Finite Group Databasebeta

← #81053

Subgroups — #81053

Order 384.

370 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 133 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#5722C211yesmaximal, characteristic, complemented
2D4×Dic6C211yesmaximal, characteristic
3C3×#476C211yesmaximal, characteristic
4#5080C211yesmaximal, characteristic, complemented
5C4×#214C211yesmaximal, characteristic
6#4855C212yesmaximal, complemented
7#4854C212yesmaximal, complemented
8#4851C211yesmaximal, characteristic, complemented
9#4830C211yesmaximal, characteristic, complemented
10#4761C212yesmaximal
11#4758C212yesmaximal
12Q8×SD16—31nomaximal, Sylow
13C22×Dic6—21no—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16#944—22no—
17D4×Dic3C2212yes—
18C2×#214C2211yescharacteristic
19#924—21no—
20#888—21no—
21#888C2212yes—
22#888C2212yes—
23#887—21no—
24#887C2212yes—
25#878—22no—
26#869C2211yescharacteristic
27C4×Dic6C2211yescharacteristic
28C4×Dic6C2211yescharacteristic
29C3×#106C2212yes—
30C3×C4⋊C8C2211yescharacteristic
31C3×#102C2212yes—
32#832C2211yescharacteristic
33#817C2212yes—
34#816C2212yes—
35Dic3⋊C8C2212yes—
36C8×Dic3C2212yes—
37#810C2211yescharacteristic
38#808C2212yes—
39#808C2212yes—
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42Q82—31no—
43D4×Q8—31no—
44#498—31no—
45#498—32no—
46#476—32no—
47#476D311yescharacteristic
48#475—32no—
49#475—31no—
50C8×Q8—31no—
51C4×SD16—31no—
52C4×SD16—32no—
53C2×C3×Q8—21no—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C22×Dic3—22no—
57C2×Dic6—21no—
58C2×Dic6—22no—
59C2×Dic6—22no—
60C2×Dic6—44no—
61C2×Dic6D411yescharacteristic
62C2×Dic6C2311yescharacteristic
63C2×C3×C8C2312yes—
64C3×C4⋊C4—21no—
65C3×C4⋊C4C2312yes—
66C3×C4⋊C4C2311yescharacteristic
67C3×C22⋊C4—21no—
68C3×C42C2311yescharacteristic
69#217—22no—
70#214Q814yescomplemented
71C4⋊Dic3—21no—
72C4⋊Dic3D411yescharacteristic
73C4⋊Dic3C2312yes—
74C4⋊Dic3C2311yescharacteristic
75Dic3⋊C4—21no—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4—22no—
79Dic3⋊C4D412yes—
80C4×Dic3—21no—
81C4×Dic3—22no—
82C4×Dic3C2312yes—
83C2×C3⋊C8C2311yescharacteristic
84C2×C3⋊C8C2311yescharacteristic
85C22×Q8—61no—
86C2×SD16—31no—
87C4⋊Q8—32no—
88C4⋊Q8—32no—
89C4⋊Q8—31no—
90C4⋊Q8—61no—
91C4⋊Q8—62no—
92C4⋊Q8C2×D311yescharacteristic
93C22⋊Q8—62no—
94C22⋊Q8—61no—
95C4×Q8—31no—
96C4×Q8—32no—
97C4×Q8—31no—
98C4×Q8—61no—
99C4×D4—32no—
100C4×D4C2×D311yescharacteristic
101#106—32no—
102#106—31no—
103#106—32no—
104#106—32no—
105#106C2×D312yes—
106C4⋊C8—32no—
107C4⋊C8C2×D311yescharacteristic
108#103—32no—
109#103—31no—
110#102—31no—
111#102C2×D312yes—
112C4×C8—32no—
113C4×C8—31no—
114C23×C3—21no—
115C3×Q8—81no—
116C3×D4—21no—
117C3×D4C2×Q812yes—
118C2×C3×C4—22no—
119C2×C3×C4—21no—
120C2×C3×C4—42no—
