Finite Group Databasebeta

← #81898

Subgroups — #81898

Order 384.

320 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#5733C211yesmaximal, characteristic
2#5729C211yesmaximal, characteristic, complemented
3C3×#498C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5081C211yesmaximal, characteristic, complemented
6#4871C211yesmaximal, characteristic, complemented
7#4870C211yesmaximal, characteristic, complemented
8#4868C211yesmaximal, characteristic, complemented
9#4866C211yesmaximal, characteristic
10#4863C211yesmaximal, characteristic
11#4862C211yesmaximal, characteristic, complemented
12#4857C211yesmaximal, characteristic, complemented
13#4854C211yesmaximal, characteristic, complemented
14#4697C211yesmaximal, characteristic, complemented
15#4690C211yesmaximal, characteristic, complemented
16#3244—31nomaximal, Sylow
17C3×C4⋊Q8C2211yescharacteristic
18C3×C4⋊Q8C2211yescharacteristic
19#947—21no—
20#946—21no—
21#946—21no—
22#944—21no—
23#898—21no—
24#897C2211yescharacteristic
25#896—21no—
26#895C2211yescharacteristic
27#894—21no—
28#893C2211yescharacteristic
29#892C2211yescharacteristic
30D3×C4⋊C4C2211yescharacteristic
31#890C2211yescharacteristic
32#889—21no—
33#888C2211yescharacteristic
34#887C2211yescharacteristic
35#875C2211yescharacteristic
36#872C2211yescharacteristic
37#870C2211yescharacteristic
38C3×#106C2211yescharacteristic
39C3×#106C2211yescharacteristic
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
59#803C2211yescharacteristic
60#544—31no—
61#542—31no—
62#501—31no—
63#498D311yescharacteristic, complemented
64#489—31no—
65#488—31no—
66#485—31no—
67#483—31no—
68#482—31no—
69#481—31no—
70#479—31no—
71#478—31no—
72#476—31no—
73#475—31no—
74#433—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8—21no—
77C2×D12—21no—
78C2×D12C2311yescharacteristic
79C2×C4×D3—21no—
80C2×C4×D3—21no—
81C2×C4×D3C2311yescharacteristic
82C2×Dic6—21no—
83C2×Dic6C2311yescharacteristic
84C2×C3×C8C2311yescharacteristic
85C2×C3×C8C2311yescharacteristic
86C3×C4⋊C4—21no—
87C3×C4⋊C4—21no—
88C3×C4⋊C4C2311yescharacteristic
89C3×C4⋊C4C2311yescharacteristic
90C3×C4⋊C4C2311yescharacteristic
91C3×C4⋊C4C2311yescharacteristic
92C3×C42C2311yescharacteristic
93(C2×D3)⋊C4—21no—
94(C2×D3)⋊C4—21no—
95(C2×D3)⋊C4—21no—
96(C2×D3)⋊C4—21no—
97(C2×D3)⋊C4—21no—
98(C2×D3)⋊C4—21no—
99(C2×D3)⋊C4D411yescharacteristic
100(C2×D3)⋊C4D411yescharacteristic
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⋊C4D411yescharacteristic
112Dic3⋊C4D411yescharacteristic
113C4×Dic3—21no—
114C4×Dic3—21no—
115C4×Dic3C2311yescharacteristic
116C2×C3⋊C8C2311yescharacteristic
117C2×C3⋊C8C2311yescharacteristic
118C4⋊Q8—31no—
119C4⋊Q8—61no—
120C4⋊Q8C2×D311yescharacteristic
121C4⋊Q8C2×D311yescharacteristic
122#126—61no—
123#125—31no—
124#125—31no—
125#125—61no—
126#124—31no—
127#124—61no—
128#123—61no—
129C22⋊Q8—31no—
130C22⋊Q8—61no—
131C22⋊Q8—61no—
132C22⋊Q8—61no—
133#121—31no—
134C4×Q8—31no—
135C4×D4—31no—
136#117—31no—
137#117—31no—
138C2×C4⋊C4—31no—
139#107—31no—
140#107—31no—
141#106—31no—
142#106—31no—
143#106C2×D311yescharacteristic
144#106C2×D311yescharacteristic
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—
159#97—31no—
160C4×C8C2×D311yescharacteristic
161C22×D3—21no—
162C22×D3C2×D411yescharacteristic
163C3×Q8—81no—
164C3×Q8—81no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4C2×D411yescharacteristic
170C2×C3×C4C2×D411yescharacteristic
171C2×C3×C4C2411yescharacteristic
172C2×Dic3—21no—
173C2×Dic3—21no—
174C2×Dic3—21no—
175C2×Dic3C2×D411yescharacteristic
176C2×Dic3C2×D411yescharacteristic
177C2×Dic3C2×D411yescharacteristic
178D12—21no—
179D12—21no—
180D12—81no—
181C4×D3—21no—
182C4×D3—21no—
183C4×D3—81no—
184C4×D3—81no—
185Dic6—21no—
186Dic6—21no—
187Dic6—81no—
188C3×C8—21no—
189C3×C8—21no—
190C3⋊C8—21no—
191C3⋊C8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—31no—
195C2×Q8—61no—
196C2×D4—31no—
197C2×D4—61no—
198C22×C4—31no—
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—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
221C4⋊C4C22×D311yescharacteristic
222C4⋊C4C22×D311yescharacteristic
223C22⋊C4—31no—
224C22⋊C4—31no—
225C22⋊C4—61no—
226C22⋊C4—61no—
227C22⋊C4—61no—
228C22⋊C4—61no—
229C22⋊C4—61no—
230C22⋊C4—61no—
231C42—31no—
232C42—61no—
233C42—61no—
234C42C22×D311yescharacteristic
235C22×C3C22×D411yescharacteristic
236C2×D3—21no—
237C2×D3—21no—
238C2×D3—21no—
239C2×D3—41no—
240C2×D3—41no—
241C2×D3—41no—
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—
260Q8—81no—
261Q8—81no—
262Q8—241no—
263D4—61no—
264D4—61no—
265D4—241no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—241no—
279C2×C4—241no—
280C2×C4D3×D411yescharacteristic
281C2×C4D3×D411yescharacteristic
282C2×C4C23×D311yescharacteristic
283C8—21no—
284C8—21no—
285C8—61no—
286C8—61no—
287C2×C3#53611yescharacteristic
288C2×C3#57711yescharacteristic
289C2×C3#57711yescharacteristic
290D3—41no—
291D3—81no—
292C22—61no—
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#324411yescharacteristic, complemented, Sylow
314C2—121no—
315C2—241no—
316C2#572411yescharacteristic
317C2#576211yescharacteristic
318C2#576211yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#498C2
socleC22×C3C22×D4
prime radical#895C22
hypercenter#498D3
hypocenterC3#3244
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #5729C2semidirect
3. C3×#498C2semidirect
4. #5089C2semidirect
5. #5081C2semidirect
6. #4871C2semidirect
7. #4870C2semidirect
8. #4868C2semidirect
11. #4862C2semidirect
12. #4857C2semidirect
13. #4854C2semidirect
14. #4697C2semidirect
15. #4690C2semidirect
63. #498D3semidirect
313. C3#3244semidirect

Maximal subgroups

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

subgroupindex
#57332
#57292
C3×#4982
#50892
#50812
#48712
#48702
#48682
#48662
#48632
#48622
#48572
#48542
#46972
#46902
#32443