Finite Group Databasebeta

← #13158901

Subgroups — #13158901

Order 1728.

692 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 168 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×#64079C211yesmaximal, characteristic
2C32×#870C211yesmaximal, characteristic
3C4×C3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
4#11722056C212yesmaximal
5#11722053C212yesmaximal
6#10592764—31nomaximal
7#10590314—34nomaximal
8C33×C42C2211yescharacteristic
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×#3750—22no—
13C32×C4⋊Dic3C2212yes—
14C32×Dic3⋊C4—22no—
15C32×#125—31no—
16#64079D311yescharacteristic
17C3×#870—34no—
18C3×#870D314yes—
19#63635—32no—
20#63634—32no—
21C4×C3⋊(C3⋊C8)—31no—
22C4×C3⋊(C3⋊C8)—34no—
23C4×C3⋊(C3⋊C8)—64no—
24#63576—38no—
25#63574—38no—
26C2×C33×C4D412yes—
27C2×C33×C4C2311yescharacteristic
28C2×C3×C3⋊Dic3—22no—
29C2×C32×Dic3—22no—
30C3⋊(C3⋊(C3⋊C8))Q812yes—
31C3⋊(C3⋊(C3⋊C8))Q812yes—
32#5065—94no—
33#5065—91no—
34C32×C4⋊C4—32no—
35C32×C4⋊C4—62no—
36C32×C42—24no—
37C32×C42C2×D314yes—
38C32×C42C2×D311yescharacteristic
39C4⋊(C3⋊Dic3)C2×D312yes—
40#3750—22no—
41C2×C3⋊(C3⋊C8)—31no—
42C2×C3⋊(C3⋊C8)—31no—
43C2×C3⋊(C3⋊C8)—34no—
44C2×C3⋊(C3⋊C8)—34no—
45C2×C3⋊(C3⋊C8)—64no—
46C2×C3⋊(C3⋊C8)—64no—
47C3×C4⋊Dic3—38no—
48C3×C4⋊Dic3C2×D318yes—
49C3×Dic3⋊C4—28no—
50C3×Dic3⋊C4—68no—
51C22×C33C2×D411yescharacteristic
52C33×C4—22no—
53C33×C4C2×Q811yescharacteristic
54C33×C4C2×Q811yescharacteristic
55C3×C3⋊Dic3—42no—
56C32×Dic3—42no—
57C3×#125—34no—
58C3×#125—91no—
59#870—34no—
60#870C3⋊D311yescharacteristic
61#808—98no—
62#808—92no—
63#807—98no—
64#807—92no—
65C4×C3⋊C8—94no—
66C4×C3⋊C8—91no—
67C4×C3⋊C8—184no—
68C2×C32×C4—28no—
69C2×C32×C4—24no—
70C2×C32×C4—62no—
71C2×C32×C4#6718yes—
72C2×C32×C4#6712yes—
73C2×C32×C4C22×D314yes—
74C2×C32×C4C22×D311yescharacteristic
75C2×C3⋊Dic3—22no—
76C2×C3×Dic3—28no—
77C2×C3×Dic3—68no—
78C3⋊(C3⋊C8)—32no—
79C3⋊(C3⋊C8)—32no—
80C3⋊(C3⋊C8)—38no—
81C3⋊(C3⋊C8)—38no—
82C3⋊(C3⋊C8)—68no—
83C3⋊(C3⋊C8)—68no—
84#499—271noSylow
85C2×C33C4⋊Q811yescharacteristic
86C2×C33#13512yes—
87C3×C4⋊C4—38no—
88C3×C4⋊C4—68no—
89C3×C4⋊C4—92no—
90C3×C4⋊C4—182no—
91C3×C42—24no—
92C3×C42D3214yes—
93C3×C42C2×C3⋊D311yescharacteristic
94C4⋊Dic3—38no—
95C4⋊Dic3C2×C3⋊D312yes—
96Dic3⋊C4—22no—
97Dic3⋊C4—68no—
98C2×C3⋊C8—94no—
99C2×C3⋊C8—94no—
100C2×C3⋊C8—91no—
101C2×C3⋊C8—91no—
102C2×C3⋊C8—184no—
103C2×C3⋊C8—184no—
104C22×C32—24no—
