Finite Group Databasebeta

← #80414

Subgroups — #80414

Order 384.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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#5721C211yesmaximal, characteristic, complemented
2D4×Dic6C211yesmaximal, characteristic
3C3×#461C211yesmaximal, characteristic
4#5162C212yesmaximal
5Dic3×SD16C212yesmaximal
6#5079C211yesmaximal, characteristic, complemented
7C4×#214C211yesmaximal, characteristic
8#4851C211yesmaximal, characteristic, complemented
9#4830C211yesmaximal, characteristic, complemented
10#4792C212yesmaximal, complemented
11#4791C212yesmaximal, complemented
12#3359—31nomaximal, Sylow
13C22×Dic6—21no—
14C2×C3×SD16C2212yes—
15C3×C4⋊Q8C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17Q8×Dic3C2212yes—
18#944C2212yes—
19D4×Dic3C2212yes—
20C2×#214C2211yescharacteristic
21#924—21no—
22#890—21no—
23#889—22no—
24#888C2212yes—
25#887—21no—
26#878—22no—
27#869C2211yescharacteristic
28C4×Dic6C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×C4⋊C8C2211yescharacteristic
31C3×#103C2212yes—
32#835C2212yes—
33#835C2212yes—
34#832C2211yescharacteristic
35#817C2212yes—
36#816C2212yes—
37Dic3⋊C8C2212yes—
38C8×Dic3C2212yes—
39#810C2211yescharacteristic
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#557—31no—
43D4×Q8—31no—
44#487—31no—
45#487—32no—
46#475—31no—
47#475—32no—
48#461—32no—
49#461D311yescharacteristic
50C8×Q8—31no—
51C4×SD16—32no—
52C4×SD16—31no—
53C2×C3×Q8C2312yes—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C22×Dic3—22no—
57C2×Dic6—21no—
58C2×Dic6—22no—
59C2×Dic6—44no—
60C2×Dic6D411yescharacteristic
61C2×Dic6C2311yescharacteristic
62C3×SD16—24no—
63C2×C3×C8C2312yes—
64C3×C4⋊C4—21no—
65C3×C4⋊C4—21no—
66C3×C4⋊C4C2311yescharacteristic
67C3×C22⋊C4—21no—
68C3×C42C2311yescharacteristic
69#217—22no—
70#214—22no—
71C4⋊Dic3—21no—
72C4⋊Dic3—22no—
73C4⋊Dic3D411yescharacteristic
74C4⋊Dic3C2311yescharacteristic
75C4⋊Dic3C2312yes—
76Dic3⋊C4—21no—
77Dic3⋊C4—22no—
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80Dic3⋊C4D412yes—
81C4×Dic3—21no—
82C4×Dic3—22no—
83C4×Dic3C2312yes—
84C2×C3⋊C8C2311yescharacteristic
85C2×C3⋊C8C2311yescharacteristic
86C22×Q8—61no—
87C2×SD16—31no—
88C2×SD16C2×D312yes—
89C4⋊Q8—32no—
90C4⋊Q8—32no—
91C4⋊Q8—31no—
92C4⋊Q8C2×D311yescharacteristic
93#125—62no—
94#125—61no—
95C22⋊Q8—62no—
96C22⋊Q8—61no—
97C4×Q8—31no—
98C4×Q8—32no—
99C4×Q8—31no—
100C4×Q8—61no—
101C4×D4—32no—
102C4×D4C2×D311yescharacteristic
103#106—32no—
104#106—31no—
105C4⋊C8—32no—
106C4⋊C8C2×D311yescharacteristic
107#103—32no—
108#103—32no—
109#103—32no—
110#103—31no—
111#103C2×D312yes—
112#102—31no—
113C4×C8—32no—
114C4×C8—31no—
115C23×C3—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—21no—
128C2×Dic3—22no—
129C2×Dic3—44no—
130C2×Dic3C2×D412yes—
131C2×Dic3C2×D412yes—
132Dic6—21no—
133Dic6—21no—
134Dic6—42no—
135Dic6—44no—
136Dic6SD1614yescomplemented
137C3×C8—22no—
138C3⋊C8—21no—
139C3⋊C8—21no—
140C2×Q8—31no—
141C2×Q8—31no—
142C2×Q8—62no—
143C2×Q8—61no—
144C2×Q8—124no—
145C2×Q8C22×D312yes—
146C2×D4C22×D311yescharacteristic
147C22×C4—21no—
148C22×C4—62no—
149SD16—24no—
150SD16—62no—
151C2×C8—31no—
152C2×C8—31no—
153C2×C8C22×D312yes—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—32no—
157C4⋊C4—31no—
158C4⋊C4—31no—
159C4⋊C4—32no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—61no—
165C4⋊C4—61no—
166C4⋊C4C22×D311yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—62no—
169C42—32no—
170C42—62no—
171C42—61no—
172C42C22×D311yescharacteristic
173C22×C3—22no—
174C22×C3—41no—
175C22×C3—41no—
176C22×C3C22×D411yescharacteristic
177C3×C4—21no—
178C3×C4—42no—
179C3×C4—41no—
180C3×C4C2×SD1612yes—
181C3×C4C2×#4111yescharacteristic
182C3×C42−1+411yescharacteristic
183Dic3—22no—
184Dic3—41no—
185Dic3—42no—
186Dic3—41no—
187Dic3C2×SD1614yes—
188C23—21no—
189Q8—22no—
190Q8—22no—
191Q8—34no—
192Q8—61no—
193Q8—61no—
194Q8—124no—
195Q8—122no—
196D4—21no—
197D4#23712yes—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—42no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—62no—
206C2×C4—124no—
207C2×C4D3×D411yescharacteristic
208C2×C4D3×D411yescharacteristic
209C2×C4C23×D311yescharacteristic
210C8—22no—
211C8—61no—
212C8—61no—
213C2×C3—42no—
214C2×C3#54811yescharacteristic
215C2×C3C22×SD1611yescharacteristic
216C2×C3#57811yescharacteristic
217C22—22no—
218C22—41no—
219C22—41no—
220C22C2×D3×D411yescharacteristic
221C4—21no—
222C4—34no—
223C4—41no—
224C4—42no—
225C4—62no—
226C4—122no—
227C4—121no—
228C4—121no—
229C4D3×SD1612yes—
230C4C2×#23711yescharacteristic
231C4#101011yescharacteristic
232C3#335911yescharacteristic, complemented, Sylow
233C2—42no—
234C2#559811yescharacteristic
235C2C2×D3×SD1611yescharacteristic
236C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radicalC2×C3×D4C23
hypercenter#461D3
hypocenterC3#3359
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. #5721C2semidirect
6. #5079C2semidirect
8. #4851C2semidirect
9. #4830C2semidirect
10. #4792C2semidirect
11. #4791C2semidirect
136. Dic6SD16semidirect
232. C3#3359semidirect

Maximal subgroups

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

subgroupindex
#57212
D4×Dic62
C3×#4612
#51622
Dic3×SD162
#50792
C4×#2142
#48512
#48302
#47922
#47912
#33593