Finite Group Databasebeta

← #80529

Subgroups — #80529

Order 384.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5671C211yesmaximal, characteristic, complemented
2#5542C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic
4#5167C211yesmaximal, characteristic, complemented
5#5166C211yesmaximal, characteristic, complemented
6#5155C211yesmaximal, characteristic
7#5153C211yesmaximal, characteristic
8#5060C211yesmaximal, characteristic, complemented
9#5018C211yesmaximal, characteristic, complemented
10#4849C211yesmaximal, characteristic, complemented
11#4828C211yesmaximal, characteristic, complemented
12#4795C211yesmaximal, characteristic, complemented
13#4790C211yesmaximal, characteristic, complemented
14#4767C211yesmaximal, characteristic, complemented
15#4757C211yesmaximal, characteristic, complemented
16#3402—31nomaximal, Sylow
17C2×C3×SD16C2211yescharacteristic
18C2×C3×D8C2211yescharacteristic
19C3×#124C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21Q8×Dic3C2211yescharacteristic
22#944C2211yescharacteristic
23C2×#216C2211yescharacteristic
24C2×#216C2211yescharacteristic
25#936C2211yescharacteristic
26#936C2211yescharacteristic
27#936C2211yescharacteristic
28D4×Dic3C2211yescharacteristic
29C2×#214C2211yescharacteristic
30C2×#214C2211yescharacteristic
31C2×#214C2211yescharacteristic
32#926—21no—
33#924—21no—
34C2×Dic12C2211yescharacteristic
35C2×Dic12C2211yescharacteristic
36#887C2211yescharacteristic
37#887C2211yescharacteristic
38#879—21no—
39#879—21no—
40#879—21no—
41#879—21no—
42#878—21no—
43#877—21no—
44#869C2211yescharacteristic
45C4×Dic6C2211yescharacteristic
46C4×Dic6C2211yescharacteristic
47C3×C4⋊C8C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#102C2211yescharacteristic
50#832C2211yescharacteristic
51#816C2211yescharacteristic
52#816C2211yescharacteristic
53#815C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#810C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#556—31no—
61#556—31no—
62#497—31no—
63#497—31no—
64#495—31no—
65#464—31no—
66#464—31no—
67#463—31no—
68#463D311yescharacteristic
69#462—31no—
70#462—31no—
71#446—31no—
72#440—31no—
73#440—31no—
74C4×Q16—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C22×Dic3—21no—
80C2×Dic6D411yescharacteristic
81C2×Dic6C2311yescharacteristic
82C2×Dic6C2311yescharacteristic
83C2×Dic6C2311yescharacteristic
84C3×SD16—41no—
85C3×D8—41no—
86C2×C3×C8C2311yescharacteristic
87C2×C3×C8C2311yescharacteristic
88C3×C4⋊C4C2311yescharacteristic
89C3×C22⋊C4—21no—
90C3×C22⋊C4—21no—
91C3×C22⋊C4—21no—
92C3×C42C2311yescharacteristic
93#217—21no—
94#217—21no—
95#217—21no—
96#217—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#216—22no—
101#216—41no—
102#214—22no—
103#214—22no—
104#214—41no—
105C4⋊Dic3—21no—
106C4⋊Dic3—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3D411yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4D411yescharacteristic
116Dic3⋊C4D411yescharacteristic
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3C2311yescharacteristic
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125Dic12—41no—
126Dic12—41no—
127C2×Q16—31no—
128C2×Q16—31no—
129C2×Q16—31no—
130C2×Q16—31no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C2×SD16C2×D311yescharacteristic
135C2×D8C2×D311yescharacteristic
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138#126—61no—
139#126—61no—
140#126—61no—
141#126—61no—
142#124—31no—
143#124—31no—
144#124—31no—
145#124C2×D311yescharacteristic
146C22⋊Q8—61no—
147C22⋊Q8—61no—
148C4×Q8—31no—
149C4×Q8—31no—
150C4×Q8—31no—
