Finite Group Databasebeta

← #81874

Subgroups — #81874

Order 384.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#5728C211yesmaximal, characteristic, complemented
2D3×#124C211yesmaximal, characteristic
3C3×#497C211yesmaximal, characteristic, complemented
4#5186C212yesmaximal, complemented
5#5185C212yesmaximal, complemented
6#5169C212yesmaximal
7#5166C212yesmaximal, complemented
8#5086C211yesmaximal, characteristic, complemented
9#5055C211yesmaximal, characteristic, complemented
10#4714C211yesmaximal, characteristic, complemented
11#4705C211yesmaximal, characteristic, complemented
12#3215—31nomaximal, Sylow
13C2×D3×Q8C2211yescharacteristic
14C2×D3×D4C2211yescharacteristic
15C2×C3×Q16C2212yescomplemented
16C2×C3×SD16C2212yes—
17C3×C4⋊Q8C2211yescharacteristic, complemented
18C3×#124C2211yescharacteristic
19#947C2211yescharacteristic
20#946C2212yes—
21Q8×Dic3C2212yescomplemented
22C2×#216C2212yescomplemented
23C2×#215C2212yes—
24#936C2211yescharacteristic
25#897—21no—
26#892—21no—
27#890—21no—
28#887—21no—
29#885—21no—
30#885—21no—
31D3×C22⋊C4—21no—
32D3×C22⋊C4—21no—
33#875C2211yescharacteristic
34C42×D3C2211yescharacteristic
35#869C2211yescharacteristic, complemented
36C3×#97C2211yescharacteristic, complemented
37#835C2212yescomplemented
38#832C2212yes—
39#821C2212yes—
40(C2×D3)⋊C8C2212yes—
41#816C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#803C2211yescharacteristic, complemented
44#533—31no—
45C2×#124—31no—
46#497—31no—
47#497D311yescharacteristic, complemented
48#489—31no—
49#489—31no—
50#464—32no—
51#464—32no—
52#451—32no—
53#450—32no—
54#432—31no—
55C23×D3—21no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×Q8C2312yes—
58C2×C3×D4C2311yescharacteristic
59C2×#67—21no—
60D3×Q8—42no—
61D3×D4—42no—
62C2×D12C2311yescharacteristic
63C2×C4×D3—21no—
64C2×C4×D3D411yescharacteristic
65C2×C4×D3D411yescharacteristic
66C2×C4×D3C2311yescharacteristic
67C2×Dic6—21no—
68C2×Dic6—21no—
69C2×Dic6C2311yescharacteristic
70C3×Q16—42no—
71C3×SD16—42no—
72C2×C3×C8C2312yes—
73C3×C4⋊C4—21no—
74C3×C4⋊C4—21no—
75C3×C22⋊C4—21no—
76C3×C22⋊C4—21no—
77C3×C42C2311yescharacteristic
78#217—21no—
79#217—21no—
80#216—42no—
81#215—42no—
82(C2×D3)⋊C4—22no—
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—22no—
89C4⋊Dic3C2312yes—
90Dic3⋊C4—22no—
91C4×Dic3—22no—
92C4×Dic3D411yescharacteristic
93C4×Dic3D411yescharacteristic
94C4×Dic3C2311yescharacteristic
95C2×C3⋊C8C2312yes—
96C22×Q8—31no—
97C22×D4—31no—
98C2×Q16—32no—
99C2×Q16C2×D312yescomplemented
100C2×SD16—32no—
101C2×SD16C2×D312yes—
102C4⋊Q8—31no—
103C4⋊Q8C2×D311yescharacteristic, complemented
104#125—61no—
105#124—31no—
106#124—31no—
107#124—31no—
108#124—61no—
109#124—61no—
110#124C2×D311yescharacteristic
111C22⋊Q8—32no—
112C22⋊Q8—61no—
113C4×Q8—32no—
114C4×Q8—61no—
115#117—61no—
116C2×C22⋊C4—61no—
117C2×C22⋊C4—61no—
118C2×C42—31no—
119C4⋊C8—32no—
120#103—32no—
121#103—32no—
122#102—32no—
123#102—32no—
124C22⋊C8—32no—
125#97—31no—
126#97C2×D311yescharacteristic, complemented
127C23×C3—21no—
128C22×D3—21no—
129C22×D3—42no—
130C22×D3—42no—
131C22×D3—42no—
132C22×D3C2×D411yescharacteristic
133C3×Q8—22no—
134C3×Q8—22no—
135C3×Q8—22no—
136C3×D4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2411yescharacteristic
142#67—42no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3C2×D411yescharacteristic
