Finite Group Databasebeta

← #83268

Subgroups — #83268

Order 384.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#876C211yesmaximal, characteristic
3#5231C212yesmaximal
4#5230C212yesmaximal, complemented
5#4993C212yesmaximal
6#4992C212yesmaximal
7#4985C212yesmaximal, complemented
8#4983C212yesmaximal, complemented
9#4937C211yesmaximal, characteristic, complemented
10#2821—31nomaximal, Sylow
11C22×C3×Q8C2212yes—
12C3×C4⋊Q8—24no—
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C42C2211yescharacteristic
16C2×(C2×D3)⋊C4C2211yescharacteristic
17C2×(C2×D3)⋊C4C2211yescharacteristic
18C2×(C2×D3)⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2214yes—
20C2×Dic3⋊C4C2211yescharacteristic
21C2×Dic3⋊C4C2211yescharacteristic
22C2×Dic3⋊C4C2211yescharacteristic
23C2×Dic3⋊C4C2214yes—
24#876—24no—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C4⋊Q8D311yescharacteristic, complemented
31C2×#126—31no—
32#400—32no—
33#399—32no—
34#398—32no—
35#397—32no—
36#395—32no—
37#393—32no—
38#383—31no—
39C23×D3C2311yescharacteristic
40C2×C3×Q8—24no—
41C2×C3×Q8—44no—
42C22×C3×C4C2314yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×Dic3C2314yes—
47C22×Dic3C2311yescharacteristic
48C22×Dic3C2311yescharacteristic
49C22×Dic3C2311yescharacteristic
50C3×C4⋊C4—24no—
51C3×C4⋊C4—24no—
52C3×C42D414yes—
53(C2×D3)⋊C4—22no—
54(C2×D3)⋊C4—22no—
55(C2×D3)⋊C4—22no—
56(C2×D3)⋊C4—44no—
57Dic3⋊C4—22no—
58Dic3⋊C4—22no—
59Dic3⋊C4—22no—
60Dic3⋊C4—44no—
61C22×Q8C2×D312yes—
62C4⋊Q8—24no—
63#126—64no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—31no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4—34no—
68C2×C4⋊C4C2×D312yes—
69C2×C4⋊C4C2×D312yes—
70C2×C22⋊C4—31no—
71C2×C22⋊C4—31no—
72C2×C22⋊C4—31no—
73C2×C22⋊C4—34no—
74C2×C42C2×D311yescharacteristic
75#95—32no—
76#95—32no—
77#95—32no—
78#95—34no—
79#95—34no—
80C23×C3C2411yescharacteristic
81C22×D3—21no—
82C22×D3—22no—
83C22×D3—22no—
84C22×D3—22no—
85C3×Q8—48no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—24no—
90C2×C3×C4—24no—
91C2×C3×C4—24no—
92C2×C3×C4C2×D412yes—
93C2×C3×C4C2×D412yes—
94C2×C3×C4C2×D412yes—
95C2×Dic3—21no—
96C2×Dic3—21no—
97C2×Dic3—21no—
98C2×Dic3—22no—
99C2×Dic3—24no—
100C2×Dic3—22no—
101C2×Dic3—22no—
102C2×Dic3—24no—
103C2×Dic3—24no—
104C24—31no—
105C2×Q8—24no—
106C2×Q8—44no—
107C22×C4—34no—
108C22×C4—31no—
109C22×C4—31no—
110C22×C4—31no—
111C22×C4C22×D314yes—
112C22×C4C22×D311yescharacteristic
113C22×C4C22×D311yescharacteristic
114C22×C4C22×D311yescharacteristic
115C4⋊C4—24no—
116C4⋊C4—24no—
117C4⋊C4—62no—
118C4⋊C4—62no—
119C4⋊C4—62no—
120C4⋊C4—124no—
121C22⋊C4—62no—
122C22⋊C4—62no—
123C22⋊C4—62no—
124C22⋊C4—124no—
125C42#6714yes—
126C22×C3C22×D411yescharacteristic
127C22×C3H(5,𝔽2)12yes—
128C22×C32−1+412yes—
129C22×C32−1+412yes—
130C2×D3—41no—
131C2×D3—41no—
132C2×D3—41no—
133C2×D3—44no—
134C3×C4—22no—
135C3×C4—22no—
136C3×C4—22no—
137C3×C4—44no—
138Dic3—41no—
139Dic3—41no—
140Dic3—41no—
141Dic3—44no—
142C23—62no—
143C23—62no—
144C23—62no—
145C23—61no—
146C23C23×D311yescharacteristic
147Q8—48no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—64no—
155C2×C4—62no—
156C2×C4—64no—
157C2×C4—64no—
158C2×C4—62no—
159C2×C4—62no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4C2×#6712yes—
164C2×C4C2×#6712yes—
165C2×C4C2×#6712yes—
166C2×C3#53511yescharacteristic
167C2×C3#53611yescharacteristic
168C2×C3#53611yescharacteristic
169C2×C3#56314yes—
170D3—81no—
171C22—124no—
172C22—121no—
173C22—121no—
174C22—121no—
175C22#100412yes—
176C22#100712yes—
177C22#100712yes—
178C22C22×#6711yescharacteristic
179C4—22no—
180C4—22no—
181C4—22no—
182C4—44no—
183C4—124no—
184C4—121no—
185C4—121no—
186C4—121no—
187C3#282111yescharacteristic, complemented, Sylow
188C2—241no—
189C2#573514yes—
190C2#580511yescharacteristic
191C2#581611yescharacteristic
192C2#581611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4⋊Q8C2
prime radicalC23×D3C23
hypercenterC2×C4⋊Q8D3
hypocenterC3#2821
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4⋊Q8C2semidirect
4. #5230C2semidirect
7. #4985C2semidirect
8. #4983C2semidirect
9. #4937C2semidirect
30. C2×C4⋊Q8D3semidirect
187. C3#2821semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Q82
C2×#8762
#52312
#52302
#49932
#49922
#49852
#49832
#49372
#28213