Finite Group Databasebeta

← #74769

Subgroups — #74769

Order 384.

584 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic3C211yesmaximal, characteristic
2C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C3×#386C211yesmaximal, characteristic
4#5220C211yesmaximal, characteristic
5#5210C211yesmaximal, characteristic, complemented
6#4950C212yesmaximal, complemented
7#4950C211yesmaximal, characteristic, complemented
8#4948C212yesmaximal, complemented
9#4942C211yesmaximal, characteristic, complemented
10#4661C212yesmaximal, complemented
11#4655C212yesmaximal, complemented
12#2579—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C23×Dic3C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2212yes—
24D4×Dic3—24no—
25C2×C4⋊Dic3—24no—
26C2×C4⋊Dic3C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2212yes—
29C2×C4⋊Dic3C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2211yescharacteristic
33C3×#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×D4—31no—
39C22×C4⋊C4—31no—
40#398—32no—
41#396—32no—
42#387—32no—
43#386—31no—
44#386D311yescharacteristic
45#385—31no—
46#385—32no—
47#385—31no—
48#380—31no—
49C24×C3C2311yescharacteristic
50C24×C3C2311yescharacteristic
51C2×C3×D4—22no—
52C2×C3×D4—22no—
53C2×C3×D4Q814yes—
54C22×C3×C4—22no—
55C22×C3×C4—24no—
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2312yes—
59C22×Dic3—22no—
60C22×Dic3—22no—
61C22×Dic3—42no—
62C22×Dic3—42no—
63C22×Dic3—42no—
64C22×Dic3D414yes—
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2312yes—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2311yescharacteristic
71C3×C22⋊C4—22no—
72C3×C22⋊C4—24no—
73#217—22no—
74#217—22no—
75#217—22no—
76#217—42no—
77C4⋊Dic3—24no—
78C4⋊Dic3—24no—
79C4⋊Dic3—22no—
80C4⋊Dic3—42no—
81Dic3⋊C4—24no—
82Dic3⋊C4—42no—
83C4×Dic3—22no—
84C22×D4C2×D311yescharacteristic
85C23×C4—31no—
86C23×C4—31no—
87C23×C4—31no—
88C23×C4C2×D311yescharacteristic
89C4×D4—64no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—64no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4C2×D311yescharacteristic
102C2×C22⋊C4C2×D312yes—
103C2×C42—31no—
104#95—32no—
105#95—32no—
106#95—32no—
107#95—32no—
108#95C2×D312yes—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—21no—
112C23×C3—22no—
113C23×C3—21no—
114C23×C3—22no—
115C23×C3C2×D412yes—
116C23×C3C2×Q812yes—
117C23×C3C2×Q812yes—
118C23×C3#4112yes—
119C23×C3C2411yescharacteristic
120C3×D4—28no—
121C2×C3×C4—28no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4C2×Q812yes—
130C2×C3×C4#4114yes—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—44no—
144C2×Dic3—44no—
145C2×Dic3—44no—
146C2×Dic3C2×D412yes—
147C2×Dic3C2×D412yes—
148C24C22×D311yescharacteristic
149C24C22×D311yescharacteristic
150C2×D4—22no—
151C2×D4—22no—
152C2×D4Dic614yes—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—34no—
158C22×C4—32no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—32no—
162C22×C4—62no—
163C22×C4—62no—
164C22×C4—122no—
165C22×C4—122no—
166C22×C4—122no—
167C22×C4C22×D312yes—
168C22×C4C22×D311yescharacteristic
169C22×C4C22×D312yes—
170C4⋊C4—64no—
171C4⋊C4—64no—
172C4⋊C4—62no—
173C4⋊C4—64no—
174C4⋊C4—122no—
175C4⋊C4—122no—
176C22⋊C4—22no—
177C22⋊C4—24no—
178C22⋊C4—62no—
179C22⋊C4—62no—
180C22⋊C4—62no—
181C22⋊C4—122no—
182C42—62no—
183C22×C3—22no—
184C22×C3—22no—
185C22×C3—24no—
186C22×C3—24no—
187C22×C3—22no—
188C22×C3—24no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—42no—
192C22×C3C22⋊Q814yes—
193C22×C3C22×D411yescharacteristic
194C22×C3C22×Q811yescharacteristic
195C22×C3C2×#4111yescharacteristic
196C22×C3C2×#4112yes—
197C22×C3H(5,𝔽2)12yes—
198C3×C4—22no—
199C3×C4—24no—
200C3×C4—42no—
201Dic3—22no—
202Dic3—22no—
203Dic3—41no—
204Dic3—42no—
205Dic3—41no—
206Dic3—42no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—22no—
211C23—21no—
212C23—22no—
213C23C2×Dic612yes—
214C23C2×Dic612yes—
215C23D3×D412yes—
216C23#23712yes—
217C23C23×D311yescharacteristic
218D4—28no—
219C2×C4—28no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—32no—
228C2×C4—32no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—124no—
242C2×C4—124no—
243C2×C4—124no—
244C2×C4C2×Dic612yes—
245C2×C4#23714yes—
246C2×C3—24no—
247C2×C3—42no—
248C2×C3C2×C22⋊Q811yescharacteristic
249C2×C3#52511yescharacteristic
250C2×C3#53411yescharacteristic
251C2×C3#55312yes—
252C2×C3#55412yes—
253C22—22no—
254C22—24no—
255C22—22no—
256C22—24no—
257C22—22no—
258C22—24no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22#87814yes—
263C22C22×Dic611yescharacteristic
264C22C2×D3×D411yescharacteristic
265C22C2×#23711yescharacteristic
266C22C2×#23712yes—
267C22#100912yes—
268C4—22no—
269C4—24no—
270C4—42no—
271C4—62no—
272C4—62no—
273C4—121no—
274C4—122no—
275C4—121no—
276C4—122no—
277C3#257911yescharacteristic, complemented, Sylow
278C2—24no—
279C2—42no—
280C2C2×#87811yescharacteristic
281C2#554312yes—
282C2#558611yescharacteristic
283C2#561812yes—
284C2#565311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC22×C3×D4C22
hypercenter#386D3
hypocenterC3#2579
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic3C2semidirect
5. #5210C2semidirect
6. #4950C2semidirect
7. #4950C2semidirect
8. #4948C2semidirect
9. #4942C2semidirect
10. #4661C2semidirect
11. #4655C2semidirect
277. C3#2579semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
C22×C4⋊Dic32
C3×#3862
#52202
#52102
#49502
#49482
#49422
#46612
#46552
#25793