151C4×Q8—31no—
152C4×Q8—31no—
153C4×D4—31no—
154C4×D4C2×D311yescharacteristic
155#117—61no—
156#117—61no—
157#107—31no—
158C4⋊C8—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic
161#103—31no—
162#103—31no—
163#103—31no—
164#103C2×D311yescharacteristic
165#102—31no—
166#102C2×D311yescharacteristic
167#97—31no—
168#97—31no—
169C4×C8—31no—
170C23×C3—21no—
171C23×C3—21no—
172C3×Q8—21no—
173C3×Q8—21no—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—21no—
177C3×D4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—42no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—42no—
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193Dic6—21no—
194Dic6—21no—
195Dic6—21no—
196Dic6—21no—
197Dic6—21no—
198Dic6—21no—
199Dic6—41no—
200C3×C8—21no—
201C3×C8—21no—
202C3⋊C8—21no—
203C3⋊C8#4112yes—
204C2×Q8—31no—
205C2×Q8—31no—
206C2×Q8—31no—
207C2×Q8—31no—
208C2×Q8C22×D311yescharacteristic
209C2×D4C22×D311yescharacteristic
210C2×D4C22×D311yescharacteristic
211C22×C4—21no—
212C22×C4—61no—
213Q16—62no—
214Q16—121no—
215Q16—121no—
216Q16—121no—
217SD16—41no—
218SD16—62no—
219SD16—62no—
220SD16—121no—
221D8—41no—
222C2×C8—31no—
223C2×C8—31no—
224C2×C8C22×D311yescharacteristic
225C2×C8C22×D311yescharacteristic
226C4⋊C4—31no—
227C4⋊C4—31no—
228C4⋊C4—31no—
229C4⋊C4—31no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4C22×D311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4—21no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C42—31no—
250C42—31no—
251C42—61no—
252C42—61no—
253C42—61no—
254C42—61no—
255C42C22×D311yescharacteristic
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3C22×D411yescharacteristic
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4C2×#4111yescharacteristic
268C3×C4H(5,𝔽2)11yescharacteristic
269Dic3—21no—
270Dic3—21no—
271Dic3—21no—
272Dic3—21no—
273Dic3—41no—
274Dic3—41no—
275Dic3—41no—
276Dic3—41no—
277C23—21no—
278C23—21no—
279Q8—21no—
280Q8—21no—
281Q8—61no—
282Q8—61no—
283Q8—61no—
284Q8—61no—
285Q8—61no—
286Q8—61no—
287Q8—121no—
288D4—21no—
289D4—21no—
290D4—21no—
291D4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—42no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—122no—
304C2×C4D3×D411yescharacteristic
305C2×C4D3×D411yescharacteristic
306C2×C4C23×D311yescharacteristic
307C8—21no—
308C8—21no—
309C8—32no—
310C8—61no—
311C2×C3—81no—
312C2×C3—81no—
313C2×C3#55011yescharacteristic
314C2×C3#57511yescharacteristic
315C2×C3#57811yescharacteristic
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22C2×D3×D411yescharacteristic
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—61no—
328C4—61no—
329C4—61no—
330C4—61no—
331C4—121no—
332C4—121no—
333C4—121no—
334C4—121no—
335C4#100411yescharacteristic
336C4D3×#4111yescharacteristic
337C3#340211yescharacteristic, complemented, Sylow
338C2—81no—
339C2—81no—
340C2#559911yescharacteristic
341C2#577111yescharacteristic
342C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
prime radicalC2×C3×D8C22
hypercenter#463D3
hypocenterC3#3402
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #5671C2semidirect
2. #5542C2semidirect
4. #5167C2semidirect
5. #5166C2semidirect
8. #5060C2semidirect
9. #5018C2semidirect
10. #4849C2semidirect
11. #4828C2semidirect
12. #4795C2semidirect
13. #4790C2semidirect
14. #4767C2semidirect
15. #4757C2semidirect
337. C3#3402semidirect

Maximal subgroups

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

subgroupindex
#56712
#55422
C3×#4632
#51672
#51662
#51552
#51532
#50602
#50182
#48492
#48282
#47952
#47902
#47672
#47572
#34023