Finite Group Databasebeta

← #80141

Subgroups — #80141

Order 384.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 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×#940C211yesmaximal, characteristic, complemented
2#5624C211yesmaximal, characteristic, complemented
3C3×#450C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic, complemented
5#5176C212yesmaximal, complemented
6#5163C212yesmaximal, complemented
7#5046C211yesmaximal, characteristic, complemented
8#4786C212yesmaximal, complemented
9#4764C212yesmaximal, complemented
10#4732C211yesmaximal, characteristic, complemented
11#4719C211yesmaximal, characteristic, complemented
12#3217—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67—21no—
15C22×#67—21no—
16C22×D12C2211yescharacteristic
17C2×C3×SD16C2212yescomplemented
18C3×C22⋊Q8C2211yescharacteristic
19#947C2211yescharacteristic, complemented
20Q8×Dic3C2211yescharacteristic, complemented
21C2×#215C2212yescomplemented
22#940—21no—
23#940—21no—
24#940C2212yes—
25#940C2212yes—
26C2×#213C2212yescomplemented
27#930C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C2×D24C2212yescomplemented
30#898—21no—
31#893C2211yescharacteristic, complemented
32#892—21no—
33#890C2211yescharacteristic, complemented
34#886—21no—
35#67⋊C4—21no—
36C3×#102C2212yescomplemented
37C3×C22⋊C8C2211yescharacteristic
38#832C2212yescomplemented
39#830C2211yescharacteristic
40#821C2212yescomplemented
41#815C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#809C2212yescomplemented
44C2×#127—31no—
45#529—31no—
46#496—32no—
47#490—32no—
48#460—32no—
49#459—32no—
50#450—31no—
51#450D311yescharacteristic, complemented
52#447—31no—
53#447—31no—
54#432—31no—
55C24×C3—21no—
56C23×D3—21no—
57C2×C3×Q8C2311yescharacteristic
58C2×C3×D4—21no—
59C2×C3×D4—42no—
60C2×C3×D4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C2×#67—21no—
63C2×#67—22no—
64C2×#67—22no—
65C2×#67—22no—
66C2×#67—22no—
67C2×#67—44no—
68C2×#67—44no—
69C22×Dic3D411yescharacteristic
70C22×Dic3D411yescharacteristic
71C2×D12—21no—
72C2×D12—42no—
73C2×D12C2312yes—
74C2×D12C2311yescharacteristic
75C2×C4×D3—21no—
76C3×SD16—42no—
77C2×C3×C8C2312yes—
78C3×C4⋊C4—21no—
79C3×C4⋊C4C2311yescharacteristic
80C3×C22⋊C4—21no—
81#215—42no—
82#213—42no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4—21no—
86(C2×D3)⋊C4—21no—
87C4⋊Dic3—21no—
88C4⋊Dic3—21no—
89C4⋊Dic3C2311yescharacteristic
90Dic3⋊C4—21no—
91C4×Dic3—21no—
92C4×Dic3D411yescharacteristic
93C4×Dic3D411yescharacteristic
94C4×Dic3C2311yescharacteristic
95C4×Dic3C2311yescharacteristic
96C2×C3⋊C8C2312yes—
97D24—42no—
98C22×D4—31no—
99C22×D4—61no—
100C22×D4—61no—
101C22×D4C2×D311yescharacteristic
102C2×SD16—32no—
103C2×SD16C2×D312yescomplemented
104C2×D8—32no—
105C2×D8—32no—
106#127—32no—
107#127—32no—
108#127—61no—
109#127—61no—
110#126—61no—
111#125—31no—
112#124—31no—
113#123—61no—
114C22⋊Q8C2×D311yescharacteristic
115#121—31no—
116C4×Q8—31no—
117C4×D4—31no—
118C4×D4—61no—
119#117—61no—
120C2×C42—31no—
121C4⋊C8—32no—
122#102—32no—
123#102—32no—
124#102—32no—
125#102C2×D312yescomplemented
126C22⋊C8—31no—
127C22⋊C8C2×D311yescharacteristic
128#97—32no—
129C23×C3—22no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3—42no—
133C23×C3C2×D411yescharacteristic
134C22×D3—21no—
135C22×D3—22no—
136C22×D3—42no—
137C22×D3—42no—
138C22×D3—42no—
139C3×Q8—22no—
140C3×D4—22no—
141C3×D4—22no—
142C3×D4—42no—
143C2×C3×C4—22no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2411yescharacteristic
148#67—44no—
149#67—44no—
150#67—44no—
151#67—42no—
