Finite Group Databasebeta

← #82960

Subgroups — #82960

Order 384.

1088 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 173 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×D4C211yesmaximal, characteristic
2C2×C3×#124C211yesmaximal, characteristic, complemented
3C2×#947C211yesmaximal, characteristic, complemented
4C2×#946C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#4959C214yesmaximal, complemented
7#4955C214yesmaximal, complemented
8#4939C211yesmaximal, characteristic, complemented
9C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
10#2435—31nomaximal, Sylow
11C24×D3C2212yes—
12C22×C3×Q8C2211yescharacteristic, complemented
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C2×D3×D4—24no—
16C2×D3×D4—24no—
17C2×D3×D4—24no—
18C22×D12C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C3×#124—24no—
21C2×C3×C22⋊C4C2214yescomplemented
22C2×C3×C42C2211yescharacteristic, complemented
23C2×#217C2214yescomplemented
24#947—24no—
25#946—24no—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2214yescomplemented
29C2×(C2×D3)⋊C4C2214yescomplemented
30C2×C4⋊Dic3C2211yescharacteristic, complemented
31C2×Dic3⋊C4C2212yescomplemented
32C2×C4×Dic3C2211yescharacteristic, complemented
33C3⋊#95C2212yescomplemented
34C23×D4—31no—
35C2×#124—31no—
36C2×#124D311yescharacteristic, complemented
37C2×C22⋊Q8—31no—
38#394—34no—
39#390—31no—
40#390—31no—
41#379—34no—
42C4×C22⋊C4—31no—
43C24×C3C2312yes—
44C23×D3—24no—
45C23×D3—24no—
46C23×D3—24no—
47C23×D3—28no—
48C23×D3—44no—
49C23×D3C2312yes—
50C23×D3C2311yescharacteristic
51C2×C3×Q8—22no—
52C2×C3×Q8—22no—
53C2×C3×Q8—22no—
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4D414yes—
57C22×C3×C4C2312yes—
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—24no—
63C22×Dic3C2312yes—
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66D3×D4—232no—
67C2×D12—22no—
68C2×D12—22no—
69C2×D12D414yes—
70C2×C4×D3—22no—
71C2×C4×D3—22no—
72C2×C4×D3D414yes—
73C3×C22⋊C4—28no—
74C3×C42—22no—
75#217—44no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—28no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—44no—
80C4⋊Dic3—22no—
81Dic3⋊C4—24no—
82C4×Dic3—22no—
83C25—32no—
84C22×Q8C2×D311yescharacteristic, complemented
85C22×D4—32no—
86C22×D4—31no—
87C22×D4—64no—
88C22×D4—64no—
89C22×D4—64no—
90C22×D4C2×D311yescharacteristic
91C23×C4—31no—
92#124—24no—
93#124—64no—
94C22⋊Q8—64no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—34no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—34no—
101C2×C22⋊C4—34no—
102C2×C22⋊C4C2×D314yescomplemented
103C2×C42—31no—
104C2×C42C2×D311yescharacteristic, complemented
105#95—32no—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3—24no—
109C23×C3—24no—
110C23×C3C2×D414yes—
111C23×C3C2411yescharacteristic
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—22no—
115C22×D3—216no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—24no—
119C22×D3—24no—
120C22×D3—416no—
121C22×D3—416no—
122C22×D3—48no—
123C22×D3—48no—
124C22×D3C2×D414yes—
125C22×D3C2×D412yes—
126C22×D3#4114yes—
127C3×Q8—44no—
128C3×D4—28no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4C2×D412yes—
136C2×C3×C4C2×D412yes—
137C2×C3×C4C2×D412yes—
138#67—216no—
139C2×Dic3—21no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—24no—
145C2×Dic3—22no—
146C2×Dic3C2×D412yes—
147D12—28no—
148C4×D3—28no—
149C24—31no—
150C24—32no—
151C24—64no—
152C24—64no—
153C24—68no—
