Finite Group Databasebeta

← #83648

Subgroups — #83648

Order 384.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#555C211yesmaximal, characteristic, complemented
2#5565C211yesmaximal, characteristic, complemented
3#5087C211yesmaximal, characteristic, complemented
4#5081C211yesmaximal, characteristic, complemented
5#5039C211yesmaximal, characteristic, complemented
6#5038C211yesmaximal, characteristic, complemented
7#5035C211yesmaximal, characteristic, complemented
8#5033C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic, complemented
10#5011C211yesmaximal, characteristic, complemented
11#5009C211yesmaximal, characteristic, complemented
12#5003C211yesmaximal, characteristic, complemented
13#5003C211yesmaximal, characteristic, complemented
14#4998C211yesmaximal, characteristic, complemented
15#4995C211yesmaximal, characteristic, complemented
16#3347—31nomaximal, Sylow
17C3×C4⋊Q8C2211yescharacteristic
18C3×#124—21no—
19C3×#124—21no—
20C3×#121—21no—
21C3×#121C2211yescharacteristic
22C3×#121C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic
24C3×C4×D4C2211yescharacteristic
25C3×#117—21no—
26C3×#117C2211yescharacteristic
27C3×#117C2211yescharacteristic
28#939—21no—
29C2×#213C2211yescharacteristic
30#930C2211yescharacteristic
31#930C2211yescharacteristic
32#926C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34C2×#208C2211yescharacteristic
35C22×C3⋊C8C2211yescharacteristic
36#898—21no—
37#883—21no—
38#67⋊C4—21no—
39C4×D12C2211yescharacteristic
40#870C2211yescharacteristic
41#832C2211yescharacteristic
42#832C2211yescharacteristic
43#832C2211yescharacteristic
44#830C2211yescharacteristic
45#830C2211yescharacteristic
46#809C2211yescharacteristic
47#809C2211yescharacteristic
48#809C2211yescharacteristic
49#808C2211yescharacteristic
50#808C2211yescharacteristic
51#808C2211yescharacteristic
52#808C2211yescharacteristic
53#807C2211yescharacteristic
54#807C2211yescharacteristic
55#807C2211yescharacteristic
56#807C2211yescharacteristic
57#807C2211yescharacteristic
58C4⋊(C3⋊C8)C2211yescharacteristic
59#803C2211yescharacteristic
60#555D311yescharacteristic, complemented
61#553—31no—
62#501—31no—
63#482—31no—
64#482—31no—
65#482—31no—
66#480—31no—
67#478—31no—
68#474—31no—
69#469—31no—
70#466—31no—
71#442—31no—
72#435—31no—
73#428—31no—
74#427—31no—
75C2×C3×Q8—21no—
76C2×C3×D4—21no—
77C2×C3×D4—21no—
78C2×C3×D4D411yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86C2×D12C2311yescharacteristic
87C2×C4×D3—21no—
88C3×C4⋊C4—21no—
89C3×C4⋊C4D411yescharacteristic
90C3×C4⋊C4C2311yescharacteristic
91C3×C4⋊C4C2311yescharacteristic
92C3×C4⋊C4C2311yescharacteristic
93C3×C22⋊C4—21no—
94C3×C22⋊C4—21no—
95C3×C22⋊C4—21no—
96C3×C22⋊C4—21no—
97C3×C22⋊C4—21no—
98C3×C22⋊C4D411yescharacteristic
99C3×C22⋊C4D411yescharacteristic
100C3×C42—21no—
101C3×C42—21no—
102C3×C42C2311yescharacteristic
103#217—21no—
104#213—22no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114C4×Dic3—21no—
115#208—22no—
116C2×C3⋊C8—22no—
117C2×C3⋊C8C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×D8—31no—
122C2×M16—31no—
123C22×C8—31no—
124C4⋊Q8C2×D311yescharacteristic
125#126—61no—
126#125—31no—
127#124—21no—
128#124—21no—
129#123—61no—
130#121—21no—
131#121—31no—
132#121—31no—
133#121—61no—
134#121C2×D311yescharacteristic
135#121C2×D311yescharacteristic
136C4×D4—31no—
137C4×D4—61no—
138C4×D4C2×D311yescharacteristic
139C4×D4C2×D311yescharacteristic
140#117—21no—
141#117—31no—
142#117C2×D311yescharacteristic
143#117C2×D311yescharacteristic
144C2×C4⋊C4—31no—
145#107—31no—
146#107—31no—
147#107—31no—
148#107—31no—
149#107—31no—
