Finite Group Databasebeta

← #79753

Subgroups — #79753

Order 384.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5585C211yesmaximal, characteristic, complemented
2#5539C211yesmaximal, characteristic, complemented
3C3×#440C211yesmaximal, characteristic, complemented
4#5165C211yesmaximal, characteristic, complemented
5#5165C211yesmaximal, characteristic, complemented
6#5022C211yesmaximal, characteristic, complemented
7#5015C211yesmaximal, characteristic
8#4891C211yesmaximal, characteristic, complemented
9#4890C211yesmaximal, characteristic, complemented
10#4800C211yesmaximal, characteristic, complemented
11#4794C211yesmaximal, characteristic, complemented
12#4785C211yesmaximal, characteristic, complemented
13#4756C211yesmaximal, characteristic, complemented
14#4713C211yesmaximal, characteristic, complemented
15#4701C211yesmaximal, characteristic, complemented
16#3370—31nomaximal, Sylow
17C2×C3×SD16C2211yescharacteristic
18C3×C4×Q8C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20#947C2211yescharacteristic
21#947C2211yescharacteristic
22D4×Dic3C2211yescharacteristic
23D4×Dic3C2211yescharacteristic
24C2×#213C2211yescharacteristic
25C2×#213C2211yescharacteristic
26#924—21no—
27C2×Dic3⋊C4—21no—
28#894—21no—
29#894—21no—
30#893C2211yescharacteristic
31#893C2211yescharacteristic
32#890C2211yescharacteristic
33#888C2211yescharacteristic
34#878—21no—
35#878—21no—
36#875—21no—
37#874C2211yescharacteristic
38C4×D12—21no—
39#870C2211yescharacteristic
40C4×Dic6C2211yescharacteristic
41C3×#107C2211yescharacteristic
42C3×#103C2211yescharacteristic
43C3×#102C2211yescharacteristic
44C3×#97C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#815C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54C8×Dic3C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#559—31no—
61#554—31no—
62#490—31no—
63#489—31no—
64#488—31no—
65#479—31no—
66#478—31no—
67#474—31no—
68#463—31no—
69#463—31no—
70#459—31no—
71#446—31no—
72#442—31no—
73#440D311yescharacteristic, complemented
74C4×D8—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C22×Dic3—21no—
79C22×Dic3—21no—
80C2×D12—21no—
81C2×D12—21no—
82C2×D12C2311yescharacteristic
83C2×D12C2311yescharacteristic
84C2×C4×D3—21no—
85C2×C4×D3—21no—
86C2×Dic6—21no—
87C2×Dic6D411yescharacteristic
88C3×SD16—22no—
89C2×C3×C8C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C3×C4⋊C4—21no—
92C3×C4⋊C4C2311yescharacteristic
93C3×C4⋊C4C2311yescharacteristic
94C3×C22⋊C4—21no—
95C3×C42—21no—
96C3×C42C2311yescharacteristic
97#217—21no—
98#217—21no—
99#213—22no—
100#213—22no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3D411yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4D411yescharacteristic
116Dic3⋊C4D411yescharacteristic
117C4×Dic3C2311yescharacteristic
118C4×Dic3C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×SD16C2×D311yescharacteristic
122C2×D8—31no—
123C2×D8—31no—
124C4⋊Q8—31no—
125#127—31no—
126#125—31no—
127#125—31no—
128#124—31no—
129#124—31no—
130#124—61no—
131#123—61no—
132#123—61no—
133C22⋊Q8—61no—
134C22⋊Q8—61no—
135C22⋊Q8—61no—
136C4×Q8—31no—
137C4×Q8C2×D311yescharacteristic
138C4×D4—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—61no—
143C4×D4C2×D311yescharacteristic
144C2×C4⋊C4—61no—
145#107—31no—
146#107C2×D311yescharacteristic
147#106—31no—
148C4⋊C8—31no—
149C4⋊C8—31no—
150C4⋊C8—31no—
151#103—31no—
152#103—31no—
153#103C2×D311yescharacteristic
154#102—31no—
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102C2×D311yescharacteristic
161#97—31no—
162#97C2×D311yescharacteristic
