Finite Group Databasebeta

← #81361

Subgroups — #81361

Order 384.

332 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5619C211yesmaximal, characteristic
2#5519C211yesmaximal, characteristic, complemented
3C3×#482C211yesmaximal, characteristic, complemented
4#5034C211yesmaximal, characteristic, complemented
5#5003C211yesmaximal, characteristic, complemented
6#4891C211yesmaximal, characteristic, complemented
7#4878C211yesmaximal, characteristic, complemented
8#4870C211yesmaximal, characteristic, complemented
9#4856C211yesmaximal, characteristic, complemented
10#4766C211yesmaximal, characteristic, complemented
11#4764C211yesmaximal, characteristic, complemented
12#4760C211yesmaximal, characteristic, complemented
13#4759C211yesmaximal, characteristic, complemented
14#4736C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3241—31nomaximal, Sylow
17C3×#121C2211yescharacteristic
18C3×#117C2211yescharacteristic
19#940C2211yescharacteristic
20#939—21no—
21#938—21no—
22D4×Dic3C2211yescharacteristic
23#930C2211yescharacteristic
24#926C2211yescharacteristic
25C2×C4⋊Dic3C2211yescharacteristic
26#893C2211yescharacteristic
27#890C2211yescharacteristic
28#890C2211yescharacteristic
29#890C2211yescharacteristic
30#888C2211yescharacteristic
31#888C2211yescharacteristic
32#886—21no—
33#883—21no—
34#878—21no—
35#878—21no—
36#870—21no—
37#869—21no—
38C3×#107C2211yescharacteristic
39C3×#106C2211yescharacteristic
40C3×#102C2211yescharacteristic
41C3×#102C2211yescharacteristic
42C3×C22⋊C8C2211yescharacteristic
43#832C2211yescharacteristic
44#832C2211yescharacteristic
45#830C2211yescharacteristic
46#821C2211yescharacteristic
47#821C2211yescharacteristic
48#818C2211yescharacteristic
49#818C2211yescharacteristic
50#815C2211yescharacteristic
51Dic3⋊C8C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53C8×Dic3C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#808C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#544—31no—
61#540—31no—
62#501—31no—
63#499—31no—
64#490—31no—
65#486—31no—
66#482—31no—
67#482D311yescharacteristic, complemented
68#481—31no—
69#480—31no—
70#478—31no—
71#478—31no—
72#476—31no—
73#474—31no—
74#433—31no—
75C2×C3×D4—21no—
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67—21no—
80C2×#67—21no—
81C22×Dic3—21no—
82C22×Dic3—21no—
83C2×D12C2311yescharacteristic
84C2×C4×D3—21no—
85C2×Dic6—21no—
86C2×Dic6—21no—
87C2×C3×C8C2311yescharacteristic
88C2×C3×C8C2311yescharacteristic
89C3×C4⋊C4C2311yescharacteristic
90C3×C4⋊C4C2311yescharacteristic
91C3×C4⋊C4C2311yescharacteristic
92C3×C22⋊C4—21no—
93C3×C22⋊C4—21no—
94C3×C42—21no—
95#217—21no—
96#217D411yescharacteristic
97#217D411yescharacteristic
98(C2×D3)⋊C4—21no—
99(C2×D3)⋊C4—21no—
100C4⋊Dic3—21no—
101C4⋊Dic3—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3—21no—
104C4⋊Dic3D411yescharacteristic
105C4⋊Dic3D411yescharacteristic
106C4⋊Dic3C2311yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114C4×Dic3C2311yescharacteristic
115C4×Dic3C2311yescharacteristic
116C2×C3⋊C8C2311yescharacteristic
117C2×C3⋊C8C2311yescharacteristic
118C4⋊Q8—31no—
119C4⋊Q8—31no—
120C4⋊Q8—61no—
121#127—31no—
122#125—31no—
123#125—31no—
124#125—31no—
125#125—61no—
126#123—61no—
127#123—61no—
128C22⋊Q8—61no—
129C22⋊Q8—61no—
130#121—31no—
131#121—61no—
132#121—61no—
133#121C2×D311yescharacteristic
134C4×D4—31no—
135C4×D4—31no—
136#117—31no—
137#117C2×D311yescharacteristic
138C2×C4⋊C4—31no—
139#107—31no—
140#107—31no—
141#107—31no—
142#107C2×D311yescharacteristic
143#106—31no—
144#106—31no—
145#106—31no—
146#106C2×D311yescharacteristic
147C4⋊C8—31no—
148C4⋊C8—31no—
149#102—31no—
150#102—31no—
151#102—31no—
152#102—31no—
