Finite Group Databasebeta

← #79568

Subgroups — #79568

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#5543C211yesmaximal, characteristic, complemented
2#5542C211yesmaximal, characteristic, complemented
3C3×C4×D8C211yesmaximal, characteristic
4#5153C212yesmaximal
5#5019C211yesmaximal, characteristic, complemented
6#5008C211yesmaximal, characteristic, complemented
7#4877C212yesmaximal, complemented
8#4768C212yesmaximal, complemented
9#4758C212yesmaximal, complemented
10#4681C211yesmaximal, characteristic, complemented
11C8×Dic6C211yesmaximal, characteristic, complemented
12#3360—31nomaximal, Sylow
13C2×C3×D8C2211yescharacteristic
14C3×C4×D4C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16#936C2212yes—
17D4×Dic3C2212yes—
18C2×#214C2212yes—
19#926—21no—
20C2×C4⋊Dic3—21no—
21#924—21no—
22#924—21no—
23#890C2212yes—
24#887C2212yes—
25#879—22no—
26#878—22no—
27#869C2211yescharacteristic
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×#107C2211yescharacteristic
31C3×#102C2211yescharacteristic
32C3×#102C2211yescharacteristic
33C3×C4×C8C2211yescharacteristic
34#832C2212yes—
35#816C2212yes—
36#816C2212yes—
37Dic3⋊C8C2212yes—
38C8×Dic3C2212yes—
39#810C2212yes—
40#808C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42#556—31no—
43#554—31no—
44#488—32no—
45#487—31no—
46#479—32no—
47#476—31no—
48#463—32no—
49#461—31no—
50C8×Q8—31no—
51C4×SD16—32no—
52C4×D8D311yescharacteristic
53C2×C3×D4C2311yescharacteristic
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C22×C3×C4—21no—
57C22×Dic3—22no—
58C2×Dic6—21no—
59C2×Dic6D411yescharacteristic
60C2×Dic6C2312yes—
61C3×D8—22no—
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2311yescharacteristic
64C3×C4⋊C4C2311yescharacteristic
65C3×C4⋊C4C2311yescharacteristic
66C3×C22⋊C4—21no—
67C3×C22⋊C4—21no—
68C3×C42C2311yescharacteristic
69#217—22no—
70#217—22no—
71#217—22no—
72#214—24no—
73C4⋊Dic3—21no—
74C4⋊Dic3—21no—
75C4⋊Dic3—21no—
76C4⋊Dic3—22no—
77C4⋊Dic3D411yescharacteristic
78C4⋊Dic3C2312yes—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81Dic3⋊C4D412yes—
82C4×Dic3—22no—
83C4×Dic3C2312yes—
84C2×C3⋊C8C2312yes—
85C2×SD16—32no—
86C2×D8C2×D311yescharacteristic
87C4⋊Q8—31no—
88C4⋊Q8—31no—
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×D4C2×D311yescharacteristic
99C4×D4C2×D311yescharacteristic
100#117—61no—
101C2×C4⋊C4—61no—
102#107C2×D311yescharacteristic
103#106—32no—
104C4⋊C8—32no—
105C4⋊C8—31no—
106#103—32no—
107#103—32no—
108#103—32no—
109#102—32no—
110#102C2×D311yescharacteristic
111#102C2×D311yescharacteristic
112C4×C8—32no—
113C4×C8C2×D311yescharacteristic
114C23×C3—21no—
115C23×C3—21no—
116C3×D4—21no—
117C3×D4—21no—
118C3×D4—21no—
119C3×D4#4112yes—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—42no—
123C2×C3×C4—42no—
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2411yescharacteristic
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—82no—
130C2×Dic3C2×D412yes—
131C2×Dic3C2×D412yes—
132Dic6—22no—
133Dic6—22no—
134Dic6—22no—
135Dic6—81no—
136C3×C8—21no—
137C3×C8—21no—
138C3⋊C8—22no—
139C2×Q8—31no—
140C2×Q8—32no—
141C2×Q8—61no—
142C2×D4C22×D311yescharacteristic
143C2×D4C22×D311yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—62no—
147SD16—64no—
148D8—22no—
149C2×C8—32no—
150C2×C8C22×D311yescharacteristic
151C2×C8C22×D311yescharacteristic
152C4⋊C4—31no—
153C4⋊C4—32no—
154C4⋊C4—32no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—61no—
159C4⋊C4—61no—
160C4⋊C4—61no—
161C4⋊C4C22×D311yescharacteristic
162C4⋊C4C22×D311yescharacteristic
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C22⋊C4—62no—
168C42—32no—
169C42—62no—
170C42C22×D311yescharacteristic
171C22×C3—22no—
172C22×C3—41no—
173C22×C3—41no—
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—41no—
177C22×C3C22×D411yescharacteristic
178C3×C4—21no—
179C3×C4—41no—
180C3×C4—41no—
181C3×C4#13512yes—
182C3×C4C2×#4111yescharacteristic
183C3×C42−1+411yescharacteristic
184Dic3—22no—
185Dic3—22no—
186Dic3—42no—
187Dic3—42no—
188C23—21no—
189C23—21no—
190Q8—62no—
191Q8—62no—
192Q8—62no—
193Q8—241no—
194D4—21no—
195D4—21no—
196D4—21no—
197D4#23512yes—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—242no—
207C2×C4D3×D411yescharacteristic
208C2×C4D3×D411yescharacteristic
209C2×C4C23×D311yescharacteristic
210C8—21no—
211C8—21no—
212C8—62no—
213C2×C3—42no—
214C2×C3—81no—
215C2×C3#54811yescharacteristic
216C2×C3C2×#13511yescharacteristic
217C2×C3#57711yescharacteristic
218C22—22no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22C2×D3×D411yescharacteristic
225C4—21no—
226C4—41no—
227C4—41no—
228C4—62no—
229C4—62no—
230C4—122no—
231C4—122no—
232C4#91212yes—
233C4C2×#23511yescharacteristic
234C4#101011yescharacteristic
235C3#336011yescharacteristic, complemented, Sylow
236C2—42no—
237C2—81no—
238C2#555211yescharacteristic
239C2C2×#91211yescharacteristic
240C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
prime radicalC2×C3×D8C22
hypercenterC4×D8D3
hypocenterC3#3360
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #5543C2semidirect
2. #5542C2semidirect
5. #5019C2semidirect
6. #5008C2semidirect
7. #4877C2semidirect
8. #4768C2semidirect
9. #4758C2semidirect
10. #4681C2semidirect
11. C8×Dic6C2semidirect
235. C3#3360semidirect

Maximal subgroups

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

subgroupindex
#55432
#55422
C3×C4×D82
#51532
#50192
#50082
#48772
#47682
#47582
#46812
C8×Dic62
#33603