Finite Group Databasebeta

← #79158

Subgroups — #79158

Order 384.

376 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#5513C211yesmaximal, characteristic, complemented
2#5507C211yesmaximal, characteristic
3C2×C3×#107C211yesmaximal, characteristic, complemented
4#5114C211yesmaximal, characteristic, complemented
5#5111C211yesmaximal, characteristic, complemented
6#4971C211yesmaximal, characteristic, complemented
7#4967C211yesmaximal, characteristic, complemented
8#4891C211yesmaximal, characteristic, complemented
9#4889C211yesmaximal, characteristic, complemented
10#4886C211yesmaximal, characteristic, complemented
11#4885C211yesmaximal, characteristic, complemented
12#4884C211yesmaximal, characteristic, complemented
13#4881C211yesmaximal, characteristic, complemented
14#4876C211yesmaximal, characteristic, complemented
15#4875C211yesmaximal, characteristic, complemented
16#3343—31nomaximal, Sylow
17C2×#235—21no—
18C22×C3×C8C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21#930C2211yescharacteristic
22#929—21no—
23#929—21no—
24C4×#67C2211yescharacteristic
25#926—21no—
26#924C2211yescharacteristic
27C2×#204C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic
29#898—21no—
30#897C2211yescharacteristic
31#896—21no—
32#895C2211yescharacteristic
33#894—21no—
34#893C2211yescharacteristic
35#892C2211yescharacteristic
36D3×C4⋊C4C2211yescharacteristic
37#890C2211yescharacteristic
38#889—21no—
39#888C2211yescharacteristic
40#887C2211yescharacteristic
41C3×#107C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#107C2211yescharacteristic
45#830C2211yescharacteristic
46#821C2211yescharacteristic
47(C2×D3)⋊C8C2211yescharacteristic
48#817C2211yescharacteristic
49#816C2211yescharacteristic
50#815C2211yescharacteristic
51Dic3⋊C8C2211yescharacteristic
52#810C2211yescharacteristic
53#810C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#808C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59#807C2211yescharacteristic
60#552—31no—
61#547—31no—
62#501—31no—
63#484—31no—
64#483—31no—
65#482—31no—
66#480—31no—
67#478—31no—
68#477—31no—
69#468—31no—
70#465—31no—
71#440—31no—
72#435—31no—
73#428—31no—
74C2×#107D311yescharacteristic, complemented
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67D411yescharacteristic
80#235—44no—
81C2×D12—21no—
82C2×D12C2311yescharacteristic
83C2×C4×D3—21no—
84C2×C4×D3—21no—
85C2×C4×D3C2311yescharacteristic
86C2×Dic6—21no—
87C2×Dic6C2311yescharacteristic
88C2×C3×C8—22no—
89C2×C3×C8C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C3×C4⋊C4—21no—
92C3×C4⋊C4—21no—
93C3×C4⋊C4C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97#217—21no—
98#217D411yescharacteristic
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4D411yescharacteristic
105C4⋊Dic3—21no—
106C4⋊Dic3—21no—
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4D411yescharacteristic
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120#204—22no—
121ΓL(1,𝔽25)—22no—
122C2×#41—61no—
123C2×SD16—31no—
124C2×M16—31no—
125C22×C8C2×D311yescharacteristic
126C4⋊Q8—31no—
127#126—61no—
128#125—31no—
129#125—61no—
130#123—61no—
131#123—61no—
132#123—61no—
133C22⋊Q8—31no—
134C22⋊Q8—31no—
135C22⋊Q8—61no—
136#121—31no—
137#121—31no—
138C4×Q8—31no—
139C4×D4—31no—
140C4×D4—31no—
141#117—31no—
142#117—61no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4C2×D311yescharacteristic
145C2×C4⋊C4C2×D311yescharacteristic
146#107—31no—
147#107—31no—
148#107C2×D311yescharacteristic
149#107C2×D311yescharacteristic
150#107C2×D311yescharacteristic
151#107C2×D311yescharacteristic
152#106—31no—
153#106—31no—
154#106—31no—
