Finite Group Databasebeta

← #82258

Subgroups — #82258

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×#117C211yesmaximal, characteristic
2C2×C4×Dic6C211yesmaximal, characteristic, complemented
3#5210C212yesmaximal
4#4981C211yesmaximal, characteristic, complemented
5#4979C211yesmaximal, characteristic, complemented
6#4978C212yesmaximal, complemented
7#4953C211yesmaximal, characteristic
8#4947C211yesmaximal, characteristic
9#4944C212yesmaximal
10#4937C211yesmaximal, characteristic, complemented
11#4933C212yesmaximal, complemented
12#2599—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C3×#117—24no—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C42C2211yescharacteristic
21C2×C3×C42C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2212yes—
29C4×Dic6—24no—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×C4×Q8—31no—
39C2×#117D311yescharacteristic
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—
49C24×C3C2311yescharacteristic
50C22×C3×C4—22no—
51C22×C3×C4—22no—
52C22×C3×C4—22no—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C2×Dic6—22no—
64C2×Dic6—24no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4—22no—
67C3×C22⋊C4—22no—
68C3×C22⋊C4—22no—
69C3×C42—22no—
70C3×C42D414yes—
71#217—24no—
72#217—24no—
73C4⋊Dic3—22no—
74Dic3⋊C4—24no—
75Dic3⋊C4—24no—
76Dic3⋊C4—24no—
77C4×Dic3—24no—
78C22×Q8—31no—
79C23×C4C2×D311yescharacteristic
80C4×Q8—64no—
81#117—24no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4C2×D311yescharacteristic
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4C2×D311yescharacteristic
91C2×C22⋊C4C2×D311yescharacteristic
92C2×C42—32no—
93C2×C42C2×D311yescharacteristic
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×C3—21no—
104C23×C3—22no—
105C23×C3—21no—
106C23×C3—22no—
107C23×C3#4112yes—
108C23×C3C2411yescharacteristic
109C2×C3×C4—28no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4C2×D412yes—
121C2×C3×C4C2×D412yes—
122C2×C3×C4C2×D412yes—
123C2×C3×C4#4112yes—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
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#4114yes—
136Dic6—28no—
137C24C22×D311yescharacteristic
138C2×Q8—64no—
139C2×Q8—62no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4—32no—
146C22×C4—32no—
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D311yescharacteristic
150C22×C4C22×D311yescharacteristic
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—64no—
156C4⋊C4—64no—
157C4⋊C4—64no—
158C4⋊C4—62no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—64no—
162C22⋊C4—64no—
163C42—22no—
164C42—64no—
165C42#6714yes—
166C22×C3—22no—
167C22×C3—24no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3—42no—
171C22×C3C22×D411yescharacteristic
172C22×C3C2×#4111yescharacteristic
173C22×C3C2×#4111yescharacteristic
174C22×C3C2×#4112yes—
175C22×C32−1+412yes—
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—22no—
179C3×C4—22no—
180C3×C4—41no—
181C3×C4—41no—
182Dic3—24no—
183Dic3—42no—
184Dic3—42no—
185Dic3—42no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23#23512yes—
191C23C23×D311yescharacteristic
192Q8—68no—
193C2×C4—28no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—34no—
205C2×C4—62no—
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—62no—
215C2×C4—62no—
216C2×C4#23512yes—
217C2×C4C2×#6712yes—
218C2×C4C2×#6712yes—
219C2×C4C2×#6712yes—
220C2×C3—42no—
221C2×C3#52511yescharacteristic
222C2×C3#52511yescharacteristic
223C2×C3#53611yescharacteristic
224C2×C3#55212yes—
225C2×C3#55612yes—
226C22—24no—
227C22—22no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22C2×#23511yescharacteristic
232C22C2×#23511yescharacteristic
233C22D3×#4112yes—
234C22#101012yes—
235C22C22×#6711yescharacteristic
236C4—22no—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—41no—
241C4—41no—
242C4—64no—
243C4—122no—
244C4—122no—
245C4—122no—
246C3#259911yescharacteristic, complemented, Sylow
247C2—42no—
248C2#553112yes—
249C2#553312yes—
250C2#579111yescharacteristic
251C2#582511yescharacteristic
252C2#582811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#117C2
prime radicalC24×C3C23
hypercenterC2×#117D3
hypocenterC3#2599
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic6C2semidirect
4. #4981C2semidirect
5. #4979C2semidirect
6. #4978C2semidirect
10. #4937C2semidirect
11. #4933C2semidirect
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×#1172
C2×C4×Dic62
#52102
#49812
#49792
#49782
#49532
#49472
#49442
#49372
#49332
#25993