Finite Group Databasebeta

← #81649

Subgroups — #81649

Order 384.

332 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5688C211yesmaximal, characteristic, complemented
2#5680C211yesmaximal, characteristic
3C3×#488C211yesmaximal, characteristic, complemented
4#5065C211yesmaximal, characteristic, complemented
5#5056C211yesmaximal, characteristic, complemented
6#4815C211yesmaximal, characteristic, complemented
7#4814C211yesmaximal, characteristic, complemented
8#4793C211yesmaximal, characteristic, complemented
9#4791C211yesmaximal, characteristic, complemented
10#4784C211yesmaximal, characteristic
11#4779C211yesmaximal, characteristic
12#4761C211yesmaximal, characteristic, complemented
13#4759C211yesmaximal, characteristic, complemented
14#4685C211yesmaximal, characteristic, complemented
15#4683C211yesmaximal, characteristic, complemented
16#3233—31nomaximal, Sylow
17C3×#125C2211yescharacteristic
18C3×#124C2211yescharacteristic
19#946C2211yescharacteristic
20Q8×Dic3C2211yescharacteristic
21#938C2211yescharacteristic
22D4×Dic3C2211yescharacteristic
23#896—21no—
24#896—21no—
25#892C2211yescharacteristic
26#892C2211yescharacteristic
27#888C2211yescharacteristic
28#888C2211yescharacteristic
29#887—21no—
30#887—21no—
31#883—21no—
32#67⋊C4—21no—
33#879—21no—
34#877—21no—
35C42×D3C2211yescharacteristic
36#870C2211yescharacteristic
37#870C2211yescharacteristic
38C3×#103C2211yescharacteristic
39C3×#103C2211yescharacteristic
40C3×#102C2211yescharacteristic
41C3×#102C2211yescharacteristic
42C3×C4×C8C2211yescharacteristic
43#835C2211yescharacteristic
44#835C2211yescharacteristic
45#832C2211yescharacteristic
46#832C2211yescharacteristic
47(C2×D3)⋊C8C2211yescharacteristic
48(C2×D3)⋊C8C2211yescharacteristic
49#818C2211yescharacteristic
50#818C2211yescharacteristic
51#817C2211yescharacteristic
52#817C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55#808C2211yescharacteristic
56#808C2211yescharacteristic
57#807C2211yescharacteristic
58#807C2211yescharacteristic
59C4×C3⋊C8C2211yescharacteristic
60#533—31no—
61#529—31no—
62#499—31no—
63#499—31no—
64#488—31no—
65#488D311yescharacteristic, complemented
66#485—31no—
67#485—31no—
68#482—31no—
69#482—31no—
70#477—31no—
71#476—31no—
72#475—31no—
73#474—31no—
74#431—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C2×#67—21no—
78C22×Dic3—21no—
79C2×C4×D3D411yescharacteristic
80C2×C4×D3D411yescharacteristic
81C2×C4×D3C2311yescharacteristic
82C2×Dic6—21no—
83C2×Dic6—21no—
84C2×C3×C8C2311yescharacteristic
85C2×C3×C8C2311yescharacteristic
86C3×C4⋊C4—21no—
87C3×C4⋊C4—21no—
88C3×C4⋊C4C2311yescharacteristic
89C3×C4⋊C4C2311yescharacteristic
90C3×C22⋊C4—21no—
91C3×C22⋊C4—21no—
92C3×C42C2311yescharacteristic
93#217—21no—
94#217—21no—
95(C2×D3)⋊C4—21no—
96(C2×D3)⋊C4—21no—
97(C2×D3)⋊C4—21no—
98C4⋊Dic3—21no—
99C4⋊Dic3C2311yescharacteristic
100C4⋊Dic3C2311yescharacteristic
101C4⋊Dic3C2311yescharacteristic
102C4⋊Dic3C2311yescharacteristic
103Dic3⋊C4—21no—
104Dic3⋊C4—21no—
105Dic3⋊C4—21no—
106Dic3⋊C4—21no—
107Dic3⋊C4—21no—
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
110C4×Dic3—21no—
111C4×Dic3—21no—
112C4×Dic3—21no—
113C4×Dic3D411yescharacteristic
114C4×Dic3D411yescharacteristic
115C4×Dic3C2311yescharacteristic
116C2×C3⋊C8C2311yescharacteristic
117C2×C3⋊C8C2311yescharacteristic
118C4⋊Q8—31no—
119C4⋊Q8—31no—
120#126—61no—
121#125—31no—
122#125—31no—
123#125C2×D311yescharacteristic
124#124C2×D311yescharacteristic
125#123—61no—
126C22⋊Q8—31no—
127C22⋊Q8—61no—
128C22⋊Q8—61no—
129#121—31no—
130C4×Q8—31no—
131C4×Q8—61no—
132C4×Q8—61no—
133C4×D4—31no—
134C4×D4—61no—
135#117—31no—
136#117—31no—
137#117—61no—
138C2×C42—31no—
139#107—31no—
140#107—31no—
141#107—31no—
142#107—31no—
143#106—31no—
144#106—31no—
145#106—31no—
146#106—31no—
147C4⋊C8—31no—
148C4⋊C8—31no—
149#103—31no—
