Finite Group Databasebeta

← #82284

Subgroups — #82284

Order 384.

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

normal subgroups only

This group has 604 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, complemented
2C2×C4×D12C211yesmaximal, characteristic, complemented
3#5213C212yesmaximal, complemented
4#4991C211yesmaximal, characteristic, complemented
5#4989C212yesmaximal, complemented
6#4986C211yesmaximal, characteristic, complemented
7#4960C211yesmaximal, characteristic
8#4956C212yesmaximal, complemented
9#4949C211yesmaximal, characteristic, complemented
10#4940C212yesmaximal, complemented
11#4937C211yesmaximal, characteristic, complemented
12#2598—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2212yes—
15C22×D12C2211yescharacteristic
16C22×C4×D3C2212yes—
17C3×#117—24no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C42C2211yescharacteristic
23C2×C3×C42C2211yescharacteristic
24C2×#217C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C4×D12—24no—
32C3⋊#95C2212yes—
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×D4—31no—
39C2×#117D311yescharacteristic, complemented
40#400—31no—
41#399—31no—
42#394—31no—
43#392—31no—
44#388—32no—
45#386—32no—
46#386—32no—
47#385—32no—
48#383—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2312yes—
51C22×C3×C4—22no—
52C22×C3×C4—22no—
53C22×C3×C4—22no—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—42no—
63C22×Dic3C2312yes—
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C2×D12—24no—
67C2×D12—22no—
68C2×C4×D3—24no—
69C2×C4×D3—42no—
70C2×C4×D3—42no—
71C3×C4⋊C4—22no—
72C3×C4⋊C4—22no—
73C3×C22⋊C4—22no—
74C3×C22⋊C4—22no—
75C3×C42—22no—
76C3×C42D414yescomplemented
77#217—42no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—42no—
82C4⋊Dic3—22no—
83Dic3⋊C4—42no—
84C22×D4—32no—
85C22×D4—31no—
86C23×C4—32no—
87C23×C4C2×D311yescharacteristic
88C4×D4—64no—
89#117—24no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4C2×D311yescharacteristic
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4C2×D311yescharacteristic
100C2×C22⋊C4C2×D311yescharacteristic
101C2×C42C2×D311yescharacteristic
102C2×C42C2×D311yescharacteristic
103#95—32no—
104#95—32no—
105#95—31no—
106#95—31no—
107#95—32no—
108#95—32no—
109C23×C3—21no—
110C23×C3—22no—
111C23×C3—21no—
112C23×C3—22no—
113C23×C3#4112yes—
114C23×C3C2411yescharacteristic
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3#4114yes—
122C2×C3×C4—28no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2×D412yes—
135C2×C3×C4C2×D412yes—
136C2×C3×C4#4112yes—
137#67—48no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147D12—28no—
148C4×D3—48no—
149C24—32no—
150C24C22×D311yescharacteristic
151C2×D4—64no—
152C2×D4—64no—
153C2×D4—64no—
154C2×D4—62no—
155C2×D4—122no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—32no—
160C22×C4—32no—
161C22×C4—32no—
162C22×C4—64no—
163C22×C4—122no—
164C22×C4—122no—
165C22×C4C22×D311yescharacteristic
166C22×C4C22×D311yescharacteristic
167C22×C4C22×D311yescharacteristic
168C22×C4C22×D311yescharacteristic
169C22×C4C22×D311yescharacteristic
170C22×C4C22×D311yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—62no—
174C4⋊C4—122no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—64no—
178C22⋊C4—64no—
179C22⋊C4—64no—
180C22⋊C4—122no—
181C22⋊C4—122no—
182C42—22no—
183C42#6714yescomplemented
184C22×C3—22no—
185C22×C3—24no—
186C22×C3—41no—
187C22×C3—41no—
188C22×C3—42no—
189C22×C3C22×D411yescharacteristic
190C22×C3C2×#4111yescharacteristic
191C22×C3C2×#4111yescharacteristic
192C22×C3C2×#4112yes—
193C22×C3H(5,𝔽2)12yes—
194C2×D3—24no—
195C2×D3—24no—
196C2×D3—24no—
197C2×D3—42no—
198C2×D3—42no—
199C2×D3—42no—
200C2×D3—42no—
201C3×C4—22no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—22no—
205C3×C4—41no—
206C3×C4—41no—
207Dic3—42no—
208Dic3—42no—
209Dic3—42no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23—21no—
214C23—34no—
215C23—62no—
216C23—62no—
217C23—62no—
218C23—62no—
219C23—62no—
220C23—62no—
221C23#23512yes—
222C23C23×D311yescharacteristic
223D4—68no—
224D4—128no—
225C2×C4—28no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—128no—
246C2×C4#23512yes—
247C2×C4C2×#6712yes—
248C2×C4C2×#6712yes—
249C2×C4C2×#6712yes—
250C2×C3—42no—
251C2×C3#52511yescharacteristic
252C2×C3#52511yescharacteristic
253C2×C3#53511yescharacteristic
254C2×C3#55112yes—
255C2×C3#55312yes—
256D3—44no—
257C22—24no—
258C22—22no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—64no—
263C22—64no—
264C22—64no—
265C22—122no—
266C22—122no—
267C22—122no—
268C22—122no—
269C22C2×#23511yescharacteristic
270C22C2×#23511yescharacteristic
271C22D3×#4112yes—
272C22#100912yes—
273C22C22×#6711yescharacteristic
274C4—22no—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—41no—
279C4—41no—
280C4—122no—
281C4—122no—
282C4—122no—
283C3#259811yescharacteristic, complemented, Sylow
284C2—42no—
285C2—124no—
286C2#552712yes—
287C2#553012yes—
288C2#579111yescharacteristic
289C2#582811yescharacteristic
290C2#583011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#117C2
prime radical#4960C2
hypercenterC2×#117D3
hypocenterC3#2598
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C3×#117C2semidirect
2. C2×C4×D12C2semidirect
3. #5213C2semidirect
4. #4991C2semidirect
5. #4989C2semidirect
6. #4986C2semidirect
8. #4956C2semidirect
9. #4949C2semidirect
10. #4940C2semidirect
11. #4937C2semidirect
39. C2×#117D3semidirect
76. C3×C42D4semidirect
183. C42#67semidirect
283. C3#2598semidirect

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×D122
#52132
#49912
#49892
#49862
#49602
#49562
#49492
#49402
#49372
#25983