Finite Group Databasebeta

← #84053

Subgroups — #84053

Order 384.

1232 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 205 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
1C23×C3×D4C211yesmaximal, characteristic, complemented
2C2×#953C212yesmaximal, complemented
3C2×#938C212yesmaximal, complemented
4C2×C4×#67C211yesmaximal, characteristic, complemented
5#5247C212yesmaximal, complemented
6#5223C212yesmaximal, complemented
7#5222C212yesmaximal, complemented
8#5221C212yesmaximal, complemented
9#5212C211yesmaximal, characteristic, complemented
10#2408—31nomaximal, Sylow
11C25×C3C2212yes—
12C22×C3×D4—28no—
13C22×C3×D4—24no—
14C22×C3×D4C2212yes—
15C22×C3×D4C2212yes—
16C23×C3×C4C2211yescharacteristic
17C22×#67C2211yescharacteristic
18C22×#67C2214yescomplemented
19C22×C4×D3C2211yescharacteristic, complemented
20#953—28no—
21C2×#217C2212yes—
22C2×#217C2211yescharacteristic
23C2×#217C2214yescomplemented
24C2×#217C2214yescomplemented
25C2×#217C2214yescomplemented
26#938—28no—
27C4×#67—24no—
28C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
29C2×C4⋊Dic3C2212yescomplemented
30C2×Dic3⋊C4C2211yescharacteristic, complemented
31C2×C4×Dic3C2211yescharacteristic, complemented
32C3⋊#95C2212yescomplemented
33C23×D4D311yescharacteristic, complemented
34C2×#121—32no—
35C2×#120—32no—
36C2×C4×D4—31no—
37#394—32no—
38#392—32no—
39#390—32no—
40#380—31no—
41#379—32no—
42C24×C3—28no—
43C24×C3—24no—
44C24×C3—24no—
45C24×C3—24no—
46C24×C3—44no—
47C24×C3C2314yes—
48C24×C3C2311yescharacteristic
49C23×D3C2311yescharacteristic
50C2×C3×D4—232no—
51C2×C3×D4—28no—
52C2×C3×D4—24no—
53C2×C3×D4—24no—
54C2×C3×D4—24no—
55C2×C3×D4D418yescomplemented
56C22×C3×C4—24no—
57C22×C3×C4D414yescomplemented
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—22no—
61C2×#67—22no—
62C2×#67—28no—
63C2×#67—28no—
64C2×#67—44no—
65C2×#67D414yescomplemented
66C22×Dic3C2314yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C2×C4×D3—22no—
71C2×C4×D3—24no—
72#217—22no—
73#217—28no—
74#217—24no—
75#217—28no—
76#217—44no—
77(C2×D3)⋊C4—22no—
78C4⋊Dic3—24no—
79Dic3⋊C4—22no—
80C4×Dic3—22no—
81C25C2×D312yes—
82C22×D4—28no—
83C22×D4—24no—
84C22×D4—31no—
85C22×D4—34no—
86C22×D4C2×D312yes—
87C22×D4C2×D312yes—
88C23×C4—31no—
89C23×C4C2×D311yescharacteristic
90#121—68no—
91#120—68no—
92C4×D4—64no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—32no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—34no—
98C2×C22⋊C4—34no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—34no—
101C2×C42—31no—
102#95—32no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—216no—
106C23×C3—24no—
107C23×C3—24no—
108C23×C3—24no—
109C23×C3—24no—
110C23×C3—24no—
111C23×C3—24no—
112C23×C3—416no—
113C23×C3—416no—
114C23×C3—48no—
115C23×C3—48no—
116C23×C3C2×D412yes—
117C23×C3C2×D418yes—
118C23×C3C2×D412yes—
119C23×C3#4112yes—
120C23×C3C2411yescharacteristic
121C22×D3—21no—
122C22×D3—22no—
123C22×D3—21no—
124C22×D3—22no—
125C22×D3C2×D412yescomplemented
126C3×D4—216no—
127C3×D4—216no—
128C2×C3×C4—22no—
129C2×C3×C4—28no—
130C2×C3×C4C2×D412yes—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×D414yes—
133#67—28no—
134#67—416no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—24no—
140C2×Dic3—22no—
141C2×Dic3—24no—
142C2×Dic3—22no—
143C2×Dic3—24no—
144C2×Dic3C2×D412yescomplemented
145C4×D3—44no—
146C24—28no—
147C24—24no—
148C24—24no—
149C24—24no—
150C24—31no—
151C24—44no—
152C24C22×D314yes—
153C24C22×D311yescharacteristic
154C2×D4—232no—
155C2×D4—28no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—34no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4—68no—
163C2×D4—68no—
