Finite Group Databasebeta

← #82568

Subgroups — #82568

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#520C211yesmaximal, characteristic, complemented
2#5535C211yesmaximal, characteristic, complemented
3#5238C211yesmaximal, characteristic, complemented
4#5238C211yesmaximal, characteristic, complemented
5#5027C211yesmaximal, characteristic, complemented
6#5026C211yesmaximal, characteristic, complemented
7C4×#215C211yesmaximal, characteristic, complemented
8#5022C211yesmaximal, characteristic, complemented
9#5015C211yesmaximal, characteristic, complemented
10#5014C211yesmaximal, characteristic, complemented
11C4×#213C211yesmaximal, characteristic, complemented
12#5009C211yesmaximal, characteristic, complemented
13#5000C211yesmaximal, characteristic, complemented
14#4968C211yesmaximal, characteristic, complemented
15#4967C211yesmaximal, characteristic, complemented
16#3119—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×#215C2211yescharacteristic
29C2×#215C2211yescharacteristic
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#930C2211yescharacteristic
33#930C2211yescharacteristic
34#894—21no—
35#894—21no—
36#884—21no—
37#884—21no—
38#874C2211yescharacteristic
39C4×D12C2211yescharacteristic
40C4×D12C2211yescharacteristic
41#870C2211yescharacteristic
42#835C2211yescharacteristic
43#835C2211yescharacteristic
44#832C2211yescharacteristic
45#832C2211yescharacteristic
46#830C2211yescharacteristic
47#830C2211yescharacteristic
48#809C2211yescharacteristic
49#809C2211yescharacteristic
50#809C2211yescharacteristic
51#809C2211yescharacteristic
52#808C2211yescharacteristic
53#808C2211yescharacteristic
54#807C2211yescharacteristic
55#807C2211yescharacteristic
56C4⋊(C3⋊C8)C2211yescharacteristic
57C4⋊(C3⋊C8)C2211yescharacteristic
58#803C2211yescharacteristic
59C4×C3⋊C8C2211yescharacteristic
60#540—31no—
61#520D311yescharacteristic, complemented
62#481—31no—
63#480—31no—
64#479—31no—
65#478—31no—
66#460—31no—
67#459—31no—
68#449—31no—
69#449—31no—
70#442—31no—
71#440—31no—
72C4×SD16—31no—
73C4×D8—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×#67—21no—
86C2×#67—21no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12C2311yescharacteristic
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4D411yescharacteristic
97C3×C4⋊C4D411yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102C3×C22⋊C4D411yescharacteristic
103C3×C22⋊C4D411yescharacteristic
104C3×C42—21no—
105C3×C42—21no—
106C3×C42C2311yescharacteristic
107C3×C42C2311yescharacteristic
108#215—22no—
109#215—22no—
110#213—22no—
111#213—22no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×#41C2×D311yescharacteristic
125C2×SD16—31no—
126C2×SD16—31no—
127C2×D8—31no—
128C2×D8—31no—
129#127—31no—
130#125—31no—
131#123—61no—
132#123—61no—
133#121—31no—
134#121—31no—
135#121—61no—
136#121—61no—
137C4×Q8C2×D311yescharacteristic
138C4×Q8C2×D311yescharacteristic
139C4×D4—21no—
140C4×D4—21no—
141C4×D4—31no—
142C4×D4—31no—
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#102—31no—
158#102—31no—
159#102—31no—
160#102—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
169C22×D3—21no—
170C22×D3—21no—
171C3×Q8—21no—
172C3×Q8#4112yes—
173C3×D4—24no—
174C3×D4—21no—
175C3×D4#4112yes—
176C2×C3×C4—24no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—81no—
183C2×C3×C4—81no—
184C2×C3×C4—81no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2411yescharacteristic
191#67—81no—
192#67—81no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195D12—21no—
196D12—21no—
197D12—21no—
198D12—21no—
199D12—81no—
200D12—81no—
201C4×D3—81no—
202C4×D3—81no—
203C3⋊C8—21no—
204C3⋊C8—21no—
205C3⋊C8—21no—
206C3⋊C8—21no—
207#41—22no—
208#41—22no—
209C2×Q8C22×D311yescharacteristic
210C2×D4—21no—
211C2×D4—31no—
212C2×D4—31no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4C22×D311yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4C22×D311yescharacteristic
225SD16—62no—
226SD16—62no—
227D8—62no—
228D8—62no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—31no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4#6711yescharacteristic
241C4⋊C4#6711yescharacteristic
242C4⋊C4C22×D311yescharacteristic
243C4⋊C4C22×D311yescharacteristic
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4#6711yescharacteristic
251C22⋊C4#6711yescharacteristic
252C42—21no—
253C42—21no—
254C42C22×D311yescharacteristic
255C42C22×D311yescharacteristic
256C22×C3—21no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3C22×D411yescharacteristic
263C2×D3—41no—
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4C2×#4111yescharacteristic
278C3×C4C2×#4111yescharacteristic
279Dic3—41no—
280Dic3—41no—
281C23—21no—
282C23—61no—
283C23—61no—
284C23C2×#6711yescharacteristic
285Q8—21no—
286Q8#23512yes—
287D4—24no—
288D4—21no—
289D4—61no—
290D4—61no—
291D4—61no—
292D4—61no—
293D4—241no—
294D4—241no—
295D4—241no—
296D4—241no—
297D4#23512yes—
298C2×C4—24no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—241no—
310C2×C4—241no—
311C2×C4C2×#6711yescharacteristic
312C2×C4C2×#6711yescharacteristic
313C2×C4C2×#6711yescharacteristic
314C2×C4C2×#6711yescharacteristic
315C2×C4C2×#6711yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—61no—
318C8—61no—
319C8—61no—
320C8—61no—
321C2×C3—41no—
322C2×C3—42no—
323C2×C3#52511yescharacteristic
324C2×C3#57611yescharacteristic
325C2×C3#57711yescharacteristic
326D3—81no—
327D3—81no—
328C22—21no—
329C22—22no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—121no—
335C22—121no—
336C22—121no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22C22×#6711yescharacteristic
341C4—21no—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—121no—
350C4—121no—
351C4C2×#23511yescharacteristic
352C4C2×#23511yescharacteristic
353C3#311911yescharacteristic, complemented, Sylow
354C2—41no—
355C2—42no—
356C2—241no—
357C2—241no—
358C2#579111yescharacteristic
359C2#583511yescharacteristic
360C2#583711yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#520C2
socleC22×C3C22×D4
hypercenter#520D3
hypocenterC3#3119
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
1. C3×#520C2semidirect
2. #5535C2semidirect
3. #5238C2semidirect
4. #5238C2semidirect
5. #5027C2semidirect
6. #5026C2semidirect
7. C4×#215C2semidirect
8. #5022C2semidirect
9. #5015C2semidirect
10. #5014C2semidirect
11. C4×#213C2semidirect
12. #5009C2semidirect
13. #5000C2semidirect
14. #4968C2semidirect
15. #4967C2semidirect
61. #520D3semidirect
353. C3#3119semidirect

Maximal subgroups

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

subgroupindex
C3×#5202
#55352
#52382
#50272
#50262
C4×#2152
#50222
#50152
#50142
C4×#2132
#50092
#50002
#49682
#49672
#31193