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2411yescharacteristic
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3—44no—
128C2×Dic3C2×D412yes—
129C2×Dic3C2×D412yes—
130Dic6—21no—
131Dic6—42no—
132Dic6—44no—
133Dic6—82no—
134Dic6SD1614yescomplemented
135Dic6C2×Q812yescomplemented
136C3×C8—22no—
137C3⋊C8—21no—
138C3⋊C8C2×Q812yescomplemented
139C2×Q8—21no—
140C2×Q8—31no—
141C2×Q8—31no—
142C2×Q8—62no—
143C2×Q8—62no—
144C2×Q8—61no—
145C2×Q8—124no—
146C2×D4C22×D311yescharacteristic
147C22×C4—21no—
148C22×C4—62no—
149SD16—34no—
150C2×C8—31no—
151C2×C8—31no—
152C2×C8C22×D312yes—
153C4⋊C4—21no—
154C4⋊C4—32no—
155C4⋊C4—31no—
156C4⋊C4—31no—
157C4⋊C4—32no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—61no—
162C4⋊C4—61no—
163C4⋊C4C22×D312yes—
164C4⋊C4C22×D311yescharacteristic
165C22⋊C4—21no—
166C22⋊C4—62no—
167C42—32no—
168C42—62no—
169C42—61no—
170C42C22×D311yescharacteristic
171C22×C3—22no—
172C22×C3—41no—
173C22×C3—41no—
174C22×C3C22×D411yescharacteristic
175C3×C4—21no—
176C3×C4—41no—
177C3×C4—42no—
178C3×C4C2×SD1612yes—
179C3×C4C22×Q811yescharacteristic
180C3×C42−1+411yescharacteristic
181Dic3—22no—
182Dic3—22no—
183Dic3—42no—
184Dic3—41no—
185Dic3C2×SD1614yes—
186C23—21no—
187Q8—34no—
188Q8—32no—
189Q8—61no—
190Q8—81no—
191Q8—124no—
192Q8—122no—
193Q8—242no—
194D4—21no—
195D4D3×Q812yes—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—32no—
199C2×C4—32no—
200C2×C4—42no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—62no—
204C2×C4—124no—
205C2×C4D3×D411yescharacteristic
206C2×C4D3×D411yescharacteristic
207C2×C4C23×D311yescharacteristic
208C8—22no—
209C8—32no—
210C8—61no—
211C2×C3—42no—
212C2×C3D4×Q811yescharacteristic
213C2×C3C22×SD1611yescharacteristic
214C2×C3#57711yescharacteristic
215C22—22no—
216C22—41no—
217C22—41no—
218C22C2×D3×D411yescharacteristic
219C4—21no—
220C4—34no—
221C4—41no—
222C4—42no—
223C4—62no—
224C4—62no—
225C4—122no—
226C4—121no—
227C4D3×SD1612yes—
228C4C2×D3×Q811yescharacteristic
229C4#101011yescharacteristic
230C3Q8×SD1611yescharacteristic, complemented, Sylow
231C2—42no—
232C2#563211yescharacteristic
233C2C2×D3×SD1611yescharacteristic
234C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#476C2
socleC22×C3C22×D4
prime radicalC2×C3×D4C23
hypercenter#476D3
hypocenterC3Q8×SD16
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #5722C2semidirect
4. #5080C2semidirect
6. #4855C2semidirect
7. #4854C2semidirect
8. #4851C2semidirect
9. #4830C2semidirect
70. #214Q8semidirect
134. Dic6SD16semidirect
135. Dic6C2×Q8semidirect
138. C3⋊C8C2×Q8semidirect
230. C3Q8×SD16semidirect

Maximal subgroups

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

subgroupindex
#57222
D4×Dic62
C3×#4762
#50802
C4×#2142
#48552
#48542
#48512
#48302
#47612
#47582
Q8×SD163