Finite Group Databasebeta

← #13138804

Subgroups — #13138804

Order 1728.

611 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 83 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×#3745⋊C3C212yesmaximal, complemented
2#11721144C212yesmaximal, complemented
3C2×#88432C212yesmaximal, complemented
4C8×#1084C211yesmaximal, characteristic, complemented
5#10590880—32nomaximal
6C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
7C2×#6198—24no—
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C4×#1084C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11#3745⋊C3—24no—
12#88432—24no—
13C2×#3745D312yescomplemented
14C2×C3×D24—32no—
15#63911—34no—
16C2×#3714—34no—
17C3×C8×Dic3—32no—
18C3×C8×Dic3—61no—
19C22×(C3⋊D3)⋊C3—24no—
20C2×C4×H(3,𝔽3)C2311yescharacteristic
21C2×#1084D412yes—
22C4⋊((C3⋊D3)⋊C3)—24no—
23C4⋊((C3⋊D3)⋊C3)—24no—
24#6198—48no—
25C8×H(3,𝔽3)D412yescomplemented
26#6180D412yescomplemented
27#5151—92no—
28#4695—91no—
29C2×C4⋊(C3⋊D3)C2×D314yes—
30C2×C3×D12—34no—
31C2×#626—68no—
32C2×C32×C8—21no—
33C2×C32×C8C2×D312yes—
34#3745—24no—
35C3×C4×Dic3—32no—
36C3×C4×Dic3—61no—
37C2×C3×C3⋊C8—32no—
38C2×C3×C3⋊C8—61no—
39C3×D24—64no—
40#3714—68no—
41C22×H(3,𝔽3)C2×D411yescharacteristic
42C2×(C3⋊D3)⋊C3—44no—
43C2×(C3⋊D3)⋊C3—48no—
44C4×H(3,𝔽3)C2×D411yescharacteristic
45C4×H(3,𝔽3)C2×D411yescharacteristic
46#1084D814yescomplemented
47C2×C3×D8—92no—
48#940—94no—
49C2×#213—94no—
50C2×D24—32no—
51C2×D24—92no—
52C2×D24—92no—
53#874—92no—
54C3×C4×C8—91no—
55C8×Dic3—32no—
56C8×Dic3—61no—
57C22×C3⋊D3—24no—
58C22×C3×D3—64no—
59C2×C32×C4—21no—
60C2×C32×C4C22×D312yes—
61C4⋊(C3⋊D3)—24no—
62C4⋊(C3⋊D3)—24no—
63C2×C3×Dic3—34no—
64C2×C3×Dic3—62no—
65C3×D12—64no—
66C3×D12—64no—
67#626—1216no—
68C32×C8—22no—
69C32×C8#6714yescomplemented
70C3×C3⋊C8—34no—
71C3×C3⋊C8—62no—
72#493—271noSylow
73C2×H(3,𝔽3)#12711yescharacteristic
74C2×H(3,𝔽3)C2×D812yes—
75(C3⋊D3)⋊C3—84no—
76C2×C3×D4—94no—
77C2×#67—188no—
78C2×D12—34no—
79C2×D12—94no—
80C2×D12—184no—
81C3×D8—184no—
82C2×C3×C8—32no—
83C2×C3×C8—61no—
84C2×C3×C8—91no—
85C2×C3×C8D3211yescharacteristic
86C3×C42—91no—
87#213—188no—
88C4×Dic3—32no—
89C4×Dic3—61no—
90C2×C3⋊C8—32no—
91C2×C3⋊C8—61no—
92D24—64no—
93D24—184no—
94D24—184no—
95C22×C32—21no—
96C22×C32D3×D412yes—
97C2×C3⋊D3—48no—
98C2×C3⋊D3—44no—
99C2×C3×D3—128no—
100C2×C3×D3—124no—
101C32×C4—21no—
102C32×C4—21no—
103C32×C4D3×D412yes—
104C32×C4C2×#6712yes—
105C3×Dic3—38no—
106C3×Dic3—64no—
107C2×D8—92no—
108C2×D8—272no—
109#127—272no—
110C4×C8—91no—
111H(3,𝔽3)#49311yescharacteristic, complemented, Sylow
112C23×C3—184no—
113C22×D3—64no—
114C22×D3—184no—
115C3×D4—184no—
116C3×D4—184no—
117C2×C3×C4—32no—
118C2×C3×C4—61no—
119C2×C3×C4—92no—
120C2×C3×C4C2×D3211yescharacteristic
121#67—3616no—
122C2×Dic3—34no—
123C2×Dic3—62no—
124D12—64no—
125D12—64no—
126D12—184no—
127D12—184no—
128D12—368no—
129C3×C8—34no—
130C3×C8—62no—
131C3×C8—92no—
132C3×C8#62512yes—
133C3⋊C8—34no—
134C3⋊C8—62no—
135C2×C32—22no—
136C2×C32—21no—
137C2×C32D3×D814yes—
138C2×C32#94012yes—
139C3⋊D3—84no—
140C3×D3—244no—
141C2×D4—94no—
142C2×D4—544no—
143D8—184no—
144D8—544no—
145C2×C8—91no—
146C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
147C42—91no—
148C22×C3—32no—
149C22×C3—61no—
150C22×C3—368no—
151C22×C3—364no—
152C22×C3#380211yescharacteristic
153C2×D3—128no—
154C2×D3—124no—
155C2×D3—364no—
156C2×D3—368no—
157C3×C4—32no—
158C3×C4—32no—
159C3×C4—61no—
160C3×C4—61no—
161C3×C4—94no—
162C3×C4#380211yescharacteristic
163C3×C4C2×#62511yescharacteristic
164Dic3—38no—
165Dic3—64no—
166C32—21no—
167C32#515112yescomplemented
168C23—184no—
169D4—184no—
170D4—184no—
171D4—1088no—
172C2×C4—92no—
173C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
174C8—92no—
175C8#619812yescomplemented
176C2×C3—34no—
177C2×C3—32no—
178C2×C3—62no—
179C2×C3—61no—
180C2×C3—724no—
181C2×C3#6379712yes—
182C2×C3#6391411yescharacteristic
183D3—244no—
184D3—724no—
185C22—368no—
186C22—364no—
187C22#8865511yescharacteristic
188C4—94no—
189C4#8865511yescharacteristic
190C4C2×#619811yescharacteristic
191C3—32no—
192C3—61no—
193C3#1059088611yescharacteristic
194C2—724no—
195C2#1172083812yes—
196C2#1172114411yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#493
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#3745⋊C3C2semidirect
2. #11721144C2semidirect
3. C2×#88432C2semidirect
4. C8×#1084C2semidirect
13. C2×#3745D3semidirect
25. C8×H(3,𝔽3)D4semidirect
26. #6180D4semidirect
46. #1084D8semidirect
69. C32×C8#67semidirect
111. H(3,𝔽3)#493semidirect
167. C32#5151semidirect
175. C8#6198semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#117211442
C2×#884322
C8×#10842
#105908803