Finite Group Databasebeta

← #84047

Subgroups — #84047

Order 384.

1116 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 197 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
2C22×#217C211yesmaximal, characteristic, complemented
3C2×#935C212yesmaximal, complemented
4#5247C212yesmaximal, complemented
5#5247C211yesmaximal, characteristic, complemented
6#5222C214yesmaximal, complemented
7#5220C212yesmaximal, complemented
8#5210C212yesmaximal, complemented
9#2405—31nomaximal, Sylow
10C25×C3C2211yescharacteristic
11C25×C3C2211yescharacteristic
12C22×C3×D4—24no—
13C22×C3×D4—28no—
14C22×C3×D4C2212yes—
15C22×C3×D4C2212yes—
16C23×C3×C4C2211yescharacteristic
17C23×Dic3C2211yescharacteristic, complemented
18C23×Dic3C2211yescharacteristic, complemented
19C2×#217—24no—
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yescomplemented
23C2×#217C2212yescomplemented
24C2×#217C2214yescomplemented
25C2×#217C2214yescomplemented
26C2×#217C2212yescomplemented
27#935—28no—
28C2×Dic3⋊C4C2214yescomplemented
29C3⋊#95C2214yescomplemented
30C23×D4D311yescharacteristic
31C2×#123—32no—
32C22×C22⋊C4—31no—
33#394—34no—
34#386—32no—
35#385—32no—
36#379—32no—
37#379—31no—
38C24×C3—24no—
39C24×C3—24no—
40C24×C3—22no—
41C24×C3—24no—
42C24×C3—24no—
43C24×C3—42no—
44C24×C3—42no—
45C24×C3D414yes—
46C24×C3C2311yescharacteristic
47C24×C3C2312yes—
48C24×C3C2311yescharacteristic
49C24×C3C2311yescharacteristic
50C2×C3×D4—232no—
51C2×C3×D4—24no—
52C2×C3×D4—24no—
53C2×C3×D4—24no—
54C2×C3×D4—24no—
55C2×C3×D4—24no—
56C2×C3×D4D418yes—
57C22×C3×C4—22no—
58C22×C3×C4—24no—
59C22×C3×C4C2312yes—
60C22×Dic3—24no—
61C22×Dic3—22no—
62C22×Dic3—22no—
63C22×Dic3—42no—
64C22×Dic3C2312yes—
65C22×Dic3C2314yes—
66C22×Dic3C2312yes—
67#217—24no—
68#217—28no—
69#217—24no—
70#217—24no—
71#217—24no—
72#217—42no—
73#217—44no—
74Dic3⋊C4—28no—
75C25C2×D311yescharacteristic
76C25C2×D311yescharacteristic
77C22×D4—24no—
78C22×D4—28no—
79C22×D4C2×D312yes—
80C22×D4C2×D312yes—
81C23×C4—31no—
82C23×C4—31no—
83C23×C4C2×D311yescharacteristic
84#123—68no—
85C2×C4⋊C4—34no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4—34no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—34no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—64no—
94#95—34no—
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—28no—
100C23×C3—28no—
101C23×C3—28no—
102C23×C3—22no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—22no—
106C23×C3—24no—
107C23×C3—22no—
108C23×C3—24no—
109C23×C3—44no—
110C23×C3—416no—
111C23×C3—416no—
112C23×C3—44no—
113C23×C3—44no—
114C23×C3C2×D412yes—
115C23×C3C2×D414yes—
116C23×C3C2×D414yes—
117C23×C3C2×D412yes—
118C23×C3C2×D412yes—
119C23×C3#4114yes—
120C23×C3C2411yescharacteristic
121C3×D4—216no—
122C3×D4—216no—
123C2×C3×C4—28no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4C2×D414yes—
127C2×Dic3—24no—
128C2×Dic3—24no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—24no—
132C2×Dic3—24no—
133C2×Dic3—24no—
134C2×Dic3—44no—
135C2×Dic3—44no—
136C24—24no—
137C24—24no—
138C24—24no—
139C24—22no—
140C24—24no—
141C24—42no—
142C24—42no—
143C24#6714yes—
144C24C22×D311yescharacteristic
145C24C22×D312yes—
146C24C22×D311yescharacteristic
