Finite Group Databasebeta

← #80970

Subgroups — #80970

Order 384.

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

normal subgroups only

This group has 332 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
1#5728C211yesmaximal, characteristic, complemented
2#5557C211yesmaximal, characteristic
3C3×#474C211yesmaximal, characteristic, complemented
4#5094C211yesmaximal, characteristic, complemented
5#5008C211yesmaximal, characteristic, complemented
6#4888C212yesmaximal, complemented
7#4878C212yesmaximal, complemented
8#4850C211yesmaximal, characteristic, complemented
9#4833C211yesmaximal, characteristic, complemented
10#4784C212yesmaximal
11#4768C212yesmaximal, complemented
12#3233—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4×D4C2211yescharacteristic
15#946—21no—
16Q8×Dic3—21no—
17#938C2211yescharacteristic
18#936C2211yescharacteristic
19C4×#67—21no—
20#926—21no—
21#897—21no—
22#897C2211yescharacteristic
23#892C2212yes—
24#890C2211yescharacteristic
25#890C2212yes—
26#887—21no—
27#883—21no—
28#879—21no—
29C42×D3C2211yescharacteristic
30#869C2211yescharacteristic
31C4×Dic6C2211yescharacteristic
32C3×#107C2212yes—
33C3×C4⋊C8C2211yescharacteristic
34C3×#102C2212yes—
35#832C2212yes—
36(C2×D3)⋊C8C2212yes—
37#818C2212yes—
38#816C2212yes—
39C8×Dic3C2212yes—
40#810C2212yes—
41#808C2212yes—
42#808C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#533—31no—
45#529—31no—
46#499—32no—
47#488—32no—
48#485—32no—
49#482—32no—
50#477—31no—
51#476—31no—
52#475—31no—
53#474D311yescharacteristic, complemented
54#431—31no—
55C2×C3×Q8—21no—
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C2×#67—21no—
59C2×C4×D3D411yescharacteristic
60C2×C4×D3D411yescharacteristic
61C2×C4×D3C2311yescharacteristic
62C2×Dic6—21no—
63C2×Dic6C2311yescharacteristic
64C2×C3×C8C2312yes—
65C3×C4⋊C4—21no—
66C3×C4⋊C4C2311yescharacteristic
67C3×C4⋊C4C2312yes—
68C3×C22⋊C4—21no—
69C3×C42C2311yescharacteristic
70#217—21no—
71#217—21no—
72#217—21no—
73(C2×D3)⋊C4—21no—
74(C2×D3)⋊C4—22no—
75C4⋊Dic3—21no—
76C4⋊Dic3—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3—22no—
79C4⋊Dic3C2311yescharacteristic
80C4⋊Dic3C2312yes—
81Dic3⋊C4—21no—
82Dic3⋊C4—21no—
83Dic3⋊C4—22no—
84C4×Dic3—21no—
85C4×Dic3—22no—
86C4×Dic3D411yescharacteristic
87C4×Dic3D411yescharacteristic
88C4×Dic3C2311yescharacteristic
89C2×C3⋊C8C2312yes—
90C4⋊Q8—31no—
91C4⋊Q8C2×D311yescharacteristic
92#126—61no—
93#125—31no—
94#125—32no—
95#124—31no—
96#123—61no—
97C22⋊Q8—31no—
98C22⋊Q8—61no—
99C22⋊Q8—61no—
100#121—31no—
101C4×Q8—31no—
102C4×Q8—61no—
103C4×Q8—61no—
104C4×D4—61no—
105C4×D4C2×D311yescharacteristic
106#117—32no—
107#117—61no—
108C2×C42—31no—
109#107—32no—
110#107C2×D312yes—
111#106—32no—
112#106—32no—
113C4⋊C8—31no—
114C4⋊C8C2×D311yescharacteristic
115#103—32no—
116#103—32no—
117#102—32no—
118#102C2×D312yes—
119C22⋊C8—32no—
120C4×C8—32no—
121C23×C3—21no—
122C22×D3C2×D411yescharacteristic
123C3×Q8—42no—
124C3×D4—21no—
