Finite Group Databasebeta

← #80412

Subgroups — #80412

Order 384.

372 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#5731C211yesmaximal, characteristic, complemented
2#5546C211yesmaximal, characteristic
3C3×#461C211yesmaximal, characteristic, complemented
4#5173C211yesmaximal, characteristic
5#5173C211yesmaximal, characteristic
6#5164C211yesmaximal, characteristic, complemented
7Dic3×SD16C211yesmaximal, characteristic, complemented
8#5090C211yesmaximal, characteristic, complemented
9#5009C211yesmaximal, characteristic, complemented
10#4840C211yesmaximal, characteristic, complemented
11#4829C211yesmaximal, characteristic, complemented
12#4811C211yesmaximal, characteristic, complemented
13#4805C211yesmaximal, characteristic, complemented
14#4790C211yesmaximal, characteristic, complemented
15#4789C211yesmaximal, characteristic, complemented
16#3120—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×C3×SD16C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C3×C4⋊Q8C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22#946C2211yescharacteristic
23#946C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25Q8×Dic3C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#216C2211yescharacteristic
28D4×Dic3C2211yescharacteristic
29C4×#67—21no—
30C2×Dic3⋊C4—21no—
31#898—21no—
32#898—21no—
33D3×C4⋊C4C2211yescharacteristic
34#889—21no—
35#889—21no—
36#887C2211yescharacteristic
37#67⋊C4—21no—
38#877—21no—
39#872C2211yescharacteristic
40#870C2211yescharacteristic
41C4×Dic6C2211yescharacteristic
42C3×C4⋊C8C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#818C2211yescharacteristic
52#817C2211yescharacteristic
53#816C2211yescharacteristic
54#816C2211yescharacteristic
55#815C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#541—31no—
61#520—31no—
62#484—31no—
63#483—31no—
64#479—31no—
65#478—31no—
66#462—31no—
67#461D311yescharacteristic, complemented
68#452—31no—
69#452—31no—
70#441—31no—
71#440—31no—
72C4×Q16—31no—
73C4×SD16—31no—
74#433—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C22×Dic3—21no—
81C22×Dic3—21no—
82#237—22no—
83#237—22no—
84C2×C4×D3—21no—
85C2×C4×D3C2311yescharacteristic
86C2×Dic6C2311yescharacteristic
87C3×SD16—22no—
88C3×SD16—22no—
89C2×C3×C8C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C3×C4⋊C4—21no—
92C3×C4⋊C4—21no—
93C3×C4⋊C4C2311yescharacteristic
94C3×C22⋊C4—21no—
95C3×C42C2311yescharacteristic
96#217—21no—
97#216—22no—
98#216—22no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4D411yescharacteristic
102(C2×D3)⋊C4D411yescharacteristic
103C4⋊Dic3—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3C2311yescharacteristic
106C4⋊Dic3C2311yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
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
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×#41—31no—
125C2×Q16—31no—
126C2×Q16—31no—
127C2×SD16C2×D311yescharacteristic
128C2×SD16C2×D311yescharacteristic
129C4⋊Q8C2×D311yescharacteristic
130#126—61no—
131#126—61no—
132#125—31no—
133#125—61no—
134#125—61no—
135C22⋊Q8—31no—
136C22⋊Q8—31no—
137C4×Q8—31no—
138C4×Q8—31no—
139C4×Q8—31no—
140C4×Q8—31no—
141C4×D4—31no—
142C4×D4—61no—
143C4×D4—61no—
144C4×D4C2×D311yescharacteristic
145#117—31no—
146#117—61no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149#107—31no—
150#107—31no—
151#106—31no—
152#106—31no—
153C4⋊C8—31no—
154C4⋊C8C2×D311yescharacteristic
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#103C2×D311yescharacteristic
160#103C2×D311yescharacteristic
161#102—31no—
162#102—31no—
163C22⋊C8—31no—
164C22⋊C8—31no—
165#97—31no—
166C4×C8—31no—
167C23×C3—21no—
168C22×D3C2×D411yescharacteristic
169C3×Q8—21no—
170C3×Q8—21no—
171C3×Q8—21no—
172C3×Q8—21no—
173C3×D4—21no—
174C3×D4#4112yes—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—81no—
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2411yescharacteristic
182#67—24no—
183C2×Dic3—24no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—81no—
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C4×D3—21no—
193C4×D3—21no—
194C4×D3—81no—
195Dic6—21no—
196Dic6#4112yes—
197C3×C8—21no—
198C3×C8—21no—
199C3⋊C8—21no—
200C3⋊C8—21no—
201#41—62no—
202#41—62no—
203C2×Q8—31no—
204C2×Q8C22×D311yescharacteristic
205C2×Q8C22×D311yescharacteristic
206C2×D4—61no—
207C2×D4C22×D311yescharacteristic
208C22×C4—21no—
209C22×C4—31no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213Q16—62no—
214Q16—62no—
215SD16—22no—
216SD16—22no—
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—31no—
226C4⋊C4—31no—
227C4⋊C4—31no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4C22×D311yescharacteristic
238C22⋊C4—21no—
239C22⋊C4—31no—
240C22⋊C4—31no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C42—31no—
245C42—61no—
246C42—61no—
247C42—61no—
248C42—61no—
249C42C22×D311yescharacteristic
250C22×C3—22no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3C22×D411yescharacteristic
254C2×D3—21no—
255C2×D3—21no—
256C2×D3—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4C2×#4111yescharacteristic
263C3×C4C2×#4111yescharacteristic
264Dic3—22no—
265Dic3—21no—
266Dic3—21no—
267Dic3—21no—
268Dic3—41no—
269Dic3—41no—
270Dic3—41no—
271C23—21no—
272C23—31no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—32no—
278Q8—61no—
279D4—21no—
280D4—64no—
281D4#23712yes—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—64no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—81no—
296C2×C4—241no—
297C2×C4—241no—
298C2×C4D3×D411yescharacteristic
299C2×C4D3×D411yescharacteristic
300C2×C4C23×D311yescharacteristic
301C8—21no—
302C8—21no—
303C8—61no—
304C8—61no—
305C2×C3—42no—
306C2×C3#52511yescharacteristic
307C2×C3#57711yescharacteristic
308C2×C3#57811yescharacteristic
309D3—41no—
310C22—22no—
311C22—41no—
312C22—41no—
313C22—61no—
314C22—61no—
315C22—61no—
316C22C2×D3×D411yescharacteristic
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—61no—
323C4—61no—
324C4—61no—
325C4—62no—
326C4—121no—
327C4—121no—
328C4—121no—
329C4C2×#23711yescharacteristic
330C4D3×#4111yescharacteristic
331C3#312011yescharacteristic, complemented, Sylow
332C2—42no—
333C2—121no—
334C2#560611yescharacteristic
335C2#576211yescharacteristic
336C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radicalC2×#237C22
hypercenter#461D3
hypocenterC3#3120
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5731C2semidirect
3. C3×#461C2semidirect
6. #5164C2semidirect
7. Dic3×SD16C2semidirect
8. #5090C2semidirect
9. #5009C2semidirect
10. #4840C2semidirect
11. #4829C2semidirect
12. #4811C2semidirect
13. #4805C2semidirect
14. #4790C2semidirect
15. #4789C2semidirect
67. #461D3semidirect
331. C3#3120semidirect

Maximal subgroups

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

subgroupindex
#57312
#55462
C3×#4612
#51732
#51642
Dic3×SD162
#50902
#50092
#48402
#48292
#48112
#48052
#47902
#47892
#31203