Finite Group Databasebeta

← #79233

Subgroups — #79233

Order 384.

336 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
1#5529C211yesmaximal, characteristic
2#5513C211yesmaximal, characteristic
3C3×#428C211yesmaximal, characteristic, complemented
4#5137C211yesmaximal, characteristic, complemented
5#5128C211yesmaximal, characteristic
6#4996C211yesmaximal, characteristic, complemented
7#4970C211yesmaximal, characteristic, complemented
8#4884C211yesmaximal, characteristic, complemented
9#4881C211yesmaximal, characteristic, complemented
10#4877C211yesmaximal, characteristic, complemented
11#4873C211yesmaximal, characteristic, complemented
12#4868C211yesmaximal, characteristic, complemented
13#4867C211yesmaximal, characteristic, complemented
14#4858C211yesmaximal, characteristic, complemented
15#4857C211yesmaximal, characteristic, complemented
16#3352—31nomaximal, Sylow
17C2×C3×M16C2211yescharacteristic
18C3×#117C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20#929—21no—
21C4×#67C2211yescharacteristic
22#926—21no—
23#924C2211yescharacteristic
24#924C2211yescharacteristic
25C2×Dic12C2211yescharacteristic
26C2×ΓL(1,𝔽25)C2211yescharacteristic
27#898—21no—
28#897C2211yescharacteristic
29#897C2211yescharacteristic
30#892C2211yescharacteristic
31D3×C4⋊C4C2211yescharacteristic
32#890C2211yescharacteristic
33#890C2211yescharacteristic
34#889—21no—
35#887C2211yescharacteristic
36#887C2211yescharacteristic
37#883—21no—
38#879—21no—
39#872—21no—
40#870—21no—
41C3×#107C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#106C2211yescharacteristic
44C3×#106C2211yescharacteristic
45#830C2211yescharacteristic
46(C2×D3)⋊C8C2211yescharacteristic
47#817C2211yescharacteristic
48#816C2211yescharacteristic
49#816C2211yescharacteristic
50Dic3⋊C8C2211yescharacteristic
51C8×Dic3C2211yescharacteristic
52#810C2211yescharacteristic
53#810C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#808C2211yescharacteristic
57#807C2211yescharacteristic
58#807C2211yescharacteristic
59#807C2211yescharacteristic
60#552—31no—
61#552—31no—
62#499—31no—
63#485—31no—
64#485—31no—
65#484—31no—
66#484—31no—
67#479—31no—
68#479—31no—
69#470—31no—
70#470—31no—
71C4×Q16—31no—
72#436—31no—
73#428—31no—
74#428D311yescharacteristic, complemented
75C22×C3×C4—21no—
76C22×C3×C4C2311yescharacteristic
77C2×#67D411yescharacteristic
78C2×C4×D3—21no—
79C2×C4×D3C2311yescharacteristic
80C2×Dic6C2311yescharacteristic
81C2×Dic6C2311yescharacteristic
82C3×M16—22no—
83C2×C3×C8C2311yescharacteristic
84C2×C3×C8C2311yescharacteristic
85C3×C4⋊C4—21no—
86C3×C4⋊C4C2311yescharacteristic
87C3×C4⋊C4C2311yescharacteristic
88C3×C4⋊C4C2311yescharacteristic
89C3×C4⋊C4C2311yescharacteristic
90C3×C22⋊C4—21no—
91C3×C42—21no—
92#217—21no—
93#217—21no—
94#217D411yescharacteristic
95(C2×D3)⋊C4—21no—
96(C2×D3)⋊C4—21no—
97(C2×D3)⋊C4—21no—
98(C2×D3)⋊C4D411yescharacteristic
99C4⋊Dic3—21no—
100C4⋊Dic3—21no—
101C4⋊Dic3—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3C2311yescharacteristic
104C4⋊Dic3C2311yescharacteristic
105Dic3⋊C4—21no—
106Dic3⋊C4—21no—
107Dic3⋊C4—21no—
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4D411yescharacteristic
113C4×Dic3—21no—
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3C2311yescharacteristic
117C2×C3⋊C8C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119Dic12—22no—
120ΓL(1,𝔽25)—22no—
121C2×Q16—31no—
122C2×M16—31no—
123C2×M16C2×D311yescharacteristic
124#126—61no—
125#126—61no—
126#125—31no—
127#125—31no—
128#125—61no—
129#125—61no—
130#123—61no—
131#123—61no—
132C22⋊Q8—31no—
133C22⋊Q8—31no—
134C22⋊Q8—31no—
135C22⋊Q8—31no—
136C4×Q8—31no—
137C4×Q8—31no—
138C4×D4—31no—
139#117—31no—
140#117—61no—
141#117—61no—
