Finite Group Databasebeta

← #76799

Subgroups — #76799

Order 384.

564 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 151 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×#944C211yesmaximal, characteristic, complemented
2C2×#67⋊C4C211yesmaximal, characteristic
3C3×#393C211yesmaximal, characteristic, complemented
4#5231C211yesmaximal, characteristic, complemented
5#4957C212yesmaximal
6#4947C211yesmaximal, characteristic, complemented
7#4942C212yesmaximal, complemented
8#4673C211yesmaximal, characteristic, complemented
9#4667C212yesmaximal, complemented
10#4650C212yesmaximal, complemented
11#4649C211yesmaximal, characteristic, complemented
12#2563—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×Dic6C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2212yes—
21#944—24no—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31#67⋊C4—24no—
32C3×#95C2211yescharacteristic
33C3×#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4⋊Q8—31no—
39C2×C4×D4—31no—
40#393—31no—
41#393—31no—
42#393—31no—
43#393D311yescharacteristic, complemented
44#390—32no—
45#382—32no—
46#380—32no—
47#380—32no—
48C4×C4⋊C4—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C2×C3×Q8—22no—
52C2×C3×Q8—42no—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C2×#67—22no—
58C2×#67—22no—
59C2×#67Q814yescomplemented
60C22×Dic3—22no—
61C22×Dic3—24no—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2312yes—
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C2×C4×D3—22no—
68C2×C4×D3—24no—
69C2×Dic6—22no—
70C2×Dic6—42no—
71C3×C22⋊C4—22no—
72C3×C22⋊C4—42no—
73#217—42no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—42no—
76(C2×D3)⋊C4—42no—
77Dic3⋊C4—22no—
78Dic3⋊C4—24no—
79Dic3⋊C4—24no—
80Dic3⋊C4—24no—
81C4×Dic3—24no—
82C4×Dic3D414yes—
83C22×Q8—31no—
84C22×Q8C2×D311yescharacteristic
85C22×D4—31no—
86C23×C4—31no—
87C23×C4—31no—
88C4⋊Q8—64no—
89C4×D4—64no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4C2×D311yescharacteristic
99C2×C22⋊C4C2×D312yes—
100C2×C42—31no—
101C2×C42—32no—
102#95—31no—
103#95—32no—
104#95—32no—
105#95—32no—
106#95C2×D311yescharacteristic
107#95C2×D312yes—
108C23×C3—21no—
109C23×C3—22no—
110C23×C3—21no—
111C23×C3—22no—
112C23×C3C2×Q812yes—
113C23×C3C2411yescharacteristic
114C22×D3—21no—
115C22×D3—22no—
116C22×D3—21no—
117C22×D3—22no—
118C22×D3C2×Q812yes—
119C3×Q8—44no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4C2×D412yes—
130#67—28no—
131C2×Dic3—22no—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—44no—
140C2×Dic3C2×D412yes—
141C2×Dic3C2×D412yes—
142C2×Dic3C2×Q812yes—
143C2×Dic3#4112yes—
144C2×Dic3#4114yes—
145C4×D3—44no—
146Dic6—44no—
147C24—31no—
148C24C22×D311yescharacteristic
149C2×Q8—22no—
150C2×Q8—42no—
151C2×Q8—62no—
152C2×Q8—122no—
153C2×D4—34no—
154C2×D4—62no—
155C2×D4—62no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—32no—
160C22×C4—32no—
161C22×C4—62no—
162C22×C4—64no—
163C22×C4—62no—
164C22×C4—64no—
165C22×C4C22×D311yescharacteristic
166C22×C4C22×D312yes—
167C22×C4C22×D312yes—
168C22×C4C22×D311yescharacteristic
169C4⋊C4—62no—
170C4⋊C4—64no—
171C4⋊C4—64no—
172C4⋊C4—64no—
173C22⋊C4—22no—
174C22⋊C4—42no—
175C22⋊C4—62no—
176C22⋊C4—122no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C42—34no—
180C42—64no—
181C22×C3—22no—
182C22×C3—24no—
183C22×C3—41no—
184C22×C3—41no—
185C22×C3—42no—
186C22×C3C22×D411yescharacteristic
187C22×C3C22×Q811yescharacteristic
188C22×C3C2×#4111yescharacteristic
189C22×C3C2×#4112yes—
190C22×C3H(5,𝔽2)12yes—
191C2×D3—22no—
192C2×D3—24no—
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—42no—
196C3×C4—22no—
197C3×C4—41no—
198C3×C4—42no—
199C3×C4—42no—
200Dic3—41no—
201Dic3—42no—
202Dic3—42no—
203Dic3C22⋊Q814yes—
204Dic3#12414yes—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—32no—
210C23—61no—
211C23—62no—
212C23—62no—
213C23—61no—
214C23D3×Q812yes—
215C23C23×D311yescharacteristic
216Q8—44no—
217Q8—124no—
218D4—68no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—32no—
229C2×C4—32no—
230C2×C4—32no—
231C2×C4—32no—
232C2×C4—34no—
233C2×C4—61no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—124no—
242C2×C4—124no—
243C2×C4D3×D412yes—
244C2×C3—42no—
245C2×C3C2×C22⋊Q811yescharacteristic
246C2×C3C2×#12411yescharacteristic
247C2×C3#53511yescharacteristic
248C2×C3#55412yes—
249C2×C3#55512yes—
250D3—42no—
251C22—24no—
252C22—22no—
253C22—42no—
254C22—41no—
255C22—41no—
256C22—64no—
257C22—62no—
258C22—122no—
259C22—121no—
260C22—121no—
261C22C2×D3×D411yescharacteristic
262C22#100412yes—
263C22C2×D3×Q811yescharacteristic
264C22D3×#4112yes—
265C22D3×#4111yescharacteristic
266C4—22no—
267C4—34no—
268C4—34no—
269C4—42no—
270C4—41no—
271C4—42no—
272C4—122no—
273C4—121no—
274C4—122no—
275C3#256311yescharacteristic, complemented, Sylow
276C2—42no—
277C2—122no—
278C2#559211yescharacteristic
279C2D3×C22⋊Q811yescharacteristic
280C2#563612yes—
281C2#567012yes—
282C2D3×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC22×#67C22
hypercenter#393D3
hypocenterC3#2563
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#944C2semidirect
3. C3×#393C2semidirect
4. #5231C2semidirect
6. #4947C2semidirect
7. #4942C2semidirect
8. #4673C2semidirect
9. #4667C2semidirect
10. #4650C2semidirect
11. #4649C2semidirect
43. #393D3semidirect
59. C2×#67Q8semidirect
275. C3#2563semidirect

Maximal subgroups

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

subgroupindex
C2×#9442
C2×#67⋊C42
C3×#3932
#52312
#49572
#49472
#49422
#46732
#46672
#46502
#46492
#25633