Finite Group Databasebeta

← #80333

Subgroups — #80333

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×#911C212yesmaximal, complemented
2D3×#127C211yesmaximal, characteristic, complemented
3C4×#237C211yesmaximal, characteristic, complemented
4C3×#459C211yesmaximal, characteristic, complemented
5#5150C212yesmaximal, complemented
6#5075C211yesmaximal, characteristic, complemented
7#5015C211yesmaximal, characteristic, complemented
8#4848C211yesmaximal, characteristic, complemented
9#4837C211yesmaximal, characteristic, complemented
10#4770C212yesmaximal, complemented
11#4762C212yesmaximal, complemented
12#3038—31nomaximal, Sylow
13C2×#237C2211yescharacteristic
14C2×D3×D4—21no—
15C2×D3×D4—21no—
16C2×D3×D4C2212yes—
17C2×C3×D8C2212yescomplemented
18C3×#127C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic, complemented
20#940—21no—
21#940—21no—
22#940C2212yes—
23D4×Dic3C2211yescharacteristic, complemented
24C2×#214C2212yescomplemented
25C2×#213C2212yescomplemented
26C4×#67—21no—
27C2×C4×Dic3—21no—
28#911—28no—
29C2×#204C2212yescomplemented
30C2×ΓL(1,𝔽25)C2212yes—
31#892C2211yescharacteristic
32#887C2211yescharacteristic, complemented
33#67⋊C4—21no—
34#877—21no—
35#874C2211yescharacteristic
36C42×D3C2211yescharacteristic
37C4×Dic6C2211yescharacteristic, complemented
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—31no—
49#460—31no—
50#460—32no—
51#459—32no—
52#459—31no—
53#459D311yescharacteristic, complemented
54#423—31no—
55#418—32no—
56C23×D3—22no—
57C2×C3×D4—21no—
58C2×C3×D4—21no—
59C2×C3×D4C2312yes—
60C2×C3×D4C2311yescharacteristic
61C22×C3×C4—21no—
62C2×#67—21no—
63C2×#67—22no—
64C2×#67—22no—
65C2×#67—22no—
66C22×Dic3—21no—
67C22×Dic3—21no—
68#237—22no—
69#237D414yescomplemented
70D3×D4—24no—
71D3×D4—24no—
72D3×D4—44no—
73D3×D4—44no—
74C2×D12—21no—
75C2×D12—21no—
76C2×D12C2312yes—
77C2×C4×D3D411yescharacteristic
78C2×C4×D3D411yescharacteristic
79C2×C4×D3C2311yescharacteristic
80C2×Dic6C2311yescharacteristic
81C3×D8—24no—
82C2×C3×C8C2312yes—
83C3×C4⋊C4C2311yescharacteristic
84C3×C22⋊C4—21no—
85C3×C42C2311yescharacteristic
86#217—21no—
87#214—24no—
88#213—24no—
89(C2×D3)⋊C4—21no—
90C4⋊Dic3C2311yescharacteristic
91Dic3⋊C4—21no—
92Dic3⋊C4—21no—
93C4×Dic3—21no—
94C4×Dic3—21no—
95C4×Dic3—21no—
96C4×Dic3D411yescharacteristic
97C4×Dic3D411yescharacteristic
98C4×Dic3C2311yescharacteristic
99C2×C3⋊C8C2312yes—
100#204—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×SD16—32no—
109C2×D8—32no—
110C2×D8C2×D312yescomplemented
111C2×M16—32no—
112#127—32no—
113#127—31no—
114#127—61no—
115#127—61no—
116#127C2×D311yescharacteristic
117C4×Q8—31no—
118C4×Q8—31no—
119C4×D4—31no—
120C4×D4—61no—
121C4×D4—61no—
122C4×D4C2×D311yescharacteristic, complemented
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—21no—
135C23×C3—22no—
136C22×D3—22no—
137C22×D3—44no—
138C22×D3—44no—
139C22×D3—44no—
140C22×D3C2×D411yescharacteristic
141C3×D4—22no—
142C3×D4—21no—
143C3×D4—22no—
144C3×D4—42no—
145C3×D4—42no—
146C3×D4C2×D412yescomplemented
147C2×C3×C4—21no—
148C2×C3×C4—42no—
149C2×C3×C4C2×D411yescharacteristic
150C2×C3×C4C2×D411yescharacteristic
151C2×C3×C4C2411yescharacteristic
152#67—24no—
153#67—44no—
154#67—44no—
155#67—44no—
156C2×Dic3—21no—
157C2×Dic3—24no—
158C2×Dic3—21no—
159C2×Dic3—42no—
160C2×Dic3C2×D411yescharacteristic
161C2×Dic3C2×D411yescharacteristic
162C2×Dic3C2×D411yescharacteristic
163D12—22no—
164D12—22no—
165D12—42no—
166D12—42no—
167C4×D3—22no—
168C4×D3—22no—
169C4×D3C2×D412yes—
170C4×D3#4112yes—
171Dic6—21no—
172Dic6C2×D412yescomplemented
173C3×C8—22no—
174C3⋊C8—22no—
175C24—62no—
176#41—34no—
177#41—62no—
178C2×Q8—31no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—32no—
