Finite Group Databasebeta

← #81003

Subgroups — #81003

Order 384.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5733C211yesmaximal, characteristic
2#5583C211yesmaximal, characteristic, complemented
3C3×#475C211yesmaximal, characteristic, complemented
4#5091C211yesmaximal, characteristic, complemented
5#5028C211yesmaximal, characteristic, complemented
6#4871C211yesmaximal, characteristic, complemented
7#4867C211yesmaximal, characteristic, complemented
8#4864C211yesmaximal, characteristic
9#4857C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4839C211yesmaximal, characteristic, complemented
12#4818C211yesmaximal, characteristic, complemented
13#4813C211yesmaximal, characteristic, complemented
14#4812C211yesmaximal, characteristic
15#4795C211yesmaximal, characteristic, complemented
16#3258—31nomaximal, Sylow
17C3×C4⋊Q8C2211yescharacteristic
18C3×C4×Q8C2211yescharacteristic
19#947—21no—
20#946—21no—
21#946C2211yescharacteristic
22#944C2211yescharacteristic
23C2×#216C2211yescharacteristic
24C2×#215C2211yescharacteristic
25C2×Dic12C2211yescharacteristic
26C2×#204C2211yescharacteristic
27C2×ΓL(1,𝔽25)C2211yescharacteristic
28C2×D3×C8C2211yescharacteristic
29#898—21no—
30#898—21no—
31#897C2211yescharacteristic
32#897C2211yescharacteristic
33#895C2211yescharacteristic
34#894—21no—
35#893C2211yescharacteristic
36#890C2211yescharacteristic
37#889—21no—
38#887C2211yescharacteristic
39#876—21no—
40#875C2211yescharacteristic
41#872C2211yescharacteristic
42#870—21no—
43C4×Dic6C2211yescharacteristic
44C3×#106C2211yescharacteristic
45C3×#106C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×#103C2211yescharacteristic
49#821C2211yescharacteristic
50#816C2211yescharacteristic
51Dic3⋊C8C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53#810C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#809C2211yescharacteristic
57#807C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#542—31no—
61#541—31no—
62#479—31no—
63#478—31no—
64#477—31no—
65#475D311yescharacteristic, complemented
66#470—31no—
67#468—31no—
68#467—31no—
69#465—31no—
70#464—31no—
71#464—31no—
72#462—31no—
73#461—31no—
74#424—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×D12—21no—
78C2×D12C2311yescharacteristic
79C2×C4×D3—21no—
80C2×C4×D3C2311yescharacteristic
81C2×Dic6C2311yescharacteristic
82C2×Dic6C2311yescharacteristic
83C2×C3×C8C2311yescharacteristic
84C2×C3×C8C2311yescharacteristic
85C3×C4⋊C4—21no—
86C3×C4⋊C4—21no—
87C3×C4⋊C4C2311yescharacteristic
88C3×C4⋊C4C2311yescharacteristic
89C3×C4⋊C4C2311yescharacteristic
90C3×C42—21no—
91C3×C42C2311yescharacteristic
92#216—41no—
93#215—41no—
94(C2×D3)⋊C4—21no—
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
102(C2×D3)⋊C4D411yescharacteristic
103C4⋊Dic3—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3—21no—
106C4⋊Dic3C2311yescharacteristic
107Dic3⋊C4—21no—
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4D411yescharacteristic
114Dic3⋊C4D411yescharacteristic
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120Dic12—41no—
121#204—41no—
122ΓL(1,𝔽25)—41no—
123D3×C8—41no—
124C2×Q16—31no—
125C2×Q16—31no—
126C2×SD16—31no—
127C2×SD16—31no—
128C2×M16—31no—
129C22×C8—31no—
130C4⋊Q8—31no—
131C4⋊Q8C2×D311yescharacteristic
132#126—61no—
133#126—61no—
134#126—61no—
135#125—31no—
136#125—61no—
137#125—61no—
138#124—31no—
139#124—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—31no—
144C22⋊Q8—61no—
145#121—31no—
146C4×Q8—31no—
147C4×Q8—31no—
148C4×Q8C2×D311yescharacteristic
149C4×D4—31no—
150#117—31no—
