Finite Group Databasebeta

← #80140

Subgroups — #80140

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×#936C211yesmaximal, characteristic
2#5622C211yesmaximal, characteristic, complemented
3C3×#450C211yesmaximal, characteristic
4#5218C211yesmaximal, characteristic
5#5167C212yesmaximal, complemented
6#5166C212yesmaximal, complemented
7#5050C211yesmaximal, characteristic, complemented
8#4767C212yesmaximal, complemented
9#4765C212yesmaximal, complemented
10#4739C211yesmaximal, characteristic, complemented
11#4719C211yesmaximal, characteristic, complemented
12#3215—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C2×C3×SD16C2212yes—
16C3×C22⋊Q8C2211yescharacteristic
17C2×#217—21no—
18C2×#217—21no—
19Q8×Dic3C2211yescharacteristic, complemented
20#944C2211yescharacteristic, complemented
21C2×#216C2212yescomplemented
22#936—21no—
23#936—21no—
24#936C2212yes—
25#936C2212yes—
26C2×#214C2212yes—
27#924C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C2×Dic12C2212yescomplemented
30#889—21no—
31#888C2211yescharacteristic, complemented
32#887—21no—
33#887C2211yescharacteristic, complemented
34#878—21no—
35#877—21no—
36C3×#102C2212yes—
37C3×C22⋊C8C2211yescharacteristic
38#832C2212yes—
39#830C2211yescharacteristic
40#816C2212yescomplemented
41#815C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#810C2212yescomplemented
44#533—31no—
45C2×#124—31no—
46#497—32no—
47#489—32no—
48#464—32no—
49#464—32no—
50#451—31no—
51#451—31no—
52#450—31no—
53#450D311yescharacteristic
54#432—31no—
55C24×C3—21no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4—21no—
58C2×C3×D4—42no—
59C2×C3×D4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C22×Dic3—21no—
62C22×Dic3D411yescharacteristic
63C22×Dic3D411yescharacteristic
64C2×Dic6—21no—
65C2×Dic6—21no—
66C2×Dic6—42no—
67C2×Dic6C2312yes—
68C2×Dic6C2311yescharacteristic
69C3×SD16—42no—
70C2×C3×C8C2312yes—
71C3×C4⋊C4—21no—
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—21no—
74#217—21no—
75#217—22no—
76#217—22no—
77#217—22no—
78#217—22no—
79#216—42no—
80#214—42no—
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—
95Dic12—42no—
96C22×Q8—31no—
97C22×D4C2×D311yescharacteristic
98C2×Q16—32no—
99C2×Q16—32no—
100C2×SD16—32no—
101C2×SD16C2×D312yes—
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#103—32no—
122#102—32no—
123#102C2×D312yes—
124C22⋊C8—31no—
125C22⋊C8C2×D311yescharacteristic
126#97—32no—
127C23×C3—22no—
128C23×C3—42no—
129C23×C3—42no—
130C23×C3—42no—
131C23×C3C2×D411yescharacteristic
132C3×Q8—22no—
133C3×D4—22no—
134C3×D4—22no—
135C3×D4—42no—
136C2×C3×C4—22no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2411yescharacteristic
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—41no—
146C2×Dic3—42no—
147C2×Dic3C2×D411yescharacteristic
148C2×Dic3C2×D411yescharacteristic
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151Dic6—22no—
152Dic6—22no—
153Dic6—22no—
154Dic6—42no—
155Dic6—42no—
156C3×C8—22no—
157C3⋊C8—22no—
158C24—21no—
159C2×Q8—32no—
160C2×Q8—31no—
161C2×Q8—61no—
162C2×Q8—61no—
163C2×Q8—122no—
164C2×Q8C22×D311yescharacteristic
165C2×D4—21no—
166C2×D4—42no—
167C2×D4C22×D312yes—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—61no—
171C22×C4C22×D311yescharacteristic
172Q16—122no—
173Q16—122no—
174SD16—42no—
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—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—84no—
205C22×C3#13712yes—
206C22×C3C22×D411yescharacteristic
207C3×C4—21no—
208C3×C4—41no—
209C3×C4—41no—
210C3×C4H(5,𝔽2)12yes—
211Dic3—21no—
212Dic3—21no—
213Dic3—21no—
214Dic3—21no—
215Dic3—42no—
216Dic3—41no—
217Dic3—41no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23D3×D411yescharacteristic
223Q8—22no—
224Q8—62no—
225Q8—62no—
226Q8—62no—
227Q8—122no—
228Q8—122no—
229D4—22no—
230D4—22no—
231D4—42no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—62no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—62no—
243C2×C4—121no—
244C2×C4—122no—
245C2×C4D3×D411yescharacteristic
246C2×C4C23×D311yescharacteristic
247C8—22no—
248C8—62no—
249C2×C3—22no—
250C2×C3—82no—
251C2×C3#53511yescharacteristic
252C2×C3C2×#13711yescharacteristic
253C2×C3#57511yescharacteristic
254C22—21no—
255C22—21no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—84no—
260C22#91512yes—
261C22C2×D3×D411yescharacteristic
262C4—21no—
263C4—41no—
264C4—41no—
265C4—61no—
266C4—61no—
267C4—61no—
268C4—61no—
269C4—121no—
270C4—122no—
271C4—121no—
272C4#100412yes—
273C3#321511yescharacteristic, complemented, Sylow
274C2—22no—
275C2—82no—
276C2#559211yescharacteristic
277C2C2×#91511yescharacteristic
278C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radicalC22×C3×D4C22
hypercenter#450D3
hypocenterC3#3215
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
2. #5622C2semidirect
5. #5167C2semidirect
6. #5166C2semidirect
7. #5050C2semidirect
8. #4767C2semidirect
9. #4765C2semidirect
10. #4739C2semidirect
11. #4719C2semidirect
19. Q8×Dic3C22semidirect
20. #944C22semidirect
21. C2×#216C22semidirect
29. C2×Dic12C22semidirect
31. #888C22semidirect
33. #887C22semidirect
40. #816C22semidirect
41. #815C22semidirect
42. Dic3⋊C8C22semidirect
43. #810C22semidirect
273. C3#3215semidirect

Maximal subgroups

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

subgroupindex
C2×#9362
#56222
C3×#4502
#52182
#51672
#51662
#50502
#47672
#47652
#47392
#47192
#32153