Finite Group Databasebeta

← #83087

Subgroups — #83087

Order 384.

582 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 141 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×#126C211yesmaximal, characteristic, complemented
2C2×C4×D12C211yesmaximal, characteristic, complemented
3#4991C211yesmaximal, characteristic, complemented
4#4990C212yesmaximal, complemented
5#4989C212yesmaximal, complemented
6#4981C211yesmaximal, characteristic, complemented
7#4960C211yesmaximal, characteristic
8#4957C212yesmaximal, complemented
9#4956C212yesmaximal, complemented
10#4953C211yesmaximal, characteristic, complemented
11#4932C211yesmaximal, characteristic, complemented
12#2603—31nomaximal, Sylow
13C22×#67C2212yes—
14C22×D12C2211yescharacteristic
15C22×C4×D3C2212yes—
16C3×#126—24no—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
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—
31C2×C4×Dic3C2212yes—
32C4×D12—24no—
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#126D311yescharacteristic, complemented
39C2×C4×D4—31no—
40#398—31no—
41#397—31no—
42#394—31no—
43#392—31no—
44#390—32no—
45#388—32no—
46#386—32no—
47#385—32no—
48#376—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C2×#67—24no—
58C2×#67—24no—
59C2×#67—42no—
60C22×Dic3C2312yes—
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C2×D12—24no—
64C2×D12—22no—
65C2×C4×D3—24no—
66C2×C4×D3—42no—
67C2×C4×D3—42no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4—22no—
70C3×C4⋊C4—22no—
71C3×C22⋊C4—22no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4—22no—
74C3×C42D414yescomplemented
75#217—42no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—42no—
80C4⋊Dic3—22no—
81Dic3⋊C4—42no—
82C4×Dic3—24no—
83C22×D4—32no—
84C22×D4—31no—
85C23×C4—32no—
86#126—24no—
87C4×D4—64no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C4⋊C4C2×D311yescharacteristic
92C2×C4⋊C4C2×D311yescharacteristic
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4C2×D311yescharacteristic
99C2×C22⋊C4C2×D311yescharacteristic
100C2×C22⋊C4C2×D311yescharacteristic
101C2×C42—32no—
102C2×C42C2×D311yescharacteristic
103#95—32no—
104#95—31no—
105#95—31no—
106#95—32no—
107#95—32no—
108C23×C3—21no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—21no—
112C23×C3—21no—
113C23×C3C2411yescharacteristic
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3#4114yes—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2×D412yes—
135#67—48no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3#4114yes—
145D12—28no—
146C4×D3—48no—
147C24—32no—
148C24C22×D311yescharacteristic
149C2×D4—64no—
150C2×D4—64no—
151C2×D4—64no—
152C2×D4—62no—
153C2×D4—122no—
154C22×C4—32no—
155C22×C4—32no—
156C22×C4—32no—
157C22×C4—64no—
158C22×C4—122no—
159C22×C4—122no—
160C22×C4C22×D311yescharacteristic
161C22×C4C22×D311yescharacteristic
162C22×C4C22×D311yescharacteristic
163C22×C4C22×D311yescharacteristic
164C22×C4C22×D311yescharacteristic
165C22×C4C22×D311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—62no—
170C4⋊C4—122no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—64no—
175C22⋊C4—64no—
176C22⋊C4—64no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C42—64no—
180C42#6714yescomplemented
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—41no—
184C22×C3—42no—
185C22×C3—42no—
186C22×C3C22×D411yescharacteristic
187C22×C3C2×#4112yes—
188C22×C3C2×#4112yes—
189C22×C3H(5,𝔽2)11yescharacteristic
190C22×C3H(5,𝔽2)11yescharacteristic
191C2×D3—24no—
192C2×D3—24no—
193C2×D3—24no—
194C2×D3—42no—
195C2×D3—42no—
196C2×D3—42no—
197C2×D3—42no—
198C3×C4—22no—
199C3×C4—22no—
200C3×C4—22no—
201C3×C4—41no—
202C3×C4—41no—
203C3×C4—41no—
204Dic3—24no—
205Dic3—42no—
206Dic3—42no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—34no—
213C23—62no—
214C23—62no—
215C23—62no—
216C23—62no—
217C23—62no—
218C23—62no—
219C23C23×D311yescharacteristic
220D4—68no—
221D4—128no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—34no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—128no—
243C2×C4C2×#6712yes—
244C2×C4C2×#6712yes—
245C2×C4C2×#6712yes—
246C2×C3—81no—
247C2×C3#52511yescharacteristic
248C2×C3#53211yescharacteristic
249C2×C3#53511yescharacteristic
250C2×C3#55112yes—
251C2×C3#55312yes—
252D3—44no—
253C22—42no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—64no—
259C22—64no—
260C22—64no—
261C22—122no—
262C22—122no—
263C22—122no—
264C22—122no—
265C22D3×#4112yes—
266C22D3×#4112yes—
267C22#100911yescharacteristic
268C22#100911yescharacteristic
269C22C22×#6711yescharacteristic
270C4—22no—
271C4—22no—
272C4—22no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—64no—
277C4—122no—
278C4—122no—
279C3#260311yescharacteristic, complemented, Sylow
280C2—81no—
281C2—124no—
282C2#570512yes—
283C2#570912yes—
284C2#582811yescharacteristic
285C2#583011yescharacteristic
286C2#583211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#126C2
prime radical#4960C2
hypercenterC2×#126D3
hypocenterC3#2603
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C3×#126C2semidirect
2. C2×C4×D12C2semidirect
3. #4991C2semidirect
4. #4990C2semidirect
5. #4989C2semidirect
6. #4981C2semidirect
8. #4957C2semidirect
9. #4956C2semidirect
10. #4953C2semidirect
11. #4932C2semidirect
38. C2×#126D3semidirect
74. C3×C42D4semidirect
180. C42#67semidirect
279. C3#2603semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1262
C2×C4×D122
#49912
#49902
#49892
#49812
#49602
#49572
#49562
#49532
#49322
#26033