Finite Group Databasebeta

← #81556

Subgroups — #81556

Order 384.

412 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 129 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#5728C211yesmaximal, characteristic, complemented
2#5716C211yesmaximal, characteristic
3C3×#486C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5#5076C211yesmaximal, characteristic, complemented
6#4778C214yesmaximal, complemented
7#4758C214yesmaximal, complemented
8C4×D3×C8C211yesmaximal, characteristic, complemented
9#4681C211yesmaximal, characteristic, complemented
10#3167—31nomaximal, Sylow
11C2×#237—21no—
12C3×C4⋊Q8C2211yescharacteristic
13C3×#127C2211yescharacteristic
14#946—21no—
15Q8×Dic3—21no—
16#938—21no—
17#938C2212yes—
18#936—21no—
19D4×Dic3—21no—
20D4×Dic3C2212yes—
21C2×#214C2214yes—
22C2×D3×C8C2212yes—
23#897C2212yes—
24#892—21no—
25#890—21no—
26#887C2212yes—
27C42×D3C2211yescharacteristic
28#869C2211yescharacteristic
29#869C2211yescharacteristic
30C3×#102C2214yes—
31C3×C4×C8C2211yescharacteristic
32#816C2214yes—
33C8×Dic3C2212yes—
34#808C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#533—31no—
37#532—31no—
38#498—31no—
39#492—31no—
40#487—31no—
41#486D311yescharacteristic, complemented
42#465—34no—
43C4×SD16—34no—
44C2×C4×C8—31no—
45C2×C3×Q8—21no—
46C2×C3×D4—21no—
47C2×C3×D4—21no—
48C2×C3×D4C2312yes—
49C2×#67—22no—
50C22×Dic3—22no—
51#237—44no—
52C2×C4×D3D411yescharacteristic
53C2×C4×D3D411yescharacteristic
54C2×C4×D3C2311yescharacteristic
55C2×Dic6—21no—
56C2×Dic6C2312yes—
57C2×C3×C8C2312yes—
58C3×C4⋊C4—21no—
59C3×C4⋊C4C2312yes—
60C3×C42C2311yescharacteristic
61#217—22no—
62#217—22no—
63#214—28no—
64(C2×D3)⋊C4—22no—
65C4⋊Dic3—21no—
66C4⋊Dic3—21no—
67C4⋊Dic3—21no—
68C4⋊Dic3—22no—
69C4⋊Dic3C2312yes—
70Dic3⋊C4—22no—
71C4×Dic3—22no—
72C4×Dic3D411yescharacteristic
73C4×Dic3D411yescharacteristic
74C4×Dic3C2311yescharacteristic
75C2×C3⋊C8C2312yes—
76D3×C8—24no—
77C2×#41—61no—
78C2×SD16—34no—
79C22×C8—32no—
80C4⋊Q8—31no—
81C4⋊Q8—31no—
82C4⋊Q8C2×D311yescharacteristic
83#127C2×D311yescharacteristic
84#125—61no—
85#124—61no—
86C22⋊Q8—32no—
87C22⋊Q8—61no—
88#121—32no—
89#121—61no—
90C4×Q8—32no—
91C4×Q8—61no—
92C4×D4—32no—
93C4×D4—61no—
94#117—61no—
95C2×C42—31no—
96#106—34no—
97#103—34no—
98#102C2×D314yes—
99C4×C8—32no—
100C4×C8—31no—
101C4×C8C2×D311yescharacteristic
102C23×C3—22no—
103C22×D3C2×D411yescharacteristic
104C3×Q8—42no—
105C3×D4—24no—
106C3×D4—42no—
107C3×D4—42no—
108C2×C3×C4—22no—
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2411yescharacteristic
112#67—44no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—44no—
116C2×Dic3C2×D411yescharacteristic
117C2×Dic3C2×D411yescharacteristic
118C2×Dic3C2×D411yescharacteristic
119C4×D3—22no—
120C4×D3—22no—
121C4×D3SD1614yescomplemented
122Dic6—24no—
123Dic6—42no—
124C3×C8—22no—
125C3⋊C8#4114yescomplemented
126#41—124no—
127C2×Q8—21no—
128C2×Q8—32no—
129C2×Q8—61no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—62no—
133C2×D4C22×D312yes—
134C22×C4—31no—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—62no—
138SD16—68no—
139C2×C8—32no—
140C2×C8—64no—
141C2×C8C22×D312yes—
142C4⋊C4—21no—
143C4⋊C4—32no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—61no—
147C4⋊C4—61no—
148C4⋊C4—61no—
149C4⋊C4C22×D312yes—
150C22⋊C4—62no—
151C22⋊C4—62no—
152C22⋊C4—62no—
153C42—31no—
154C42—31no—
155C42—31no—
156C42—62no—
157C42C22×D311yescharacteristic
158C22×C3—42no—
159C22×C3—42no—
160C22×C3—42no—
161C22×C3C22×D411yescharacteristic
162C2×D3—21no—
163C2×D3—21no—
164C2×D3C2×SD1612yes—
165C3×C4—42no—
166C3×C4C2×SD1612yes—
167C3×C4#13512yes—
168C3×C4C2×#4112yes—
169Dic3—21no—
170Dic3—21no—
171Dic3—42no—
172Dic3—42no—
173Dic3C2×SD1612yes—
174C23—22no—
175C23—31no—
176Q8—42no—
177Q8—64no—
178Q8—122no—
179D4—24no—
180D4—42no—
181D4—42no—
182D4—124no—
183C2×C4—22no—
184C2×C4—31no—
185C2×C4—34no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—124no—
193C2×C4D3×D411yescharacteristic
194C2×C4D3×D411yescharacteristic
195C2×C4C23×D311yescharacteristic
196C8—22no—
197C8—34no—
198C2×C3—82no—
199C2×C3#53211yescharacteristic
200C2×C3C22×SD1611yescharacteristic
201C2×C3C2×#13511yescharacteristic
202D3—22no—
203C22—32no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—61no—
208C22—61no—
209C22C2×D3×D411yescharacteristic
210C4—32no—
211C4—42no—
212C4—61no—
213C4—61no—
214C4—122no—
215C4—122no—
216C4#91212yes—
217C4D3×SD1612yes—
218C4D3×#4112yes—
219C3#316711yescharacteristic, complemented, Sylow
220C2—62no—
221C2—82no—
222C2#566811yescharacteristic
223C2C2×#91211yescharacteristic
224C2C2×D3×SD1611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
prime radical#5716C2
hypercenter#486D3
hypocenterC3#3167
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #5728C2semidirect
3. C3×#486C2semidirect
4. #5080C2semidirect
5. #5076C2semidirect
6. #4778C2semidirect
7. #4758C2semidirect
8. C4×D3×C8C2semidirect
9. #4681C2semidirect
41. #486D3semidirect
121. C4×D3SD16semidirect
125. C3⋊C8#41semidirect
219. C3#3167semidirect

Maximal subgroups

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

subgroupindex
#57282
#57162
C3×#4862
#50802
#50762
#47782
#47582
C4×D3×C82
#46812
#31673