Finite Group Databasebeta

← #83005

Subgroups — #83005

Order 384.

688 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 137 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×C3×#124C211yesmaximal, characteristic, complemented
2C2×#937C211yesmaximal, characteristic
3#5231C212yesmaximal, complemented
4#5220C211yesmaximal, characteristic, complemented
5#4959C212yesmaximal, complemented
6#4958C212yesmaximal, complemented
7#4955C212yesmaximal, complemented
8#4952C212yesmaximal, complemented
9#4940C212yesmaximal, complemented
10#2684—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C22×C3×Q8C2211yescharacteristic, complemented
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C23×Dic3C2211yescharacteristic, complemented
16C3×#124—24no—
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2212yescomplemented
19C2×C3×C42C2211yescharacteristic, complemented
20C2×#217C2211yescharacteristic
21C2×#217C2212yes—
22C2×#217C2211yescharacteristic, complemented
23#937—24no—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×Dic3⋊C4C2212yescomplemented
30C3⋊#95C2212yescomplemented
31C3⋊#95C2212yescomplemented
32C3⋊#95C2212yescomplemented
33C3⋊#95C2212yescomplemented
34C2×#124D311yescharacteristic, complemented
35C2×#120—31no—
36#397—32no—
37#394—32no—
38#393—32no—
39#388—32no—
40#386—31no—
41#386—32no—
42#379—32no—
43C24×C3C2311yescharacteristic
44C24×C3C2311yescharacteristic
45C23×D3—22no—
46C23×D3—24no—
47C23×D3—41no—
48C23×D3—41no—
49C23×D3—42no—
50C23×D3C2312yes—
51C2×C3×Q8—22no—
52C2×C3×Q8—41no—
53C2×C3×Q8—41no—
54C2×C3×D4—41no—
55C2×C3×D4—41no—
56C2×C3×D4D414yes—
57C22×C3×C4C2312yes—
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—42no—
63C22×Dic3—41no—
64C22×Dic3—42no—
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C22×Dic3C2312yes—
68C3×C22⋊C4—24no—
69C3×C22⋊C4—24no—
70C3×C42—22no—
71#217—22no—
72#217—42no—
73#217—41no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—42no—
77(C2×D3)⋊C4—42no—
78(C2×D3)⋊C4—42no—
79Dic3⋊C4—42no—
80C25—31no—
81C22×Q8C2×D311yescharacteristic, complemented
82C22×D4—32no—
83C22×D4C2×D311yescharacteristic
84C23×C4—31no—
85#124—24no—
86#120—64no—
87C2×C4⋊C4—32no—
88C2×C22⋊C4—31no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—31no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4C2×D312yes—
97C2×C22⋊C4C2×D312yescomplemented
98C2×C42C2×D311yescharacteristic, complemented
99#95—32no—
100#95—32no—
101#95—32no—
102#95—32no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—21no—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3—21no—
109C23×C3—21no—
110C23×C3—21no—
111C23×C3C2×D412yes—
112C23×C3C2×D412yes—
113C23×C3C2411yescharacteristic
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—44no—
121C22×D3—48no—
122C22×D3—48no—
123C22×D3—44no—
124C22×D3—44no—
125C22×D3#4114yes—
126C3×Q8—44no—
127C3×D4—44no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4C2×D412yes—
137#67—48no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—44no—
148C24—32no—
149C24—64no—
150C24—62no—
151C24—121no—
152C24—121no—
153C24—122no—
154C24C22×D311yescharacteristic
155C24C22×D311yescharacteristic
156C2×Q8—22no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—64no—
162C2×D4—64no—
163C2×D4—122no—
164C2×D4#6714yes—
165C22×C4—32no—
166C22×C4—32no—
167C22×C4—32no—
168C22×C4—121no—
169C22×C4—122no—
170C22×C4C22×D312yes—
171C22×C4C22×D312yes—
172C22×C4C22×D311yescharacteristic
173C4⋊C4—122no—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—64no—
177C22⋊C4—62no—
178C22⋊C4—64no—
179C22⋊C4—122no—
180C22⋊C4—122no—
181C22⋊C4—122no—
182C22⋊C4—121no—
183C22⋊C4—122no—
184C42—22no—
185C22×C3—22no—
186C22×C3—22no—
187C22×C3—24no—
188C22×C3—24no—
189C22×C3—42no—
190C22×C3—42no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3C22×D411yescharacteristic
196C22×C3C2×#4112yes—
197C22×C3H(5,𝔽2)11yescharacteristic
198C22×C3H(5,𝔽2)11yescharacteristic
199C22×C3H(5,𝔽2)12yes—
200C2×D3—24no—
201C2×D3—24no—
202C2×D3—24no—
203C2×D3—416no—
204C2×D3—42no—
205C2×D3—42no—
206C2×D3—42no—
207C2×D3—42no—
208C3×C4—22no—
209C3×C4—42no—
210C3×C4—42no—
211Dic3—42no—
212Dic3—42no—
213Dic3—42no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—34no—
223C23—62no—
224C23—62no—
225C23—62no—
226C23—62no—
227C23—62no—
228C23—62no—
229C23—128no—
230C23—124no—
231C23—124no—
232C23—128no—
233C23—124no—
234C23C2×#6712yes—
235C23C2×#6712yes—
236C23C23×D311yescharacteristic
237Q8—44no—
238D4—44no—
239D4—128no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—124no—
258C2×C4C2×#6712yes—
259C2×C3—42no—
260C2×C3—42no—
261C2×C3#53411yescharacteristic
262C2×C3#53812yes—
263C2×C3#54612yes—
264C2×C3#55111yescharacteristic
265C2×C3H(3,𝔽4)11yescharacteristic
266D3—44no—
267C22—22no—
268C22—22no—
269C22—24no—
270C22—24no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—64no—
278C22—64no—
279C22—64no—
280C22—1216no—
281C22—122no—
282C22—122no—
283C22—122no—
284C22—122no—
285C22#100411yescharacteristic
286C22#100411yescharacteristic
287C22D3×#4112yes—
288C22#100912yes—
289C22C22×#6711yescharacteristic
290C4—22no—
291C4—42no—
292C4—42no—
293C4—122no—
294C4—122no—
295C4—122no—
296C3#268411yescharacteristic, complemented, Sylow
297C2—42no—
298C2—42no—
299C2—124no—
300C2#567812yes—
301C2#567911yescharacteristic
302C2#568311yescharacteristic
303C2#580411yescharacteristic
304C2#582912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#124C2
prime radicalC2×#937C2
hypercenterC2×#124D3
hypocenterC3#2684
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×#124C2semidirect
3. #5231C2semidirect
4. #5220C2semidirect
5. #4959C2semidirect
6. #4958C2semidirect
7. #4955C2semidirect
8. #4952C2semidirect
9. #4940C2semidirect
12. C22×C3×Q8C22semidirect
15. C23×Dic3C22semidirect
18. C2×C3×C22⋊C4C22semidirect
19. C2×C3×C42C22semidirect
22. C2×#217C22semidirect
29. C2×Dic3⋊C4C22semidirect
30. C3⋊#95C22semidirect
31. C3⋊#95C22semidirect
32. C3⋊#95C22semidirect
33. C3⋊#95C22semidirect
34. C2×#124D3semidirect
81. C22×Q8C2×D3semidirect
97. C2×C22⋊C4C2×D3semidirect
98. C2×C42C2×D3semidirect
296. C3#2684semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C2×#9372
#52312
#52202
#49592
#49582
#49552
#49522
#49402
#26843