Finite Group Databasebeta

← #82274

Subgroups — #82274

Order 384.

560 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 157 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, complemented
2C2×C4×#67C211yesmaximal, characteristic
3C4×#217C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4992C211yesmaximal, characteristic, complemented
6#4985C211yesmaximal, characteristic, complemented
7#4983C211yesmaximal, characteristic, complemented
8#4959C212yesmaximal
9#4952C212yesmaximal, complemented
10C4×(C2×D3)⋊C4C212yesmaximal, complemented
11C4×Dic3⋊C4C212yesmaximal, complemented
12#2395—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C3×#117—24no—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C42C2212yes—
21C2×#217C2211yescharacteristic
22C2×#217C2212yes—
23C4×#67—24no—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31C2×C4×Dic3C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2211yescharacteristic
38C2×C4×D4—31no—
39C2×#117D311yescharacteristic, complemented
40#400—31no—
41#399—31no—
42#398—31no—
43#397—31no—
44#397—32no—
45#394—32no—
46C4×C4⋊C4—32no—
47C4×C22⋊C4—31no—
48C4×C22⋊C4—32no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4—24no—
52C22×C3×C4D414yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C2×#67—22no—
58C2×#67—22no—
59C2×#67—22no—
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2312yes—
65C2×C4×D3—22no—
66C2×C4×D3—24no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—22no—
69C3×C22⋊C4—24no—
70C3×C42—24no—
71#217—22no—
72#217—24no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—24no—
76Dic3⋊C4—22no—
77Dic3⋊C4—24no—
78Dic3⋊C4—24no—
79C4×Dic3—22no—
80C4×Dic3—24no—
81C22×D4—31no—
82C23×C4—31no—
83C23×C4C2×D311yescharacteristic
84C4×D4—64no—
85#117—24no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4C2×D311yescharacteristic
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C22⋊C4—31no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4C2×D312yes—
97C2×C42—31no—
98C2×C42—32no—
99C2×C42C2×D312yes—
100#95—32no—
101#95—31no—
102#95—32no—
103#95—31no—
104#95—32no—
105#95—32no—
106C23×C3—22no—
107C23×C3—21no—
108C23×C3—21no—
109C23×C3—22no—
110C23×C3C2×D412yes—
111C23×C3C2411yescharacteristic
112C22×D3—21no—
113C22×D3—22no—
114C22×D3—22no—
115C22×D3—21no—
116C22×D3#4112yes—
117C2×C3×C4—22no—
118C2×C3×C4—28no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4C2×D412yes—
126C2×C3×C4C2×D412yes—
127C2×C3×C4#4114yes—
128#67—28no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3#4112yes—
141C2×Dic3#4114yes—
142C4×D3—44no—
143C24—31no—
144C24C22×D311yescharacteristic
145C2×D4—62no—
146C2×D4—62no—
147C2×D4—62no—
148C22×C4—24no—
149C22×C4—32no—
150C22×C4—31no—
151C22×C4—32no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—64no—
155C22×C4—62no—
156C22×C4#6714yes—
157C22×C4C22×D312yes—
158C22×C4C22×D312yes—
159C22×C4C22×D311yescharacteristic
160C22×C4C22×D311yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—64no—
164C4⋊C4—64no—
165C4⋊C4—62no—
166C22⋊C4—24no—
167C22⋊C4—64no—
168C22⋊C4—64no—
169C22⋊C4—64no—
170C22⋊C4—62no—
171C22⋊C4—62no—
172C42—24no—
173C42—64no—
174C42—62no—
175C22×C3—22no—
176C22×C3—22no—
177C22×C3—22no—
178C22×C3—42no—
179C22×C3—42no—
180C22×C3C22×D411yescharacteristic
181C22×C3C2×#4112yes—
182C22×C3C2×#4111yescharacteristic
183C22×C3C2×#4111yescharacteristic
184C22×C3C2×#4112yes—
185C2×D3—22no—
186C2×D3—24no—
187C2×D3—41no—
188C2×D3—42no—
189C2×D3—41no—
190C3×C4—22no—
191C3×C4—42no—
192C3×C4—42no—
193C3×C4#12314yes—
194Dic3—22no—
195Dic3—24no—
196Dic3—41no—
197Dic3—42no—
198Dic3—41no—
199C23—22no—
200C23—22no—
201C23—21no—
202C23—21no—
203C23—32no—
204C23—62no—
205C23—62no—
206C23—61no—
207C23—61no—
208C23C2×#6712yes—
209C23C23×D311yescharacteristic
210D4—68no—
211C2×C4—22no—
212C2×C4—28no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—34no—
220C2×C4—32no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4—124no—
233C2×C4#23514yes—
234C2×C4C2×#6712yes—
235C2×C4C2×#6712yes—
236C2×C3—42no—
237C2×C3C2×#12311yescharacteristic
238C2×C3#52511yescharacteristic
239C2×C3#52912yes—
240C2×C3#52912yes—
241C2×C3#53211yescharacteristic
242D3—42no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—64no—
249C22—62no—
250C22—122no—
251C22—121no—
252C22—121no—
253C22C2×#23512yes—
254C22D3×#4111yescharacteristic
255C22D3×#4111yescharacteristic
256C22D3×#4112yes—
257C22C22×#6711yescharacteristic
258C4—22no—
259C4—42no—
260C4—42no—
261C4—64no—
262C4—62no—
263C4—122no—
264C4—121no—
265C4—121no—
266C4#92914yes—
267C3#239511yescharacteristic, complemented, Sylow
268C2—42no—
269C2—122no—
270C2#552812yes—
271C2#552812yes—
272C2C2×#92911yescharacteristic
273C2#582811yescharacteristic
274C2#583211yescharacteristic
subgroupG/H
centerC22×C4C22×D3
commutator, socle, cosocleC23×C3C24
FrattiniC23C23×D3
FittingC2×C3×#117C2
prime radicalC22×#67C22
hypercenterC2×#117D3
hypocenterC3#2395
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C3×#117C2semidirect
3. C4×#217C2semidirect
4. #4993C2semidirect
5. #4992C2semidirect
6. #4985C2semidirect
7. #4983C2semidirect
9. #4952C2semidirect
10. C4×(C2×D3)⋊C4C2semidirect
11. C4×Dic3⋊C4C2semidirect
39. C2×#117D3semidirect
267. C3#2395semidirect

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×#672
C4×#2172
#49932
#49922
#49852
#49832
#49592
#49522
C4×(C2×D3)⋊C42
C4×Dic3⋊C42
#23953