182C2×D4—64no—
183C2×D4—64no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—62no—
187C2×D4—61no—
188C2×D4—61no—
189C2×D4—61no—
190C2×D4—124no—
191C2×D4—124no—
192C2×D4C22×D312yes—
193C2×D4C22×D311yescharacteristic
194C22×C4—21no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—61no—
199C22×C4—61no—
200SD16—64no—
201SD16—64no—
202D8—24no—
203D8—64no—
204M16—64no—
205C2×C8—32no—
206C2×C8C22×D312yes—
207C4⋊C4—31no—
208C4⋊C4—61no—
209C4⋊C4—61no—
210C4⋊C4C22×D311yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—61no—
213C22⋊C4—61no—
214C42—31no—
215C42—31no—
216C42—31no—
217C42—61no—
218C42—61no—
219C42—61no—
220C42C22×D311yescharacteristic
221C22×C3—22no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3C22×D411yescharacteristic
228C2×D3—21no—
229C2×D3—21no—
230C2×D3—42no—
231C2×D3—42no—
232C2×D3—42no—
233C2×D3—88no—
234C2×D3#12112yes—
235C3×C4—21no—
236C3×C4—41no—
237C3×C4AGL(1,ℤ/8)12yescomplemented
238C3×C4C22×D411yescharacteristic
239C3×C4C2×#4111yescharacteristic
240Dic3—22no—
241Dic3—21no—
242Dic3—41no—
243Dic3#12112yes—
244Dic3AGL(1,ℤ/8)12yescomplemented
245C23—21no—
246C23—22no—
247C23—31no—
248C23—62no—
249C23—124no—
250C23—124no—
251C23—124no—
252Q8—32no—
253Q8—61no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—42no—
259D4—64no—
260D4—62no—
261D4—62no—
262D4—122no—
263D4—124no—
264D4—124no—
265D4—124no—
266D4—122no—
267D4D3×D412yescomplemented
268C2×C4—21no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—32no—
272C2×C4—32no—
273C2×C4—31no—
274C2×C4—42no—
275C2×C4—62no—
276C2×C4—64no—
277C2×C4—61no—
278C2×C4—62no—
279C2×C4—61no—
280C2×C4—122no—
281C2×C4D3×D411yescharacteristic
282C2×C4D3×D411yescharacteristic
283C2×C4C23×D311yescharacteristic
284C8—22no—
285C8—62no—
286C2×C3—42no—
287C2×C3—82no—
288C2×C3C2×#12111yescharacteristic
289C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
290C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
291D3—22no—
292D3—82no—
293C22—22no—
294C22—32no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—61no—
301C22—61no—
302C22—122no—
303C22—122no—
304C22—122no—
305C22—248no—
306C22C2×D3×D411yescharacteristic
307C4—21no—
308C4—32no—
309C4—32no—
310C4—41no—
311C4—61no—
312C4—62no—
313C4—121no—
314C4#91112yescomplemented
315C4C2×D3×D411yescharacteristic
316C4D3×#4111yescharacteristic
317C3#303811yescharacteristic, complemented, Sylow
318C2—42no—
319C2—62no—
320C2—82no—
321C2—242no—
322C2D3×#12111yescharacteristic
323C2C2×#91111yescharacteristic
324C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3038
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. D3×#127C2semidirect
3. C4×#237C2semidirect
4. C3×#459C2semidirect
5. #5150C2semidirect
6. #5075C2semidirect
7. #5015C2semidirect
8. #4848C2semidirect
9. #4837C2semidirect
10. #4770C2semidirect
11. #4762C2semidirect
17. C2×C3×D8C22semidirect
19. C3×C4×D4C22semidirect
23. D4×Dic3C22semidirect
24. C2×#214C22semidirect
25. C2×#213C22semidirect
29. C2×#204C22semidirect
32. #887C22semidirect
37. C4×Dic6C22semidirect
38. C3×C4⋊C8C22semidirect
39. C3×#102C22semidirect
40. #832C22semidirect
41. #821C22semidirect
42. Dic3⋊C8C22semidirect
43. #809C22semidirect
44. C4⋊(C3⋊C8)C22semidirect
53. #459D3semidirect
69. #237D4semidirect
110. C2×D8C2×D3semidirect
122. C4×D4C2×D3semidirect
129. C4⋊C8C2×D3semidirect
133. #102C2×D3semidirect
146. C3×D4C2×D4semidirect
172. Dic6C2×D4semidirect
237. C3×C4AGL(1,ℤ/8)semidirect
244. Dic3AGL(1,ℤ/8)semidirect
267. D4D3×D4semidirect
314. C4#911semidirect
317. C3#3038semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
D3×#1272
C4×#2372
C3×#4592
#51502
#50752
#50152
#48482
#48372
#47702
#47622
#30383