150#106—31no—
151#106—31no—
152#106—31no—
153#106—31no—
154C4⋊C8—31no—
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161C22⋊C8—31no—
162C22⋊C8—31no—
163#97—31no—
164C23×C3—21no—
165C23×C3C2×D411yescharacteristic
166C23×C3C2×D411yescharacteristic
167C22×D3—21no—
168C3×Q8—81no—
169C3×D4—21no—
170C3×D4—21no—
171C3×D4—41no—
172C3×D4—81no—
173C3×D4—81no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—42no—
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67—42no—
188#67—81no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—81no—
193D12—21no—
194D12—21no—
195C4×D3—81no—
196C3⋊C8—21no—
197C3⋊C8—21no—
198C3⋊C8—21no—
199C3⋊C8#4112yes—
200C2×Q8—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—31no—
204C2×D4—61no—
205C2×D4—61no—
206C2×D4#6711yescharacteristic
207C2×D4C22×D311yescharacteristic
208C22×C4—21no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4C22×D311yescharacteristic
212C22×C4C22×D311yescharacteristic
213D8—62no—
214M16—62no—
215C2×C8—31no—
216C2×C8—31no—
217C2×C8—31no—
218C2×C8—31no—
219C2×C8—62no—
220C4⋊C4—21no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4#6711yescharacteristic
228C4⋊C4C22×D311yescharacteristic
229C4⋊C4C22×D311yescharacteristic
230C4⋊C4C22×D311yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—21no—
233C22⋊C4—21no—
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4#6711yescharacteristic
241C22⋊C4#6711yescharacteristic
242C42—21no—
243C42—21no—
244C42—61no—
245C42C22×D311yescharacteristic
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3C22×D411yescharacteristic
256C2×D3—41no—
257C2×D3—41no—
258C2×D3—41no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4C2×#4111yescharacteristic
267C3×C42−1+411yescharacteristic
268Dic3—41no—
269Dic3—41no—
270Dic3—41no—
271C23—21no—
272C23—61no—
273C23C2×#6711yescharacteristic
274C23C2×#6711yescharacteristic
275Q8—81no—
276D4—21no—
277D4—21no—
278D4—41no—
279D4—61no—
280D4—61no—
281D4—81no—
282D4—81no—
283D4—122no—
284D4—241no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—241no—
297C2×C4—241no—
298C2×C4C2×#6711yescharacteristic
299C2×C4C2×#6711yescharacteristic
300C2×C4C2×#6711yescharacteristic
301C2×C4C2×#6711yescharacteristic
302C2×C4C23×D311yescharacteristic
303C8—32no—
304C8—61no—
305C8—61no—
306C8—61no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—81no—
310C2×C3#54711yescharacteristic
311C2×C3#57511yescharacteristic
312C2×C3#57611yescharacteristic
313D3—81no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—121no—
324C22—121no—
325C22—121no—
326C22C22×#6711yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—121no—
335C4—121no—
336C4—121no—
337C4D3×#4111yescharacteristic
338C4#101011yescharacteristic
339C3#334711yescharacteristic, complemented, Sylow
340C2—41no—
341C2—41no—
342C2—81no—
343C2—241no—
344C2#582211yescharacteristic
345C2#582411yescharacteristic
346C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#555C2
socleC22×C3C22×D4
hypercenter#555D3
hypocenterC3#3347
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. C3×#555C2semidirect
2. #5565C2semidirect
3. #5087C2semidirect
4. #5081C2semidirect
5. #5039C2semidirect
6. #5038C2semidirect
7. #5035C2semidirect
8. #5033C2semidirect
9. #5014C2semidirect
10. #5011C2semidirect
11. #5009C2semidirect
12. #5003C2semidirect
13. #5003C2semidirect
14. #4998C2semidirect
15. #4995C2semidirect
60. #555D3semidirect
339. C3#3347semidirect

Maximal subgroups

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

subgroupindex
C3×#5552
#55652
#50872
#50812
#50392
#50382
#50352
#50332
#50142
#50112
#50092
#50032
#49982
#49952
#33473