155C4⋊C8—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162C22⋊C8—31no—
163C22⋊C8—31no—
164#97—31no—
165C23×C3C2×D411yescharacteristic
166C22×D3—21no—
167C22×D3C2×D411yescharacteristic
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2411yescharacteristic
178#67—22no—
179#67—42no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3C2×D411yescharacteristic
184C2×Dic3C2×D411yescharacteristic
185C2×Dic3C2×D411yescharacteristic
186D12—21no—
187D12—21no—
188D12—42no—
189C4×D3—21no—
190C4×D3—21no—
191C4×D3—42no—
192C4×D3—42no—
193Dic6—21no—
194Dic6—21no—
195Dic6—42no—
196C3×C8—21no—
197C3×C8#4112yescomplemented
198C3⋊C8—21no—
199C3⋊C8—21no—
200#41—124no—
201C2×Q8—31no—
202C2×Q8—61no—
203C2×D4—31no—
204C2×D4—31no—
205C2×D4—61no—
206C2×D4—61no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—31no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4C22×D311yescharacteristic
213SD16—62no—
214M16—62no—
215C2×C8—22no—
216C2×C8—31no—
217C2×C8—31no—
218C2×C8C22×D311yescharacteristic
219C2×C8C22×D311yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—31no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4C22×D311yescharacteristic
233C4⋊C4C22×D311yescharacteristic
234C4⋊C4C22×D311yescharacteristic
235C4⋊C4C22×D311yescharacteristic
236C22⋊C4—31no—
237C22⋊C4—31no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C42—31no—
245C42—61no—
246C42—61no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3AGL(1,ℤ/8)12yes—
250C22×C3C22×D411yescharacteristic
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—21no—
254C2×D3—41no—
255C2×D3—41no—
256C2×D3—41no—
257C3×C4—21no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4C2×#4111yescharacteristic
263C3×C42−1+411yescharacteristic
264Dic3—21no—
265Dic3—21no—
266Dic3—21no—
267Dic3—41no—
268Dic3—41no—
269Dic3—41no—
270C23—31no—
271C23—61no—
272C23D3×D411yescharacteristic
273Q8—61no—
274Q8—61no—
275Q8—122no—
276D4—62no—
277D4—61no—
278D4—61no—
279D4—122no—
280D4—122no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4D3×D411yescharacteristic
300C2×C4C23×D311yescharacteristic
301C8—21no—
302C8—61no—
303C8—61no—
304C8#23512yescomplemented
305C2×C3—22no—
306C2×C3#54711yescharacteristic
307C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
308C2×C3#57811yescharacteristic
309D3—41no—
310D3—81no—
311C22—21no—
312C22—21no—
313C22—61no—
314C22—61no—
315C22—61no—
316C22—121no—
317C22—121no—
318C22—121no—
319C22#91112yes—
320C22C2×D3×D411yescharacteristic
321C4—21no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—61no—
327C4—61no—
328C4—61no—
329C4—121no—
330C4—121no—
331C4—121no—
332C4C2×#23511yescharacteristic
333C4#100711yescharacteristic
334C3#334311yescharacteristic, complemented, Sylow
335C2—22no—
336C2—121no—
337C2—241no—
338C2#550711yescharacteristic
339C2C2×#91111yescharacteristic
340C2#576611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#107C2
socleC22×C3C22×D4
prime radical#5507C2
hypercenterC2×#107D3
hypocenterC3#3343
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
1. #5513C2semidirect
3. C2×C3×#107C2semidirect
4. #5114C2semidirect
5. #5111C2semidirect
6. #4971C2semidirect
7. #4967C2semidirect
8. #4891C2semidirect
9. #4889C2semidirect
10. #4886C2semidirect
11. #4885C2semidirect
12. #4884C2semidirect
13. #4881C2semidirect
14. #4876C2semidirect
15. #4875C2semidirect
74. C2×#107D3semidirect
197. C3×C8#41semidirect
304. C8#235semidirect
334. C3#3343semidirect

Maximal subgroups

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

subgroupindex
#55132
#55072
C2×C3×#1072
#51142
#51112
#49712
#49672
#48912
#48892
#48862
#48852
#48842
#48812
#48762
#48752
#33433