Finite Group Databasebeta

← #81920

Subgroups — #81920

Order 384.

334 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#5691C211yesmaximal, characteristic
2D3×#125C211yesmaximal, characteristic, complemented
3C3×#499C211yesmaximal, characteristic, complemented
4#5068C211yesmaximal, characteristic, complemented
5#5065C211yesmaximal, characteristic, complemented
6#4891C211yesmaximal, characteristic, complemented
7#4884C211yesmaximal, characteristic, complemented
8#4882C211yesmaximal, characteristic
9#4877C211yesmaximal, characteristic, complemented
10#4871C211yesmaximal, characteristic, complemented
11#4863C211yesmaximal, characteristic
12#4862C211yesmaximal, characteristic, complemented
13#4857C211yesmaximal, characteristic, complemented
14#4697C211yesmaximal, characteristic, complemented
15#4685C211yesmaximal, characteristic, complemented
16#3232—31nomaximal, Sylow
17C3×#125C2211yescharacteristic
18C3×#125C2211yescharacteristic
19#898—21no—
20#897C2211yescharacteristic
21#895C2211yescharacteristic
22#894—21no—
23#893—21no—
24#893C2211yescharacteristic
25#892—21no—
26D3×C4⋊C4—21no—
27D3×C4⋊C4—21no—
28D3×C4⋊C4C2211yescharacteristic
29D3×C4⋊C4C2211yescharacteristic
30#890C2211yescharacteristic
31#890C2211yescharacteristic
32#889—21no—
33#889—21no—
34#887C2211yescharacteristic
35#875C2211yescharacteristic
36C42×D3C2211yescharacteristic
37#870C2211yescharacteristic
38C3×#107C2211yescharacteristic
39C3×#107C2211yescharacteristic
40C3×#106C2211yescharacteristic
41C3×#106C2211yescharacteristic
42C3×C4×C8C2211yescharacteristic
43#821C2211yescharacteristic
44#821C2211yescharacteristic
45(C2×D3)⋊C8C2211yescharacteristic
46(C2×D3)⋊C8C2211yescharacteristic
47#816C2211yescharacteristic
48#816C2211yescharacteristic
49Dic3⋊C8C2211yescharacteristic
50Dic3⋊C8C2211yescharacteristic
51#810C2211yescharacteristic
52#810C2211yescharacteristic
53#809C2211yescharacteristic
54#809C2211yescharacteristic
55#808C2211yescharacteristic
56#808C2211yescharacteristic
57#807C2211yescharacteristic
58#807C2211yescharacteristic
59C4×C3⋊C8C2211yescharacteristic
60#529—31no—
61C2×#125—31no—
62#499—31no—
63#499D311yescharacteristic, complemented
64#488—31no—
65#488—31no—
66#484—31no—
67#483—31no—
68#481—31no—
69#480—31no—
70#479—31no—
71#479—31no—
72#478—31no—
73#478—31no—
74#431—31no—
75C2×D12—21no—
76C2×D12C2311yescharacteristic
77C2×C4×D3—21no—
78C2×C4×D3—21no—
79C2×C4×D3—21no—
80C2×C4×D3D411yescharacteristic
81C2×C4×D3D411yescharacteristic
82C2×C4×D3C2311yescharacteristic
83C2×Dic6C2311yescharacteristic
84C2×C3×C8C2311yescharacteristic
85C2×C3×C8C2311yescharacteristic
86C3×C4⋊C4—21no—
87C3×C4⋊C4—21no—
88C3×C4⋊C4—21no—
89C3×C4⋊C4—21no—
90C3×C4⋊C4C2311yescharacteristic
91C3×C4⋊C4C2311yescharacteristic
92C3×C4⋊C4C2311yescharacteristic
93C3×C4⋊C4C2311yescharacteristic
94C3×C42C2311yescharacteristic
95(C2×D3)⋊C4—21no—
96(C2×D3)⋊C4—21no—
97(C2×D3)⋊C4—21no—
98(C2×D3)⋊C4—21no—
99(C2×D3)⋊C4—21no—
100C4⋊Dic3—21no—
101C4⋊Dic3—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3C2311yescharacteristic
104C4⋊Dic3C2311yescharacteristic
105Dic3⋊C4—21no—
106Dic3⋊C4—21no—
107Dic3⋊C4—21no—
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112C4×Dic3—21no—
113C4×Dic3D411yescharacteristic
114C4×Dic3D411yescharacteristic
115C4×Dic3C2311yescharacteristic
116C2×C3⋊C8C2311yescharacteristic
117C2×C3⋊C8C2311yescharacteristic
118#126—61no—
119#125—31no—
120#125—31no—
121#125—31no—
122#125—61no—
123#125—61no—
124#125C2×D311yescharacteristic
125#125C2×D311yescharacteristic
126#124—31no—
127#123—61no—
128C22⋊Q8—31no—
129#121—31no—
130C4×Q8—31no—
131C4×D4—31no—
132C4×D4—61no—
133#117—61no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C42—31no—
139#107—31no—
140#107—31no—
