Finite Group Databasebeta

← #80384

Subgroups — #80384

Order 384.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#914C212yesmaximal, complemented
2D3×#127C211yesmaximal, characteristic, complemented
3C4×#239C211yesmaximal, characteristic, complemented
4C3×#460C211yesmaximal, characteristic, complemented
5#5163C212yesmaximal, complemented
6#5072C211yesmaximal, characteristic, complemented
7#5027C211yesmaximal, characteristic, complemented
8#4843C211yesmaximal, characteristic, complemented
9#4837C211yesmaximal, characteristic, complemented
10#4786C212yesmaximal, complemented
11#4770C212yesmaximal, complemented
12#3038—31nomaximal, Sylow
13C2×#239C2211yescharacteristic
14C2×D3×D4—21no—
15C2×D3×D4—21no—
16C2×D3×D4C2212yes—
17C2×C3×SD16C2212yescomplemented
18C3×#127C2211yescharacteristic
19C3×C4×Q8C2211yescharacteristic, complemented
20Q8×Dic3C2211yescharacteristic, complemented
21C2×#215C2212yescomplemented
22#940—21no—
23#940—21no—
24#940C2212yes—
25C2×#213C2212yescomplemented
26#914—28no—
27C2×D24C2212yescomplemented
28C2×ΓL(1,𝔽25)C2212yes—
29#893—21no—
30#893C2211yescharacteristic, complemented
31#892—21no—
32#892C2211yescharacteristic
33#874C2211yescharacteristic
34C4×D12—21no—
35C4×D12C2211yescharacteristic, complemented
36C42×D3—21no—
37C42×D3C2211yescharacteristic
38C3×C4⋊C8C2211yescharacteristic, complemented
39C3×#102C2212yescomplemented
40#832C2212yescomplemented
41#821C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#809C2212yescomplemented
44C4⋊(C3⋊C8)C2211yescharacteristic, complemented
45C2×AGL(1,ℤ/8)—32no—
46C2×#127—31no—
47C4×#41—31no—
48#460—32no—
49#460—31no—
50#460D311yescharacteristic, complemented
51#459—31no—
52#459—31no—
53#459—32no—
54#423—31no—
55#418—32no—
56C23×D3—22no—
57C2×C3×Q8C2311yescharacteristic
58C2×C3×D4—21no—
59C2×C3×D4—21no—
60C2×C3×D4C2312yes—
61C2×#67—22no—
62C2×#67—22no—
63C2×#67—22no—
64#239—22no—
65#239D414yescomplemented
66D3×D4—24no—
67D3×D4—24no—
68D3×D4—44no—
69D3×D4—44no—
70C2×D12—21no—
71C2×D12—21no—
72C2×D12—21no—
73C2×D12C2312yes—
74C2×D12C2311yescharacteristic
75C2×C4×D3—21no—
76C2×C4×D3—21no—
77C2×C4×D3—21no—
78C2×C4×D3D411yescharacteristic
79C2×C4×D3D411yescharacteristic
80C2×C4×D3C2311yescharacteristic
81C3×SD16—24no—
82C2×C3×C8C2312yes—
83C3×C4⋊C4—21no—
84C3×C4⋊C4C2311yescharacteristic
85C3×C42—21no—
86C3×C42C2311yescharacteristic
87#215—24no—
88#213—24no—
89(C2×D3)⋊C4—21no—
90(C2×D3)⋊C4—21no—
91(C2×D3)⋊C4—21no—
92C4⋊Dic3—21no—
93C4⋊Dic3C2311yescharacteristic
94C4×Dic3—21no—
95C4×Dic3—21no—
96C4×Dic3D411yescharacteristic
97C4×Dic3D411yescharacteristic
98C4×Dic3C2311yescharacteristic
99C2×C3⋊C8C2312yes—
100D24—24no—
101ΓL(1,𝔽25)—24no—
102C2×#41—31no—
103C22×D4—32no—
104C22×D4—61no—
105C22×D4—61no—
106AGL(1,ℤ/8)—68no—
107C2×SD16—32no—
108C2×SD16C2×D312yescomplemented
109C2×D8—32no—
110C2×D8—32no—
111C2×M16—32no—
112#127—32no—
113#127—31no—
114#127—61no—
115#127—61no—
116#127C2×D311yescharacteristic
117C4×Q8—31no—
118C4×Q8C2×D311yescharacteristic, complemented
119C4×D4—31no—
120C4×D4—31no—
121C4×D4—61no—
122C4×D4—61no—
123#117—31no—
124#117—61no—
125C2×C42—31no—
126C2×C42—61no—
127C4⋊C8—32no—
128C4⋊C8—31no—
129C4⋊C8C2×D311yescharacteristic, complemented
130#102—32no—
131#102—32no—
132#102—32no—
133#102C2×D312yescomplemented
134C23×C3—22no—
135C22×D3—21no—
136C22×D3—22no—
137C22×D3—44no—
138C22×D3—44no—
139C22×D3—44no—
140C22×D3C2×D411yescharacteristic
141C3×Q8—21no—
142C3×Q8C2×D412yescomplemented
143C3×D4—22no—
144C3×D4—22no—
145C3×D4—42no—
146C3×D4—42no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4C2×D411yescharacteristic
150C2×C3×C4C2×D411yescharacteristic
151C2×C3×C4C2411yescharacteristic
152#67—44no—
153#67—44no—
154#67—44no—
155C2×Dic3—21no—
156C2×Dic3C2×D411yescharacteristic
157C2×Dic3C2×D411yescharacteristic
158C2×Dic3C2×D411yescharacteristic
159D12—24no—
160D12—22no—
161D12—21no—
162D12—22no—
163D12—42no—
164D12—42no—
165D12C2×D412yescomplemented
166C4×D3—22no—
167C4×D3—24no—
168C4×D3—22no—
