Finite Group Databasebeta

← #80855

Subgroups — #80855

Order 384.

430 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 113 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#5596C212yesmaximal, complemented
2C3×#469C211yesmaximal, characteristic, complemented
3#5150C212yesmaximal, complemented
4#5135C211yesmaximal, characteristic, complemented
5#5118C211yesmaximal, characteristic, complemented
6#5039C212yesmaximal, complemented
7#4870C212yesmaximal, complemented
8#4786C212yesmaximal, complemented
9#4761C212yesmaximal, complemented
10#3253—31nomaximal, Sylow
11C2×#237—22no—
12C2×C3×D8C2211yescharacteristic
13C2×C3×M16C2211yescharacteristic
14C3×#121C2212yes—
15#940C2212yes—
16#939—22no—
17D4×Dic3C2212yes—
18C2×#213C2212yes—
19#930C2212yes—
20C2×C4×Dic3C2211yescharacteristic
21C2×#208C2211yescharacteristic
22C2×D24C2211yescharacteristic
23#893C2212yes—
24#888C2212yes—
25#885—22no—
26#67⋊C4—22no—
27C3×#106C2211yescharacteristic
28C3×#102C2212yes—
29#832C2212yes—
30#821C2212yes—
31#817C2211yescharacteristic
32Dic3⋊C8C2212yes—
33Dic3⋊C8C2212yes—
34#809C2212yes—
35#808C2212yes—
36#532—32no—
37#476—32no—
38#476—32no—
39#469—31no—
40#469—32no—
41#469D311yescharacteristic, complemented
42#459—32no—
43#459—32no—
44#423—31no—
45C2×C3×D4—22no—
46C2×C3×D4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C2×#67—22no—
49C2×#67—22no—
50C2×#67—22no—
51C22×Dic3—22no—
52C22×Dic3D412yes—
53#237—48no—
54C2×D12C2312yes—
55C2×C4×D3—22no—
56C2×Dic6—22no—
57C3×D8—41no—
58C3×M16—41no—
59C2×C3×C8C2311yescharacteristic
60C2×C3×C8C2311yescharacteristic
61C3×C4⋊C4C2312yes—
62C3×C22⋊C4—22no—
63#217—22no—
64#213—42no—
65(C2×D3)⋊C4—22no—
66(C2×D3)⋊C4—22no—
67C4⋊Dic3C2312yes—
68Dic3⋊C4—22no—
69C4×Dic3D411yescharacteristic
70C4×Dic3D411yescharacteristic
71C4×Dic3C2312yes—
72#208—41no—
73C2×C3⋊C8C2312yes—
74D24—41no—
75C2×#41—62no—
76C2×D8—32no—
77C2×D8—31no—
78C2×D8C2×D311yescharacteristic
79C2×M16—31no—
80C2×M16C2×D311yescharacteristic
81C4⋊Q8—32no—
82#127—32no—
83#124—62no—
84#121—32no—
85#121—62no—
86#121C2×D312yes—
87C4×D4—32no—
88C4×D4—32no—
89C4×D4—62no—
90C2×C42—31no—
91#106—31no—
92#106—32no—
93#106C2×D311yescharacteristic
94C4⋊C8—32no—
95C4⋊C8—32no—
96#102—32no—
97#102—32no—
98#102—32no—
99#102C2×D312yes—
100C23×C3—22no—
101C23×C3C2×D411yescharacteristic
102C22×D3—22no—
103C3×D4—22no—
104C3×D4—22no—
105C3×D4—44no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—22no—
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2411yescharacteristic
111#67—44no—
112#67—44no—
113#67—44no—
114C2×Dic3—24no—
115C2×Dic3—22no—
116C2×Dic3—44no—
117C2×Dic3C2×D412yes—
118C2×Dic3C2×D412yes—
119D12—22no—
120D12—22no—
121C4×D3—44no—
122Dic6—44no—
123C3×C8—21no—
124C3×C8—21no—
125C3⋊C8—22no—
126#41—128no—
127C2×Q8—62no—
128C2×D4—22no—
129C2×D4—32no—
130C2×D4—62no—
131C2×D4—62no—
132C2×D4—62no—
133C2×D4C22×D312yes—
134C22×C4—32no—
135C22×C4—62no—
136C22×C4—62no—
137C22×C4C22×D311yescharacteristic
138D8—41no—
139D8—121no—
140D8—122no—
141M16—41no—
142M16—121no—
143C2×C8—32no—
144C2×C8C22×D311yescharacteristic
145C2×C8C22×D311yescharacteristic
146C4⋊C4—32no—
147C4⋊C4—62no—
148C4⋊C4C22×D312yes—
149C22⋊C4—22no—
150C22⋊C4—62no—
151C22⋊C4—62no—
152C22⋊C4—62no—
153C42—31no—
154C42—31no—
155C42—32no—
156C22×C3—21no—
157C22×C3—22no—
158C22×C3—42no—
159C22×C3—42no—
160C22×C3—42no—
161C22×C3C22×D411yescharacteristic
162C2×D3—42no—
163C2×D3—42no—
164C2×D3—42no—
165C3×C4—21no—
166C3×C4—42no—
167C3×C4H(5,𝔽2)11yescharacteristic
168C3×C42−1+411yescharacteristic
169Dic3—22no—
170Dic3—42no—
171Dic3AGL(1,ℤ/8)14yes—
172C23—22no—
173C23—62no—
174C23D3×D411yescharacteristic
175Q8—124no—
176D4—22no—
177D4—22no—
178D4—44no—
179D4—62no—
180D4—62no—
181D4—124no—
182D4—124no—
183D4—124no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—32no—
188C2×C4—32no—
189C2×C4—64no—
190C2×C4—62no—
191C2×C4—124no—
192C2×C4—124no—
193C2×C4D3×D411yescharacteristic
194C2×C4C23×D311yescharacteristic
195C8—21no—
196C8—21no—
197C8—62no—
198C2×C3—41no—
199C2×C3—82no—
200C2×C3#53711yescharacteristic
201C2×C3C2×AGL(1,ℤ/8)12yes—
202D3—82no—
203C22—22no—
204C22—21no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—122no—
209C22—122no—
210C22—122no—
211C22C2×D3×D411yescharacteristic
212C4—21no—
213C4—34no—
214C4—42no—
215C4—62no—
216C4—122no—
217C4#100411yescharacteristic
218C4#101011yescharacteristic
219C3#325311yescharacteristic, complemented, Sylow
220C2—41no—
221C2—82no—
222C2—242no—
223C2#559711yescharacteristic
224C2D3×AGL(1,ℤ/8)12yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3253
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #5596C2semidirect
2. C3×#469C2semidirect
3. #5150C2semidirect
4. #5135C2semidirect
5. #5118C2semidirect
6. #5039C2semidirect
7. #4870C2semidirect
8. #4786C2semidirect
9. #4761C2semidirect
41. #469D3semidirect
219. C3#3253semidirect

Maximal subgroups

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

subgroupindex
#55962
C3×#4692
#51502
#51352
#51182
#50392
#48702
#47862
#47612
#32533