164C2×D4—124no—
165C2×D4#6718yescomplemented
166C22×C4—24no—
167C22×C4—31no—
168C22×C4—34no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—62no—
172C22×C4—64no—
173C22×C4#6714yescomplemented
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D311yescharacteristic
176C4⋊C4—62no—
177C4⋊C4—64no—
178C22⋊C4—68no—
179C22⋊C4—64no—
180C22⋊C4—62no—
181C22⋊C4—68no—
182C22⋊C4—62no—
183C22⋊C4—124no—
184C42—62no—
185C22×C3—22no—
186C22×C3—22no—
187C22×C3—24no—
188C22×C3—24no—
189C22×C3—28no—
190C22×C3—28no—
191C22×C3—28no—
192C22×C3—432no—
193C22×C3—44no—
194C22×C3—44no—
195C22×C3—44no—
196C22×C3—44no—
197C22×C3#12114yescomplemented
198C22×C3C22×D411yescharacteristic
199C22×C3C22×D411yescharacteristic
200C22×C3C22×D412yes—
201C22×C3C2×#4111yescharacteristic
202C22×C3H(5,𝔽2)12yes—
203C2×D3—22no—
204C2×D3—24no—
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—42no—
208C3×C4—22no—
209C3×C4#12014yescomplemented
210Dic3—22no—
211Dic3—41no—
212Dic3—41no—
213Dic3—44no—
214C23—22no—
215C23—22no—
216C23—216no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—32no—
224C23—416no—
225C23—416no—
226C23—48no—
227C23—48no—
228C23—61no—
229C23—62no—
230C23—61no—
231C23—62no—
232C23D3×D412yes—
233C23#23712yes—
234C23C2×#6718yes—
235C23C2×#6712yes—
236C23C23×D311yescharacteristic
237D4—216no—
238D4—216no—
239D4—68no—
240D4—1216no—
241C2×C4—22no—
242C2×C4—28no—
243C2×C4—32no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—64no—
247C2×C4—62no—
248C2×C4—64no—
249C2×C4—61no—
250C2×C4—62no—
251C2×C4—61no—
252C2×C4—64no—
253C2×C4—124no—
254C2×C4C2×#6712yes—
255C2×C4C2×#6712yes—
256C2×C4C2×#6714yes—
257C2×C3—24no—
258C2×C3—48no—
259C2×C3C2×#12011yescharacteristic
260C2×C3C2×#12111yescharacteristic
261C2×C3#53511yescharacteristic
262C2×C3D4212yes—
263C2×C3#54612yes—
264D3—42no—
265C22—24no—
266C22—22no—
267C22—24no—
268C22—22no—
269C22—28no—
270C22—28no—
271C22—28no—
272C22—432no—
273C22—44no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—62no—
278C22—64no—
279C22—121no—
280C22—122no—
281C22—121no—
282C22#93814yescomplemented
283C22C2×D3×D411yescharacteristic
284C22C2×#23711yescharacteristic
285C22#100412yes—
286C22C22×#6711yescharacteristic
287C22C22×#6712yes—
288C4—22no—
289C4—62no—
290C4—124no—
291C4—121no—
292C4—121no—
293C4#95314yescomplemented
294C3#240811yescharacteristic, complemented, Sylow
295C2—24no—
296C2—48no—
297C2—122no—
298C2C2×#93811yescharacteristic
299C2D4×#6712yes—
300C2#580511yescharacteristic
301C2#580612yes—
302C2C2×#95311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC23×C3×D4C2
hypercenterC23×D4D3
hypocenterC3#2408
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3×D4C2semidirect
2. C2×#953C2semidirect
3. C2×#938C2semidirect
4. C2×C4×#67C2semidirect
5. #5247C2semidirect
6. #5223C2semidirect
7. #5222C2semidirect
8. #5221C2semidirect
9. #5212C2semidirect
18. C22×#67C22semidirect
19. C22×C4×D3C22semidirect
23. C2×#217C22semidirect
24. C2×#217C22semidirect
25. C2×#217C22semidirect
28. C2×(C2×D3)⋊C4C22semidirect
29. C2×C4⋊Dic3C22semidirect
30. C2×Dic3⋊C4C22semidirect
31. C2×C4×Dic3C22semidirect
32. C3⋊#95C22semidirect
33. C23×D4D3semidirect
55. C2×C3×D4D4semidirect
57. C22×C3×C4D4semidirect
65. C2×#67D4semidirect
125. C22×D3C2×D4semidirect
144. C2×Dic3C2×D4semidirect
165. C2×D4#67semidirect
173. C22×C4#67semidirect
197. C22×C3#121semidirect
209. C3×C4#120semidirect
282. C22#938semidirect
293. C4#953semidirect
294. C3#2408semidirect

Maximal subgroups

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

subgroupindex
C23×C3×D42
C2×#9532
C2×#9382
C2×C4×#672
#52472
#52232
#52222
#52212
#52122
#24083