Finite Group Databasebeta

← #80057

Subgroups — #80057

Order 384.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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×#947C211yesmaximal, characteristic, complemented
2#5600C211yesmaximal, characteristic, complemented
3C3×#448C211yesmaximal, characteristic, complemented
4#5228C211yesmaximal, characteristic, complemented
5#5168C212yesmaximal, complemented
6#5165C212yesmaximal, complemented
7#5036C211yesmaximal, characteristic
8#4800C212yesmaximal, complemented
9#4798C212yesmaximal, complemented
10#4731C211yesmaximal, characteristic, complemented
11#4718C211yesmaximal, characteristic, complemented
12#3211—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×D12C2211yescharacteristic
15C2×C3×SD16C2212yescomplemented
16C3×#121C2211yescharacteristic
17#947—21no—
18#947—21no—
19#947C2212yes—
20#947C2212yes—
21C2×#216C2212yescomplemented
22#936C2211yescharacteristic, complemented
23#935—21no—
24D4×Dic3—21no—
25D4×Dic3C2211yescharacteristic, complemented
26C2×#213C2212yes—
27C2×(C2×D3)⋊C4—21no—
28C2×(C2×D3)⋊C4—21no—
29#924C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31C2×#204C2212yes—
32#890C2211yescharacteristic, complemented
33#887C2211yescharacteristic, complemented
34#879—21no—
35#877—21no—
36C3×#103C2212yescomplemented
37C3×C22⋊C8C2211yescharacteristic
38#835C2212yescomplemented
39#830C2211yescharacteristic
40#821C2212yes—
41Dic3⋊C8C2212yescomplemented
42C8×Dic3C2212yescomplemented
43#809C2212yes—
44#529—31no—
45C2×#124—31no—
46#495—32no—
47#488—32no—
48#464—32no—
49#463—32no—
50#451—31no—
51#450—31no—
52#448D311yescharacteristic, complemented
53#447—31no—
54#431—31no—
55C23×D3—21no—
56C2×C3×Q8—21no—
57C2×C3×Q8—42no—
58C2×C3×Q8C2312yes—
59C2×C3×D4—21no—
60C2×C3×D4C2311yescharacteristic
61C22×C3×C4—21no—
62C22×C3×C4C2311yescharacteristic
63C22×Dic3—21no—
64C22×Dic3D411yescharacteristic
65C22×Dic3D411yescharacteristic
66C2×D12—21no—
67C2×D12—42no—
68C2×D12C2312yes—
69C2×Dic6C2311yescharacteristic
70C3×SD16—42no—
71C2×C3×C8C2312yes—
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—21no—
74#217—21no—
75#217—21no—
76#217—21no—
77#217—21no—
78#216—42no—
79#213—42no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—22no—
84C4⋊Dic3—21no—
85C4⋊Dic3C2311yescharacteristic
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88Dic3⋊C4—21no—
89C4×Dic3—21no—
90C4×Dic3D411yescharacteristic
91C4×Dic3D411yescharacteristic
92C4×Dic3C2311yescharacteristic
93C4×Dic3C2311yescharacteristic
94C2×C3⋊C8C2312yes—
95#204—42no—
96C22×Q8C2×D311yescharacteristic
97C22×D4—31no—
98C2×Q16—32no—
99C2×SD16—32no—
100C2×SD16C2×D312yescomplemented
101C2×D8—32no—
102#126—61no—
103#125—31no—
104#124—32no—
105#124—32no—
106#124—31no—
107#124—61no—
108#124—61no—
109#123—61no—
110C22⋊Q8—31no—
111#121C2×D311yescharacteristic
112C4×Q8—31no—
113C4×D4—31no—
114C4×D4—61no—
115#117—61no—
116C2×C22⋊C4—61no—
117C2×C22⋊C4—61no—
118C2×C42—31no—
119C4⋊C8—32no—
120#103—32no—
121#103C2×D312yescomplemented
122#102—32no—
123#102—32no—
124C22⋊C8—31no—
125C22⋊C8C2×D311yescharacteristic
126C4×C8—32no—
127C23×C3—21no—
128C23×C3C2×D411yescharacteristic
129C22×D3—22no—
130C22×D3—42no—
131C22×D3—42no—
132C22×D3—42no—
133C3×Q8—22no—
134C3×Q8—22no—
135C3×Q8—42no—
136C3×D4—22no—
137C3×D4—42no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—22no—
141C2×C3×C4—42no—
