Finite Group Databasebeta

← #82575

Subgroups — #82575

Order 384.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#520C211yesmaximal, characteristic
2#5518C211yesmaximal, characteristic, complemented
3#5239C211yesmaximal, characteristic
4#5239C211yesmaximal, characteristic
5#5031C211yesmaximal, characteristic, complemented
6#5030C211yesmaximal, characteristic, complemented
7C4×#216C211yesmaximal, characteristic, complemented
8#5022C211yesmaximal, characteristic, complemented
9#5019C211yesmaximal, characteristic, complemented
10#5018C211yesmaximal, characteristic, complemented
11C4×#214C211yesmaximal, characteristic, complemented
12#5009C211yesmaximal, characteristic, complemented
13#5000C211yesmaximal, characteristic, complemented
14#4971C211yesmaximal, characteristic, complemented
15#4970C211yesmaximal, characteristic, complemented
16#3120—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C3×C4×Q8C2211yescharacteristic
19C3×C4×Q8C2211yescharacteristic
20C3×C4×D4—21no—
21C3×C4×D4—21no—
22C3×C4×D4C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic
24C3×#117—21no—
25C3×#117C2211yescharacteristic
26C2×C3×C4⋊C4—21no—
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×#216C2211yescharacteristic
29C2×#216C2211yescharacteristic
30C2×#214C2211yescharacteristic
31C2×#214C2211yescharacteristic
32#924C2211yescharacteristic
33#924C2211yescharacteristic
34#889—21no—
35#889—21no—
36#879—21no—
37#879—21no—
38#870C2211yescharacteristic
39#869C2211yescharacteristic
40C4×Dic6C2211yescharacteristic
41C4×Dic6C2211yescharacteristic
42#835C2211yescharacteristic
43#835C2211yescharacteristic
44#832C2211yescharacteristic
45#832C2211yescharacteristic
46#830C2211yescharacteristic
47#830C2211yescharacteristic
48#810C2211yescharacteristic
49#810C2211yescharacteristic
50#810C2211yescharacteristic
51#810C2211yescharacteristic
52#808C2211yescharacteristic
53#808C2211yescharacteristic
54#807C2211yescharacteristic
55#807C2211yescharacteristic
56C4⋊(C3⋊C8)C2211yescharacteristic
57C4⋊(C3⋊C8)C2211yescharacteristic
58#803C2211yescharacteristic
59C4×C3⋊C8C2211yescharacteristic
60#541—31no—
61#520D311yescharacteristic
62#484—31no—
63#483—31no—
64#479—31no—
65#478—31no—
66#462—31no—
67#461—31no—
68#452—31no—
69#452—31no—
70#441—31no—
71#440—31no—
72C4×Q16—31no—
73C4×SD16—31no—
74#433—31no—
75C3×#41—22no—
76C3×#41—22no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4C2311yescharacteristic
85C2×Dic6C2311yescharacteristic
86C2×Dic6C2311yescharacteristic
87C3×C4⋊C4—21no—
88C3×C4⋊C4—21no—
89C3×C4⋊C4—21no—
90C3×C4⋊C4D411yescharacteristic
91C3×C4⋊C4D411yescharacteristic
92C3×C4⋊C4C2311yescharacteristic
93C3×C4⋊C4C2311yescharacteristic
94C3×C22⋊C4—21no—
95C3×C22⋊C4—21no—
96C3×C22⋊C4D411yescharacteristic
97C3×C22⋊C4D411yescharacteristic
98C3×C42—21no—
99C3×C42—21no—
100C3×C42C2311yescharacteristic
101C3×C42C2311yescharacteristic
102#217—21no—
103#217—21no—
104#216—22no—
105#216—22no—
106#214—22no—
107#214—22no—
108C4⋊Dic3—21no—
109C4⋊Dic3—21no—
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×#41C2×D311yescharacteristic
125C2×Q16—31no—
126C2×Q16—31no—
127C2×SD16—31no—
128C2×SD16—31no—
129C4⋊Q8—31no—
130#126—61no—
131#126—61no—
132#125—31no—
133#125—61no—
134#125—61no—
135C22⋊Q8—31no—
136C22⋊Q8—31no—
137C4×Q8—31no—
138C4×Q8—31no—
139C4×Q8C2×D311yescharacteristic
140C4×Q8C2×D311yescharacteristic
141C4×D4—21no—
142C4×D4—21no—
143C4×D4C2×D311yescharacteristic
144C4×D4C2×D311yescharacteristic
145#117—21no—
