Finite Group Databasebeta

← #80417

Subgroups — #80417

Order 384.

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

normal subgroups only

This group has 340 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#5721C211yesmaximal, characteristic, complemented
2#5543C211yesmaximal, characteristic
3C3×#461C211yesmaximal, characteristic
4#5164C212yesmaximal
5#5162C212yesmaximal
6#5079C211yesmaximal, characteristic, complemented
7C4×#214C211yesmaximal, characteristic
8#4850C211yesmaximal, characteristic, complemented
9#4831C211yesmaximal, characteristic, complemented
10#4797C212yesmaximal, complemented
11#4796C212yesmaximal, complemented
12#3372—31nomaximal, Sylow
13C2×C3×SD16C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16Q8×Dic3C2212yes—
17#944C2212yes—
18D4×Dic3C2212yes—
19C2×#214C2211yescharacteristic
20C2×C4⋊Dic3—21no—
21#924—21no—
22#890—21no—
23#890C2212yes—
24#889—22no—
25#887—21no—
26#878—22no—
27#869C2211yescharacteristic
28C4×Dic6C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×C4⋊C8C2211yescharacteristic
31C3×#103C2212yes—
32#835C2212yes—
33#835C2212yes—
34#832C2211yescharacteristic
35#818C2212yes—
36#816C2212yes—
37#815C2212yes—
38Dic3⋊C8C2212yes—
39#810C2211yescharacteristic
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#557—31no—
43#554—31no—
44#491—32no—
45#487—31no—
46#479—32no—
47#477—31no—
48#461—32no—
49#461D311yescharacteristic
50#446—31no—
51#440—32no—
52C4×SD16—31no—
53C2×C3×Q8C2312yes—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C22×Dic3—22no—
57C2×Dic6—21no—
58C2×Dic6D411yescharacteristic
59C2×Dic6C2311yescharacteristic
60C3×SD16—24no—
61C2×C3×C8C2312yes—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—21no—
64C3×C4⋊C4C2311yescharacteristic
65C3×C22⋊C4—21no—
66C3×C42C2311yescharacteristic
67#217—22no—
68#214—22no—
69C4⋊Dic3—21no—
70C4⋊Dic3—22no—
71C4⋊Dic3—22no—
72C4⋊Dic3D411yescharacteristic
73C4⋊Dic3C2311yescharacteristic
74C4⋊Dic3C2312yes—
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
85C2×SD16—31no—
86C2×SD16C2×D312yes—
87C4⋊Q8—32no—
88C4⋊Q8—31no—
89C4⋊Q8C2×D311yescharacteristic
90#125—32no—
91#125—62no—
92#125—61no—
93C22⋊Q8—62no—
94C22⋊Q8—61no—
95C4×Q8—31no—
96C4×Q8—32no—
97C4×Q8—31no—
98C4×Q8—61no—
99C4×D4—32no—
100C4×D4C2×D311yescharacteristic
101C2×C4⋊C4—61no—
102#107—32no—
103#106—31no—
104C4⋊C8—32no—
105C4⋊C8C2×D311yescharacteristic
106#103—32no—
107#103—32no—
108#103—32no—
109#103—31no—
110#103C2×D312yes—
111#102—31no—
112#97—32no—
113C4×C8—31no—
114C23×C3—21no—
115C3×Q8—22no—
116C3×Q8—22no—
117C3×D4—21no—
118C3×D4#4112yes—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—42no—
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2411yescharacteristic
125C2×Dic3—21no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—82no—
129C2×Dic3C2×D412yes—
130C2×Dic3C2×D412yes—
131Dic6—22no—
132Dic6—21no—
133Dic6—21no—
134Dic6—81no—
135C3×C8—22no—
136C3⋊C8—21no—
137C3⋊C8—21no—
138C2×Q8—31no—
139C2×Q8—31no—
140C2×Q8—61no—
141C2×Q8C22×D312yes—
142C2×D4C22×D311yescharacteristic
143C22×C4—21no—
144C22×C4—62no—
145SD16—24no—
146SD16—62no—
147C2×C8—31no—
148C2×C8—31no—
149C2×C8C22×D312yes—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—31no—
153C4⋊C4—32no—
154C4⋊C4—31no—
155C4⋊C4—32no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—61no—
162C4⋊C4—61no—
163C4⋊C4C22×D311yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—62no—
166C42—32no—
167C42—62no—
168C42—61no—
169C42C22×D311yescharacteristic
170C22×C3—22no—
171C22×C3—41no—
172C22×C3—41no—
173C22×C3C22×D411yescharacteristic
174C3×C4—21no—
175C3×C4—42no—
176C3×C4—41no—
177C3×C4#13712yes—
178C3×C4C2×#4111yescharacteristic
179C3×C42−1+411yescharacteristic
180Dic3—22no—
181Dic3—22no—
182Dic3—41no—
183Dic3—42no—
184Dic3—41no—
185C23—21no—
186Q8—22no—
187Q8—22no—
188Q8—62no—
189Q8—61no—
190Q8—61no—
191Q8—241no—
192D4—21no—
193D4#23712yes—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—32no—
197C2×C4—32no—
198C2×C4—42no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—62no—
202C2×C4—242no—
203C2×C4D3×D411yescharacteristic
204C2×C4D3×D411yescharacteristic
205C2×C4C23×D311yescharacteristic
206C8—22no—
207C8—61no—
208C8—61no—
209C2×C3—42no—
210C2×C3#54811yescharacteristic
211C2×C3C2×#13711yescharacteristic
212C2×C3#57711yescharacteristic
213C22—22no—
214C22—41no—
215C22—41no—
216C22C2×D3×D411yescharacteristic
217C4—21no—
218C4—41no—
219C4—42no—
220C4—62no—
221C4—62no—
222C4—122no—
223C4—121no—
224C4—121no—
225C4#91512yes—
226C4C2×#23711yescharacteristic
227C4#101011yescharacteristic
228C3#337211yescharacteristic, complemented, Sylow
229C2—42no—
230C2#559811yescharacteristic
231C2C2×#91511yescharacteristic
232C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radicalC2×C3×D4C23
hypercenter#461D3
hypocenterC3#3372
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. #4850C2semidirect
9. #4831C2semidirect
10. #4797C2semidirect
11. #4796C2semidirect
228. C3#3372semidirect

Maximal subgroups

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

subgroupindex
#57212
#55432
C3×#4612
#51642
#51622
#50792
C4×#2142
#48502
#48312
#47972
#47962
#33723