154C24—64no—
155C24—124no—
156C24C22×D312yes—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—34no—
163C2×D4—64no—
164C2×D4—632no—
165C2×D4—64no—
166C2×D4—64no—
167C2×D4—62no—
168C2×D4—62no—
169C2×D4#6714yes—
170C22×C4—31no—
171C22×C4—34no—
172C22×C4—32no—
173C22×C4—32no—
174C22×C4—62no—
175C22×C4—62no—
176C22×C4C22×D312yes—
177C22×C4C22×D312yes—
178C22×C4C22×D311yescharacteristic
179C4⋊C4—64no—
180C4⋊C4—62no—
181C22⋊C4—28no—
182C22⋊C4—64no—
183C22⋊C4—68no—
184C22⋊C4—64no—
185C22⋊C4—124no—
186C22⋊C4—124no—
187C42—22no—
188C42—62no—
189C22×C3—24no—
190C22×C3—28no—
191C22×C3—42no—
192C22×C3—42no—
193C22×C3—44no—
194C22×C3C22×D411yescharacteristic
195C22×C3C22×D411yescharacteristic
196C22×C3C22×D411yescharacteristic
197C22×C3C2×#4112yes—
198C22×C3H(5,𝔽2)12yes—
199C2×D3—22no—
200C2×D3—22no—
201C2×D3—24no—
202C2×D3—24no—
203C2×D3—24no—
204C2×D3—28no—
205C2×D3—432no—
206C2×D3—42no—
207C2×D3—42no—
208C2×D3—44no—
209C2×D3#12414yescomplemented
210C3×C4—42no—
211C3×C4—42no—
212C3×C4#12014yes—
213Dic3—22no—
214Dic3—42no—
215Dic3—42no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—24no—
220C23—32no—
221C23—34no—
222C23—34no—
223C23—616no—
224C23—61no—
225C23—62no—
226C23—61no—
227C23—62no—
228C23—64no—
229C23—64no—
230C23—62no—
231C23—1216no—
232C23—128no—
233C23—1216no—
234C23—128no—
235C23C2×#6714yes—
236C23C23×D311yescharacteristic
237Q8—44no—
238D4—28no—
239D4—616no—
240D4—68no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—32no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—68no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—64no—
255C2×C4—62no—
256C2×C4D3×D412yes—
257C2×C4D3×D412yes—
258C2×C4C2×#6712yes—
259C2×C3—44no—
260C2×C3C2×#12011yescharacteristic
261C2×C3C2×#12411yescharacteristic
262C2×C3#53511yescharacteristic
263C2×C3#54612yes—
264C2×C3#54612yes—
265D3—24no—
266D3—44no—
267C22—24no—
268C22—28no—
269C22—34no—
270C22—42no—
271C22—42no—
272C22—44no—
273C22—62no—
274C22—64no—
275C22—62no—
276C22—64no—
277C22—68no—
278C22—64no—
279C22—1232no—
280C22—122no—
281C22—122no—
282C22—124no—
283C22C2×D3×D411yescharacteristic
284C22C2×D3×D411yescharacteristic
285C22D3×#4112yes—
286C22#100912yes—
287C22C22×#6711yescharacteristic
288C4—42no—
289C4—42no—
290C4—62no—
291C4—122no—
292C4—122no—
293C4#93714yes—
294C3#243511yescharacteristic, complemented, Sylow
295C2—44no—
296C2—64no—
297C2—124no—
298C2D3×#12411yescharacteristic
299C2#567411yescharacteristic
300C2#567612yes—
301C2C2×#93711yescharacteristic
302C2#582912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#124C2
prime radicalC22×D3×D4C2
hypercenterC2×#124D3
hypocenterC3#2435
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×C3×#124C2semidirect
3. C2×#947C2semidirect
4. C2×#946C2semidirect
5. #5221C2semidirect
6. #4959C2semidirect
7. #4955C2semidirect
8. #4939C2semidirect
9. C4×(C2×D3)⋊C4C2semidirect
12. C22×C3×Q8C22semidirect
21. C2×C3×C22⋊C4C22semidirect
22. C2×C3×C42C22semidirect
23. C2×#217C22semidirect
28. C2×(C2×D3)⋊C4C22semidirect
29. C2×(C2×D3)⋊C4C22semidirect
30. C2×C4⋊Dic3C22semidirect
31. C2×Dic3⋊C4C22semidirect
32. C2×C4×Dic3C22semidirect
33. C3⋊#95C22semidirect
36. C2×#124D3semidirect
84. C22×Q8C2×D3semidirect
102. C2×C22⋊C4C2×D3semidirect
104. C2×C42C2×D3semidirect
209. C2×D3#124semidirect
294. C3#2435semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×C3×#1242
C2×#9472
C2×#9462
#52212
#49592
#49552
#49392
C4×(C2×D3)⋊C42
#24353