Finite Group Databasebeta

← #76780

Subgroups — #76780

Order 384.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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×D3×Q8C211yesmaximal, characteristic, complemented
2C2×#67⋊C4C211yesmaximal, characteristic
3C2×#878C212yesmaximal, complemented
4C3×#393C211yesmaximal, characteristic, complemented
5#5230C211yesmaximal, characteristic, complemented
6#4956C212yesmaximal
7#4947C211yesmaximal, characteristic, complemented
8#4668C212yesmaximal, complemented
9#4667C211yesmaximal, characteristic, complemented
10#4656C211yesmaximal, characteristic, complemented
11#4646C212yesmaximal, complemented
12#2409—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C2×D3×Q8—24no—
17C2×D3×Q8—28no—
18C22×C4×D3C2212yes—
19C22×C4×D3C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C22×Dic6C2212yes—
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2212yes—
24C2×#217C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×C4⋊Dic3C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32#67⋊C4—24no—
33#878—28no—
34C3×#95C2212yes—
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2211yescharacteristic
40C23×Q8—31no—
41C2×C22⋊Q8—32no—
42C2×C4×D4—31no—
43#395—31no—
44#395—31no—
45#393—31no—
46#393D311yescharacteristic, complemented
47#391—32no—
48#386—32no—
49#385—32no—
50#380—31no—
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C2×C3×Q8—22no—
54C2×C3×Q8—24no—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2312yes—
59C2×#67—22no—
60C2×#67—22no—
61C2×#67Q814yescomplemented
62C22×Dic3—24no—
63C22×Dic3—22no—
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69D3×Q8—232no—
70C2×C4×D3—24no—
71C2×C4×D3—24no—
72C2×C4×D3—24no—
73C2×C4×D3—24no—
74C2×C4×D3D414yes—
75C2×Dic6—22no—
76C2×Dic6—24no—
77C2×Dic6—24no—
78C2×Dic6—24no—
79C2×Dic6D418yescomplemented
80C3×C22⋊C4—24no—
81C3×C22⋊C4—22no—
82#217—24no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—24no—
85C4⋊Dic3—24no—
86Dic3⋊C4—24no—
87Dic3⋊C4—22no—
88Dic3⋊C4—24no—
89C4×Dic3—22no—
90C22×Q8—32no—
91C22×Q8—31no—
92C22×Q8—64no—
93C22×Q8—68no—
94C22×Q8C2×D311yescharacteristic
95C22×D4—31no—
96C23×C4—31no—
97C23×C4—31no—
98C23×C4—32no—
99C22⋊Q8—68no—
100C4×D4—64no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4C2×D311yescharacteristic
109C2×C22⋊C4C2×D312yes—
110C2×C42—31no—
111#95—31no—
112#95—32no—
113#95—32no—
114#95—32no—
115#95C2×D311yescharacteristic
116#95C2×D312yes—
117C23×C3—22no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—22no—
121C23×C3C2×Q812yescomplemented
122C23×C3C2411yescharacteristic
123C22×D3—22no—
124C22×D3—22no—
125C22×D3C2×D412yes—
126C22×D3C2×Q812yes—
127C22×D3#4112yes—
128C3×Q8—28no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4C2×D412yes—
138C2×C3×C4C2×D414yes—
139#67—28no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—44no—
150C2×Dic3C2×D414yes—
151C2×Dic3C2×D412yes—
152C2×Dic3C2×D414yes—
153C2×Dic3C2×Q812yescomplemented
154C4×D3—216no—
155C4×D3—28no—
156Dic6—28no—
157Dic6—216no—
158C24—31no—
159C24C22×D311yescharacteristic
160C2×Q8—22no—
161C2×Q8—24no—
