Finite Group Databasebeta

← C3×Q8×ΓL(1,𝔽25) #11861494

Subgroups — C3×Q8×ΓL(1,𝔽25) #11861494

Order 1152.

827 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×D3×Q8C211yesmaximal, characteristic
2C32×C8×Q8C211yesmaximal, characteristic, complemented
3C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×#4837C213yesmaximal
5C3×#4831C213yesmaximal, complemented
6C3×C4×ΓL(1,𝔽25)C213yesmaximal
7C3×#4679C213yesmaximal, complemented
8C3×Q8×M16—31nomaximal
9Q8×ΓL(1,𝔽25)C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×Q8C411yescharacteristic
11C32×C4×Q8C2211yescharacteristic
12C3×Q8×Dic3C411yescharacteristic
13C2×C3×ΓL(1,𝔽25)C2213yes—
14C2×C3×ΓL(1,𝔽25)C2211yescharacteristic
15C3×D3×C4⋊C4C413yes—
16C3×#887C413yes—
17C3×C42×D3C2213yes—
18C3×C4×Dic6C2213yes—
19C32×C4⋊C8C2213yes—
20C32×C4×C8C2213yes—
21C3×#815C2213yes—
22C3×#815C2213yes—
23C3×Dic3⋊C8C2216yes—
24C3×C4⋊(C3⋊C8)C2213yes—
25C3×C4×C3⋊C8C2213yes—
26C2×C3×C4×Q8—31no—
27C4×D3×Q8C2×C311yescharacteristic
28C3×#446—33no—
29C3×#446—33no—
30C3×C8×Q8—21no—
31C3×C8×Q8—31no—
32C3×C8×Q8D311yescharacteristic, complemented
33C3×C8×Q8C2×C311yescharacteristic, complemented
34C3×#423—33no—
35C3×C4×M16—33no—
36Q8×C3⋊C8C2×C311yescharacteristic, complemented
37#4837C2×C313yes—
38#4831C2×C313yescomplemented
39C4×ΓL(1,𝔽25)C2×C313yes—
40#4679C2×C313yescomplemented
41C2×C32×Q8C2×C411yescharacteristic
42C3×D3×Q8—24no—
43C2×C3×C4×D3C2×C413yes—
44C2×C3×C4×D3C2×C413yes—
45C2×C3×C4×D3C2311yescharacteristic
46C2×C3×Dic6C2×C413yes—
47C2×C32×C8C2313yes—
48C2×C32×C8C2311yescharacteristic
49C32×C4⋊C4C2×C413yes—
50C32×C42C2313yes—
51C3×C4⋊Dic3C2×C413yes—
52C3×Dic3⋊C4C2×C416yes—
53C3×C4×Dic3C2×C413yes—
54C3×C4×Dic3C2×C413yes—
55C3×C4×Dic3C2313yes—
56C2×C3×C3⋊C8C2313yes—
57C2×C3×C3⋊C8C2311yescharacteristic
58C3×ΓL(1,𝔽25)—26no—
59C3×ΓL(1,𝔽25)Q814yesdirect product factor
60Q8×M16—31noSylow
61C22×C3×Q8—31no—
62C2×D3×Q8C3×C411yescharacteristic
63C2×C3×M16—31no—
64C2×C3×M16—33no—
65C3×C4×Q8—21no—
66C3×C4×Q8—31no—
67C3×C4×Q8—33no—
68C3×C4×Q8—33no—
69C3×C4×Q8C2×D311yescharacteristic
70C3×C4×Q8C22×C311yescharacteristic
71C2×C3×C4⋊C4—33no—
72C2×C3×C42—33no—
73Q8×Dic3C3×C411yescharacteristic
74C2×ΓL(1,𝔽25)C22×C313yes—
75C2×ΓL(1,𝔽25)C22×C311yescharacteristic
76D3×C4⋊C4C3×C413yes—
77#887C3×C413yes—
78C42×D3C22×C313yes—
79C4×Dic6C22×C313yes—
80C3×C4⋊C8—23no—
81C3×C4⋊C8—36no—
82C3×C4⋊C8—33no—
83C3×C4⋊C8C2×D313yes—
84C3×C4⋊C8C22×C313yes—
85C3×#97—33no—
86C3×#97—33no—
87C3×C4×C8—23no—
88C3×C4×C8—33no—
89C3×C4×C8C2×D313yes—
90C3×C4×C8C22×C313yes—
91#815C22×C313yes—
92#815C22×C313yes—
93Dic3⋊C8C22×C316yes—
94C4⋊(C3⋊C8)C22×C313yes—
95C4×C3⋊C8C22×C313yes—
96C22×C3×D3C22×C411yescharacteristic
97C32×Q8M1614yescomplemented
98C2×C32×C4C22×C413yes—
