—24no—
210C2×C3×C4—28no—
211C2×C3×C4—28no—
212C2×C3×C4—416no—
213C2×C3×C4C3×#41116yes—
214C2×C3×C4C3×#41116yes—
215C2×C32—42no—
216C2×C32#52912yes—
217C2×C32#53911yescharacteristic
218C2×C32#55111yescharacteristic
219C2×C32#55111yescharacteristic
220C2×C32#55612yes—
221C24C23×C3211yescharacteristic
222C2×Q8—24no—
223C2×Q8—41no—
224C22×C4—24no—
225C22×C4—41no—
226C22×C4C23×C3212yes—
227C22×C4C23×C3211yescharacteristic
228C22×C4C23×C3212yes—
229C22×C4C23×C3212yes—
230C22×C4C23×C3212yes—
231C22×C4C23×C3211yescharacteristic
232C22×C4C23×C3212yes—
233C22×C4C23×C3212yes—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—41no—
237C22⋊C4—24no—
238C22⋊C4—22no—
239C22⋊C4—24no—
240C22⋊C4—41no—
241C42—24no—
242C42—42no—
243C22×C3—216no—
244C22×C3—28no—
245C22×C3—44no—
246C22×C3—44no—
247C22×C3—48no—
248C22×C3C2×C3×#4114yes—
249C22×C3C2×C3×#4118yes—
250C22×C3C2×C3×#4118yes—
251C22×C3C3×H(5,𝔽2)14yes—
252C22×C3C3×2−1+414yes—
253C3×C4—216no—
254C3×C4—216no—
255C3×C4—48no—
256C3×C4—48no—
257C3×C4—48no—
258C3×C4—44no—
259C3×C4—44no—
260C3×C4—48no—
261C32#252311yescharacteristic, direct product factor, Sylow
262C23—22no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23C32×#4112yes—
267C23C24×C3211yescharacteristic
268Q8—44no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—44no—
290C2×C4C32×#4114yes—
291C2×C4C32×#4114yes—
292C2×C3—48no—
293C2×C3C3×#52918yes—
294C2×C3C3×#53914yes—
295C2×C3C3×#55114yes—
296C2×C3C3×#55114yes—
297C2×C3C3×#55618yes—
298C22—24no—
299C22—22no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22C2×C32×#4112yes—
304C22C2×C32×#4112yes—
305C22C2×C32×#4111yescharacteristic
306C22C32×H(5,𝔽2)11yescharacteristic
307C22C32×2−1+411yescharacteristic
308C4—24no—
309C4—24no—
310C4—41no—
311C4—41no—
312C4—42no—
313C4—42no—
314C4—42no—
315C4—42no—
316C3C3×#252314yesdirect product factor
317C2—42no—
318C2C32×#52912yes—
319C2C32×#53911yescharacteristic
320C2C32×#55111yescharacteristic
321C2C32×#55111yescharacteristic
322C2C32×#55612yes—
subgroupG/H
center, socleC23×C32C24
commutator, Frattini, cosocleC23C24×C32
prime radicalC24×C32C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C32 ◁ G

Complements

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

NH ≅ G/Ntype
4. C32×#391C2semidirect
5. C32×#387C2semidirect
6. C32×#387C2semidirect
8. C32×#383C2semidirect
10. C3×#2523C3direct (N × H)
35. C3×#391C2×C3semidirect
36. C3×#387C2×C3semidirect
37. C3×#387C2×C3semidirect
39. C3×#383C2×C3semidirect
63. #2523C32direct (N × H)
118. #391C2×C32semidirect
119. #387C2×C32semidirect
120. #387C2×C32semidirect
122. #383C2×C32semidirect
261. C32#2523direct (N × H)
316. C3C3×#2523direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#3972
C32×#3932
C32×#3912
C32×#3872
C32×#3852
C32×#3832
C32×C4×C22⋊C42
C3×#25233