Finite Group Databasebeta

← #80900

Subgroups — #80900

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#5642C211yesmaximal, characteristic
2#5639C211yesmaximal, characteristic
3C3×#470C211yesmaximal, characteristic, complemented
4#5189C211yesmaximal, characteristic, complemented
5Dic3×Q16C211yesmaximal, characteristic, complemented
6#5128C211yesmaximal, characteristic
7#5121C211yesmaximal, characteristic, complemented
8#5045C211yesmaximal, characteristic, complemented
9#5044C211yesmaximal, characteristic, complemented
10#4861C211yesmaximal, characteristic, complemented
11#4858C211yesmaximal, characteristic, complemented
12#4815C211yesmaximal, characteristic, complemented
13#4811C211yesmaximal, characteristic, complemented
14#4799C211yesmaximal, characteristic, complemented
15#4796C211yesmaximal, characteristic, complemented
16#3352—31nomaximal, Sylow
17C2×C3×Q16C2211yescharacteristic
18C2×C3×M16C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×C22⋊Q8C2211yescharacteristic
21#946C2211yescharacteristic
22#946C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25C4×#67C2211yescharacteristic
26#926C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic
29#898—21no—
30#892—21no—
31#892C2211yescharacteristic
32D3×C4⋊C4C2211yescharacteristic
33#890—21no—
34#890C2211yescharacteristic
35#890C2211yescharacteristic
36#889—21no—
37#886—21no—
38#883—21no—
39#879—21no—
40#877—21no—
41C3×#106C2211yescharacteristic
42C3×#103C2211yescharacteristic
43C3×#103C2211yescharacteristic
44#835C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#830C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50#818C2211yescharacteristic
51#818C2211yescharacteristic
52#817C2211yescharacteristic
53#817C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
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#470D311yescharacteristic, complemented
71C4×Q16—31no—
72#436—31no—
73#428—31no—
74#428—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67D411yescharacteristic
79C22×Dic3—21no—
80C2×C4×D3—21no—
81C2×C4×D3C2311yescharacteristic
82C3×Q16—22no—
83C3×M16—22no—
84C2×C3×C8C2311yescharacteristic
85C2×C3×C8C2311yescharacteristic
86C3×C4⋊C4—21no—
87C3×C4⋊C4—21no—
88C3×C4⋊C4C2311yescharacteristic
89C3×C4⋊C4C2311yescharacteristic
90C3×C22⋊C4—21no—
91C3×C22⋊C4—21no—
92#217—21no—
93#217D411yescharacteristic
94(C2×D3)⋊C4—21no—
95(C2×D3)⋊C4—21no—
96(C2×D3)⋊C4—21no—
97(C2×D3)⋊C4D411yescharacteristic
98C4⋊Dic3—21no—
99C4⋊Dic3—21no—
100C4⋊Dic3—21no—
101C4⋊Dic3—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3C2311yescharacteristic
104C4⋊Dic3C2311yescharacteristic
105C4⋊Dic3C2311yescharacteristic
106C4⋊Dic3C2311yescharacteristic
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×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120ΓL(1,𝔽25)—22no—
121C2×Q16C2×D311yescharacteristic
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⋊Q8C2×D311yescharacteristic
135C22⋊Q8C2×D311yescharacteristic
136C4×Q8—31no—
137C4×Q8—31no—
138C4×D4—31no—
139#117—31no—
140#117—31no—
141#117—61no—
142#117—61no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145#107—31no—
146#107—31no—
147#107—31no—
148#107—31no—
149#107—31no—
150#106—31no—
151#106—31no—
152#106—31no—
153#106C2×D311yescharacteristic
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#103C2×D311yescharacteristic
160#103C2×D311yescharacteristic
161C22⋊C8—31no—
162C22⋊C8—31no—
163C4×C8—31no—
164C23×C3C2×D411yescharacteristic
165C22×D3C2×D411yescharacteristic
166C3×Q8—21no—
167C3×Q8—21no—
168C3×Q8—21no—
169C3×Q8—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2411yescharacteristic
178#67—41no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—42no—
184C2×Dic3C2×D411yescharacteristic
185C2×Dic3C2×D411yescharacteristic
186C2×Dic3C2×D411yescharacteristic
187C4×D3—21no—
188C4×D3—21no—
189C4×D3—42no—
190C3×C8—21no—
191C3×C8#4112yes—
192C3⋊C8—21no—
193C3⋊C8—21no—
194C2×Q8C22×D311yescharacteristic
195C2×Q8C22×D311yescharacteristic
196C2×D4—31no—
197C22×C4—31no—
198C22×C4—61no—
199C22×C4—61no—
200C22×C4C22×D311yescharacteristic
201Q16—22no—
202M16—22no—
203M16—62no—
204C2×C8—31no—
205C2×C8—31no—
206C2×C8C22×D311yescharacteristic
207C2×C8C22×D311yescharacteristic
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—31no—
211C4⋊C4—31no—
212C4⋊C4—31no—
213C4⋊C4—31no—
214C4⋊C4—31no—
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⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4C22×D311yescharacteristic
226C4⋊C4C22×D311yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—21no—
229C22⋊C4—31no—
230C22⋊C4—31no—
231C22⋊C4—61no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C42—31no—
236C42—61no—
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—21no—
264Q8—21no—
265Q8—21no—
266Q8—21no—
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—61no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4D3×D411yescharacteristic
286C2×C4C23×D311yescharacteristic
287C8—21no—
288C8—61no—
289C8—61no—
290C8#23712yes—
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×#23711yescharacteristic
316C4#101011yescharacteristic
317C3#335211yescharacteristic, complemented, Sylow
318C2—41no—
319C2—121no—
320C2#561111yescharacteristic
321C2#577711yescharacteristic
322C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
prime radicalC2×#67D4
hypercenter#470D3
hypocenterC3#3352
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
3. C3×#470C2semidirect
4. #5189C2semidirect
5. Dic3×Q16C2semidirect
7. #5121C2semidirect
8. #5045C2semidirect
9. #5044C2semidirect
10. #4861C2semidirect
11. #4858C2semidirect
12. #4815C2semidirect
13. #4811C2semidirect
14. #4799C2semidirect
15. #4796C2semidirect
70. #470D3semidirect
317. C3#3352semidirect

Maximal subgroups

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

subgroupindex
#56422
#56392
C3×#4702
#51892
Dic3×Q162
#51282
#51212
#50452
#50442
#48612
#48582
#48152
#48112
#47992
#47962
#33523