141#107C2×D311yescharacteristic
142#107C2×D311yescharacteristic
143#106—31no—
144#106—31no—
145#106C2×D311yescharacteristic
146#106C2×D311yescharacteristic
147C4⋊C8—31no—
148C4⋊C8—31no—
149#103—31no—
150#103—31no—
151#103—31no—
152#103—31no—
153#102—31no—
154#102—31no—
155#102—31no—
156#102—31no—
157C22⋊C8—31no—
158C22⋊C8—31no—
159C4×C8—31no—
160C4×C8C2×D311yescharacteristic
161C22×D3—21no—
162C22×D3C2×D411yescharacteristic
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4C2×D411yescharacteristic
168C2×C3×C4C2×D411yescharacteristic
169C2×C3×C4C2411yescharacteristic
170C2×Dic3—21no—
171C2×Dic3—21no—
172C2×Dic3—21no—
173C2×Dic3C2×D411yescharacteristic
174C2×Dic3C2×D411yescharacteristic
175C2×Dic3C2×D411yescharacteristic
176D12—21no—
177D12—21no—
178D12—42no—
179C4×D3—22no—
180C4×D3—21no—
181C4×D3—21no—
182C4×D3—41no—
183C4×D3—42no—
184C4×D3—42no—
185C4×D3—42no—
186Dic6—21no—
187Dic6—21no—
188C3×C8—21no—
189C3×C8—21no—
190C3⋊C8—21no—
191C3⋊C8—21no—
192C2×Q8—31no—
193C2×D4—31no—
194C2×D4—61no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—61no—
199C22×C4—61no—
200C22×C4—61no—
201C2×C8—31no—
202C2×C8—31no—
203C2×C8C22×D311yescharacteristic
204C2×C8C22×D311yescharacteristic
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
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⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4C22×D311yescharacteristic
222C4⋊C4C22×D311yescharacteristic
223C4⋊C4C22×D311yescharacteristic
224C4⋊C4C22×D311yescharacteristic
225C22⋊C4—61no—
226C22⋊C4—61no—
227C22⋊C4—61no—
228C22⋊C4—61no—
229C22⋊C4—61no—
230C42—31no—
231C42—31no—
232C42—31no—
233C42—61no—
234C42C22×D311yescharacteristic
235C22×C3C22×D411yescharacteristic
236C2×D3—21no—
237C2×D3—21no—
238C2×D3—41no—
239C2×D3—41no—
240C2×D3—41no—
241C2×D3#13512yes—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—41no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C42−1+411yescharacteristic
249C3×C42−1+411yescharacteristic
250Dic3—21no—
251Dic3—21no—
252Dic3—21no—
253Dic3—41no—
254Dic3—41no—
255Dic3—41no—
256C23—31no—
257C23—61no—
258Q8—61no—
259Q8—61no—
260D4—61no—
261D4—61no—
262D4—122no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—62no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—121no—
277C2×C4—122no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4D3×D411yescharacteristic
281C2×C4D3×D411yescharacteristic
282C2×C4C23×D311yescharacteristic
283C8—21no—
284C8—21no—
285C8—61no—
286C8—61no—
287C2×C3#53611yescharacteristic
288C2×C3C2×#13511yescharacteristic
289C2×C3#57511yescharacteristic
290D3—22no—
291D3—81no—
292C22—32no—
293C22—61no—
294C22—61no—
295C22—121no—
296C22—121no—
297C22—121no—
298C22C2×D3×D411yescharacteristic
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—61no—
306C4—61no—
307C4—61no—
308C4—121no—
309C4—121no—
310C4—121no—
311C4#100711yescharacteristic
312C4#100711yescharacteristic
313C3#323211yescharacteristic, complemented, Sylow
314C2—62no—
315C2—241no—
316C2#572411yescharacteristic
317C2D3×#13511yescharacteristic
318C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#499C2
socleC22×C3C22×D4
prime radical#895C22
hypercenter#499D3
hypocenterC3#3232
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
2. D3×#125C2semidirect
3. C3×#499C2semidirect
4. #5068C2semidirect
5. #5065C2semidirect
6. #4891C2semidirect
7. #4884C2semidirect
9. #4877C2semidirect
10. #4871C2semidirect
12. #4862C2semidirect
13. #4857C2semidirect
14. #4697C2semidirect
15. #4685C2semidirect
63. #499D3semidirect
313. C3#3232semidirect

Maximal subgroups

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

subgroupindex
#56912
D3×#1252
C3×#4992
#50682
#50652
#48912
#48842
#48822
#48772
#48712
#48632
#48622
#48572
#46972
#46852
#32323