Finite Group Databasebeta

← #81464

Subgroups — #81464

Order 384.

354 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5642C211yesmaximal, characteristic, complemented
2#5514C211yesmaximal, characteristic
3C3×#484C211yesmaximal, characteristic, complemented
4#5048C211yesmaximal, characteristic, complemented
5C2×#808C211yesmaximal, characteristic, complemented
6#4891C211yesmaximal, characteristic, complemented
7#4885C211yesmaximal, characteristic, complemented
8#4880C211yesmaximal, characteristic, complemented
9#4878C211yesmaximal, characteristic, complemented
10#4814C211yesmaximal, characteristic, complemented
11#4807C211yesmaximal, characteristic, complemented
12#4796C211yesmaximal, characteristic, complemented
13#4788C211yesmaximal, characteristic, complemented
14#4736C211yesmaximal, characteristic, complemented
15#67⋊C8C211yesmaximal, characteristic, complemented
16#3337—31nomaximal, Sylow
17C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19#946C2211yescharacteristic
20Q8×Dic3C2211yescharacteristic
21C2×#215C2211yescharacteristic
22#930—21no—
23#930C2211yescharacteristic
24C4×#67—21no—
25C4×#67C2211yescharacteristic
26C2×C4⋊Dic3C2211yescharacteristic
27C22×C3⋊C8C2211yescharacteristic
28C2×D3×C8C2211yescharacteristic
29#898—21no—
30#898—21no—
31#895C2211yescharacteristic
32#894—21no—
33#893C2211yescharacteristic
34#892C2211yescharacteristic
35#892C2211yescharacteristic
36#890—21no—
37#890C2211yescharacteristic
38#890C2211yescharacteristic
39#886—21no—
40#877—21no—
41C3×#107C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50#818C2211yescharacteristic
51#818C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53C8×Dic3C2211yescharacteristic
54#809C2211yescharacteristic
55#808C2211yescharacteristic
56#808C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59#808C2211yescharacteristic
60#553—31no—
61#552—31no—
62#499—31no—
63#485—31no—
64#484D311yescharacteristic, complemented
65#482—31no—
66#480—31no—
67#479—31no—
68#478—31no—
69#465—31no—
70#465—31no—
71C4×SD16—31no—
72C8×D4—31no—
73#427—31no—
74C2×#106—31no—
75C2×C3×Q8C2311yescharacteristic
76C22×C3×C4—21no—
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67D411yescharacteristic
80C22×Dic3—21no—
81C2×D12—21no—
82C2×D12C2311yescharacteristic
83C2×C4×D3—21no—
84C2×C4×D3C2311yescharacteristic
85C2×C3×C8C2311yescharacteristic
86C2×C3×C8C2311yescharacteristic
87C3×C4⋊C4—21no—
88C3×C4⋊C4—21no—
89C3×C4⋊C4C2311yescharacteristic
90C3×C4⋊C4C2311yescharacteristic
91C3×C4⋊C4C2311yescharacteristic
92C3×C22⋊C4—21no—
93#217D411yescharacteristic
94#215—22no—
95(C2×D3)⋊C4—21no—
96(C2×D3)⋊C4—21no—
97(C2×D3)⋊C4—21no—
98(C2×D3)⋊C4—21no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4D411yescharacteristic
102C4⋊Dic3—21no—
103C4⋊Dic3—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3—21no—
106C4⋊Dic3C2311yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4D411yescharacteristic
113C4×Dic3—21no—
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3C2311yescharacteristic
117C2×C3⋊C8—22no—
118C2×C3⋊C8C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120D3×C8—22no—
121C2×SD16—31no—
122C22×C8—31no—
123C22×C8—31no—
124#126—61no—
125#126—61no—
126#125—31no—
127#125—31no—
128#125—61no—
129#123—61no—
130#123—61no—
131C22⋊Q8—31no—
132C22⋊Q8C2×D311yescharacteristic
133#121—31no—
134#121—31no—
135#121—61no—
136C4×Q8—31no—
137C4×D4—31no—
138C4×D4—31no—
139C4×D4—61no—
140#117—31no—
141#117—31no—
142#117—61no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4C2×D311yescharacteristic
145#107—31no—
146#107—31no—
147#107C2×D311yescharacteristic
148#107C2×D311yescharacteristic
149#106—31no—
150#106—31no—
151#106—31no—
152#106—31no—
153#106—31no—
