Finite Group Databasebeta

← #76469

Subgroups — #76469

Order 384.

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

normal subgroups only

This group has 680 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×#937C211yesmaximal, characteristic
2C2×#935C211yesmaximal, characteristic, complemented
3C3×#390C211yesmaximal, characteristic, complemented
4#4994C212yesmaximal, complemented
5#4961C212yesmaximal, complemented
6#4958C212yesmaximal, complemented
7#4955C212yesmaximal, complemented
8#4953C212yesmaximal, complemented
9#4940C212yesmaximal, complemented
10#2697—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C22×C3×D4C2211yescharacteristic
13C22×#67C2212yes—
14C23×Dic3C2211yescharacteristic, complemented
15C2×C3×C4⋊C4C2211yescharacteristic, complemented
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C22⋊C4C2212yescomplemented
18C2×C3×C42C2211yescharacteristic, complemented
19C2×#217C2211yescharacteristic
20C2×#217C2212yes—
21C2×#217C2211yescharacteristic, complemented
22#937—24no—
23#935—24no—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2212yescomplemented
30C2×Dic3⋊C4C2212yescomplemented
31C3⋊#95C2212yescomplemented
32C3⋊#95C2212yescomplemented
33C3⋊#95C2212yescomplemented
34C2×#123—31no—
35C2×#120—31no—
36#397—32no—
37#396—32no—
38#394—32no—
39#390D311yescharacteristic, complemented
40#388—32no—
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×D4—41no—
52C2×C3×D4—41no—
53C2×C3×D4D414yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C2×#67—24no—
58C2×#67—24no—
59C2×#67—42no—
60C22×Dic3—22no—
61C22×Dic3—41no—
62C22×Dic3—41no—
63C22×Dic3C2312yes—
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C3×C4⋊C4—41no—
67C3×C22⋊C4—42no—
68C3×C22⋊C4—42no—
69C3×C42—41no—
70#217—24no—
71#217—22no—
72#217—22no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—42no—
76(C2×D3)⋊C4—42no—
77(C2×D3)⋊C4—42no—
78C4⋊Dic3—42no—
79Dic3⋊C4—24no—
80C25—31no—
81C22×D4—32no—
82C22×D4C2×D311yescharacteristic
83C23×C4—31no—
84#123—64no—
85#120—64no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4C2×D311yescharacteristic, complemented
89C2×C22⋊C4—31no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—31no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4C2×D312yes—
98C2×C22⋊C4C2×D312yescomplemented
99C2×C42C2×D311yescharacteristic, complemented
100#95—32no—
101#95—32no—
102#95—32no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—22no—
106C23×C3—22no—
107C23×C3—21no—
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×D4—44no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
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—
136#67—48no—
137C2×Dic3—22no—
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—44no—
147C24—32no—
148C24—64no—
149C24—62no—
150C24—122no—
151C24—121no—
152C24—121no—
153C24C22×D311yescharacteristic
154C24C22×D311yescharacteristic
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—64no—
158C2×D4—64no—
159C2×D4—122no—
160C2×D4D1214yes—
161C22×C4—32no—
162C22×C4—32no—
163C22×C4—32no—
164C22×C4—62no—
165C22×C4—121no—
166C22×C4—121no—
167C22×C4C22×D312yes—
168C22×C4C22×D312yes—
169C22×C4C22×D311yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—64no—
172C4⋊C4—122no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—64no—
176C22⋊C4—62no—
177C22⋊C4—62no—
178C22⋊C4—64no—
179C22⋊C4—64no—
180C22⋊C4—122no—
181C22⋊C4—122no—
182C22⋊C4—122no—
183C42—41no—
184C22×C3—22no—
185C22×C3—22no—
186C22×C3—24no—
187C22×C3—24no—
188C22×C3—42no—
189C22×C3—42no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3C22×D411yescharacteristic
195C22×C3C2×#4112yes—
196C22×C3H(5,𝔽2)11yescharacteristic
197C22×C3H(5,𝔽2)11yescharacteristic
198C22×C3H(5,𝔽2)12yes—
199C2×D3—24no—
200C2×D3—24no—
201C2×D3—24no—
202C2×D3—416no—
203C2×D3—42no—
204C2×D3—42no—
205C2×D3—42no—
206C2×D3—42no—
207C3×C4—22no—
208C3×C4—42no—
209C3×C4—42no—
210Dic3—42no—
211Dic3—42no—
212Dic3—42no—
213C23—22no—
214C23—22no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—22no—
221C23—34no—
222C23—62no—
223C23—62no—
224C23—62no—
225C23—62no—
226C23—62no—
227C23—62no—
228C23—128no—
229C23—124no—
230C23—124no—
231C23—128no—
232C23—124no—
233C23C2×D1212yes—
234C23C2×D1212yes—
235C23C23×D311yescharacteristic
236D4—44no—
237D4—128no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—124no—
256C2×C4C2×D1212yes—
257C2×C3—42no—
258C2×C3—42no—
259C2×C3#53411yescharacteristic
260C2×C3#53812yes—
261C2×C3#54612yes—
262C2×C3#55111yescharacteristic
263C2×C3#56011yescharacteristic
264D3—44no—
265C22—22no—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—42no—
275C22—64no—
276C22—64no—
277C22—64no—
278C22—1216no—
279C22—122no—
280C22—122no—
281C22—122no—
282C22—122no—
283C22C22×D1211yescharacteristic
284C22#100411yescharacteristic
285C22#100412yes—
286C22#100411yescharacteristic
287C22D3×#4112yes—
288C4—22no—
289C4—42no—
290C4—42no—
291C4—122no—
292C4—122no—
293C4—122no—
294C3#269711yescharacteristic, complemented, Sylow
295C2—42no—
296C2—42no—
297C2—124no—
298C2#549311yescharacteristic
299C2#555412yes—
300C2#561312yes—
301C2#567911yescharacteristic
302C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC2×#937C2
hypercenter#390D3
hypocenterC3#2697
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#935C2semidirect
3. C3×#390C2semidirect
4. #4994C2semidirect
5. #4961C2semidirect
6. #4958C2semidirect
7. #4955C2semidirect
8. #4953C2semidirect
9. #4940C2semidirect
14. C23×Dic3C22semidirect
15. C2×C3×C4⋊C4C22semidirect
17. C2×C3×C22⋊C4C22semidirect
18. C2×C3×C42C22semidirect
21. C2×#217C22semidirect
29. C2×C4⋊Dic3C22semidirect
30. C2×Dic3⋊C4C22semidirect
31. C3⋊#95C22semidirect
32. C3⋊#95C22semidirect
33. C3⋊#95C22semidirect
39. #390D3semidirect
88. C2×C4⋊C4C2×D3semidirect
98. C2×C22⋊C4C2×D3semidirect
99. C2×C42C2×D3semidirect
294. C3#2697semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#9352
C3×#3902
#49942
#49612
#49582
#49552
#49532
#49402
#26973