146C2×Dic3C2×D411yescharacteristic
147C2×Dic3C2×D411yescharacteristic
148D12—22no—
149C4×D3—22no—
150C4×D3—22no—
151C4×D3—41no—
152C4×D3—42no—
153Dic6—22no—
154Dic6—42no—
155Dic6—42no—
156C3×C8—22no—
157C3⋊C8—22no—
158C24—61no—
159C2×Q8—31no—
160C2×Q8—61no—
161C2×Q8—61no—
162C2×Q8—122no—
163C2×Q8C22×D311yescharacteristic
164C2×Q8C22×D312yes—
165C2×D4—31no—
166C2×D4—61no—
167C2×D4—122no—
168C2×D4C22×D311yescharacteristic
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—61no—
173Q16—42no—
174Q16—122no—
175SD16—42no—
176SD16—122no—
177C2×C8—32no—
178C2×C8C22×D312yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—32no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C4⋊C4—61no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—61no—
188C22⋊C4—61no—
189C22⋊C4—62no—
190C22⋊C4—61no—
191C22⋊C4—61no—
192C22⋊C4—61no—
193C22⋊C4—61no—
194C42—31no—
195C42—31no—
196C42—31no—
197C42—62no—
198C42C22×D311yescharacteristic
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—41no—
202C22×C3C22×D411yescharacteristic
203C2×D3—21no—
204C2×D3—21no—
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—41no—
208C2×D3—84no—
209C2×D3#13712yes—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—41no—
213C3×C4—42no—
214C3×C4H(5,𝔽2)12yes—
215Dic3—21no—
216Dic3—21no—
217Dic3—21no—
218Dic3—41no—
219Dic3—42no—
220C23—21no—
221C23—31no—
222C23—61no—
223C23—122no—
224C23—122no—
225C23—122no—
226Q8—22no—
227Q8—22no—
228Q8—22no—
229Q8—62no—
230Q8—122no—
231Q8—122no—
232D4—22no—
233D4—62no—
234D4—122no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—121no—
245C2×C4—122no—
246C2×C4D3×D411yescharacteristic
247C2×C4D3×D411yescharacteristic
248C2×C4C23×D311yescharacteristic
249C8—22no—
250C8—62no—
251C2×C3—81no—
252C2×C3#53511yescharacteristic
253C2×C3C2×#13711yescharacteristic
254C2×C3#57511yescharacteristic
255D3—22no—
256D3—81no—
257C22—32no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—61no—
262C22—61no—
263C22—121no—
264C22—121no—
265C22—121no—
266C22—244no—
267C22C2×D3×D411yescharacteristic
268C4—21no—
269C4—21no—
270C4—41no—
271C4—42no—
272C4—61no—
273C4—61no—
274C4—61no—
275C4—121no—
276C4—122no—
277C4#100412yes—
278C3#321511yescharacteristic, complemented, Sylow
279C2—62no—
280C2—81no—
281C2—241no—
282C2#571511yescharacteristic
283C2D3×#13711yescharacteristic
284C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#497C2
socleC22×C3C22×D4
prime radicalC2×D3×D4C22
hypercenter#497D3
hypocenterC3#3215
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #5728C2semidirect
3. C3×#497C2semidirect
4. #5186C2semidirect
5. #5185C2semidirect
7. #5166C2semidirect
8. #5086C2semidirect
9. #5055C2semidirect
10. #4714C2semidirect
11. #4705C2semidirect
15. C2×C3×Q16C22semidirect
17. C3×C4⋊Q8C22semidirect
21. Q8×Dic3C22semidirect
22. C2×#216C22semidirect
35. #869C22semidirect
36. C3×#97C22semidirect
37. #835C22semidirect
41. #816C22semidirect
42. Dic3⋊C8C22semidirect
43. #803C22semidirect
47. #497D3semidirect
99. C2×Q16C2×D3semidirect
103. C4⋊Q8C2×D3semidirect
126. #97C2×D3semidirect
278. C3#3215semidirect

Maximal subgroups

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

subgroupindex
#57282
D3×#1242
C3×#4972
#51862
#51852
#51692
#51662
#50862
#50552
#47142
#47052
#32153