154C4⋊C8—31no—
155#103—31no—
156#103C2×D311yescharacteristic
157#103C2×D311yescharacteristic
158#102—31no—
159#102—31no—
160#102—31no—
161C22⋊C8—31no—
162C22⋊C8C2×D311yescharacteristic
163C4×C8—31no—
164C23×C3C2×D411yescharacteristic
165C22×D3—21no—
166C22×D3C2×D411yescharacteristic
167C3×Q8—21no—
168C3×Q8—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—81no—
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2411yescharacteristic
178#67—22no—
179#67—42no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—42no—
184C2×Dic3C2×D411yescharacteristic
185C2×Dic3C2×D411yescharacteristic
186C2×Dic3C2×D411yescharacteristic
187D12—21no—
188D12—21no—
189D12—81no—
190C4×D3—21no—
191C4×D3—21no—
192C4×D3—81no—
193C3×C8—21no—
194C3×C8—21no—
195C3⋊C8—21no—
196C3⋊C8#4112yes—
197C2×Q8C22×D311yescharacteristic
198C2×D4—31no—
199C2×D4—31no—
200C2×D4—61no—
201C2×D4—61no—
202C22×C4—21no—
203C22×C4—31no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4C22×D311yescharacteristic
207SD16—62no—
208C2×C8—31no—
209C2×C8—31no—
210C2×C8—62no—
211C2×C8—62no—
212C2×C8C22×D311yescharacteristic
213C2×C8C22×D311yescharacteristic
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—31no—
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—31no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4C22×D311yescharacteristic
228C4⋊C4C22×D311yescharacteristic
229C4⋊C4C22×D311yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—31no—
232C22⋊C4—31no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C42—31no—
240C42—61no—
241C42—61no—
242C42—61no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3#13512yes—
246C22×C3C22×D411yescharacteristic
247C2×D3—21no—
248C2×D3—21no—
249C2×D3—21no—
250C2×D3—41no—
251C2×D3—41no—
252C2×D3—41no—
253C3×C4—21no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4C2×#4111yescharacteristic
259C3×C42−1+411yescharacteristic
260Dic3—21no—
261Dic3—21no—
262Dic3—21no—
263Dic3—41no—
264Dic3—41no—
265Dic3—41no—
266C23—31no—
267C23—61no—
268C23D3×D411yescharacteristic
269Q8—21no—
270Q8—21no—
271D4—62no—
272D4—61no—
273D4—61no—
274D4—122no—
275D4—241no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—81no—
291C2×C4—122no—
292C2×C4—241no—
293C2×C4D3×D411yescharacteristic
294C2×C4C23×D311yescharacteristic
295C8—21no—
296C8—21no—
297C8—32no—
298C8—61no—
299C2×C3—22no—
300C2×C3#54711yescharacteristic
301C2×C3C2×#13511yescharacteristic
302C2×C3#57711yescharacteristic
303D3—41no—
304D3—81no—
305C22—21no—
306C22—21no—
307C22—61no—
308C22—61no—
309C22—61no—
310C22—121no—
311C22—121no—
312C22—121no—
313C22#91912yes—
314C22C2×D3×D411yescharacteristic
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—61no—
321C4—61no—
322C4—61no—
323C4—121no—
324C4—121no—
325C4—121no—
326C4D3×#4111yescharacteristic
327C4#101011yescharacteristic
328C3#333711yescharacteristic, complemented, Sylow
329C2—22no—
330C2—121no—
331C2—241no—
332C2#565511yescharacteristic
333C2C2×#91911yescharacteristic
334C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#484C2
socleC22×C3C22×D4
prime radical#5514C2
hypercenter#484D3
hypocenterC3#3337
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #5642C2semidirect
3. C3×#484C2semidirect
4. #5048C2semidirect
5. C2×#808C2semidirect
6. #4891C2semidirect
7. #4885C2semidirect
8. #4880C2semidirect
9. #4878C2semidirect
10. #4814C2semidirect
11. #4807C2semidirect
12. #4796C2semidirect
13. #4788C2semidirect
14. #4736C2semidirect
15. #67⋊C8C2semidirect
64. #484D3semidirect
328. C3#3337semidirect

Maximal subgroups

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

subgroupindex
#56422
#55142
C3×#4842
#50482
C2×#8082
#48912
#48852
#48802
#48782
#48142
#48072
#47962
#47882
#47362
#67⋊C82
#33373