Finite Group Databasebeta

← #76695

Subgroups — #76695

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×D4×Dic3C211yesmaximal, characteristic, complemented
2C2×#879C211yesmaximal, characteristic, complemented
3C3×#392C211yesmaximal, characteristic
4#5222C211yesmaximal, characteristic, complemented
5#5221C212yesmaximal, complemented
6#5220C212yesmaximal, complemented
7#4953C211yesmaximal, characteristic, complemented
8#4950C212yesmaximal, complemented
9#4944C212yesmaximal, complemented
10#4661C211yesmaximal, characteristic, complemented
11#4648C211yesmaximal, characteristic, complemented
12#2603—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C23×Dic3C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24D4×Dic3—24no—
25C2×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2212yes—
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×C4×Dic3C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32#879—24no—
33C3×#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#126—31no—
39C2×C4×D4—31no—
40#398—31no—
41#397—31no—
42#394—31no—
43#392D311yescharacteristic
44#390—32no—
45#388—32no—
46#386—32no—
47#385—32no—
48#376—31no—
49C24×C3C2311yescharacteristic
50C24×C3C2312yes—
51C2×C3×D4—24no—
52C2×C3×D4—24no—
53C2×C3×D4—22no—
54C2×C3×D4—24no—
55C2×C3×D4—42no—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2312yes—
59C22×Dic3—24no—
60C22×Dic3—42no—
61C22×Dic3—42no—
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C3×C22⋊C4—22no—
69C3×C22⋊C4—24no—
70#217—24no—
71#217—22no—
72#217—22no—
73#217—24no—
74#217—42no—
75#217—42no—
76C4⋊Dic3—22no—
77C4⋊Dic3—22no—
78C4⋊Dic3—42no—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81C4×Dic3—24no—
82C4×Dic3D414yescomplemented
83C22×D4C2×D311yescharacteristic
84C22×D4C2×D312yes—
85C23×C4—32no—
86#126—64no—
87C4×D4—64no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4C2×D311yescharacteristic
100C2×C22⋊C4C2×D312yes—
101C2×C42—31no—
102C2×C42—32no—
103#95—32no—
104#95—31no—
105#95—32no—
106#95—32no—
107#95C2×D311yescharacteristic
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—21no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3#4114yes—
120C23×C3C2411yescharacteristic
121C3×D4—28no—
122C3×D4—48no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4C2×D412yes—
129C2×C3×C4#4114yes—
130C2×Dic3—22no—
131C2×Dic3—21no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—21no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—48no—
145C2×Dic3C2×D412yes—
146C2×Dic3C2×D412yes—
147C24C22×D311yescharacteristic
148C24C22×D312yes—
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—22no—
152C2×D4—24no—
153C2×D4—42no—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—32no—
157C22×C4—31no—
158C22×C4—32no—
159C22×C4—31no—
160C22×C4—64no—
161C22×C4—122no—
162C22×C4—122no—
163C22×C4C22×D311yescharacteristic
164C22×C4C22×D311yescharacteristic
165C22×C4C22×D312yes—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—122no—
171C22⋊C4—22no—
172C22⋊C4—24no—
173C22⋊C4—64no—
174C22⋊C4—62no—
175C22⋊C4—62no—
176C22⋊C4—64no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C42—34no—
180C42—64no—
181C22×C3—24no—
182C22×C3—24no—
183C22×C3—24no—
184C22×C3—42no—
185C22×C3—42no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—41no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—42no—
192C22×C3—42no—
193C22×C3C22×D411yescharacteristic
194C22×C3C2×#4112yes—
195C22×C3C2×#4112yes—
196C22×C3H(5,𝔽2)11yescharacteristic
197C22×C3H(5,𝔽2)11yescharacteristic
198C3×C4—22no—
199C3×C4—24no—
200C3×C4—41no—
201Dic3—22no—
202Dic3—22no—
203Dic3—41no—
204Dic3—41no—
205Dic3—42no—
206Dic3—42no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—21no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23#23714yes—
219C23C23×D311yescharacteristic
220D4—28no—
221D4—48no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—32no—
228C2×C4—32no—
229C2×C4—62no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—61no—
239C2×C4—62no—
240C2×C4—61no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—128no—
244C2×C4D3×D412yes—
245C2×C4#23714yes—
246C2×C3—44no—
247C2×C3—81no—
248C2×C3#52511yescharacteristic
249C2×C3#53211yescharacteristic
250C2×C3#53511yescharacteristic
251C2×C3#55112yes—
252C2×C3#55312yes—
253C22—24no—
254C22—24no—
255C22—24no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—42no—
263C22—41no—
264C22—42no—
265C22C2×D3×D411yescharacteristic
266C22C2×#23712yes—
267C22C2×#23712yes—
268C22#100411yescharacteristic
269C22#100911yescharacteristic
270C4—22no—
271C4—24no—
272C4—41no—
273C4—62no—
274C4—62no—
275C4—121no—
276C4—122no—
277C4—121no—
278C4—122no—
279C3#260311yescharacteristic, complemented, Sylow
280C2—44no—
281C2—81no—
282C2#558611yescharacteristic
283C2#558712yes—
284C2#560711yescharacteristic
285C2#561812yes—
286C2#571611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC3×#392C2
hypercenter#392D3
hypocenterC3#2603
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic3C2semidirect
2. C2×#879C2semidirect
4. #5222C2semidirect
5. #5221C2semidirect
6. #5220C2semidirect
7. #4953C2semidirect
8. #4950C2semidirect
9. #4944C2semidirect
10. #4661C2semidirect
11. #4648C2semidirect
82. C4×Dic3D4semidirect
279. C3#2603semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
C2×#8792
C3×#3922
#52222
#52212
#52202
#49532
#49502
#49442
#46612
#46482
#26033