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2411yescharacteristic
144C2×Dic3—22no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—41no—
148C2×Dic3—42no—
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151C2×Dic3C2×D411yescharacteristic
152C2×Dic3C2×D411yescharacteristic
153D12—22no—
154D12—22no—
155D12—42no—
156Dic6—22no—
157C3×C8—22no—
158C3⋊C8—22no—
159C24—61no—
160C2×Q8—21no—
161C2×Q8—31no—
162C2×Q8—42no—
163C2×Q8C22×D312yes—
164C2×D4—21no—
165C2×D4—32no—
166C2×D4—61no—
167C2×D4—122no—
168C2×D4C22×D311yescharacteristic
169C22×C4—21no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—61no—
173C22×C4C22×D311yescharacteristic
174Q16—122no—
175SD16—42no—
176SD16—122no—
177D8—122no—
178C2×C8—32no—
179C2×C8C22×D312yes—
180C4⋊C4—31no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C4⋊C4—61no—
185C4⋊C4C22×D311yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—62no—
188C22⋊C4—62no—
189C22⋊C4—62no—
190C22⋊C4—62no—
191C22⋊C4—61no—
192C22⋊C4—61no—
193C22⋊C4—61no—
194C22⋊C4—61no—
195C42—31no—
196C42—31no—
197C42—31no—
198C42—31no—
199C42—61no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3#13512yes—
206C22×C3C22×D411yescharacteristic
207C2×D3—42no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—84no—
211C3×C4—21no—
212C3×C4—42no—
213C3×C4—41no—
214C3×C4H(5,𝔽2)12yes—
215Dic3—21no—
216Dic3—21no—
217Dic3—21no—
218Dic3—21no—
219Dic3—41no—
220Dic3—41no—
221C23—21no—
222C23—62no—
223C23—122no—
224C23—122no—
225C23—122no—
226C23D3×D411yescharacteristic
227Q8—22no—
228Q8—22no—
229Q8—42no—
230Q8—62no—
231D4—22no—
232D4—42no—
233D4—62no—
234D4—62no—
235D4—122no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—31no—
243C2×C4—42no—
244C2×C4—62no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—121no—
248C2×C4—122no—
249C2×C4D3×D411yescharacteristic
250C2×C4C23×D311yescharacteristic
251C8—22no—
252C8—62no—
253C2×C3—22no—
254C2×C3—81no—
255C2×C3#53511yescharacteristic
256C2×C3C2×#13511yescharacteristic
257C2×C3#57511yescharacteristic
258D3—82no—
259C22—21no—
260C22—21no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—122no—
265C22—122no—
266C22—122no—
267C22—244no—
268C22#91612yes—
269C22C2×D3×D411yescharacteristic
270C4—21no—
271C4—41no—
272C4—42no—
273C4—61no—
274C4—61no—
275C4—61no—
276C4—61no—
277C4—121no—
278C4—121no—
279C4#100412yes—
280C3#321111yescharacteristic, complemented, Sylow
281C2—22no—
282C2—81no—
283C2—242no—
284C2#559211yescharacteristic
285C2C2×#91611yescharacteristic
286C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radical#5036C2
hypercenter#448D3
hypocenterC3#3211
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#947C2semidirect
2. #5600C2semidirect
3. C3×#448C2semidirect
4. #5228C2semidirect
5. #5168C2semidirect
6. #5165C2semidirect
8. #4800C2semidirect
9. #4798C2semidirect
10. #4731C2semidirect
11. #4718C2semidirect
15. C2×C3×SD16C22semidirect
21. C2×#216C22semidirect
22. #936C22semidirect
25. D4×Dic3C22semidirect
32. #890C22semidirect
33. #887C22semidirect
36. C3×#103C22semidirect
38. #835C22semidirect
41. Dic3⋊C8C22semidirect
42. C8×Dic3C22semidirect
52. #448D3semidirect
100. C2×SD16C2×D3semidirect
121. #103C2×D3semidirect
280. C3#3211semidirect

Maximal subgroups

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

subgroupindex
C2×#9472
#56002
C3×#4482
#52282
#51682
#51652
#50362
#48002
#47982
#47312
#47182
#32113