142#117C2×D311yescharacteristic
143C2×C4⋊C4—31no—
144C2×C4⋊C4C2×D311yescharacteristic
145#107—31no—
146#107—31no—
147#107—31no—
148#107C2×D311yescharacteristic
149#107C2×D311yescharacteristic
150#106—31no—
151#106—31no—
152#106C2×D311yescharacteristic
153#106C2×D311yescharacteristic
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161C22⋊C8—31no—
162C22⋊C8—31no—
163C4×C8—31no—
164C23×C3C2×D411yescharacteristic
165C22×D3C2×D411yescharacteristic
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—42no—
173C2×C3×C4C2×D411yescharacteristic
174C2×C3×C4C2411yescharacteristic
175#67—41no—
176C2×Dic3—21no—
177C2×Dic3—21no—
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3C2×D411yescharacteristic
181C2×Dic3C2×D411yescharacteristic
182C2×Dic3C2×D411yescharacteristic
183C4×D3—21no—
184C4×D3—21no—
185C4×D3—42no—
186Dic6—21no—
187Dic6—21no—
188Dic6—21no—
189Dic6—21no—
190C3×C8—21no—
191C3×C8#4112yes—
192C3⋊C8—21no—
193C3⋊C8—21no—
194C2×Q8—31no—
195C2×Q8—31no—
196C2×D4—31no—
197C22×C4—21no—
198C22×C4—31no—
199C22×C4—61no—
200C22×C4C22×D311yescharacteristic
201Q16—62no—
202M16—22no—
203M16—62no—
204C2×C8—31no—
205C2×C8—31no—
206C2×C8C22×D311yescharacteristic
207C2×C8C22×D311yescharacteristic
208C4⋊C4—21no—
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—31no—
212C4⋊C4—61no—
213C4⋊C4—61no—
214C4⋊C4—61no—
215C4⋊C4—61no—
216C4⋊C4—61no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4C22×D311yescharacteristic
224C4⋊C4C22×D311yescharacteristic
225C4⋊C4C22×D311yescharacteristic
226C4⋊C4C22×D311yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—31no—
229C22⋊C4—31no—
230C22⋊C4—61no—
231C22⋊C4—61no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C42—21no—
236C42—31no—
237C42—61no—
238C42—61no—
239C42—61no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3C22×D411yescharacteristic
244C2×D3—21no—
245C2×D3—21no—
246C2×D3—21no—
247C3×C4—21no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4C2×#4111yescharacteristic
253C3×C42−1+411yescharacteristic
254Dic3—21no—
255Dic3—21no—
256Dic3—21no—
257Dic3—41no—
258Dic3—41no—
259Dic3—41no—
260Dic3—41no—
261C23—31no—
262C23D3×D411yescharacteristic
263Q8—61no—
264Q8—61no—
265Q8—61no—
266Q8—61no—
267D4—121no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—42no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—122no—
285C2×C4D3×D411yescharacteristic
286C2×C4C23×D311yescharacteristic
287C8—21no—
288C8—61no—
289C8—61no—
290C8#23512yes—
291C2×C3—41no—
292C2×C3#54711yescharacteristic
293C2×C3#57511yescharacteristic
294C2×C3#57811yescharacteristic
295D3—41no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—61no—
300C22—61no—
301C22—61no—
302C22C2×D3×D411yescharacteristic
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—61no—
309C4—61no—
310C4—61no—
311C4—121no—
312C4—121no—
313C4—121no—
314C4—121no—
315C4C2×#23511yescharacteristic
316C4#100711yescharacteristic
317C3#335211yescharacteristic, complemented, Sylow
318C2—41no—
319C2—121no—
320C2#550711yescharacteristic
321C2#577311yescharacteristic
322C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radicalC2×#67D4
hypercenter#428D3
hypocenterC3#3352
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. C3×#428C2semidirect
4. #5137C2semidirect
6. #4996C2semidirect
7. #4970C2semidirect
8. #4884C2semidirect
9. #4881C2semidirect
10. #4877C2semidirect
11. #4873C2semidirect
12. #4868C2semidirect
13. #4867C2semidirect
14. #4858C2semidirect
15. #4857C2semidirect
74. #428D3semidirect
317. C3#3352semidirect

Maximal subgroups

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

subgroupindex
#55292
#55132
C3×#4282
#51372
#51282
#49962
#49702
#48842
#48812
#48772
#48732
#48682
#48672
#48582
#48572
#33523