105C22×C32C2×#6714yes—
106C22×C32C2×#6711yescharacteristic
107C32×C4—216no—
108C32×C4—28no—
109C32×C4—22no—
110C32×C4—24no—
111C32×C4—24no—
112C32×C4—122no—
113C32×C4D3×Q814yes—
114C32×C4D3×Q814yes—
115C32×C4D3×Q811yescharacteristic
116C32×C4D3×Q811yescharacteristic
117C3⋊Dic3—42no—
118C3×Dic3—48no—
119C3×Dic3—128no—
120#125—31no—
121#125—91no—
122#107—272no—
123#106—272no—
124C4×C8—271no—
125C33#49911yescharacteristic, complemented, Sylow
126C2×C3×C4—28no—
127C2×C3×C4—24no—
128C2×C3×C4—68no—
129C2×C3×C4—182no—
130C2×C3×C4#62518yes—
131C2×C3×C4#63812yes—
132C2×C3×C4C2×D3214yes—
133C2×C3×C4C22×C3⋊D311yescharacteristic
134C2×Dic3—22no—
135C2×Dic3—68no—
136C3⋊C8—98no—
137C3⋊C8—98no—
138C3⋊C8—92no—
139C3⋊C8—92no—
140C3⋊C8—188no—
141C3⋊C8—188no—
142C2×C32—28no—
143C2×C32—24no—
144C2×C32#94414yes—
145C2×C32#94411yescharacteristic
146C2×C32#95018yes—
147C2×C32#95012yes—
148C2×C8—271no—
149C2×C8—271no—
150C4⋊C4—32no—
151C4⋊C4—62no—
152C4⋊C4—92no—
153C4⋊C4—182no—
154C42D3×C3⋊D311yescharacteristic
155C22×C3—24no—
156C22×C3C2×#62514yes—
157C22×C3C2×#63811yescharacteristic
158C3×C4—216no—
159C3×C4—28no—
160C3×C4—22no—
161C3×C4—24no—
162C3×C4—24no—
163C3×C4—128no—
164C3×C4—362no—
165C3×C4#379714yes—
166C3×C4#379714yes—
167C3×C4Q8×C3⋊D311yescharacteristic
168C3×C4Q8×C3⋊D311yescharacteristic
169Dic3—42no—
170Dic3—128no—
171C32—24no—
172C32#506514yescomplemented
173C32#506511yescharacteristic, complemented
174C2×C4—62no—
175C2×C4—182no—
176C2×C4#629012yes—
177C2×C4C2×D3×C3⋊D311yescharacteristic
178C8—272no—
179C8—272no—
180C2×C3—28no—
181C2×C3—24no—
182C2×C3#6382218yes—
183C2×C3#6387314yes—
184C2×C3#6415311yescharacteristic
185C2×C3#6415912yes—
186C22C2×#629011yescharacteristic
187C4—22no—
188C4—122no—
189C4—362no—
190C4#8901911yescharacteristic
191C4#8901911yescharacteristic
192C3—24no—
193C3#1059031414yescomplemented
194C3#1059276411yescharacteristic, complemented
195C2#1172336312yes—
196C2#1172345111yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC42D3×C3⋊D3
hypocenterC33#499
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
125. C33#499semidirect
172. C32#5065semidirect
173. C32#5065semidirect
193. C3#10590314semidirect
194. C3#10592764semidirect

Maximal subgroups

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

subgroupindex
C3×#640792
C32×#8702
C4×C3⋊(C3⋊(C3⋊C8))2
#117220562
#117220532
#105927643
#105903143