150#103—31no—
151#103C2×D311yescharacteristic
152#103C2×D311yescharacteristic
153#102—31no—
154#102—31no—
155#102C2×D311yescharacteristic
156#102C2×D311yescharacteristic
157C22⋊C8—31no—
158C22⋊C8—31no—
159C4×C8—31no—
160C4×C8C2×D311yescharacteristic
161C23×C3—21no—
162C22×D3C2×D411yescharacteristic
163C3×Q8—21no—
164C3×Q8—21no—
165C3×D4—21no—
166C3×D4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4C2×D411yescharacteristic
171C2×C3×C4C2×D411yescharacteristic
172C2×C3×C4C2411yescharacteristic
173#67—42no—
174C2×Dic3—21no—
175C2×Dic3—21no—
176C2×Dic3—21no—
177C2×Dic3—21no—
178C2×Dic3—42no—
179C2×Dic3C2×D411yescharacteristic
180C2×Dic3C2×D411yescharacteristic
181C2×Dic3C2×D411yescharacteristic
182C4×D3—21no—
183C4×D3—21no—
184C4×D3—22no—
185C4×D3—41no—
186Dic6—42no—
187Dic6—42no—
188C3×C8—21no—
189C3×C8—21no—
190C3⋊C8—21no—
191C3⋊C8—21no—
192C2×Q8—61no—
193C2×Q8—61no—
194C2×Q8C22×D311yescharacteristic
195C2×D4—61no—
196C2×D4C22×D311yescharacteristic
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—61no—
201C2×C8—31no—
202C2×C8—31no—
203C2×C8C22×D311yescharacteristic
204C2×C8C22×D311yescharacteristic
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—31no—
208C4⋊C4—31no—
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—61no—
212C4⋊C4—61no—
213C4⋊C4—61no—
214C4⋊C4—61no—
215C4⋊C4—61no—
216C4⋊C4—61no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4C22×D311yescharacteristic
220C4⋊C4C22×D311yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—21no—
223C22⋊C4—61no—
224C22⋊C4—61no—
225C22⋊C4—61no—
226C22⋊C4—61no—
227C22⋊C4—61no—
228C42—31no—
229C42—31no—
230C42—31no—
231C42—61no—
232C42—61no—
233C42—61no—
234C42C22×D311yescharacteristic
235C22×C3—41no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3C22×D411yescharacteristic
239C2×D3—21no—
240C2×D3—21no—
241C2×D3—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—41no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C42−1+411yescharacteristic
248C3×C42−1+411yescharacteristic
249Dic3—21no—
250Dic3—21no—
251Dic3—41no—
252Dic3—41no—
253Dic3—41no—
254Dic3—41no—
255Dic3#13512yes—
256C23—21no—
257C23—31no—
258Q8—21no—
259Q8—21no—
260Q8—122no—
261Q8—122no—
262D4—21no—
263D4—21no—
264D4—122no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—62no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—121no—
279C2×C4—122no—
280C2×C4D3×D411yescharacteristic
281C2×C4D3×D411yescharacteristic
282C2×C4C23×D311yescharacteristic
283C8—21no—
284C8—21no—
285C8—61no—
286C8—61no—
287C2×C3—81no—
288C2×C3#53611yescharacteristic
289C2×C3C2×#13511yescharacteristic
290C2×C3#57511yescharacteristic
291D3—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—61no—
296C22—61no—
297C22—61no—
298C22C2×D3×D411yescharacteristic
299C4—21no—
300C4—21no—
301C4—32no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—61no—
306C4—61no—
307C4—121no—
308C4—121no—
309C4—121no—
310C4—121no—
311C4#101011yescharacteristic
312C4#101011yescharacteristic
313C3#323311yescharacteristic, complemented, Sylow
314C2—81no—
315C2—121no—
316C2#567511yescharacteristic
317C2D3×#13511yescharacteristic
318C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#488C2
socleC22×C3C22×D4
prime radical#938C22
hypercenter#488D3
hypocenterC3#3233
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #5688C2semidirect
3. C3×#488C2semidirect
4. #5065C2semidirect
5. #5056C2semidirect
6. #4815C2semidirect
7. #4814C2semidirect
8. #4793C2semidirect
9. #4791C2semidirect
12. #4761C2semidirect
13. #4759C2semidirect
14. #4685C2semidirect
15. #4683C2semidirect
65. #488D3semidirect
313. C3#3233semidirect

Maximal subgroups

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

subgroupindex
#56882
#56802
C3×#4882
#50652
#50562
#48152
#48142
#47932
#47912
#47842
#47792
#47612
#47592
#46852
#46832
#32333