Finite Group Databasebeta

← #80651

Subgroups — #80651

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#5642C211yesmaximal, characteristic, complemented
2#5611C211yesmaximal, characteristic
3C3×#465C211yesmaximal, characteristic, complemented
4#5174C211yesmaximal, characteristic, complemented
5#5164C211yesmaximal, characteristic, complemented
6#5111C211yesmaximal, characteristic, complemented
7C2×#818C211yesmaximal, characteristic, complemented
8#5043C211yesmaximal, characteristic, complemented
9#5033C211yesmaximal, characteristic, complemented
10#4861C211yesmaximal, characteristic, complemented
11#4856C211yesmaximal, characteristic, complemented
12#4811C211yesmaximal, characteristic, complemented
13#4793C211yesmaximal, characteristic, complemented
14#4784C211yesmaximal, characteristic, complemented
15#4766C211yesmaximal, characteristic, complemented
16#3343—31nomaximal, Sylow
17C2×#237—21no—
18C2×C3×SD16C2211yescharacteristic
19C22×C3×C8C2211yescharacteristic
20C3×C22⋊Q8C2211yescharacteristic
21C3×#121C2211yescharacteristic
22#946C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24#938C2211yescharacteristic
25#935—21no—
26D4×Dic3C2211yescharacteristic
27C4×#67C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×ΓL(1,𝔽25)C2211yescharacteristic
31#898—21no—
32#892C2211yescharacteristic
33D3×C4⋊C4C2211yescharacteristic
34#890—21no—
35#890C2211yescharacteristic
36#888C2211yescharacteristic
37#886—21no—
38#883—21no—
39#878—21no—
40#877—21no—
41C3×#106C2211yescharacteristic
42C3×#103C2211yescharacteristic
43C3×#102C2211yescharacteristic
44#835C2211yescharacteristic
45#835C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#830C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50#818C2211yescharacteristic
51#818C2211yescharacteristic
52#818C2211yescharacteristic
53#818C2211yescharacteristic
54#815C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
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#465D311yescharacteristic, complemented
71#440—31no—
72#435—31no—
73#428—31no—
74C2×#107—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67D411yescharacteristic
81C22×Dic3—21no—
82C22×Dic3—21no—
83C22×Dic3—21no—
84#237—44no—
85C2×C4×D3—21no—
86C2×C4×D3C2311yescharacteristic
87C2×Dic6—21no—
88C3×SD16—22no—
89C2×C3×C8—22no—
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C3×C4⋊C4—21no—
93C3×C4⋊C4C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C22⋊C4—21no—
96C3×C22⋊C4—21no—
97#217—21no—
98#217—21no—
99#217D411yescharacteristic
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4D411yescharacteristic
103C4⋊Dic3—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3—21no—
106C4⋊Dic3—21no—
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4D411yescharacteristic
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121ΓL(1,𝔽25)—22no—
122C2×#41—61no—
123C2×SD16C2×D311yescharacteristic
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—61no—
135C22⋊Q8C2×D311yescharacteristic
136#121—31no—
137#121C2×D311yescharacteristic
138C4×Q8—31no—
139C4×D4—31no—
140C4×D4—31no—
141#117—31no—
142#117—61no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146#107—31no—
147#107—31no—
148#107—31no—
149#107—31no—
150#107—31no—
151#107—31no—
152#106—31no—
153#106—31no—
154#106C2×D311yescharacteristic
155C4⋊C8—31no—
156#103—31no—
157#103—31no—
158#103C2×D311yescharacteristic
159#102—31no—
160#102—31no—
161#102C2×D311yescharacteristic
162C22⋊C8—31no—
163C22⋊C8—31no—
164#97—31no—
165C23×C3—21no—
166C23×C3C2×D411yescharacteristic
167C22×D3C2×D411yescharacteristic
168C3×Q8—21no—
169C3×Q8—21no—
170C3×D4—21no—
171C3×D4—21no—
172C3×D4—42no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2411yescharacteristic
180#67—22no—
181#67—42no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C2×Dic3—42no—
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C4×D3—21no—
193C4×D3—21no—
194C4×D3—42no—
195Dic6—42no—
196C3×C8—21no—
197C3×C8#4112yescomplemented
198C3⋊C8—21no—
199C3⋊C8—21no—
200#41—124no—
201C2×Q8—61no—
202C2×Q8C22×D311yescharacteristic
203C2×D4—21no—
204C2×D4—31no—
205C2×D4—61no—
206C2×D4C22×D311yescharacteristic
207C22×C4—31no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4C22×D311yescharacteristic
213SD16—22no—
214M16—62no—
215C2×C8—22no—
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⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4C22×D311yescharacteristic
235C4⋊C4C22×D311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—31no—
239C22⋊C4—31no—
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×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3AGL(1,ℤ/8)12yes—
253C22×C3C22×D411yescharacteristic
254C2×D3—21no—
255C2×D3—21no—
256C2×D3—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—41no—
267Dic3—41no—
268Dic3—41no—
269Dic3—41no—
270C23—21no—
271C23—31no—
272C23D3×D411yescharacteristic
273Q8—21no—
274Q8—21no—
275Q8—122no—
276D4—21no—
277D4—21no—
278D4—42no—
279D4—62no—
280D4—122no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4D3×D411yescharacteristic
300C2×C4C23×D311yescharacteristic
301C8—21no—
302C8—61no—
303C8—61no—
304C8#23712yescomplemented
305C2×C3—22no—
306C2×C3—81no—
307C2×C3#54711yescharacteristic
308C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
309C2×C3#57811yescharacteristic
310D3—41no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—61no—
317C22—61no—
318C22—61no—
319C22#91412yes—
320C22C2×D3×D411yescharacteristic
321C4—21no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—61no—
326C4—61no—
327C4—61no—
328C4—121no—
329C4—121no—
330C4—121no—
331C4—121no—
332C4C2×#23711yescharacteristic
333C4#101011yescharacteristic
334C3#334311yescharacteristic, complemented, Sylow
335C2—22no—
336C2—81no—
337C2—121no—
338C2#561111yescharacteristic
339C2C2×#91411yescharacteristic
340C2#577111yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
prime radical#5611C2
hypercenter#465D3
hypocenterC3#3343
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #5642C2semidirect
3. C3×#465C2semidirect
4. #5174C2semidirect
5. #5164C2semidirect
6. #5111C2semidirect
7. C2×#818C2semidirect
8. #5043C2semidirect
9. #5033C2semidirect
10. #4861C2semidirect
11. #4856C2semidirect
12. #4811C2semidirect
13. #4793C2semidirect
14. #4784C2semidirect
15. #4766C2semidirect
70. #465D3semidirect
197. C3×C8#41semidirect
304. C8#237semidirect
334. C3#3343semidirect

Maximal subgroups

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

subgroupindex
#56422
#56112
C3×#4652
#51742
#51642
#51112
C2×#8182
#50432
#50332
#48612
#48562
#48112
#47932
#47842
#47662
#33433