Finite Group Databasebeta

← #73835

Subgroups — #73835

Order 384.

672 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#217C211yesmaximal, characteristic
2C2×#878C211yesmaximal, characteristic, complemented
3C3×#379C211yesmaximal, characteristic
4#5247C211yesmaximal, characteristic
5#4952C212yesmaximal, complemented
6#4951C212yesmaximal, complemented
7#4947C212yesmaximal, complemented
8#4944C212yesmaximal, complemented
9#4943C211yesmaximal, characteristic, complemented
10#4943C211yesmaximal, characteristic, complemented
11Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
12#2561—31nomaximal, Sylow
13C25×C3C2211yescharacteristic
14C23×Dic3C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic, complemented
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C22⋊C4C2212yes—
21C2×#217—24no—
22C2×#217C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×#217C2212yes—
25C2×#217C2212yes—
26C2×#217C2212yes—
27C2×#217C2212yes—
28C2×C4⋊Dic3C2211yescharacteristic, complemented
29C2×Dic3⋊C4C2211yescharacteristic, complemented
30C2×Dic3⋊C4C2211yescharacteristic, complemented
31C2×C4×Dic3C2212yescomplemented
32#878—24no—
33C3⋊#95C2212yescomplemented
34C3⋊#95C2212yescomplemented
35C3⋊#95C2212yescomplemented
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C2×C22⋊Q8—31no—
39C22×C22⋊C4—31no—
40#397—32no—
41#393—32no—
42#388—32no—
43#388—32no—
44#381—31no—
45#381—31no—
46#379—31no—
47#379D311yescharacteristic
48C4×C22⋊C4—31no—
49C24×C3—22no—
50C24×C3—24no—
51C24×C3—22no—
52C24×C3—22no—
53C24×C3—42no—
54C24×C3C2312yes—
55C24×C3C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×Dic3—22no—
60C22×Dic3—22no—
61C22×Dic3—22no—
62C22×Dic3—42no—
63C22×Dic3D414yes—
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C2×Dic6—22no—
71C2×Dic6—42no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4—24no—
74C3×C22⋊C4—22no—
75C3×C22⋊C4—42no—
76#217—22no—
77#217—22no—
78#217—24no—
79#217—24no—
80#217—24no—
81#217—42no—
82C4⋊Dic3—22no—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85C4×Dic3—42no—
86C25C2×D311yescharacteristic
87C22×Q8—31no—
88C23×C4—31no—
89C23×C4—31no—
90C22⋊Q8—64no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—31no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—64no—
101C2×C22⋊C4C2×D311yescharacteristic
102C2×C22⋊C4C2×D311yescharacteristic
103C2×C22⋊C4C2×D312yes—
104C2×C22⋊C4C2×D312yes—
105C2×C42—32no—
106#95—32no—
107#95—32no—
108#95—32no—
109#95—32no—
110#95—32no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—28no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—48no—
121C23×C3—48no—
122C23×C3—44no—
123C23×C3—44no—
124C23×C3C2×D412yes—
125C23×C3#4112yes—
126C23×C3#4114yes—
127C23×C3#4112yes—
128C23×C3C2411yescharacteristic
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—21no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—44no—
149C2×Dic3—44no—
150C2×Dic3C2×D412yes—
151C2×Dic3C2×D412yes—
152Dic6—44no—
153C24—22no—
154C24—24no—
155C24—22no—
156C24—22no—
157C24—42no—
158C24C22×D312yes—
159C24C22×D311yescharacteristic
160C2×Q8—62no—
161C2×Q8—122no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—34no—
165C22×C4—32no—
166C22×C4—32no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—62no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4—122no—
173C22×C4C22×D312yes—
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D311yescharacteristic
176C4⋊C4—62no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C22⋊C4—22no—
180C22⋊C4—24no—
181C22⋊C4—22no—
182C22⋊C4—42no—
183C22⋊C4—64no—
184C22⋊C4—64no—
185C22⋊C4—62no—
186C22⋊C4—62no—
187C22⋊C4—64no—
188C22⋊C4—122no—
189C42—122no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—24no—
193C22×C3—24no—
194C22×C3—24no—
195C22×C3—24no—
196C22×C3—24no—
197C22×C3—416no—
198C22×C3—42no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3#12414yes—
203C22×C3C22×D411yescharacteristic
204C22×C3C2×#4111yescharacteristic
205C22×C3C2×#4112yes—
206C22×C3C2×#4111yescharacteristic
207C22×C3H(5,𝔽2)12yes—
208C3×C4—41no—
209C3×C4—41no—
210C3×C4—42no—
211Dic3—22no—
212Dic3—22no—
213Dic3—41no—
214Dic3—42no—
215Dic3—41no—
216Dic3—42no—
217C23—22no—
218C23—22no—
219C23—28no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—48no—
227C23—48no—
228C23—44no—
229C23—44no—
230C23#23512yes—
231C23D3×D412yes—
232C23#23714yes—
233C23#23712yes—
234C23C23×D311yescharacteristic
235Q8—124no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—32no—
244C2×C4—32no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—61no—
251C2×C4—61no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—124no—
258C2×C4—124no—
259C2×C3—24no—
260C2×C3—44no—
261C2×C3#52511yescharacteristic
262C2×C3C2×#12411yescharacteristic
263C2×C3#53411yescharacteristic
264C2×C3#55112yes—
265C2×C3#55112yes—
266C22—24no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—24no—
271C22—24no—
272C22—24no—
273C22—416no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22#88514yes—
279C22C2×#23511yescharacteristic
280C22C2×D3×D411yescharacteristic
281C22C2×#23711yescharacteristic
282C22C2×#23712yes—
283C22#100412yes—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—62no—
288C4—62no—
289C4—122no—
290C4—122no—
291C4—121no—
292C4—121no—
293C3#256111yescharacteristic, complemented, Sylow
294C2—24no—
295C2—44no—
296C2C2×#88511yescharacteristic
297C2#549512yes—
298C2#558611yescharacteristic
299C2#558712yes—
300C2#558911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#379C2
prime radicalC25×C3C22
hypercenter#379D3
hypocenterC3#2561
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#878C2semidirect
5. #4952C2semidirect
6. #4951C2semidirect
7. #4947C2semidirect
8. #4944C2semidirect
9. #4943C2semidirect
10. #4943C2semidirect
11. Dic3×C22⋊C4C2semidirect
16. C22×Dic6C22semidirect
28. C2×C4⋊Dic3C22semidirect
29. C2×Dic3⋊C4C22semidirect
30. C2×Dic3⋊C4C22semidirect
31. C2×C4×Dic3C22semidirect
33. C3⋊#95C22semidirect
34. C3⋊#95C22semidirect
35. C3⋊#95C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
293. C3#2561semidirect

Maximal subgroups

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

subgroupindex
C22×#2172
C2×#8782
C3×#3792
#52472
#49522
#49512
#49472
#49442
#49432
Dic3×C22⋊C42
#25613