Finite Group Databasebeta

← #80994

Subgroups — #80994

Order 384.

336 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 115 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#5542C211yesmaximal, characteristic
3C3×#474C211yesmaximal, characteristic
4#5093C211yesmaximal, characteristic, complemented
5C4×#214C211yesmaximal, characteristic
6#4877C212yesmaximal, complemented
7#4873C212yesmaximal, complemented
8#4849C211yesmaximal, characteristic, complemented
9#4830C211yesmaximal, characteristic, complemented
10#4768C212yesmaximal
11#4767C212yesmaximal
12#3364—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4×D4C2211yescharacteristic
15Q8×Dic3—21no—
16#944—21no—
17#936C2212yes—
18C2×#214C2211yescharacteristic
19#926—21no—
20#924—21no—
21C2×Dic12C2212yes—
22#890C2212yes—
23#889—22no—
24#887C2212yes—
25#887C2212yes—
26#879—22no—
27#869C2211yescharacteristic
28C4×Dic6C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×#107C2212yes—
31C3×C4⋊C8C2211yescharacteristic
32C3×#102C2212yes—
33#832C2211yescharacteristic
34#816C2212yes—
35Dic3⋊C8C2212yes—
36C8×Dic3C2212yes—
37#810C2211yescharacteristic
38#810C2212yes—
39#810C2212yes—
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#557—31no—
43#556—31no—
44#488—32no—
45#487—31no—
46#479—32no—
47#474D311yescharacteristic
48#464—32no—
49#462—31no—
50C8×Q8—31no—
51C4×Q16—32no—
52C4×SD16—31no—
53C2×C3×Q8—21no—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×Dic6D411yescharacteristic
57C2×Dic6C2312yes—
58C2×Dic6C2311yescharacteristic
59C2×C3×C8C2312yes—
60C3×C4⋊C4—21no—
61C3×C4⋊C4C2311yescharacteristic
62C3×C4⋊C4C2312yes—
63C3×C22⋊C4—21no—
64C3×C42C2311yescharacteristic
65#217—22no—
66#217—22no—
67#214—22no—
68C4⋊Dic3—21no—
69C4⋊Dic3—21no—
70C4⋊Dic3—22no—
71C4⋊Dic3D411yescharacteristic
72C4⋊Dic3C2311yescharacteristic
73Dic3⋊C4—21no—
74Dic3⋊C4—22no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4D412yes—
78C4×Dic3—21no—
79C4×Dic3—22no—
80C4×Dic3—22no—
81C4×Dic3C2312yes—
82C2×C3⋊C8C2311yescharacteristic
83C2×C3⋊C8C2311yescharacteristic
84Dic12—24no—
85C2×Q16—32no—
86C2×SD16—31no—
87C4⋊Q8—31no—
88C4⋊Q8—61no—
89C4⋊Q8C2×D311yescharacteristic
90#126—62no—
91#125—32no—
92#125—62no—
93#124—32no—
94C22⋊Q8—61no—
95C4×Q8—31no—
96C4×Q8—32no—
97C4×Q8—32no—
98C4×Q8—31no—
99C4×Q8—61no—
100C4×D4C2×D311yescharacteristic
101#117—61no—
102#107C2×D312yes—
103#106—31no—
104C4⋊C8—32no—
105C4⋊C8C2×D311yescharacteristic
106#103—32no—
107#103—31no—
108#103—32no—
109#103—32no—
110#102—31no—
111#102C2×D312yes—
112C4×C8—32no—
113C4×C8—31no—
114C23×C3—21no—
115C3×Q8—42no—
116C3×D4—21no—
117C3×D4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—22no—
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×Dic3C2×D412yes—
128C2×Dic3C2×D412yes—
129Dic6—22no—
130Dic6—21no—
131Dic6—22no—
132Dic6—22no—
133Dic6#4112yes—
134C3×C8—22no—
135C3⋊C8—21no—
136C3⋊C8—21no—
137C2×Q8—21no—
138C2×Q8—31no—
139C2×Q8—31no—
140C2×Q8—32no—
141C2×D4C22×D311yescharacteristic
142C22×C4—21no—
143Q16—64no—
144SD16—62no—
145C2×C8—31no—
146C2×C8—31no—
147C2×C8C22×D312yes—
148C4⋊C4—21no—
149C4⋊C4—31no—
150C4⋊C4—32no—
151C4⋊C4—31no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—61no—
157C4⋊C4—61no—
158C4⋊C4—61no—
159C4⋊C4C22×D312yes—
160C4⋊C4C22×D311yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C42—32no—
165C42—62no—
166C42—62no—
167C42—61no—
168C42C22×D311yescharacteristic
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3C22×D411yescharacteristic
173C3×C4—21no—
174C3×C4—41no—
175C3×C4—42no—
176C3×C4#13512yes—
177C3×C4C2×#4111yescharacteristic
178C3×C42−1+411yescharacteristic
179Dic3—22no—
180Dic3—22no—
181Dic3—22no—
182Dic3—42no—
183Dic3—41no—
184C23—21no—
185Q8—32no—
186Q8—42no—
187Q8—62no—
188Q8—61no—
189Q8—62no—
190Q8—62no—
191D4—21no—
192D4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—32no—
196C2×C4—32no—
197C2×C4—42no—
198C2×C4—61no—
199C2×C4—62no—
200C2×C4—61no—
201C2×C4D3×D411yescharacteristic
202C2×C4D3×D411yescharacteristic
203C2×C4C23×D311yescharacteristic
204C8—22no—
205C8—61no—
206C8—61no—
207C2×C3—81no—
208C2×C3#54811yescharacteristic
209C2×C3C2×#13511yescharacteristic
210C2×C3#57811yescharacteristic
211C22—41no—
212C22—41no—
213C22—41no—
214C22C2×D3×D411yescharacteristic
215C4—21no—
216C4—41no—
217C4—42no—
218C4—62no—
219C4—62no—
220C4—62no—
221C4—122no—
222C4—121no—
223C4#91212yes—
224C4#100711yescharacteristic
225C4D3×#4111yescharacteristic
226C3#336411yescharacteristic, complemented, Sylow
227C2—81no—
228C2#563311yescharacteristic
229C2C2×#91211yescharacteristic
230C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#474C2
socleC22×C3C22×D4
prime radicalC2×C3×D4C23
hypercenter#474D3
hypocenterC3#3364
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #5721C2semidirect
4. #5093C2semidirect
6. #4877C2semidirect
7. #4873C2semidirect
8. #4849C2semidirect
9. #4830C2semidirect
226. C3#3364semidirect

Maximal subgroups

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

subgroupindex
#57212
#55422
C3×#4742
#50932
C4×#2142
#48772
#48732
#48492
#48302
#47682
#47672
#33643