Finite Group Databasebeta

← #76696

Subgroups — #76696

Order 384.

576 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 139 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×#936C211yesmaximal, characteristic, complemented
2C2×#877C211yesmaximal, characteristic, complemented
3C3×#392C211yesmaximal, characteristic
4#5222C211yesmaximal, characteristic, complemented
5#5221C212yesmaximal, complemented
6#5220C212yesmaximal, complemented
7#4951C212yesmaximal, complemented
8#4947C211yesmaximal, characteristic, complemented
9#4944C212yesmaximal, complemented
10#4656C211yesmaximal, characteristic, complemented
11#4648C211yesmaximal, characteristic, complemented
12#2610—31nomaximal, Sylow
13C22×C3×D4C2212yes—
14C22×C3×D4C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×#217C2211yescharacteristic
23C2×#217C2212yes—
24C2×#217C2212yes—
25#936—24no—
26C2×C4⋊Dic3C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×C4×Dic3C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31C2×C4×Dic3C2211yescharacteristic
32#877—24no—
33C3×#95C2211yescharacteristic
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#124—31no—
39C2×#117—31no—
40#395—31no—
41#394—31no—
42#393—31no—
43#392D311yescharacteristic
44#390—32no—
45#388—32no—
46#388—32no—
47#386—32no—
48#376—31no—
49C24×C3C2312yes—
50C24×C3C2311yescharacteristic
51C2×C3×D4—24no—
52C2×C3×D4—22no—
53C2×C3×D4—24no—
54C2×C3×D4—42no—
55C2×C3×D4—42no—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2312yes—
59C22×Dic3—22no—
60C22×Dic3—22no—
61C22×Dic3—22no—
62C22×Dic3C2312yes—
63C22×Dic3C2312yes—
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C2×Dic6—22no—
69C2×Dic6—42no—
70C3×C22⋊C4—22no—
71C3×C22⋊C4—24no—
72#217—22no—
73#217—24no—
74#217—24no—
75#217—24no—
76#217—42no—
77#217—42no—
78C4⋊Dic3—42no—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81C4×Dic3—24no—
82C4×Dic3—22no—
83C4×Dic3D414yescomplemented
84C22×Q8—31no—
85C22×D4C2×D312yes—
86C22×D4C2×D311yescharacteristic
87C23×C4—31no—
88#124—64no—
89#117—64no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C22⋊C4—31no—
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×D312yes—
101C2×C42—31no—
102C2×C42—31no—
103C2×C42—32no—
104#95—32no—
105#95—31no—
106#95—32no—
107#95—32no—
108#95C2×D311yescharacteristic
109C23×C3—21no—
110C23×C3—21no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3#4112yes—
121C23×C3C2411yescharacteristic
122C3×D4—48no—
123C3×D4—44no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4C2×D412yes—
130C2×C3×C4#4114yes—
131C2×Dic3—28no—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3C2×D412yes—
146C2×Dic3C2×D412yes—
147C2×Dic3#4112yes—
148Dic6—44no—
149C24C22×D312yes—
150C24C22×D311yescharacteristic
151C2×Q8—62no—
152C2×Q8—122no—
153C2×D4—24no—
154C2×D4—22no—
155C2×D4—24no—
156C2×D4—42no—
157C2×D4—42no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—32no—
162C22×C4—32no—
163C22×C4—31no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4—62no—
167C22×C4C22×D311yescharacteristic
168C22×C4C22×D311yescharacteristic
169C22×C4C22×D312yes—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C22⋊C4—22no—
174C22⋊C4—24no—
175C22⋊C4—62no—
176C22⋊C4—64no—
177C22⋊C4—64no—
178C22⋊C4—64no—
179C22⋊C4—122no—
180C22⋊C4—122no—
181C42—34no—
182C42—62no—
183C42—64no—
184C22×C3—22no—
185C22×C3—24no—
186C22×C3—41no—
187C22×C3—41no—
188C22×C3—42no—
189C22×C3—42no—
190C22×C3—42no—
191C22×C3—42no—
192C22×C3—42no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3C22×D411yescharacteristic
197C22×C3C2×#4111yescharacteristic
198C22×C3C2×#4112yes—
199C22×C3C2×#4111yescharacteristic
200C22×C3H(5,𝔽2)12yes—
201C3×C4—22no—
202C3×C4—24no—
203C3×C4—41no—
204Dic3—22no—
205Dic3—22no—
206Dic3—22no—
207Dic3—42no—
208Dic3—41no—
209Dic3—42no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23—21no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23#23712yes—
222C23C23×D311yescharacteristic
223Q8—124no—
224D4—48no—
225D4—44no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—68no—
235C2×C4—61no—
236C2×C4—62no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4D3×D412yes—
249C2×C4#23714yes—
250C2×C3—42no—
251C2×C3—82no—
252C2×C3#52511yescharacteristic
253C2×C3#53211yescharacteristic
254C2×C3#53511yescharacteristic
255C2×C3#55112yes—
256C2×C3#55512yes—
257C22—24no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22C2×D3×D411yescharacteristic
270C22C2×#23712yes—
271C22C2×#23711yescharacteristic
272C22#100412yes—
273C22D3×#4111yescharacteristic
274C4—24no—
275C4—22no—
276C4—41no—
277C4—62no—
278C4—62no—
279C4—62no—
280C4—122no—
281C4—121no—
282C4—122no—
283C3#261011yescharacteristic, complemented, Sylow
284C2—42no—
285C2—82no—
286C2#558712yes—
287C2#559211yescharacteristic
288C2#560611yescharacteristic
289C2#561612yes—
290C2#571611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC3×#392C2
hypercenter#392D3
hypocenterC3#2610
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#936C2semidirect
2. C2×#877C2semidirect
4. #5222C2semidirect
5. #5221C2semidirect
6. #5220C2semidirect
7. #4951C2semidirect
8. #4947C2semidirect
9. #4944C2semidirect
10. #4656C2semidirect
11. #4648C2semidirect
83. C4×Dic3D4semidirect
283. C3#2610semidirect

Maximal subgroups

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

subgroupindex
C2×#9362
C2×#8772
C3×#3922
#52222
#52212
#52202
#49512
#49472
#49442
#46562
#46482
#26103