99C2×C32×C4C22×C413yes—
100C2×C32×C4C2411yescharacteristic
101C2×C3×Dic3C22×C413yes—
102C2×C3×Dic3C22×C413yes—
103C2×C3×Dic3C22×C411yescharacteristic
104C3×C4×D3—26no—
105C3×C4×D3—43no—
106C3×C4×D3C2×Q812yes—
107C3×C4×D3#4112yes—
108C3×Dic6—26no—
109C32×C8—23no—
110C32×C8C2×Q812yescomplemented
111C3×C3⋊C8—23no—
112C3×C3⋊C8C2×Q812yescomplemented
113C2×C4×Q8—31no—
114#446—33no—
115#446—33no—
116C8×Q8—31no—
117C8×Q8C3×D311yescharacteristic, complemented
118#423—33no—
119C4×M16—33no—
120C2×C3×Q8—21no—
121C2×C3×Q8—33no—
122C2×C3×Q8—64no—
123C2×C3×Q8C4×D311yescharacteristic
124C2×C3×Q8C2×C3×C411yescharacteristic
125C22×C3×C4—31no—
126C22×C3×C4—33no—
127C22×C3×C4—33no—
128D3×Q8—24no—
129C2×C4×D3C2×C3×C413yes—
130C2×C4×D3C2×C3×C413yes—
131C2×C4×D3C23×C311yescharacteristic
132C2×Dic6C2×C3×C413yes—
133C3×M16—34no—
134C3×M16—66no—
135C2×C3×C8—21no—
136C2×C3×C8—23no—
137C2×C3×C8—31no—
138C2×C3×C8—33no—
139C2×C3×C8C22×D313yes—
140C2×C3×C8C22×D311yescharacteristic
141C2×C3×C8C23×C313yes—
142C2×C3×C8C23×C311yescharacteristic
143C3×C4⋊C4—23no—
144C3×C4⋊C4—36no—
145C3×C4⋊C4—33no—
146C3×C4⋊C4C4×D313yes—
147C3×C4⋊C4C2×C3×C413yes—
148C3×C42—23no—
149C3×C42—33no—
150C3×C42—33no—
151C3×C42—33no—
152C3×C42C22×D313yes—
153C3×C42C23×C313yes—
154C4⋊Dic3C2×C3×C413yes—
155Dic3⋊C4C2×C3×C416yes—
156C4×Dic3C2×C3×C413yes—
157C4×Dic3C2×C3×C413yes—
158C4×Dic3C23×C313yes—
159C2×C3⋊C8C23×C313yes—
160C2×C3⋊C8C23×C311yescharacteristic
161ΓL(1,𝔽25)—26no—
162ΓL(1,𝔽25)C3×Q814yesdirect product factor
163C22×C32C23×C411yescharacteristic
164C2×C3×D3—21no—
165C2×C3×D3—21no—
166C2×C3×D3C4×Q812yes—
167C32×C4—23no—
168C32×C4C2×M1616yes—
169C32×C4C22×Q811yescharacteristic
170C32×C4C2×#4111yescharacteristic
171C3×Dic3—23no—
172C3×Dic3—23no—
173C3×Dic3C4×Q812yes—
174C22×Q8—31no—
175C2×M16—31no—
176C2×M16—33no—
177C4×Q8—31no—
178C4×Q8—33no—
179C4×Q8—33no—
180C4×Q8C2×C3×D311yescharacteristic
181C2×C4⋊C4—33no—
182C2×C42—33no—
183C4⋊C8—36no—
184C4⋊C8—33no—
185C4⋊C8C2×C3×D313yes—
186#97—33no—
187#97—33no—
188C4×C8—33no—
189C4×C8C2×C3×D313yes—
190C23×C3—31no—
191C22×D3C22×C3×C411yescharacteristic
192C3×Q8—24no—
193C3×Q8—66no—
194C3×Q8ΓL(1,𝔽25)14yesdirect product factor
195C3×Q8C3×M1614yescomplemented
196C2×C3×C4—21no—
197C2×C3×C4—23no—
198C2×C3×C4—23no—
199C2×C3×C4—31no—
200C2×C3×C4—32no—
201C2×C3×C4—32no—
202C2×C3×C4—33no—
203C2×C3×C4—33no—
204C2×C3×C4—66no—
205C2×C3×C4—123no—
206C2×C3×C4C2×C4×D313yes—
207C2×C3×C4C2×C4×D313yes—
208C2×C3×C4C22×C3×C413yes—
209C2×C3×C4C22×C3×C413yes—
210C2×C3×C4C23×D311yescharacteristic
211C2×C3×C4C24×C311yescharacteristic
212C2×Dic3C22×C3×C413yes—
213C2×Dic3C22×C3×C413yes—
214C2×Dic3C22×C3×C411yescharacteristic
215C4×D3—26no—
216C4×D3—43no—
217C4×D3C2×C3×Q812yes—
218C4×D3C3×#4112yes—