147C24C22×D311yescharacteristic
148C2×D4—232no—
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—24no—
153C2×D4—24no—
154C2×D4#6718yes—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—32no—
158C22×C4—34no—
159C22×C4—32no—
160C22×C4—62no—
161C22×C4—62no—
162C22×C4—64no—
163C22×C4—122no—
164C22×C4C22×D312yes—
165C4⋊C4—68no—
166C22⋊C4—64no—
167C22⋊C4—64no—
168C22⋊C4—64no—
169C22⋊C4—64no—
170C22⋊C4—68no—
171C22⋊C4—122no—
172C22⋊C4—124no—
173C22×C3—22no—
174C22×C3—22no—
175C22×C3—22no—
176C22×C3—22no—
177C22×C3—28no—
178C22×C3—28no—
179C22×C3—22no—
180C22×C3—24no—
181C22×C3—22no—
182C22×C3—22no—
183C22×C3—28no—
184C22×C3—416no—
185C22×C3—416no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—44no—
189C22×C3—44no—
190C22×C3#12014yes—
191C22×C3#12314yescomplemented
192C22×C3C22×D411yescharacteristic
193C22×C3C22×D412yes—
194C22×C3C2×#4112yes—
195C22×C3H(5,𝔽2)12yes—
196C3×C4—24no—
197Dic3—42no—
198Dic3—42no—
199Dic3—44no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—28no—
205C23—28no—
206C23—28no—
207C23—22no—
208C23—24no—
209C23—22no—
210C23—24no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—44no—
215C23—416no—
216C23—416no—
217C23—44no—
218C23—44no—
219C23#23714yes—
220C23C2×#6712yes—
221C23C2×#6714yes—
222C23C2×#6712yes—
223C23C2×#6714yes—
224C23C2×#6712yes—
225C23C23×D311yescharacteristic
226D4—216no—
227D4—216no—
228C2×C4—28no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—64no—
232C2×C4—64no—
233C2×C4—64no—
234C2×C4—64no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—64no—
238C2×C4—124no—
239C2×C4—124no—
240C2×C4C2×#6714yes—
241C2×C3—24no—
242C2×C3—24no—
243C2×C3—42no—
244C2×C3—44no—
245C2×C3C2×#12011yescharacteristic
246C2×C3C2×#12311yescharacteristic
247C2×C3#53411yescharacteristic
248C2×C3#54614yes—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—28no—
253C22—28no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—28no—
258C22—22no—
259C22—24no—
260C22—416no—
261C22—416no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—44no—
266C22#93514yescomplemented
267C22#95314yes—
268C22C2×#23712yes—
269C22#100412yes—
270C22C22×#6711yescharacteristic
271C22C22×#6712yes—
272C4—24no—
273C4—122no—
274C4—124no—
275C4—122no—
276C3#240511yescharacteristic, complemented, Sylow
277C2—24no—
278C2—24no—
279C2—42no—
280C2—44no—
281C2C2×#93511yescharacteristic
282C2#580411yescharacteristic
283C2#580614yes—
284C2C2×#95311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC23×C3×D4C2
hypercenterC23×D4D3
hypocenterC3#2405
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
2. C22×#217C2semidirect
3. C2×#935C2semidirect
4. #5247C2semidirect
5. #5247C2semidirect
6. #5222C2semidirect
7. #5220C2semidirect
8. #5210C2semidirect
17. C23×Dic3C22semidirect
18. C23×Dic3C22semidirect
22. C2×#217C22semidirect
23. C2×#217C22semidirect
24. C2×#217C22semidirect
25. C2×#217C22semidirect
26. C2×#217C22semidirect
28. C2×Dic3⋊C4C22semidirect
29. C3⋊#95C22semidirect
191. C22×C3#123semidirect
266. C22#935semidirect
276. C3#2405semidirect

Maximal subgroups

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

subgroupindex
C23×C3×D42
C22×#2172
C2×#9352
#52472
#52222
#52202
#52102
#24053