Finite Group Databasebeta

← #79242

Subgroups — #79242

Order 384.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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#5521C211yesmaximal, characteristic
2#5500C211yesmaximal, characteristic, complemented
3C3×#428C211yesmaximal, characteristic, complemented
4#5131C211yesmaximal, characteristic
5#5119C211yesmaximal, characteristic, complemented
6#4998C211yesmaximal, characteristic, complemented
7C2×#809C211yesmaximal, characteristic
8#4891C212yesmaximal, complemented
9#4890C212yesmaximal, complemented
10#4871C212yesmaximal, complemented
11#4870C212yesmaximal, complemented
12#3081—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C2×C3×M16C2211yescharacteristic
15C3×#117C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17#926C2211yescharacteristic
18#926C2211yescharacteristic
19#924—22no—
20C22×C3⋊C8C2211yescharacteristic
21#893C2212yes—
22#893C2212yes—
23#890—21no—
24#890C2212yescomplemented
25#888—21no—
26#888C2212yescomplemented
27D3×C22⋊C4—22no—
28C4×D12—21no—
29C4×D12—21no—
30C3×#107C2212yescomplemented
31C3×#106C2212yescomplemented
32#821C2212yes—
33#821C2212yes—
34Dic3⋊C8C2212yescomplemented
35Dic3⋊C8C2212yescomplemented
36#809C2212yes—
37#809C2212yes—
38#808C2211yescharacteristic, complemented
39#808C2211yescharacteristic, complemented
40#807C2211yescharacteristic, complemented
41#807C2211yescharacteristic, complemented
42#544—31no—
43#518—31no—
44#478—32no—
45#478—32no—
46#476—32no—
47#474—32no—
48#428D311yescharacteristic, complemented
49#427—31no—
50#424—31no—
51#418—31no—
52C2×#102—31no—
53C23×D3—21no—
54C22×C3×C4—21no—
55C22×C3×C4C2311yescharacteristic
56C2×D12—21no—
57C2×D12—22no—
58C2×D12Q814yescomplemented
59C2×D12C2312yes—
60C2×C4×D3—22no—
61C2×C4×D3—22no—
62C2×Dic6—22no—
63C3×M16—22no—
64C2×C3×C8C2312yes—
65C3×C4⋊C4—21no—
66C3×C4⋊C4C2311yescharacteristic
67C3×C4⋊C4C2311yescharacteristic
68C3×C4⋊C4C2312yes—
69C3×C22⋊C4—21no—
70C3×C42—21no—
71#217—21no—
72#217D412yes—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75C4⋊Dic3—22no—
76C4⋊Dic3D411yescharacteristic
77C4⋊Dic3D411yescharacteristic
78C4⋊Dic3C2311yescharacteristic
79C4⋊Dic3C2311yescharacteristic
80Dic3⋊C4—22no—
81Dic3⋊C4—22no—
82C4×Dic3—21no—
83C4×Dic3C2312yes—
84C2×C3⋊C8—22no—
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87C22×D4—31no—
88C2×M16C2×D311yescharacteristic
89C22×C8—31no—
90C4⋊Q8—32no—
91C4⋊Q8—61no—
92#125—32no—
93#125—61no—
94C22⋊Q8—62no—
95C4×D4—32no—
96C4×D4—32no—
97C4×D4—61no—
98C4×D4—61no—
99#117—31no—
100#117—31no—
101#117C2×D311yescharacteristic
102C2×C4⋊C4C2×D311yescharacteristic
103C2×C22⋊C4—62no—
104#107—31no—
105#107—31no—
106#107C2×D312yescomplemented
107#106—31no—
108#106—31no—
109#106C2×D312yescomplemented
110C4⋊C8—32no—
111C4⋊C8—32no—
112#102—32no—
113#102—32no—
114#102—32no—
115#102—32no—
116C23×C3C2×D411yescharacteristic
117C22×D3—22no—
118C22×D3—24no—
119C22×D3—81no—
120C22×D3—81no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—81no—
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×Q812yes—
127C2×C3×C4#4112yes—
