Finite Group Databasebeta

← #80202

Subgroups — #80202

Order 384.

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

normal subgroups only

This group has 400 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
2#5622C211yesmaximal, characteristic, complemented
3C3×#451C211yesmaximal, characteristic, complemented
4#5228C211yesmaximal, characteristic, complemented
5#5190C212yesmaximal, complemented
6#5185C212yesmaximal, complemented
7#5046C211yesmaximal, characteristic
8#4798C212yesmaximal, complemented
9#4794C212yesmaximal, complemented
10#4731C211yesmaximal, characteristic
11#4719C211yesmaximal, characteristic, complemented
12#3215—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×D12C2211yescharacteristic
15C2×C3×Q16C2212yescomplemented
16C3×C22⋊Q8C2211yescharacteristic
17#947—21no—
18#947—21no—
19#947C2212yes—
20#947C2212yes—
21Q8×Dic3C2211yescharacteristic, complemented
22#944C2211yescharacteristic, complemented
23C2×#216C2212yescomplemented
24C2×#215C2212yes—
25C2×(C2×D3)⋊C4—21no—
26C2×(C2×D3)⋊C4—21no—
27#924C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C2×#204C2212yes—
30#889—21no—
31#888C2211yescharacteristic, complemented
32#887—21no—
33#887C2211yescharacteristic, complemented
34#878—21no—
35#877—21no—
36C3×#103C2212yescomplemented
37C3×C22⋊C8C2211yescharacteristic
38#835C2212yescomplemented
39#830C2211yescharacteristic
40#821C2212yes—
41#815C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#809C2212yes—
44#533—31no—
45C2×#124—31no—
46#497—32no—
47#489—32no—
48#464—32no—
49#464—32no—
50#451—31no—
51#451D311yescharacteristic, complemented
52#450—31no—
53#450—31no—
54#432—31no—
55C23×D3—21no—
56C2×C3×Q8—21no—
57C2×C3×Q8—42no—
58C2×C3×Q8C2312yes—
59C2×C3×Q8C2311yescharacteristic
60C22×C3×C4—21no—
61C22×C3×C4C2311yescharacteristic
62C22×Dic3D411yescharacteristic
63C22×Dic3D411yescharacteristic
64C2×D12—21no—
65C2×D12—42no—
66C2×D12C2312yes—
67C2×Dic6—21no—
68C2×Dic6C2311yescharacteristic
69C3×Q16—42no—
70C2×C3×C8C2312yes—
71C3×C4⋊C4—21no—
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—21no—
74#217—21no—
75#216—42no—
76#215—42no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81C4⋊Dic3—21no—
82C4⋊Dic3C2311yescharacteristic
83Dic3⋊C4—21no—
84Dic3⋊C4—21no—
85Dic3⋊C4—21no—
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88C4×Dic3—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×Q16C2×D312yescomplemented
100C2×SD16—32no—
101C2×SD16—32no—
102C4⋊Q8—31no—
103C4⋊Q8—31no—
104#125—61no—
105#124—32no—
106#124—32no—
107#124—61no—
108#124—61no—
109C22⋊Q8—31no—
110C22⋊Q8—61no—
111C22⋊Q8C2×D311yescharacteristic
112C4×Q8—31no—
113C4×Q8—31no—
114C4×Q8—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
126#97—32no—
127C23×C3C2×D411yescharacteristic
128C22×D3—22no—
129C22×D3—42no—
130C22×D3—42no—
131C22×D3—42no—
132C3×Q8—22no—
133C3×Q8—22no—
134C3×Q8—22no—
135C3×Q8—42no—
136C2×C3×C4—22no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—42no—
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2411yescharacteristic
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3—41no—
147C2×Dic3C2×D411yescharacteristic
148C2×Dic3C2×D411yescharacteristic
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151D12—22no—
152D12—22no—
153D12—42no—
154Dic6—22no—
155Dic6—42no—
156C3×C8—22no—
157C3⋊C8—22no—
158C24—61no—
159C2×Q8—21no—
160C2×Q8—31no—
161C2×Q8—42no—
162C2×Q8—61no—
163C2×Q8C22×D312yes—
164C2×Q8C22×D311yescharacteristic
165C2×D4—32no—
166C2×D4—61no—
167C2×D4—122no—
168C22×C4—21no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4C22×D311yescharacteristic
172Q16—42no—
173Q16—122no—
174SD16—122no—
175SD16—122no—
176C2×C8—32no—
177C2×C8C22×D312yes—
178C4⋊C4—21no—
179C4⋊C4—31no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C4⋊C4—61no—
185C4⋊C4—61no—
186C4⋊C4C22×D311yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—62no—
189C22⋊C4—62no—
190C22⋊C4—62no—
191C22⋊C4—62no—
192C22⋊C4—61no—
193C42—31no—
194C42—31no—
195C42—31no—
196C42—31no—
197C42—61no—
198C42—61no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3#13712yes—
202C22×C3C22×D411yescharacteristic
203C2×D3—42no—
204C2×D3—42no—
205C2×D3—42no—
206C2×D3—84no—
207C3×C4—21no—
208C3×C4—42no—
209C3×C4—41no—
210C3×C4—41no—
211C3×C4H(5,𝔽2)12yes—
212Dic3—21no—
213Dic3—21no—
214Dic3—21no—
215Dic3—21no—
216Dic3—41no—
217Dic3—41no—
218C23—62no—
219C23—122no—
220C23—122no—
221C23—122no—
222C23D3×D411yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—22no—
226Q8—42no—
227Q8—62no—
228Q8—122no—
229D4—62no—
230D4—62no—
231D4—122no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—42no—
241C2×C4—62no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—121no—
245C2×C4D3×D411yescharacteristic
246C2×C4C23×D311yescharacteristic
247C8—22no—
248C8—62no—
249C2×C3—22no—
250C2×C3#53511yescharacteristic
251C2×C3C2×#13711yescharacteristic
252C2×C3#57511yescharacteristic
253D3—82no—
254C22—21no—
255C22—21no—
256C22—122no—
257C22—122no—
258C22—122no—
259C22—244no—
260C22#91812yes—
261C22C2×D3×D411yescharacteristic
262C4—21no—
263C4—41no—
264C4—42no—
265C4—41no—
266C4—61no—
267C4—61no—
268C4—61no—
269C4—61no—
270C4—121no—
271C4—121no—
272C4#100412yes—
273C3#321511yescharacteristic, complemented, Sylow
274C2—22no—
275C2—242no—
276C2#559211yescharacteristic
277C2C2×#91811yescharacteristic
278C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radicalC22×D12C22
hypercenter#451D3
hypocenterC3#3215
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #5622C2semidirect
3. C3×#451C2semidirect
4. #5228C2semidirect
5. #5190C2semidirect
6. #5185C2semidirect
8. #4798C2semidirect
9. #4794C2semidirect
11. #4719C2semidirect
15. C2×C3×Q16C22semidirect
21. Q8×Dic3C22semidirect
22. #944C22semidirect
23. C2×#216C22semidirect
31. #888C22semidirect
33. #887C22semidirect
36. C3×#103C22semidirect
38. #835C22semidirect
41. #815C22semidirect
42. Dic3⋊C8C22semidirect
51. #451D3semidirect
99. C2×Q16C2×D3semidirect
121. #103C2×D3semidirect
273. C3#3215semidirect

Maximal subgroups

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

subgroupindex
C2×#9472
#56222
C3×#4512
#52282
#51902
#51852
#50462
#47982
#47942
#47312
#47192
#32153