Finite Group Databasebeta

← #71880

Subgroups — #71880

Order 384.

282 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C3×#537C211yesmaximal, characteristic
2#5040C212yesmaximal, complemented
3#4979C211yesmaximal, characteristic, complemented
4#4537C212yesmaximal, complemented
5#4528C211yesmaximal, characteristic, complemented
6#1604—31nomaximal, Sylow
7C2×C3×#41—21no—
8C22×Dic6C2211yescharacteristic
9C3×C22⋊Q8—22no—
10C3×#121—22no—
11C3×#121—21no—
12C3×#121C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×#214—22no—
15C2×Dic3⋊C4—21no—
16C3⋊#95—21no—
17C3⋊#95C2211yescharacteristic
18#830C2212yes—
19#810—22no—
20#537D311yescharacteristic
21#448—32no—
22#395—31no—
23#327—32no—
24#324—31no—
25C3×#41—42no—
26C3×#41—42no—
27C2×C3×Q8—21no—
28C2×C3×D4—21no—
29C2×C3×D4—22no—
30C2×C3×D4—21no—
31C2×C3×D4D412yes—
32C22×C3×C4—22no—
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4C2311yescharacteristic
36C22×Dic3—21no—
37C22×Dic3—21no—
38C2×Dic6—22no—
39C2×Dic6—42no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4—22no—
42C3×C4⋊C4D412yes—
43C3×C22⋊C4—22no—
44C3×C22⋊C4—22no—
45#214—44no—
46Dic3⋊C4—42no—
47C2×C3⋊C8—22no—
48C2×#41—21no—
49C22×Q8—31no—
50C2×SD16—62no—
51C22⋊Q8—22no—
52#121—22no—
53#121—21no—
54#121C2×D312yes—
55C2×C4⋊C4—61no—
56C2×C4⋊C4C2×D311yescharacteristic
57#103—62no—
58C22⋊C8—32no—
59#95—31no—
60#95—61no—
61C23×C3—22no—
62C23×C3C2×D411yescharacteristic
63C3×Q8—42no—
64C3×D4—42no—
65C3×D4—42no—
66C3×D4—42no—
67C3×D4—42no—
68C3×D4—81no—
69C2×C3×C4—21no—
70C2×C3×C4—21no—
71C2×C3×C4—21no—
72C2×C3×C4—22no—
73C2×C3×C4—41no—
74C2×C3×C4—44no—
75C2×C3×C4—41no—
76C2×C3×C4SD1612yescomplemented
77C2×C3×C4C2×D412yes—
78C2×Dic3—22no—
79C2×Dic3—22no—
80C2×Dic3—41no—
81C2×Dic3—41no—
82C2×Dic3—41no—
83C2×Dic3—41no—
84Dic6—24no—
85Dic6—42no—
86C3⋊C8—42no—
87#41—42no—
88#41—42no—
89C2×Q8—21no—
90C2×Q8—62no—
91C2×Q8—122no—
92C2×D4—21no—
93C2×D4—22no—
94C2×D4—21no—
95C2×D4#6712yes—
96C22×C4—22no—
97C22×C4—61no—
98C22×C4—61no—
99C22×C4#6711yescharacteristic
100C22×C4#6711yescharacteristic
101C22×C4C22×D311yescharacteristic
102SD16—124no—
103C2×C8—62no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—122no—
107C4⋊C4#6712yes—
108C22⋊C4—22no—
109C22⋊C4—22no—
110C22×C3—21no—
111C22×C3—21no—
112C22×C3—42no—
113C22×C3—42no—
114C22×C3—42no—
115C22×C3#12011yescharacteristic
116C22×C3C2×SD1611yescharacteristic
117C22×C3AGL(1,ℤ/8)11yescharacteristic
118C3×C4—22no—
119C3×C4—22no—
120C3×C4—42no—
121C3×C4—41no—
122Dic3—42no—
123Dic3—42no—
124C23—22no—
125C23C2×#6711yescharacteristic
126Q8—42no—
127Q8—64no—
128Q8—122no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—42no—
133D4—81no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—44no—
140C2×C4—41no—
141C2×C4—62no—
142C2×C4—62no—
143C2×C4—121no—
144C2×C4—121no—
145C2×C4—121no—
146C2×C4—121no—
147C2×C4#21412yescomplemented
148C2×C4C2×#6712yes—
149C8—122no—
150C2×C3—21no—
151C2×C3—21no—
152C2×C3—82no—
153C2×C3#45011yescharacteristic
154C2×C3#45611yescharacteristic
155C2×C3B(4,𝔽2)11yescharacteristic
156C22—21no—
157C22—21no—
158C22—42no—
159C22—42no—
160C22—42no—
161C22#93211yescharacteristic
162C22C2×#21411yescharacteristic
163C22#95311yescharacteristic
164C4—22no—
165C4—22no—
166C4—42no—
167C4—41no—
168C4—122no—
169C4—122no—
170C3#160411yescharacteristic, complemented, Sylow
171C2—21no—
172C2—21no—
173C2—82no—
174C2#521811yescharacteristic
175C2#524411yescharacteristic
176C2#524611yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
Fitting, prime radicalC3×#537C2
socleC22×C3#120
hypercenter#537D3
hypocenterC3#1604
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
2. #5040C2semidirect
3. #4979C2semidirect
4. #4537C2semidirect
5. #4528C2semidirect
76. C2×C3×C4SD16semidirect
147. C2×C4#214semidirect
170. C3#1604semidirect

Maximal subgroups

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

subgroupindex
C3×#5372
#50402
#49792
#45372
#45282
#16043