Finite Group Databasebeta

← #74245

Subgroups — #74245

Order 384.

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

normal subgroups only

This group has 562 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×#876C211yesmaximal, characteristic, complemented
2C2×C4×D12C211yesmaximal, characteristic, complemented
3C3×#382C211yesmaximal, characteristic, complemented
4#4994C212yesmaximal, complemented
5#4991C212yesmaximal, complemented
6#4677C211yesmaximal, characteristic, complemented
7#4672C211yesmaximal, characteristic, complemented
8#4671C211yesmaximal, characteristic
9#4668C212yesmaximal, complemented
10#4665C212yesmaximal, complemented
11#4659C211yesmaximal, characteristic, complemented
12#2636—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×D12C2212yes—
15C22×C4×D3C2212yes—
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C42C2211yescharacteristic
18C2×(C2×D3)⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2212yes—
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31#876—24no—
32C4×D12—24no—
33C3×#95C2211yescharacteristic
34C3×#95C2212yes—
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#126—31no—
39C2×C4×D4—31no—
40#400—31no—
41#399—31no—
42#396—32no—
43#394—32no—
44#394—31no—
45#392—31no—
46#386—32no—
47#385—32no—
48#382D311yescharacteristic, complemented
49C23×D3C2311yescharacteristic
50C23×D3C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C2×D12—24no—
61C2×D12—22no—
62C2×D12—24no—
63C2×D12—42no—
64C2×D12—42no—
65C2×C4×D3—24no—
66C2×C4×D3—42no—
67C2×C4×D3—42no—
68C3×C4⋊C4—42no—
69C3×C42D414yescomplemented
70(C2×D3)⋊C4—24no—
71(C2×D3)⋊C4—24no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—42no—
76(C2×D3)⋊C4—42no—
77(C2×D3)⋊C4—42no—
78C4⋊Dic3—22no—
79C4⋊Dic3—42no—
80Dic3⋊C4—22no—
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83C22×D4—31no—
84C22×D4—32no—
85C23×C4—32no—
86#126—64no—
87C4×D4—64no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4C2×D312yes—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C42C2×D311yescharacteristic
103#95—32no—
104#95—32no—
105#95C2×D312yes—
106#95C2×D311yescharacteristic
107#95C2×D311yescharacteristic
108C23×C3C2411yescharacteristic
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—22no—
115C22×D3—21no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—21no—
119C22×D3—21no—
120C22×D3#4114yes—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—21no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D412yes—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—21no—
142C2×Dic3—22no—
143C2×Dic3—21no—
144D12—28no—
145D12—48no—
146C4×D3—48no—
147C24—32no—
148C24—31no—
149C2×D4—64no—
150C2×D4—64no—
151C2×D4—62no—
152C2×D4—122no—
153C2×D4—122no—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—32no—
157C22×C4—31no—
158C22×C4—64no—
159C22×C4—122no—
160C22×C4—122no—
161C22×C4C22×D312yes—
162C22×C4C22×D312yes—
163C22×C4C22×D311yescharacteristic
164C22×C4C22×D311yescharacteristic
165C22×C4C22×D311yescharacteristic
166C4⋊C4—42no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—122no—
172C22⋊C4—64no—
173C22⋊C4—64no—
174C22⋊C4—62no—
175C22⋊C4—62no—
176C22⋊C4—62no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C22⋊C4—122no—
180C42D1214yescomplemented
181C22×C3C22×D411yescharacteristic
182C22×C3C2×#4112yes—
183C22×C3H(5,𝔽2)11yescharacteristic
184C22×C3H(5,𝔽2)12yes—
185C22×C32−1+411yescharacteristic
186C2×D3—24no—
187C2×D3—24no—
188C2×D3—24no—
189C2×D3—42no—
190C2×D3—42no—
191C2×D3—42no—
192C2×D3—42no—
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—41no—
196C2×D3—42no—
197C2×D3—42no—
198C3×C4—22no—
199C3×C4—22no—
200C3×C4—22no—
201C3×C4—42no—
202C3×C4—42no—
203Dic3—41no—
204Dic3—42no—
205Dic3—41no—
206Dic3—41no—
207C23—34no—
208C23—62no—
209C23—62no—
210C23—62no—
211C23—62no—
212C23—62no—
213C23—62no—
214C23—61no—
215C23—61no—
216C23—61no—
217C23—62no—
218C23—62no—
219C23C23×D311yescharacteristic
220D4—68no—
221D4—128no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—62no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—128no—
243C2×C4C2×D1212yes—
244C2×C4C2×D1212yes—
245C2×C4C2×D1212yes—
246C2×C3#52511yescharacteristic
247C2×C3#53511yescharacteristic
248C2×C3#53711yescharacteristic
249C2×C3#53812yes—
250C2×C3#53912yes—
251D3—44no—
252D3—81no—
253C22—64no—
254C22—64no—
255C22—64no—
256C22—122no—
257C22—122no—
258C22—122no—
259C22—122no—
260C22—122no—
261C22—122no—
262C22—121no—
263C22—121no—
264C22—121no—
265C22C22×D1211yescharacteristic
266C22#100411yescharacteristic
267C22#100711yescharacteristic
268C22D3×#4112yes—
269C22#100912yes—
270C4—22no—
271C4—22no—
272C4—22no—
273C4—42no—
274C4—42no—
275C4—121no—
276C4—121no—
277C4—122no—
278C4—121no—
279C3#263611yescharacteristic, complemented, Sylow
280C2—124no—
281C2—241no—
282C2#551011yescharacteristic
283C2#552411yescharacteristic
284C2#552511yescharacteristic
285C2#567812yes—
286C2#569212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#382C2
prime radical#4671C2
hypercenter#382D3
hypocenterC3#2636
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C2×#876C2semidirect
2. C2×C4×D12C2semidirect
3. C3×#382C2semidirect
4. #4994C2semidirect
5. #4991C2semidirect
6. #4677C2semidirect
7. #4672C2semidirect
9. #4668C2semidirect
10. #4665C2semidirect
11. #4659C2semidirect
48. #382D3semidirect
69. C3×C42D4semidirect
180. C42D12semidirect
279. C3#2636semidirect

Maximal subgroups

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

subgroupindex
C2×#8762
C2×C4×D122
C3×#3822
#49942
#49912
#46772
#46722
#46712
#46682
#46652
#46592
#26363