Finite Group Databasebeta

← #83103

Subgroups — #83103

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×#126C211yesmaximal, characteristic
2C2×#869C211yesmaximal, characteristic, complemented
3#4986C212yesmaximal, complemented
4#4981C212yesmaximal, complemented
5#4979C212yesmaximal, complemented
6#4953C212yesmaximal
7#4949C212yesmaximal
8#4947C212yesmaximal
9#4937C211yesmaximal, characteristic, complemented
10#2821—31nomaximal, Sylow
11C22×Dic6C2212yes—
12C3×#126—24no—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×#217C2214yes—
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2212yes—
23C2×Dic3⋊C4C2214yes—
24#869—24no—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C4⋊Q8—31no—
31C2×#126D311yescharacteristic
32#400—32no—
33#399—32no—
34#398—32no—
35#397—32no—
36#395—32no—
37#393—32no—
38#383—31no—
39C24×C3C2311yescharacteristic
40C22×C3×C4C2311yescharacteristic
41C22×C3×C4C2311yescharacteristic
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×Dic3C2314yes—
47C22×Dic3C2314yes—
48C2×Dic6—24no—
49C2×Dic6—44no—
50C3×C4⋊C4—22no—
51C3×C4⋊C4—22no—
52C3×C4⋊C4—22no—
53C3×C22⋊C4—22no—
54C3×C22⋊C4—22no—
55C3×C22⋊C4—22no—
56C3×C42D414yes—
57#217—44no—
58C4⋊Dic3—24no—
59C4⋊Dic3—24no—
60Dic3⋊C4—44no—
61C22×Q8—32no—
62C4⋊Q8—64no—
63#126—24no—
64C2×C4⋊C4—32no—
65C2×C4⋊C4—32no—
66C2×C4⋊C4—34no—
67C2×C4⋊C4C2×D311yescharacteristic
68C2×C4⋊C4C2×D311yescharacteristic
69C2×C4⋊C4C2×D311yescharacteristic
70C2×C22⋊C4—34no—
71C2×C22⋊C4C2×D311yescharacteristic
72C2×C22⋊C4C2×D311yescharacteristic
73C2×C22⋊C4C2×D311yescharacteristic
74C2×C42C2×D311yescharacteristic
75#95—32no—
76#95—32no—
77#95—32no—
78#95—34no—
79#95—34no—
80C23×C3—21no—
81C23×C3—22no—
82C23×C3—22no—
83C23×C3—22no—
84C23×C3C2411yescharacteristic
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—21no—
89C2×C3×C4—21no—
90C2×C3×C4—21no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4C2×D412yes—
95C2×C3×C4C2×D412yes—
96C2×C3×C4C2×D412yes—
97C2×Dic3—24no—
98C2×Dic3—24no—
99C2×Dic3—24no—
100C2×Dic3—24no—
101C2×Dic3—24no—
102C2×Dic3—24no—
103Dic6—48no—
104C24C22×D311yescharacteristic
105C2×Q8—64no—
106C2×Q8—124no—
107C22×C4—34no—
108C22×C4—34no—
109C22×C4C22×D311yescharacteristic
110C22×C4C22×D311yescharacteristic
111C22×C4C22×D311yescharacteristic
112C22×C4C22×D311yescharacteristic
113C22×C4C22×D311yescharacteristic
114C22×C4C22×D311yescharacteristic
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—64no—
119C4⋊C4—64no—
120C4⋊C4—124no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C22⋊C4—124no—
125C42#6714yes—
126C22×C3—41no—
127C22×C3—41no—
128C22×C3—41no—
129C22×C3—44no—
130C22×C3C22×D411yescharacteristic
131C22×C3H(5,𝔽2)12yes—
132C22×C32−1+412yes—
133C22×C32−1+412yes—
134C3×C4—22no—
135C3×C4—22no—
136C3×C4—22no—
137C3×C4—41no—
138C3×C4—41no—
139C3×C4—41no—
140Dic3—44no—
141Dic3—44no—
142C23—22no—
143C23—22no—
144C23—22no—
145C23—21no—
146C23C23×D311yescharacteristic
147Q8—128no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—64no—
158C2×C4—64no—
159C2×C4—64no—
160C2×C4—64no—
161C2×C4—64no—
162C2×C4—64no—
163C2×C4C2×#6712yes—
164C2×C4C2×#6712yes—
165C2×C4C2×#6712yes—
166C2×C3—81no—
167C2×C3#53511yescharacteristic
168C2×C3#53611yescharacteristic
169C2×C3#53611yescharacteristic
170C2×C3#56314yes—
171C22—44no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22#100912yes—
176C22#101012yes—
177C22#101012yes—
178C22C22×#6711yescharacteristic
179C4—22no—
180C4—22no—
181C4—22no—
182C4—41no—
183C4—41no—
184C4—41no—
185C4—124no—
186C4—124no—
187C3#282111yescharacteristic, complemented, Sylow
188C2—81no—
189C2#571214yes—
190C2#582511yescharacteristic
191C2#582511yescharacteristic
192C2#583011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#126C2
prime radicalC24×C3C23
hypercenterC2×#126D3
hypocenterC3#2821
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
2. C2×#869C2semidirect
3. #4986C2semidirect
4. #4981C2semidirect
5. #4979C2semidirect
9. #4937C2semidirect
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×#1262
C2×#8692
#49862
#49812
#49792
#49532
#49492
#49472
#49372
#28213