163C4×C8—31no—
164C23×C3—21no—
165C22×D3—21no—
166C22×D3—21no—
167C3×Q8—21no—
168C3×Q8—21no—
169C3×D4—21no—
170C3×D4#4112yes—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—81no—
175C2×C3×C4C2×D411yescharacteristic
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2411yescharacteristic
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3—42no—
181C2×Dic3—81no—
182C2×Dic3C2×D411yescharacteristic
183C2×Dic3C2×D411yescharacteristic
184C2×Dic3C2×D411yescharacteristic
185C2×Dic3C2×D411yescharacteristic
186D12—21no—
187D12—21no—
188D12—21no—
189D12—21no—
190D12—42no—
191D12—81no—
192C4×D3—81no—
193C4×D3—81no—
194Dic6—41no—
195Dic6—81no—
196C3×C8—21no—
197C3×C8—21no—
198C3⋊C8—21no—
199C3⋊C8—21no—
200C2×Q8—31no—
201C2×Q8—61no—
202C2×Q8C22×D311yescharacteristic
203C2×D4—31no—
204C2×D4—31no—
205C2×D4—61no—
206C2×D4—61no—
207C2×D4C22×D311yescharacteristic
208C22×C4—21no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213SD16—22no—
214D8—62no—
215D8—62no—
216C2×C8—31no—
217C2×C8—31no—
218C2×C8C22×D311yescharacteristic
219C2×C8C22×D311yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4C22×D311yescharacteristic
232C4⋊C4C22×D311yescharacteristic
233C22⋊C4—21no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C42—21no—
243C42—31no—
244C42—31no—
245C42C22×D311yescharacteristic
246C22×C3—22no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3C22×D411yescharacteristic
250C2×D3—41no—
251C2×D3—41no—
252C2×D3—41no—
253C2×D3—41no—
254C2×D3—41no—
255C2×D3—41no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4C2×#4111yescharacteristic
262C3×C42−1+411yescharacteristic
263Dic3—21no—
264Dic3—21no—
265Dic3—21no—
266Dic3—21no—
267Dic3—41no—
268Dic3—41no—
269C23—21no—
270C23—61no—
271C23—61no—
272Q8—21no—
273Q8—21no—
274Q8—121no—
275Q8—241no—
276D4—21no—
277D4—61no—
278D4—61no—
279D4—61no—
280D4—61no—
281D4—122no—
282D4—241no—
283D4#23512yes—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—81no—
294C2×C4—122no—
295C2×C4—241no—
296C2×C4—241no—
297C2×C4—241no—
298C2×C4D3×D411yescharacteristic
299C2×C4D3×D411yescharacteristic
300C2×C4C23×D311yescharacteristic
301C8—21no—
302C8—21no—
303C8—61no—
304C8—61no—
305C2×C3—42no—
306C2×C3#54811yescharacteristic
307C2×C3#57511yescharacteristic
308C2×C3#57611yescharacteristic
309D3—81no—
310D3—81no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—121no—
315C22—121no—
316C22—121no—
317C22—121no—
318C22—121no—
319C22—121no—
320C22C2×D3×D411yescharacteristic
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—61no—
327C4—61no—
328C4—61no—
329C4—61no—
330C4—121no—
331C4—121no—
332C4C2×#23511yescharacteristic
333C4#101011yescharacteristic
334C3#337011yescharacteristic, complemented, Sylow
335C2—42no—
336C2—241no—
337C2—241no—
338C2#555211yescharacteristic
339C2#577411yescharacteristic
340C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
prime radical#5015C2
hypercenter#440D3
hypocenterC3#3370
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #5585C2semidirect
2. #5539C2semidirect
3. C3×#440C2semidirect
4. #5165C2semidirect
5. #5165C2semidirect
6. #5022C2semidirect
8. #4891C2semidirect
9. #4890C2semidirect
10. #4800C2semidirect
11. #4794C2semidirect
12. #4785C2semidirect
13. #4756C2semidirect
14. #4713C2semidirect
15. #4701C2semidirect
73. #440D3semidirect
334. C3#3370semidirect

Maximal subgroups

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

subgroupindex
#55852
#55392
C3×#4402
#51652
#50222
#50152
#48912
#48902
#48002
#47942
#47852
#47562
#47132
#47012
#33703