Finite Group Databasebeta

← #80897

Subgroups — #80897

Order 384.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5643C211yesmaximal, characteristic
2#5625C211yesmaximal, characteristic, complemented
3C3×#470C211yesmaximal, characteristic, complemented
4#5185C211yesmaximal, characteristic, complemented
5#5182C211yesmaximal, characteristic, complemented
6#5134C211yesmaximal, characteristic
7#5119C211yesmaximal, characteristic, complemented
8#5051C211yesmaximal, characteristic, complemented
9#5048C211yesmaximal, characteristic
10#4871C211yesmaximal, characteristic, complemented
11#4854C211yesmaximal, characteristic, complemented
12#4818C211yesmaximal, characteristic, complemented
13#4798C211yesmaximal, characteristic, complemented
14#4796C211yesmaximal, characteristic, complemented
15#4793C211yesmaximal, characteristic, complemented
16#3258—31nomaximal, Sylow
17C2×C3×Q16C2211yescharacteristic
18C2×C3×M16C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×C22⋊Q8C2211yescharacteristic
21#947C2211yescharacteristic
22Q8×Dic3C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24#944C2211yescharacteristic
25C2×#216C2211yescharacteristic
26C2×#215C2211yescharacteristic
27#930C2211yescharacteristic
28#926C2211yescharacteristic
29#924C2211yescharacteristic
30C22×C3⋊C8C2211yescharacteristic
31C2×#204C2211yescharacteristic
32#898—21no—
33#897—21no—
34#893C2211yescharacteristic
35#890—21no—
36#890C2211yescharacteristic
37#889—21no—
38#888C2211yescharacteristic
39#887C2211yescharacteristic
40#885—21no—
41#883—21no—
42#879—21no—
43#879—21no—
44C3×#106C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#103C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#821C2211yescharacteristic
50#818C2211yescharacteristic
51#816C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56#810C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#542—31no—
61#541—31no—
62#479—31no—
63#478—31no—
64#477—31no—
65#475—31no—
66#470D311yescharacteristic, complemented
67#468—31no—
68#467—31no—
69#465—31no—
70#464—31no—
71#464—31no—
72#462—31no—
73#461—31no—
74#424—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×D12C2311yescharacteristic
80C2×C4×D3—21no—
81C2×Dic6—21no—
82C2×Dic6C2311yescharacteristic
83C3×Q16—41no—
84C3×M16—41no—
85C2×C3×C8C2311yescharacteristic
86C2×C3×C8C2311yescharacteristic
87C3×C4⋊C4—21no—
88C3×C4⋊C4—21no—
89C3×C4⋊C4C2311yescharacteristic
90C3×C4⋊C4C2311yescharacteristic
91C3×C22⋊C4—21no—
92C3×C22⋊C4—21no—
93#217—21no—
94#217D411yescharacteristic
95#217D411yescharacteristic
96#216—41no—
97#215—41no—
98(C2×D3)⋊C4—21no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3D411yescharacteristic
106C4⋊Dic3D411yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8—41no—
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123#204—41no—
124C2×Q16—31no—
125C2×Q16C2×D311yescharacteristic
126C2×SD16—31no—
127C2×SD16—31no—
128C2×M16C2×D311yescharacteristic
129C22×C8—31no—
130C4⋊Q8—31no—
131C4⋊Q8—31no—
132#126—61no—
133#126—61no—
134#126—61no—
135#125—31no—
136#125—61no—
137#125—61no—
138#124—31no—
139#124—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143C22⋊Q8C2×D311yescharacteristic
144C22⋊Q8C2×D311yescharacteristic
145#121—31no—
146C4×Q8—31no—
147C4×Q8—31no—
148C4×Q8—31no—
149C4×D4—31no—
150#117—31no—
151#107—31no—
152#107—31no—
153#106—31no—
154#106C2×D311yescharacteristic
155C4⋊C8—31no—
156C4⋊C8—31no—
157C4⋊C8—31no—
158C4⋊C8—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103C2×D311yescharacteristic
164#103C2×D311yescharacteristic
165#102—31no—
166#102—31no—
167C23×C3C2×D411yescharacteristic
168C22×D3—21no—
169C3×Q8—21no—
170C3×Q8—21no—
171C3×Q8—21no—
172C3×Q8—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2411yescharacteristic
181#67—81no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3C2×D411yescharacteristic
186C2×Dic3C2×D411yescharacteristic
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189D12—21no—
190D12—21no—
191C4×D3—81no—
192Dic6—21no—
193Dic6—21no—
194Dic6—81no—
195C3×C8—21no—
196C3×C8—21no—
197C3⋊C8—21no—
198C3⋊C8—21no—
199C2×Q8—31no—
200C2×Q8—61no—
201C2×Q8C22×D311yescharacteristic
202C2×Q8C22×D311yescharacteristic
203C2×D4—31no—
204C2×D4—61no—
205C22×C4—61no—
206C22×C4C22×D311yescharacteristic
207Q16—41no—
208Q16—121no—
209SD16—121no—
210SD16—121no—
211M16—41no—
212C2×C8—31no—
213C2×C8—31no—
214C2×C8—121no—
215C2×C8C22×D311yescharacteristic
216C2×C8C22×D311yescharacteristic
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—31no—
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4C22×D311yescharacteristic
233C4⋊C4C22×D311yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—31no—
237C22⋊C4—31no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C42—31no—
244C42—31no—
245C42—61no—
246C42—61no—
247C42—61no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3C22×D411yescharacteristic
252C2×D3—41no—
253C2×D3—41no—
254C2×D3—41no—
255C3×C4—21no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4H(5,𝔽2)11yescharacteristic
261C3×C42−1+411yescharacteristic
262Dic3—21no—
263Dic3—21no—
264Dic3—21no—
265Dic3—21no—
266Dic3—41no—
267Dic3—41no—
268Dic3—41no—
269C23—61no—
270C23D3×D411yescharacteristic
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—61no—
276Q8—61no—
277Q8—241no—
278D4—61no—
279D4—61no—
280D4—241no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—241no—
295C2×C4D3×D411yescharacteristic
296C2×C4C23×D311yescharacteristic
297C8—21no—
298C8—21no—
299C8—61no—
300C8—61no—
301C2×C3—41no—
302C2×C3#53711yescharacteristic
303C2×C3#57711yescharacteristic
304C2×C3#57811yescharacteristic
305D3—81no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—121no—
310C22—121no—
311C22—121no—
312C22C2×D3×D411yescharacteristic
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—61no—
319C4—61no—
320C4—61no—
321C4—61no—
322C4—121no—
323C4—121no—
324C4—121no—
325C4#100411yescharacteristic
326C4#101011yescharacteristic
327C3#325811yescharacteristic, complemented, Sylow
328C2—41no—
329C2—241no—
330C2#559711yescharacteristic
331C2#577811yescharacteristic
332C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
prime radical#930C22
hypercenter#470D3
hypocenterC3#3258
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #5625C2semidirect
3. C3×#470C2semidirect
4. #5185C2semidirect
5. #5182C2semidirect
7. #5119C2semidirect
8. #5051C2semidirect
10. #4871C2semidirect
11. #4854C2semidirect
12. #4818C2semidirect
13. #4798C2semidirect
14. #4796C2semidirect
15. #4793C2semidirect
66. #470D3semidirect
327. C3#3258semidirect

Maximal subgroups

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

subgroupindex
#56432
#56252
C3×#4702
#51852
#51822
#51342
#51192
#50512
#50482
#48712
#48542
#48182
#47982
#47962
#47932
#32583