219Dic6—26no—
220C3×C8—23no—
221C3×C8—23no—
222C3×C8—22no—
223C3×C8—32no—
224C3×C8—43no—
225C3×C8—63no—
226C3×C8D3×Q812yescomplemented
227C3×C8C2×C3×Q812yescomplemented
228C3⋊C8—23no—
229C3⋊C8C2×C3×Q812yescomplemented
230C2×C32C2×C4×Q811yescharacteristic
231C2×C32C22×M1611yescharacteristic
232C2×C32#56811yescharacteristic
233C3×D3—22no—
234C2×Q8—33no—
235C2×Q8—64no—
236C2×Q8C3×C4×D311yescharacteristic
237C22×C4—31no—
238C22×C4—33no—
239C22×C4—33no—
240M16—34no—
241M16—66no—
242C2×C8—31no—
243C2×C8—33no—
244C2×C8C22×C3×D313yes—
245C2×C8C22×C3×D311yescharacteristic
246C4⋊C4—36no—
247C4⋊C4—33no—
248C4⋊C4C3×C4×D313yes—
249C42—33no—
250C42—33no—
251C42—33no—
252C42C22×C3×D313yes—
253C22×C3—21no—
254C22×C3—32no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3C22×C4×D311yescharacteristic
258C22×C3C23×C3×C411yescharacteristic
259C2×D3—21no—
260C2×D3—21no—
261C2×D3C3×C4×Q812yes—
262C3×C4—23no—
263C3×C4—23no—
264C3×C4—26no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—32no—
268C3×C4—43no—
269C3×C4—63no—
270C3×C4—63no—
271C3×C4C2×ΓL(1,𝔽25)16yes—
272C3×C4C2×C3×M1616yes—
273C3×C4C2×D3×Q811yescharacteristic
274C3×C4D3×#4111yescharacteristic
275C3×C4C22×C3×Q811yescharacteristic
276C3×C4C2×C3×#4111yescharacteristic
277Dic3—23no—
278Dic3—23no—
279Dic3C3×C4×Q812yes—
280C32Q8×M1611yescharacteristic, complemented, Sylow
281C23—31no—
282Q8—66no—
283Q8C3×ΓL(1,𝔽25)14yesdirect product factor
284C2×C4—31no—
285C2×C4—32no—
286C2×C4—32no—
287C2×C4—33no—
288C2×C4—33no—
289C2×C4—66no—
290C2×C4—123no—
291C2×C4C2×C3×C4×D313yes—
292C2×C4C2×C3×C4×D313yes—
293C2×C4C23×C3×D311yescharacteristic
294C8—23no—
295C8—32no—
296C8—63no—
297C8C3×D3×Q812yescomplemented
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—62no—
302C2×C3C4×D3×Q811yescharacteristic
303C2×C3C22×ΓL(1,𝔽25)11yescharacteristic
304C2×C3#575011yescharacteristic
305C2×C3C2×C3×C4×Q811yescharacteristic
306C2×C3C22×C3×M1611yescharacteristic
307C2×C3C3×#56811yescharacteristic
308D3—22no—
309C22—32no—
310C22—61no—
311C22—61no—
312C22C22×C3×C4×D311yescharacteristic
313C4—23no—
314C4—32no—
315C4—63no—
316C4—63no—
317C4C2×C3×ΓL(1,𝔽25)16yes—
318C4C2×C3×D3×Q811yescharacteristic
319C4C3×D3×#4111yescharacteristic
320C3—21no—
321C3Q8×ΓL(1,𝔽25)11yescharacteristic, direct product factor
322C3C3×Q8×M1611yescharacteristic, complemented
323C2—62no—
324C2C3×C4×D3×Q811yescharacteristic
325C2C22×C3×ΓL(1,𝔽25)11yescharacteristic
326C2C3×#575011yescharacteristic
subgroupG/H
centerC2×C3×C4C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×C8×Q8C2
socleC22×C32C23×C4
prime radicalC22×C3×D3C22×C4
cosocleC2×C3×C4C24×C3
hypercenterC3×C8×Q8D3
hypocenterC3C3×Q8×M16
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C3×C8×Q8