153#102—31no—
154#102—31no—
155#102C2×D311yescharacteristic
156#102C2×D311yescharacteristic
157C22⋊C8—31no—
158C22⋊C8C2×D311yescharacteristic
159#97—31no—
160C4×C8—31no—
161C23×C3—21no—
162C23×C3C2×D411yescharacteristic
163C22×D3—21no—
164C3×D4—21no—
165C3×D4—21no—
166C3×D4—81no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4C2×D411yescharacteristic
173C2×C3×C4C2411yescharacteristic
174#67—81no—
175#67—81no—
176#67—81no—
177C2×Dic3—21no—
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3—81no—
181C2×Dic3—81no—
182C2×Dic3C2×D411yescharacteristic
183C2×Dic3C2×D411yescharacteristic
184C2×Dic3C2×D411yescharacteristic
185C2×Dic3C2×D411yescharacteristic
186D12—21no—
187D12—21no—
188C4×D3—81no—
189Dic6—81no—
190Dic6—81no—
191C3×C8—21no—
192C3×C8—21no—
193C3⋊C8—21no—
194C3⋊C8—21no—
195C2×Q8—61no—
196C2×Q8—61no—
197C2×D4—21no—
198C2×D4—31no—
199C2×D4—61no—
200C2×D4—61no—
201C2×D4—61no—
202C2×D4C22×D311yescharacteristic
203C22×C4—61no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4C22×D311yescharacteristic
207C2×C8—31no—
208C2×C8—31no—
209C2×C8C22×D311yescharacteristic
210C2×C8C22×D311yescharacteristic
211C4⋊C4—31no—
212C4⋊C4—31no—
213C4⋊C4—31no—
214C4⋊C4—31no—
215C4⋊C4—31no—
216C4⋊C4—61no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4C22×D311yescharacteristic
226C4⋊C4C22×D311yescharacteristic
227C4⋊C4C22×D311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—21no—
230C22⋊C4—31no—
231C22⋊C4—31no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C42—21no—
236C42—31no—
237C42—31no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3C22×D411yescharacteristic
245C2×D3—41no—
246C2×D3—41no—
247C2×D3—41no—
248C3×C4—21no—
249C3×C4—41no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C42−1+411yescharacteristic
253C3×C42−1+411yescharacteristic
254Dic3—21no—
255Dic3—21no—
256Dic3—21no—
257Dic3—21no—
258Dic3—41no—
259Dic3—41no—
260Dic3—41no—
261C23—21no—
262C23—61no—
263C23D3×D411yescharacteristic
264Q8—241no—
265Q8—241no—
266D4—21no—
267D4—21no—
268D4—61no—
269D4—61no—
270D4—81no—
271D4—241no—
272D4—241no—
273D4—241no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—241no—
287C2×C4—241no—
288C2×C4—241no—
289C2×C4D3×D411yescharacteristic
290C2×C4C23×D311yescharacteristic
291C8—21no—
292C8—21no—
293C8—61no—
294C8—61no—
295C2×C3—41no—
296C2×C3—81no—
297C2×C3#53611yescharacteristic
298C2×C3#57611yescharacteristic
299C2×C3#57711yescharacteristic
300D3—81no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—121no—
308C22—121no—
309C22—121no—
310C22C2×D3×D411yescharacteristic
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—61no—
316C4—61no—
317C4—61no—
318C4—61no—
319C4—121no—
320C4—121no—
321C4—121no—
322C4#101011yescharacteristic
323C4#101011yescharacteristic
324C3#324111yescharacteristic, complemented, Sylow
325C2—41no—
326C2—81no—
327C2—241no—
328C2#564811yescharacteristic
329C2#576911yescharacteristic
330C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#482C2
socleC22×C3C22×D4
prime radical#5619C2
hypercenter#482D3
hypocenterC3#3241
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #5519C2semidirect
3. C3×#482C2semidirect
4. #5034C2semidirect
5. #5003C2semidirect
6. #4891C2semidirect
7. #4878C2semidirect
8. #4870C2semidirect
9. #4856C2semidirect
10. #4766C2semidirect
11. #4764C2semidirect
12. #4760C2semidirect
13. #4759C2semidirect
14. #4736C2semidirect
15. #4720C2semidirect
67. #482D3semidirect
324. C3#3241semidirect

Maximal subgroups

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

subgroupindex
#56192
#55192
C3×#4822
#50342
#50032
#48912
#48782
#48702
#48562
#47662
#47642
#47602
#47592
#47362
#47202
#32413