169C4×D3—42no—
170C4×D3—42no—
171C4×D3C2×D412yes—
172C4×D3#4112yes—
173C3×C8—22no—
174C3⋊C8—22no—
175C24—62no—
176#41—34no—
177#41—62no—
178C2×Q8C22×D311yescharacteristic
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—31no—
182C2×D4—32no—
183C2×D4—61no—
184C2×D4—64no—
185C2×D4—64no—
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—62no—
189C2×D4—61no—
190C2×D4—61no—
191C2×D4—124no—
192C2×D4—124no—
193C2×D4C22×D312yes—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4—61no—
200SD16—24no—
201SD16—64no—
202D8—64no—
203D8—64no—
204M16—64no—
205C2×C8—32no—
206C2×C8C22×D312yes—
207C4⋊C4—21no—
208C4⋊C4—31no—
209C4⋊C4—61no—
210C4⋊C4C22×D311yescharacteristic
211C22⋊C4—61no—
212C22⋊C4—61no—
213C22⋊C4—61no—
214C42—21no—
215C42—31no—
216C42—31no—
217C42—31no—
218C42—61no—
219C42—61no—
220C42C22×D311yescharacteristic
221C22×C3—42no—
222C22×C3—42no—
223C22×C3—42no—
224C22×C3C22×D411yescharacteristic
225C2×D3—21no—
226C2×D3—21no—
227C2×D3—22no—
228C2×D3—41no—
229C2×D3—41no—
230C2×D3—42no—
231C2×D3—42no—
232C2×D3—42no—
233C2×D3—88no—
234C2×D3#12112yes—
235C3×C4—22no—
236C3×C4—21no—
237C3×C4—41no—
238C3×C4AGL(1,ℤ/8)12yescomplemented
239C3×C4C22×D411yescharacteristic
240C3×C4C2×#4111yescharacteristic
241Dic3—21no—
242Dic3—41no—
243Dic3#12112yes—
244Dic3AGL(1,ℤ/8)12yescomplemented
245C23—22no—
246C23—31no—
247C23—61no—
248C23—62no—
249C23—124no—
250C23—124no—
251C23—124no—
252Q8—21no—
253Q8D3×D412yescomplemented
254D4—22no—
255D4—22no—
256D4—32no—
257D4—42no—
258D4—42no—
259D4—64no—
260D4—62no—
261D4—61no—
262D4—62no—
263D4—122no—
264D4—124no—
265D4—124no—
266D4—124no—
267D4—122no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—32no—
273C2×C4—32no—
274C2×C4—31no—
275C2×C4—62no—
276C2×C4—64no—
277C2×C4—62no—
278C2×C4—61no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4D3×D411yescharacteristic
282C2×C4D3×D411yescharacteristic
283C2×C4C23×D311yescharacteristic
284C8—22no—
285C8—62no—
286C2×C3—82no—
287C2×C3C2×#12111yescharacteristic
288C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
289C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
290D3—22no—
291D3—42no—
292D3—82no—
293C22—32no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—61no—
298C22—61no—
299C22—62no—
300C22—121no—
301C22—121no—
302C22—122no—
303C22—122no—
304C22—122no—
305C22—248no—
306C22C2×D3×D411yescharacteristic
307C4—21no—
308C4—22no—
309C4—32no—
310C4—32no—
311C4—41no—
312C4—61no—
313C4—121no—
314C4#91412yescomplemented
315C4C2×D3×D411yescharacteristic
316C4D3×#4111yescharacteristic
317C3#303811yescharacteristic, complemented, Sylow
318C2—62no—
319C2—82no—
320C2—122no—
321C2—242no—
322C2D3×#12111yescharacteristic
323C2C2×#91411yescharacteristic
324C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
hypercenter#460D3
hypocenterC3#3038
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#914C2semidirect
2. D3×#127C2semidirect
3. C4×#239C2semidirect
4. C3×#460C2semidirect
5. #5163C2semidirect
6. #5072C2semidirect
7. #5027C2semidirect
8. #4843C2semidirect
9. #4837C2semidirect
10. #4786C2semidirect
11. #4770C2semidirect
17. C2×C3×SD16C22semidirect
19. C3×C4×Q8C22semidirect
20. Q8×Dic3C22semidirect
21. C2×#215C22semidirect
25. C2×#213C22semidirect
27. C2×D24C22semidirect
30. #893C22semidirect
35. C4×D12C22semidirect
38. C3×C4⋊C8C22semidirect
39. C3×#102C22semidirect
40. #832C22semidirect
41. #821C22semidirect
42. Dic3⋊C8C22semidirect
43. #809C22semidirect
44. C4⋊(C3⋊C8)C22semidirect
50. #460D3semidirect
65. #239D4semidirect
108. C2×SD16C2×D3semidirect
118. C4×Q8C2×D3semidirect
129. C4⋊C8C2×D3semidirect
133. #102C2×D3semidirect
142. C3×Q8C2×D4semidirect
165. D12C2×D4semidirect
238. C3×C4AGL(1,ℤ/8)semidirect
244. Dic3AGL(1,ℤ/8)semidirect
253. Q8D3×D4semidirect
314. C4#914semidirect
317. C3#3038semidirect

Maximal subgroups

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

subgroupindex
C2×#9142
D3×#1272
C4×#2392
C3×#4602
#51632
#50722
#50272
#48432
#48372
#47862
#47702
#30383