Finite Group Databasebeta

← #82502

Subgroups — #82502

Order 384.

492 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 145 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×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#872C211yesmaximal, characteristic
3#5231C211yesmaximal, characteristic
4#5230C211yesmaximal, characteristic, complemented
5#4992C211yesmaximal, characteristic
6#4989C212yesmaximal
7#4983C211yesmaximal, characteristic, complemented
8#4978C212yesmaximal, complemented
9#4940C212yesmaximal
10#4937C211yesmaximal, characteristic, complemented
11#4933C212yesmaximal, complemented
12#2599—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C3×C4×Q8—24no—
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C42C2212yes—
19C2×C3×C42C2211yescharacteristic
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29#872—24no—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×Q8D311yescharacteristic, complemented
39C2×#117—31no—
40#398—31no—
41#397—31no—
42#395—31no—
43#393—31no—
44#388—32no—
45#387—32no—
46#387—32no—
47#385—32no—
48#383—31no—
49C23×D3C2311yescharacteristic
50C2×C3×Q8—22no—
51C2×C3×Q8—24no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2311yescharacteristic
62C2×C4×D3—22no—
63C2×C4×D3—22no—
64C2×C4×D3—22no—
65C3×C4⋊C4—24no—
66C3×C4⋊C4—22no—
67C3×C42—24no—
68C3×C42D414yes—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—24no—
73Dic3⋊C4—22no—
74Dic3⋊C4—22no—
75Dic3⋊C4—24no—
76Dic3⋊C4—24no—
77C4×Dic3—22no—
78C22×Q8C2×D311yescharacteristic
79C23×C4—31no—
80C4×Q8—24no—
81#117—64no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4C2×D312yes—
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C22⋊C4—31no—
89C2×C22⋊C4—31no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—32no—
92C2×C42—31no—
93C2×C42C2×D312yes—
94C2×C42C2×D311yescharacteristic
95#95—32no—
96#95—32no—
97#95—31no—
98#95—31no—
99#95—32no—
100#95—32no—
101#95—32no—
102#95—32no—
103C23×C3C2411yescharacteristic
104C22×D3—21no—
105C22×D3—22no—
106C22×D3—21no—
107C22×D3—22no—
108C22×D3#4112yes—
109C3×Q8—28no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4C2×D412yes—
120C2×C3×C4C2×D412yes—
121C2×C3×C4C2×D412yes—
122C2×C3×C4#4114yes—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3#4112yes—
136C4×D3—28no—
137C24—31no—
138C2×Q8—24no—
139C2×Q8—22no—
140C22×C4—31no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—62no—
146C22×C4—62no—
147C22×C4—62no—
148C22×C4C22×D312yes—
149C22×C4C22×D312yes—
150C22×C4C22×D311yescharacteristic
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D311yescharacteristic
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—64no—
156C4⋊C4—64no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C22⋊C4—64no—
160C22⋊C4—64no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C42—24no—
164C42—62no—
165C42#6714yes—
166C22×C3C22×D411yescharacteristic
167C22×C3C2×#4112yes—
168C22×C3C2×#4111yescharacteristic
169C22×C3C2×#4111yescharacteristic
170C22×C32−1+412yes—
171C2×D3—22no—
172C2×D3—24no—
173C2×D3—41no—
174C2×D3—41no—
175C2×D3—42no—
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—24no—
179C3×C4—22no—
180C3×C4—42no—
181Dic3—22no—
182Dic3—41no—
183Dic3—41no—
184Dic3—42no—
185Dic3—42no—
186C23—32no—
187C23—62no—
188C23—62no—
189C23—61no—
190C23—61no—
191C23C23×D311yescharacteristic
192Q8—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—32no—
203C2×C4—68no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4#23514yes—
217C2×C4C2×#6712yes—
218C2×C4C2×#6712yes—
219C2×C4C2×#6712yes—
220C2×C3#52511yescharacteristic
221C2×C3#52511yescharacteristic
222C2×C3#53611yescharacteristic
223C2×C3#55212yes—
224C2×C3#55612yes—
225D3—42no—
226C22—62no—
227C22—64no—
228C22—121no—
229C22—121no—
230C22—122no—
231C22C2×#23512yes—
232C22#100712yes—
233C22D3×#4111yescharacteristic
234C22D3×#4111yescharacteristic
235C22C22×#6711yescharacteristic
236C4—24no—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—62no—
242C4—121no—
243C4—121no—
244C4—122no—
245C4—122no—
246C3#259911yescharacteristic, complemented, Sylow
247C2—122no—
248C2#556812yes—
249C2#558112yes—
250C2#579111yescharacteristic
251C2#581611yescharacteristic
252C2#582811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×Q8C2
prime radicalC23×D3C23
hypercenterC2×C4×Q8D3
hypocenterC3#2599
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×Q8C2semidirect
4. #5230C2semidirect
7. #4983C2semidirect
8. #4978C2semidirect
10. #4937C2semidirect
11. #4933C2semidirect
38. C2×C4×Q8D3semidirect
246. C3#2599semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q82
C2×#8722
#52312
#52302
#49922
#49892
#49832
#49782
#49402
#49372
#49332
#25993