151#107—31no—
152#107—31no—
153#106C2×D311yescharacteristic
154#106C2×D311yescharacteristic
155C4⋊C8—31no—
156C4⋊C8—31no—
157C4⋊C8—31no—
158C4⋊C8C2×D311yescharacteristic
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103C2×D311yescharacteristic
164#103C2×D311yescharacteristic
165#102—31no—
166#102—31no—
167C22×D3—21no—
168C22×D3C2×D411yescharacteristic
169C3×Q8—21no—
170C3×Q8—21no—
171C3×Q8—81no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2×D411yescharacteristic
178C2×C3×C4C2411yescharacteristic
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3C2×D411yescharacteristic
183C2×Dic3C2×D411yescharacteristic
184C2×Dic3C2×D411yescharacteristic
185D12—21no—
186D12—21no—
187D12—81no—
188C4×D3—21no—
189C4×D3—21no—
190C4×D3—81no—
191Dic6—21no—
192Dic6—21no—
193Dic6—21no—
194Dic6—21no—
195C3×C8—21no—
196C3×C8—21no—
197C3⋊C8—21no—
198C3⋊C8—21no—
199C2×Q8—21no—
200C2×Q8—31no—
201C2×Q8—31no—
202C2×Q8C22×D311yescharacteristic
203C2×D4—31no—
204C2×D4—61no—
205C22×C4—31no—
206C22×C4—61no—
207Q16—121no—
208Q16—121no—
209SD16—121no—
210SD16—121no—
211M16—121no—
212C2×C8—31no—
213C2×C8—31no—
214C2×C8—121no—
215C2×C8C22×D311yescharacteristic
216C2×C8C22×D311yescharacteristic
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—31no—
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4C22×D311yescharacteristic
232C4⋊C4C22×D311yescharacteristic
233C4⋊C4C22×D311yescharacteristic
234C22⋊C4—31no—
235C22⋊C4—31no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C42—21no—
244C42—31no—
245C42—61no—
246C42—61no—
247C42C22×D311yescharacteristic
248C22×C3C22×D411yescharacteristic
249C2×D3—21no—
250C2×D3—21no—
251C2×D3—21no—
252C2×D3—41no—
253C2×D3—41no—
254C2×D3—41no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4H(5,𝔽2)11yescharacteristic
262C3×C42−1+411yescharacteristic
263Dic3—21no—
264Dic3—21no—
265Dic3—21no—
266Dic3—41no—
267Dic3—41no—
268Dic3—41no—
269C23—31no—
270C23—61no—
271Q8—21no—
272Q8—21no—
273Q8—61no—
274Q8—61no—
275Q8—61no—
276Q8—61no—
277Q8—81no—
278D4—61no—
279D4—61no—
280D4—241no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—241no—
294C2×C4D3×D411yescharacteristic
295C2×C4D3×D411yescharacteristic
296C2×C4C23×D311yescharacteristic
297C8—21no—
298C8—21no—
299C8—61no—
300C8—61no—
301C2×C3#53711yescharacteristic
302C2×C3#57711yescharacteristic
303C2×C3#57811yescharacteristic
304D3—41no—
305D3—81no—
306C22—61no—
307C22—61no—
308C22—61no—
309C22—121no—
310C22—121no—
311C22—121no—
312C22C2×D3×D411yescharacteristic
313C4—21no—
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—61no—
320C4—61no—
321C4—61no—
322C4—121no—
323C4—121no—
324C4—121no—
325C4#100711yescharacteristic
326C4#100911yescharacteristic
327C3#325811yescharacteristic, complemented, Sylow
328C2—121no—
329C2—241no—
330C2#562911yescharacteristic
331C2#576211yescharacteristic
332C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#475C2
socleC22×C3C22×D4
prime radical#895C22
hypercenter#475D3
hypocenterC3#3258
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #5583C2semidirect
3. C3×#475C2semidirect
4. #5091C2semidirect
5. #5028C2semidirect
6. #4871C2semidirect
7. #4867C2semidirect
9. #4857C2semidirect
10. #4845C2semidirect
11. #4839C2semidirect
12. #4818C2semidirect
13. #4813C2semidirect
15. #4795C2semidirect
65. #475D3semidirect
327. C3#3258semidirect

Maximal subgroups

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

subgroupindex
#57332
#55832
C3×#4752
#50912
#50282
#48712
#48672
#48642
#48572
#48452
#48392
#48182
#48132
#48122
#47952
#32583