Finite Group Databasebeta

← #80415

Subgroups — #80415

Order 384.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#5732C211yesmaximal, characteristic, complemented
2#5543C211yesmaximal, characteristic, complemented
3C3×#461C211yesmaximal, characteristic, complemented
4#5165C212yesmaximal
5Dic3×SD16C212yesmaximal, complemented
6#5079C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic
8#4842C211yesmaximal, characteristic, complemented
9#4830C211yesmaximal, characteristic, complemented
10#4800C212yesmaximal, complemented
11#4799C212yesmaximal, complemented
12#3360—31nomaximal, Sylow
13C2×C3×SD16C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16#947C2212yes—
17Q8×Dic3C2212yes—
18D4×Dic3C2212yes—
19C2×#213C2211yescharacteristic
20C2×C4⋊Dic3—21no—
21#924—21no—
22#898—22no—
23#897—21no—
24#892—21no—
25#890C2212yes—
26#878—22no—
27C4×D12C2211yescharacteristic
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×C4⋊C8C2211yescharacteristic
31C3×#103C2212yes—
32#835C2212yes—
33#835C2212yes—
34#832C2211yescharacteristic
35#821C2212yes—
36#817C2212yes—
37Dic3⋊C8C2212yes—
38C8×Dic3C2212yes—
39#809C2211yescharacteristic
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43#554—31no—
44#488—32no—
45#487—31no—
46#479—32no—
47#476—31no—
48#463—32no—
49#461D311yescharacteristic, complemented
50C8×Q8—31no—
51C4×SD16—32no—
52C4×D8—31no—
53C2×C3×Q8C2312yes—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C22×Dic3—22no—
57C2×D12C2311yescharacteristic
58C2×C4×D3—21no—
59C2×Dic6—21no—
60C2×Dic6D411yescharacteristic
61C3×SD16—24no—
62C2×C3×C8C2312yes—
63C3×C4⋊C4—21no—
64C3×C4⋊C4—21no—
65C3×C4⋊C4C2311yescharacteristic
66C3×C22⋊C4—21no—
67C3×C42C2311yescharacteristic
68#217—22no—
69#213—22no—
70(C2×D3)⋊C4—21no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73C4⋊Dic3—21no—
74C4⋊Dic3—22no—
75C4⋊Dic3—22no—
76C4⋊Dic3D411yescharacteristic
77C4⋊Dic3C2311yescharacteristic
78C4⋊Dic3C2312yes—
79Dic3⋊C4—22no—
80Dic3⋊C4D412yes—
81C4×Dic3—22no—
82C4×Dic3C2312yes—
83C2×C3⋊C8C2311yescharacteristic
84C2×C3⋊C8C2311yescharacteristic
85C2×SD16C2×D312yes—
86C2×D8—31no—
87C4⋊Q8—31no—
88C4⋊Q8C2×D311yescharacteristic
89#126—62no—
90#125—32no—
91#124—32no—
92C22⋊Q8—62no—
93C22⋊Q8—61no—
94C22⋊Q8—61no—
95C4×Q8—31no—
96C4×Q8—32no—
97C4×D4—32no—
98C4×D4—31no—
99C4×D4C2×D311yescharacteristic
100#117—61no—
101C2×C4⋊C4—61no—
102#107—31no—
103#106—32no—
104C4⋊C8—32no—
105C4⋊C8C2×D311yescharacteristic
106#103—32no—
107#103—32no—
108#103C2×D312yes—
109#102—31no—
110#102—32no—
111#102—31no—
112C4×C8—32no—
113C4×C8—31no—
114C23×C3—21no—
115C22×D3—21no—
116C3×Q8—22no—
117C3×Q8—22no—
118C3×D4—21no—
119C3×D4#4112yes—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—42no—
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2411yescharacteristic
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—82no—
129C2×Dic3C2×D412yes—
130C2×Dic3C2×D412yes—
131D12—21no—
132D12—21no—
133C4×D3—42no—
134Dic6—22no—
135Dic6—81no—
136C3×C8—22no—
137C3⋊C8—21no—
138C3⋊C8—21no—
139C2×Q8—31no—
140C2×Q8—61no—
141C2×Q8C22×D312yes—
142C2×D4—31no—
143C2×D4C22×D311yescharacteristic
144C22×C4—21no—
145C22×C4—62no—
146C22×C4—61no—
147SD16—24no—
148D8—62no—
149C2×C8—31no—
150C2×C8—31no—
151C2×C8C22×D312yes—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—31no—
155C4⋊C4—32no—
156C4⋊C4—32no—
157C4⋊C4—31no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—61no—
162C4⋊C4C22×D311yescharacteristic
163C22⋊C4—21no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C22⋊C4—61no—
168C42—32no—
169C42—62no—
170C42C22×D311yescharacteristic
171C22×C3—22no—
172C22×C3—41no—
173C22×C3—41no—
174C22×C3C22×D411yescharacteristic
175C2×D3—41no—
176C2×D3—41no—
177C2×D3—41no—
178C3×C4—21no—
179C3×C4—42no—
180C3×C4—41no—
181C3×C4#13512yes—
182C3×C4C2×#4111yescharacteristic
183C3×C42−1+411yescharacteristic
184Dic3—22no—
185Dic3—22no—
186Dic3—41no—
187Dic3—42no—
188C23—21no—
189C23—61no—
190Q8—22no—
191Q8—22no—
192Q8—62no—
193Q8—241no—
194D4—21no—
195D4—61no—
196D4—61no—
197D4#23712yes—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—42no—
203C2×C4—61no—
204C2×C4—62no—
205C2×C4—122no—
206C2×C4—242no—
207C2×C4D3×D411yescharacteristic
208C2×C4D3×D411yescharacteristic
209C2×C4C23×D311yescharacteristic
210C8—22no—
211C8—61no—
212C8—61no—
213C2×C3—42no—
214C2×C3#54811yescharacteristic
215C2×C3C2×#13511yescharacteristic
216C2×C3#57711yescharacteristic
217D3—81no—
218C22—22no—
219C22—41no—
220C22—41no—
221C22—121no—
222C22—121no—
223C22—121no—
224C22C2×D3×D411yescharacteristic
225C4—21no—
226C4—41no—
227C4—42no—
228C4—62no—
229C4—62no—
230C4—121no—
231C4—122no—
232C4#91612yes—
233C4C2×#23711yescharacteristic
234C4#101011yescharacteristic
235C3#336011yescharacteristic, complemented, Sylow
236C2—42no—
237C2—241no—
238C2#559811yescharacteristic
239C2C2×#91611yescharacteristic
240C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radicalC2×#213C22
hypercenter#461D3
hypocenterC3#3360
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5732C2semidirect
2. #5543C2semidirect
3. C3×#461C2semidirect
5. Dic3×SD16C2semidirect
6. #5079C2semidirect
8. #4842C2semidirect
9. #4830C2semidirect
10. #4800C2semidirect
11. #4799C2semidirect
49. #461D3semidirect
235. C3#3360semidirect

Maximal subgroups

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

subgroupindex
#57322
#55432
C3×#4612
#51652
Dic3×SD162
#50792
C4×#2132
#48422
#48302
#48002
#47992
#33603