128C2×C3×C4C2411yescharacteristic
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3C2×D412yes—
132C2×Dic3C2×D412yes—
133D12—24no—
134D12—22no—
135D12C2×Q814yes—
136C4×D3—82no—
137C4×D3—82no—
138Dic6—82no—
139C3×C8—22no—
140C3⋊C8—21no—
141C3⋊C8—21no—
142C24—61no—
143C2×Q8—62no—
144C2×D4—34no—
145C2×D4—32no—
146C2×D4—62no—
147C2×D4—61no—
148C22×C4—21no—
149C22×C4—62no—
150C22×C4—62no—
151C22×C4C22×D311yescharacteristic
152M16—22no—
153C2×C8—31no—
154C2×C8—31no—
155C2×C8—62no—
156C2×C8C22×D312yes—
157C4⋊C4—21no—
158C4⋊C4—31no—
159C4⋊C4—31no—
160C4⋊C4—31no—
161C4⋊C4—31no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4C22×D312yes—
166C4⋊C4C22×D311yescharacteristic
167C4⋊C4C22×D311yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—32no—
170C22⋊C4—62no—
171C22⋊C4—62no—
172C22⋊C4—61no—
173C42—21no—
174C42—32no—
175C42—61no—
176C22×C3—21no—
177C22×C3—21no—
178C22×C3C22⋊Q812yes—
179C22×C3C22×D411yescharacteristic
180C2×D3—24no—
181C2×D3—42no—
182C2×D3—42no—
183C2×D3—84no—
184C3×C4—42no—
185C3×C4—41no—
186C3×C4—41no—
187C3×C4C22⋊Q812yes—
188C3×C4C22×Q811yescharacteristic
189C3×C4C2×#4111yescharacteristic
190Dic3—22no—
191Dic3—22no—
192Dic3—41no—
193Dic3—41no—
194C23—62no—
195C23—64no—
196C23—241no—
197C23—241no—
198C23D3×D411yescharacteristic
199Q8—242no—
200D4—34no—
201D4—64no—
202D4—62no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—32no—
207C2×C4—32no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—81no—
211C2×C4—242no—
212C2×C4—242no—
213C2×C4#23512yes—
214C2×C4D3×D411yescharacteristic
215C2×C4D3×Q812yes—
216C2×C4C23×D311yescharacteristic
217C8—22no—
218C8—61no—
219C8—61no—
220C2×C3—22no—
221C2×C3C2×C22⋊Q811yescharacteristic
222C2×C3#57611yescharacteristic
223C2×C3#57711yescharacteristic
224D3—44no—
225C22—21no—
226C22—21no—
227C22—64no—
228C22—122no—
229C22—122no—
230C22—244no—
231C22#89612yes—
232C22C2×D3×D411yescharacteristic
233C4—42no—
234C4—41no—
235C4—41no—
236C4—62no—
237C4—62no—
238C4—121no—
239C4—121no—
240C4#89612yes—
241C4C2×#23511yescharacteristic
242C4C2×D3×Q811yescharacteristic
243C3#308111yescharacteristic, complemented, Sylow
244C2—22no—
245C2—124no—
246C2C2×#89611yescharacteristic
247C2#577411yescharacteristic
248C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radicalC22×D12C22
hypercenter#428D3
hypocenterC3#3081
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #5500C2semidirect
3. C3×#428C2semidirect
5. #5119C2semidirect
6. #4998C2semidirect
8. #4891C2semidirect
9. #4890C2semidirect
10. #4871C2semidirect
11. #4870C2semidirect
24. #890C22semidirect
26. #888C22semidirect
30. C3×#107C22semidirect
31. C3×#106C22semidirect
34. Dic3⋊C8C22semidirect
35. Dic3⋊C8C22semidirect
38. #808C22semidirect
39. #808C22semidirect
40. #807C22semidirect
41. #807C22semidirect
48. #428D3semidirect
58. C2×D12Q8semidirect
106. #107C2×D3semidirect
109. #106C2×D3semidirect
243. C3#3081semidirect

Maximal subgroups

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

subgroupindex
#55212
#55002
C3×#4282
#51312
#51192
#49982
C2×#8092
#48912
#48902
#48712
#48702
#30813