125C3×D4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—42no—
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2411yescharacteristic
132#67—42no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3C2×D411yescharacteristic
137C2×Dic3C2×D411yescharacteristic
138C2×Dic3C2×D411yescharacteristic
139C4×D3—22no—
140C4×D3—21no—
141C4×D3—21no—
142C4×D3—41no—
143Dic6—21no—
144Dic6—21no—
145Dic6—42no—
146C3×C8—22no—
147C3⋊C8—22no—
148C2×Q8—21no—
149C2×Q8—31no—
150C2×Q8—61no—
151C2×D4—61no—
152C2×D4C22×D311yescharacteristic
153C22×C4—21no—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—31no—
157C2×C8—32no—
158C2×C8C22×D312yes—
159C4⋊C4—21no—
160C4⋊C4—32no—
161C4⋊C4—31no—
162C4⋊C4—61no—
163C4⋊C4—61no—
164C4⋊C4—61no—
165C4⋊C4—62no—
166C4⋊C4—62no—
167C4⋊C4—61no—
168C4⋊C4—61no—
169C4⋊C4C22×D312yes—
170C4⋊C4C22×D311yescharacteristic
171C22⋊C4—21no—
172C22⋊C4—62no—
173C22⋊C4—61no—
174C22⋊C4—61no—
175C22⋊C4—61no—
176C22⋊C4—61no—
177C42—31no—
178C42—31no—
179C42—31no—
180C42—62no—
181C42—61no—
182C42C22×D311yescharacteristic
183C22×C3—41no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3C22×D411yescharacteristic
187C2×D3—21no—
188C2×D3—21no—
189C2×D3—21no—
190C3×C4—21no—
191C3×C4—41no—
192C3×C4—42no—
193C3×C4#13512yes—
194C3×C42−1+411yescharacteristic
195C3×C42−1+411yescharacteristic
196Dic3—21no—
197Dic3—21no—
198Dic3—21no—
199Dic3—41no—
200Dic3—42no—
201Dic3—41no—
202C23—21no—
203C23—31no—
204Q8—42no—
205Q8—61no—
206Q8—61no—
207Q8—122no—
208D4—21no—
209D4—21no—
210D4—122no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—42no—
217C2×C4—62no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—62no—
223C2×C4—121no—
224C2×C4D3×D411yescharacteristic
225C2×C4D3×D411yescharacteristic
226C2×C4C23×D311yescharacteristic
227C8—22no—
228C8—62no—
229C2×C3—81no—
230C2×C3#53611yescharacteristic
231C2×C3C2×#13511yescharacteristic
232C2×C3#57511yescharacteristic
233D3—41no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—61no—
238C22—61no—
239C22—61no—
240C22C2×D3×D411yescharacteristic
241C4—21no—
242C4—41no—
243C4—42no—
244C4—61no—
245C4—61no—
246C4—61no—
247C4—121no—
248C4—122no—
249C4—121no—
250C4#91212yes—
251C4#100711yescharacteristic
252C4#101011yescharacteristic
253C3#323311yescharacteristic, complemented, Sylow
254C2—81no—
255C2—121no—
256C2#562811yescharacteristic
257C2C2×#91211yescharacteristic
258C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#474C2
socleC22×C3C22×D4
prime radical#938C22
hypercenter#474D3
hypocenterC3#3233
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #5728C2semidirect
3. C3×#474C2semidirect
4. #5094C2semidirect
5. #5008C2semidirect
6. #4888C2semidirect
7. #4878C2semidirect
8. #4850C2semidirect
9. #4833C2semidirect
11. #4768C2semidirect
53. #474D3semidirect
253. C3#3233semidirect

Maximal subgroups

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

subgroupindex
#57282
#55572
C3×#4742
#50942
#50082
#48882
#48782
#48502
#48332
#47842
#47682
#32333