Finite Group Databasebeta

← #80894

Subgroups — #80894

Order 384.

348 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5624C211yesmaximal, characteristic
2#5622C211yesmaximal, characteristic, complemented
3C3×#470C211yesmaximal, characteristic, complemented
4#5185C211yesmaximal, characteristic, complemented
5#5182C211yesmaximal, characteristic, complemented
6#5134C211yesmaximal, characteristic
7#5118C211yesmaximal, characteristic, complemented
8#5052C211yesmaximal, characteristic, complemented
9#5049C211yesmaximal, characteristic
10#4870C211yesmaximal, characteristic, complemented
11#4857C211yesmaximal, characteristic, complemented
12#4818C211yesmaximal, characteristic, complemented
13#4798C211yesmaximal, characteristic, complemented
14#4796C211yesmaximal, characteristic, complemented
15#4793C211yesmaximal, characteristic, complemented
16#3252—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#924C2211yescharacteristic
29C2×C4×Dic3C2211yescharacteristic
30C2×#208C2211yescharacteristic
31C2×#204C2211yescharacteristic
32#898—21no—
33#893C2211yescharacteristic
34#892—21no—
35#890C2211yescharacteristic
36#889—21no—
37#888C2211yescharacteristic
38#887—21no—
39#887C2211yescharacteristic
40#886—21no—
41#67⋊C4—21no—
42#878—21no—
43#877—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#533—31no—
61#529—31no—
62#479—31no—
63#479—31no—
64#477—31no—
65#476—31no—
66#470—31no—
67#470D311yescharacteristic, complemented
68#468—31no—
69#468—31no—
70#464—31no—
71#464—31no—
72#462—31no—
73#461—31no—
74#423—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C22×Dic3D411yescharacteristic
80C22×Dic3D411yescharacteristic
81C2×D12C2311yescharacteristic
82C2×C4×D3—21no—
83C2×Dic6—21no—
84C2×Dic6C2311yescharacteristic
85C3×Q16—41no—
86C3×M16—41no—
87C2×C3×C8C2311yescharacteristic
88C2×C3×C8C2311yescharacteristic
89C3×C4⋊C4—21no—
90C3×C4⋊C4—21no—
91C3×C4⋊C4C2311yescharacteristic
92C3×C4⋊C4C2311yescharacteristic
93C3×C22⋊C4—21no—
94C3×C22⋊C4—21no—
95#217—21no—
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⋊Dic3C2311yescharacteristic
106C4⋊Dic3C2311yescharacteristic
107Dic3⋊C4—21no—
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113C4×Dic3—21no—
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3D411yescharacteristic
117C4×Dic3D411yescharacteristic
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120#208—41no—
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123#204—41no—
124C2×Q16—31no—
125C2×Q16C2×D311yescharacteristic
126C2×SD16—31no—
127C2×SD16—31no—
128C2×M16—31no—
129C2×M16C2×D311yescharacteristic
130C4⋊Q8—31no—
131C4⋊Q8—31no—
132#126—61no—
133#125—31no—
134#125—61no—
135#124—31no—
136#123—61no—
137C22⋊Q8—31no—
138C22⋊Q8—61no—
139C22⋊Q8C2×D311yescharacteristic
140C22⋊Q8C2×D311yescharacteristic
141#121—31no—
142C4×Q8—31no—
143C4×Q8—31no—
144C4×Q8—31no—
145C4×Q8—61no—
146C4×D4—31no—
147C4×D4—61no—
148#117—61no—
149#117—61no—
150C2×C42—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—42no—
182C2×Dic3—22no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—41no—
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191D12—21no—
192D12—21no—
193C4×D3—42no—
194Dic6—21no—
195Dic6—21no—
196Dic6—42no—
197C3×C8—21no—
198C3×C8—21no—
199C3⋊C8—21no—
200C3⋊C8—21no—
201C2×Q8—31no—
202C2×Q8—61no—
203C2×Q8C22×D311yescharacteristic
204C2×Q8C22×D311yescharacteristic
205C2×D4—31no—
206C2×D4—61no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—61no—
210C22×C4C22×D311yescharacteristic
211Q16—41no—
212Q16—121no—
213SD16—121no—
214SD16—121no—
215M16—41no—
216M16—121no—
217C2×C8—31no—
218C2×C8—31no—
219C2×C8C22×D311yescharacteristic
220C2×C8C22×D311yescharacteristic
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—31no—
224C4⋊C4—31no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4C22×D311yescharacteristic
235C4⋊C4C22×D311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C42—31no—
244C42—31no—
245C42—31no—
246C42—31no—
247C42—61no—
248C42—61no—
249C42—61no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3C22×D411yescharacteristic
254C2×D3—41no—
255C2×D3—41no—
256C2×D3—41no—
257C3×C4—21no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4H(5,𝔽2)11yescharacteristic
263C3×C42−1+411yescharacteristic
264Dic3—21no—
265Dic3—21no—
266Dic3—21no—
267Dic3—41no—
268Dic3—41no—
269Dic3—41no—
270Dic3#13712yes—
271C23—61no—
272C23D3×D411yescharacteristic
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—61no—
278Q8—61no—
279Q8—122no—
280D4—61no—
281D4—61no—
282D4—122no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—62no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—122no—
298C2×C4—121no—
299C2×C4D3×D411yescharacteristic
300C2×C4C23×D311yescharacteristic
301C8—21no—
302C8—21no—
303C8—61no—
304C8—61no—
305C2×C3—41no—
306C2×C3#53711yescharacteristic
307C2×C3C2×#13711yescharacteristic
308C2×C3#57511yescharacteristic
309D3—81no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—121no—
314C22—121no—
315C22—121no—
316C22C2×D3×D411yescharacteristic
317C4—21no—
318C4—32no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—61no—
324C4—61no—
325C4—61no—
326C4—121no—
327C4—121no—
328C4—121no—
329C4#100411yescharacteristic
330C4#101011yescharacteristic
331C3#325211yescharacteristic, complemented, Sylow
332C2—41no—
333C2—241no—
334C2#559711yescharacteristic
335C2D3×#13711yescharacteristic
336C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
prime radical#930C22
hypercenter#470D3
hypocenterC3#3252
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. #5622C2semidirect
3. C3×#470C2semidirect
4. #5185C2semidirect
5. #5182C2semidirect
7. #5118C2semidirect
8. #5052C2semidirect
10. #4870C2semidirect
11. #4857C2semidirect
12. #4818C2semidirect
13. #4798C2semidirect
14. #4796C2semidirect
15. #4793C2semidirect
67. #470D3semidirect
331. C3#3252semidirect

Maximal subgroups

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

subgroupindex
#56242
#56222
C3×#4702
#51852
#51822
#51342
#51182
#50522
#50492
#48702
#48572
#48182
#47982
#47962
#47932
#32523