152#67—44no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—21no—
156C2×Dic3C2×D411yescharacteristic
157C2×Dic3C2×D411yescharacteristic
158C2×Dic3C2×D411yescharacteristic
159C2×Dic3C2×D411yescharacteristic
160D12—22no—
161D12—22no—
162D12—22no—
163D12—42no—
164C4×D3—42no—
165C3×C8—22no—
166C3⋊C8—22no—
167C24—21no—
168C24—61no—
169C2×Q8C22×D311yescharacteristic
170C2×D4—21no—
171C2×D4—32no—
172C2×D4—31no—
173C2×D4—42no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—62no—
178C2×D4—61no—
179C2×D4—61no—
180C2×D4—122no—
181C2×D4—124no—
182C2×D4—124no—
183C2×D4C22×D312yes—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—61no—
187C22×C4C22×D311yescharacteristic
188SD16—42no—
189SD16—122no—
190D8—122no—
191D8—122no—
192C2×C8—32no—
193C2×C8C22×D312yes—
194C4⋊C4—21no—
195C4⋊C4—31no—
196C4⋊C4—61no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4C22×D311yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—61no—
202C22⋊C4—61no—
203C22⋊C4—61no—
204C22⋊C4—61no—
205C42—31no—
206C42—31no—
207C42—31no—
208C42—31no—
209C42—61no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—84no—
216C22×C3AGL(1,ℤ/8)12yes—
217C22×C3C22×D411yescharacteristic
218C2×D3—42no—
219C2×D3—41no—
220C2×D3—41no—
221C2×D3—42no—
222C2×D3—42no—
223C2×D3—41no—
224C2×D3—84no—
225C3×C4—21no—
226C3×C4—41no—
227C3×C4—41no—
228C3×C4H(5,𝔽2)12yes—
229Dic3—21no—
230Dic3—21no—
231Dic3—21no—
232Dic3—41no—
233Dic3AGL(1,ℤ/8)12yes—
234C23—22no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—61no—
239C23—62no—
240C23—122no—
241C23—122no—
242C23—122no—
243C23D3×D411yescharacteristic
244Q8—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—62no—
249D4—62no—
250D4—62no—
251D4—122no—
252D4—124no—
253D4—124no—
254D4—124no—
255D4—124no—
256D4—122no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—122no—
268C2×C4D3×D411yescharacteristic
269C2×C4C23×D311yescharacteristic
270C8—22no—
271C8—62no—
272C2×C3—22no—
273C2×C3—82no—
274C2×C3#53511yescharacteristic
275C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
276C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
277D3—81no—
278D3—82no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—122no—
286C22—121no—
287C22—121no—
288C22—122no—
289C22—122no—
290C22—121no—
291C22—244no—
292C22#91412yes—
293C22C2×D3×D411yescharacteristic
294C4—21no—
295C4—32no—
296C4—41no—
297C4—41no—
298C4—61no—
299C4—61no—
300C4—61no—
301C4—121no—
302C4#100412yes—
303C3#321711yescharacteristic, complemented, Sylow
304C2—22no—
305C2—82no—
306C2—242no—
307C2—241no—
308C2#559211yescharacteristic
309C2C2×#91411yescharacteristic
310C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
hypercenter#450D3
hypocenterC3#3217
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#940C2semidirect
2. #5624C2semidirect
3. C3×#450C2semidirect
4. #5217C2semidirect
5. #5176C2semidirect
6. #5163C2semidirect
7. #5046C2semidirect
8. #4786C2semidirect
9. #4764C2semidirect
10. #4732C2semidirect
11. #4719C2semidirect
17. C2×C3×SD16C22semidirect
19. #947C22semidirect
20. Q8×Dic3C22semidirect
21. C2×#215C22semidirect
26. C2×#213C22semidirect
29. C2×D24C22semidirect
31. #893C22semidirect
33. #890C22semidirect
36. C3×#102C22semidirect
38. #832C22semidirect
40. #821C22semidirect
41. #815C22semidirect
42. Dic3⋊C8C22semidirect
43. #809C22semidirect
51. #450D3semidirect
103. C2×SD16C2×D3semidirect
125. #102C2×D3semidirect
303. C3#3217semidirect

Maximal subgroups

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

subgroupindex
C2×#9402
#56242
C3×#4502
#52172
#51762
#51632
#50462
#47862
#47642
#47322
#47192
#32173