Complements

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

NH ≅ G/Ntype
2. C32×C8×Q8C2semidirect
3. C3×Q8×C3⋊C8C2semidirect
5. C3×#4831C2semidirect
7. C3×#4679C2semidirect
9. Q8×ΓL(1,𝔽25)C3direct (N × H)
32. C3×C8×Q8D3semidirect
33. C3×C8×Q8C2×C3semidirect
36. Q8×C3⋊C8C2×C3semidirect
38. #4831C2×C3semidirect
40. #4679C2×C3semidirect
59. C3×ΓL(1,𝔽25)Q8direct (N × H)
97. C32×Q8M16semidirect
110. C32×C8C2×Q8semidirect
112. C3×C3⋊C8C2×Q8semidirect
117. C8×Q8C3×D3semidirect
162. ΓL(1,𝔽25)C3×Q8direct (N × H)
194. C3×Q8ΓL(1,𝔽25)direct (N × H)
195. C3×Q8C3×M16semidirect
226. C3×C8D3×Q8semidirect
227. C3×C8C2×C3×Q8semidirect
229. C3⋊C8C2×C3×Q8semidirect
280. C32Q8×M16semidirect
283. Q8C3×ΓL(1,𝔽25)direct (N × H)
297. C8C3×D3×Q8semidirect
321. C3Q8×ΓL(1,𝔽25)direct (N × H)
322. C3C3×Q8×M16semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D3×Q82
C32×C8×Q82
C3×Q8×C3⋊C82
C3×#48372
C3×#48312
C3×C4×ΓL(1,𝔽25)2
C3×#46792
C3×Q8×M163
Q8×ΓL(1,𝔽25)3