Finite Group Databasebeta

← #80706

Subgroups — #80706

Order 384.

370 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 109 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#5619C212yesmaximal, complemented
2C3×#466C211yesmaximal, characteristic, complemented
3#5150C212yesmaximal, complemented
4#5115C211yesmaximal, characteristic, complemented
5#5102C211yesmaximal, characteristic, complemented
6#5039C212yesmaximal, complemented
7#4891C212yesmaximal, complemented
8#4786C212yesmaximal, complemented
9#4766C212yesmaximal, complemented
10#3259—31nomaximal, Sylow
11C2×C3×D8C2211yescharacteristic
12C22×C3×C8C2211yescharacteristic
13C3×#121C2212yes—
14#940C2212yes—
15#939—22no—
16#938—22no—
17D4×Dic3C2212yes—
18C2×#213C2212yes—
19#930C2212yes—
20#926C2211yescharacteristic
21C2×#208C2211yescharacteristic
22C2×D24C2211yescharacteristic
23#893C2212yes—
24#890C2212yes—
25#886—22no—
26#883—22no—
27C3×#107C2211yescharacteristic
28C3×#102C2212yes—
29#832C2212yes—
30#821C2212yes—
31#818C2211yescharacteristic
32Dic3⋊C8C2212yes—
33Dic3⋊C8C2212yes—
34#809C2212yes—
35#808C2212yes—
36#540—32no—
37#478—32no—
38#478—32no—
39#469—32no—
40#466—31no—
41#466D311yescharacteristic, complemented
42#459—32no—
43#459—32no—
44#424—31no—
45C2×C3×D4—22no—
46C2×C3×D4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C2×#67—22no—
49C2×#67—22no—
50C2×#67—22no—
51C22×Dic3—22no—
52C2×D12C2312yes—
53C2×C4×D3—22no—
54C3×D8—41no—
55C2×C3×C8—41no—
56C2×C3×C8C2311yescharacteristic
57C2×C3×C8C2311yescharacteristic
58C3×C4⋊C4C2312yes—
59C3×C22⋊C4—22no—
60#217—22no—
61#217D412yes—
62#213—42no—
63(C2×D3)⋊C4—22no—
64(C2×D3)⋊C4—22no—
65C4⋊Dic3—22no—
66C4⋊Dic3D411yescharacteristic
67C4⋊Dic3D411yescharacteristic
68C4⋊Dic3C2312yes—
69Dic3⋊C4—22no—
70C4×Dic3C2312yes—
71#208—41no—
72C2×C3⋊C8C2312yes—
73D24—41no—
74C2×D8—32no—
75C2×D8—31no—
76C2×D8C2×D311yescharacteristic
77C2×M16—31no—
78C22×C8C2×D311yescharacteristic
79#127—32no—
80#125—32no—
81#123—62no—
82#123—62no—
83#121—32no—
84#121—62no—
85#121—62no—
86#121C2×D312yes—
87C4×D4—32no—
88C4×D4—32no—
89#117—31no—
90#107—31no—
91#107C2×D311yescharacteristic
92#106—32no—
93C4⋊C8—32no—
94C4⋊C8—32no—
95#102—32no—
96#102—32no—
97#102—32no—
98#102C2×D312yes—
99C23×C3—22no—
100C23×C3C2×D411yescharacteristic
101C22×D3—22no—
102C3×D4—22no—
103C3×D4—22no—
104C3×D4—82no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—22no—
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2411yescharacteristic
110#67—82no—
111#67—82no—
112#67—82no—
113C2×Dic3—22no—
114C2×Dic3—82no—
115C2×Dic3C2×D412yes—
116C2×Dic3C2×D412yes—
117D12—22no—
118D12—22no—
119C4×D3—82no—
120C3×C8—21no—
121C3×C8—21no—
122C3⋊C8—22no—
123C2×D4—22no—
124C2×D4—32no—
125C2×D4—62no—
126C2×D4—62no—
127C2×D4—62no—
128C2×D4C22×D312yes—
129C22×C4—62no—
130C22×C4—62no—
131C22×C4C22×D311yescharacteristic
132D8—41no—
133D8—121no—
134D8—122no—
135M16—121no—
136C2×C8—32no—
137C2×C8—41no—
138C2×C8C22×D311yescharacteristic
139C2×C8C22×D311yescharacteristic
140C4⋊C4—31no—
141C4⋊C4—31no—
142C4⋊C4—32no—
143C4⋊C4—62no—
144C4⋊C4—62no—
145C4⋊C4C22×D312yes—
146C22⋊C4—22no—
147C22⋊C4—32no—
148C22⋊C4—62no—
149C22⋊C4—62no—
150C22⋊C4—62no—
151C42—32no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3—21no—
155C22×C3—42no—
156C22×C3—42no—
157C22×C3—42no—
158C22×C3C22×D411yescharacteristic
159C2×D3—42no—
160C2×D3—42no—
161C2×D3—42no—
162C3×C4—21no—
163C3×C4—42no—
164C3×C4H(5,𝔽2)11yescharacteristic
165C3×C42−1+411yescharacteristic
166Dic3—22no—
167Dic3—22no—
168Dic3—42no—
169C23—22no—
170C23—62no—
171C23D3×D411yescharacteristic
172D4—22no—
173D4—22no—
174D4—62no—
175D4—62no—
176D4—82no—
177D4—242no—
178D4—242no—
179D4—242no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—32no—
184C2×C4—32no—
185C2×C4—62no—
186C2×C4—242no—
187C2×C4—242no—
188C2×C4D3×D411yescharacteristic
189C2×C4C23×D311yescharacteristic
190C8—21no—
191C8—21no—
192C8—62no—
193C2×C3—41no—
194C2×C3—82no—
195C2×C3#53711yescharacteristic
196C2×C3#57611yescharacteristic
197C2×C3#57611yescharacteristic
198D3—82no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—122no—
206C22—122no—
207C22—122no—
208C22C2×D3×D411yescharacteristic
209C4—21no—
210C4—42no—
211C4—62no—
212C4—62no—
213C4—122no—
214C4#100411yescharacteristic
215C4#101011yescharacteristic
216C3#325911yescharacteristic, complemented, Sylow
217C2—41no—
218C2—82no—
219C2—242no—
220C2#559711yescharacteristic
221C2#575711yescharacteristic
222C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#466C2
socleC22×C3C22×D4
hypercenter#466D3
hypocenterC3#3259
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #5619C2semidirect
2. C3×#466C2semidirect
3. #5150C2semidirect
4. #5115C2semidirect
5. #5102C2semidirect
6. #5039C2semidirect
7. #4891C2semidirect
8. #4786C2semidirect
9. #4766C2semidirect
41. #466D3semidirect
216. C3#3259semidirect

Maximal subgroups

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

subgroupindex
#56192
C3×#4662
#51502
#51152
#51022
#50392
#48912
#47862
#47662
#32593