162C2×Q8—38no—
163C2×Q8—64no—
164C2×Q8—64no—
165C2×Q8—64no—
166C2×Q8—632no—
167C2×Q8—62no—
168C2×D4—34no—
169C2×D4—62no—
170C2×D4—62no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—32no—
174C22×C4—34no—
175C22×C4—32no—
176C22×C4—31no—
177C22×C4—64no—
178C22×C4—64no—
179C22×C4—64no—
180C22×C4—62no—
181C22×C4—64no—
182C22×C4—64no—
183C22×C4C22×D311yescharacteristic
184C22×C4C22×D312yes—
185C22×C4C22×D311yescharacteristic
186C22×C4C22×D312yes—
187C4⋊C4—64no—
188C4⋊C4—64no—
189C4⋊C4—62no—
190C4⋊C4—64no—
191C22⋊C4—22no—
192C22⋊C4—24no—
193C22⋊C4—62no—
194C22⋊C4—64no—
195C22⋊C4—64no—
196C42—62no—
197C22×C3—22no—
198C22×C3—24no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—42no—
202C22×C3C22×D411yescharacteristic
203C22×C3C22×D412yes—
204C22×C3C22×Q811yescharacteristic
205C22×C3C2×#4111yescharacteristic
206C22×C32−1+412yes—
207C2×D3—22no—
208C2×D3—22no—
209C2×D3—24no—
210C2×D3—24no—
211C2×D3C22⋊Q814yescomplemented
212C3×C4—22no—
213C3×C4—24no—
214C3×C4—41no—
215C3×C4—42no—
216Dic3—24no—
217Dic3—22no—
218Dic3—41no—
219Dic3—42no—
220Dic3#12014yes—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—32no—
226C23—32no—
227C23—32no—
228C23—62no—
229C23—62no—
230C23D3×Q812yescomplemented
231C23C23×D311yescharacteristic
232Q8—28no—
233Q8—616no—
234Q8—68no—
235D4—68no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—32no—
245C2×C4—34no—
246C2×C4—34no—
247C2×C4—32no—
248C2×C4—68no—
249C2×C4—616no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—61no—
254C2×C4—62no—
255C2×C4—61no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—124no—
260C2×C4D3×D412yes—
261C2×C4D3×D414yes—
262C2×C3—42no—
263C2×C3C2×#12011yescharacteristic
264C2×C3C2×C22⋊Q811yescharacteristic
265C2×C3#53611yescharacteristic
266C2×C3#54812yes—
267C2×C3D4×Q812yes—
268D3—24no—
269C22—24no—
270C22—22no—
271C22—34no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—64no—
276C22—62no—
277C22—64no—
278C22—62no—
279C22C2×D3×D412yes—
280C22C2×D3×D411yescharacteristic
281C22C2×D3×Q811yescharacteristic
282C22D3×#4111yescharacteristic
283C22#101012yes—
284C4—22no—
285C4—24no—
286C4—34no—
287C4—41no—
288C4—42no—
289C4—64no—
290C4—62no—
291C4—122no—
292C4—121no—
293C3#240911yescharacteristic, complemented, Sylow
294C2—42no—
295C2—64no—
296C2D3×#12011yescharacteristic
297C2D3×C22⋊Q811yescharacteristic
298C2#563212yes—
299C2#567511yescharacteristic
300C2#567712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC22×#67C22
hypercenter#393D3
hypocenterC3#2409
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×D3×Q8C2semidirect
3. C2×#878C2semidirect
4. C3×#393C2semidirect
5. #5230C2semidirect
7. #4947C2semidirect
8. #4668C2semidirect
9. #4667C2semidirect
10. #4656C2semidirect
11. #4646C2semidirect
46. #393D3semidirect
61. C2×#67Q8semidirect
79. C2×Dic6D4semidirect
121. C23×C3C2×Q8semidirect
153. C2×Dic3C2×Q8semidirect
211. C2×D3C22⋊Q8semidirect
230. C23D3×Q8semidirect
293. C3#2409semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q82
C2×#67⋊C42
C2×#8782
C3×#3932
#52302
#49562
#49472
#46682
#46672
#46562
#46462
#24093