146#117C2×D311yescharacteristic
147C2×C4⋊C4—21no—
148C2×C4⋊C4C2×D311yescharacteristic
149#107—31no—
150#107—31no—
151#106—31no—
152#106—31no—
153C4⋊C8—31no—
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163C22⋊C8—31no—
164C22⋊C8—31no—
165#97—31no—
166C4×C8—31no—
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C3×Q8—21no—
170C3×Q8#4112yes—
171C3×D4—24no—
172C3×D4—21no—
173C3×D4#4112yes—
174C2×C3×C4—24no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—81no—
181C2×C3×C4—81no—
182C2×C3×C4—81no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193Dic6—21no—
194Dic6—21no—
195Dic6—21no—
196Dic6—21no—
197C3⋊C8—21no—
198C3⋊C8—21no—
199C3⋊C8—21no—
200C3⋊C8—21no—
201#41—22no—
202#41—22no—
203C2×Q8—31no—
204C2×Q8—31no—
205C2×Q8C22×D311yescharacteristic
206C2×D4—21no—
207C2×D4C22×D311yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4C22×D311yescharacteristic
213Q16—62no—
214Q16—62no—
215SD16—62no—
216SD16—62no—
217C2×C8—31no—
218C2×C8—31no—
219C2×C8—31no—
220C2×C8—31no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
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⋊C4#6711yescharacteristic
235C4⋊C4#6711yescharacteristic
236C4⋊C4C22×D311yescharacteristic
237C4⋊C4C22×D311yescharacteristic
238C22⋊C4—21no—
239C22⋊C4—21no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4#6711yescharacteristic
243C22⋊C4#6711yescharacteristic
244C42—21no—
245C42—21no—
246C42—61no—
247C42—61no—
248C42C22×D311yescharacteristic
249C42C22×D311yescharacteristic
250C22×C3—21no—
251C22×C3—22no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3C22×D411yescharacteristic
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4C2×#4111yescharacteristic
266C3×C4C2×#4111yescharacteristic
267Dic3—41no—
268Dic3—41no—
269Dic3—41no—
270Dic3—41no—
271C23—21no—
272C23C2×#6711yescharacteristic
273Q8—21no—
274Q8—61no—
275Q8—61no—
276Q8—61no—
277Q8—61no—
278Q8#23512yes—
279D4—24no—
280D4—21no—
281D4#23512yes—
282C2×C4—24no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4C2×#6711yescharacteristic
296C2×C4C2×#6711yescharacteristic
297C2×C4C2×#6711yescharacteristic
298C2×C4C2×#6711yescharacteristic
299C2×C4C2×#6711yescharacteristic
300C2×C4C23×D311yescharacteristic
301C8—61no—
302C8—61no—
303C8—61no—
304C8—61no—
305C2×C3—41no—
306C2×C3—42no—
307C2×C3#52511yescharacteristic
308C2×C3#57711yescharacteristic
309C2×C3#57811yescharacteristic
310C22—21no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22C22×#6711yescharacteristic
317C4—21no—
318C4—22no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—121no—
326C4—121no—
327C4—121no—
328C4—121no—
329C4C2×#23511yescharacteristic
330C4C2×#23511yescharacteristic
331C3#312011yescharacteristic, complemented, Sylow
332C2—41no—
333C2—42no—
334C2#579111yescharacteristic
335C2#583611yescharacteristic
336C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#520C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
hypercenter#520D3
hypocenterC3#3120
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
2. #5518C2semidirect
5. #5031C2semidirect
6. #5030C2semidirect
7. C4×#216C2semidirect
8. #5022C2semidirect
9. #5019C2semidirect
10. #5018C2semidirect
11. C4×#214C2semidirect
12. #5009C2semidirect
13. #5000C2semidirect
14. #4971C2semidirect
15. #4970C2semidirect
331. C3#3120semidirect

Maximal subgroups

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

subgroupindex
C3×#5202
#55182
#52392
#50312
#50302
C4×#2162
#50222
#50192
#50182
C4×#2142
#50092
#50002
#49712
#49702
#31203