Finite Group Databasebeta

← #13146597

Subgroups — #13146597

Order 1728.

459 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 49 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×#88588C214yesmaximal, complemented
2#11721144C212yesmaximal, complemented
3C4×#6188C211yesmaximal, characteristic, complemented
4C2×C4⋊((C3⋊D3)⋊C3)—22no—
5C2×#6198—22no—
6C2×#6198C2214yes—
7C2×#6188C2212yes—
8#88588—28no—
9C4×#1084C2211yescharacteristic
10#63911—62no—
11C22×(C3⋊D3)⋊C3—24no—
12C2×C4×H(3,𝔽3)D411yescharacteristic
13C2×#1084D411yescharacteristic
14C2×#1084C2311yescharacteristic
15C4⋊((C3⋊D3)⋊C3)—44no—
16#6198—28no—
17#6198—44no—
18#6188D414yescomplemented
19#4695—91nomaximal
20C2×C4⋊(C3⋊D3)—22no—
21C2×C3×D12—62no—
22C2×#626—64no—
23C2×#626—64no—
24C3×C4×Dic3—62no—
25C22×H(3,𝔽3)C2×D411yescharacteristic
26C2×(C3⋊D3)⋊C3—44no—
27C2×(C3⋊D3)⋊C3—44no—
28C2×(C3⋊D3)⋊C3—44no—
29C4×H(3,𝔽3)D812yescomplemented
30#1084D812yescomplemented
31#1084C2×D412yes—
32#940—182no—
33C2×D24—94no—
34#874—92no—
35C3×C4×C8—91no—
36C22×C3⋊D3—24no—
37C22×C3×D3—64no—
38C2×C32×C4—22no—
39C4⋊(C3⋊D3)—44no—
40C2×C3×Dic3—62no—
41C2×C3×Dic3—62no—
42C3×D12—124no—
43#626—128no—
44#626—128no—
45#493—271noSylow
46C2×H(3,𝔽3)#12711yescharacteristic
47C2×H(3,𝔽3)C2×D811yescharacteristic
48C2×H(3,𝔽3)C2×D811yescharacteristic
49(C3⋊D3)⋊C3—84no—
50C2×C3×D4—182no—
51C2×#67—184no—
52C2×#67—184no—
53C2×D12—62no—
54C2×D12—94no—
55C2×D12—182no—
56C2×D12—182no—
57C2×C3×C8—92no—
58C3×C42—91no—
59C4×Dic3—62no—
60D24—188no—
61C22×C32—22no—
62C2×C3⋊D3—44no—
63C2×C3⋊D3—44no—
64C2×C3⋊D3—44no—
65C2×C3×D3—124no—
66C2×C3×D3—124no—
67C2×C3×D3—124no—
68C32×C4—24no—
69C3×Dic3—64no—
70C3×Dic3—64no—
71C2×D8—274no—
72#127—272no—
73C4×C8—91no—
74H(3,𝔽3)#49311yescharacteristic, complemented, Sylow
75C23×C3—184no—
76C22×D3—64no—
77C22×D3—184no—
78C3×D4—364no—
79C2×C3×C4—62no—
80C2×C3×C4—91no—
81C2×C3×C4—91no—
82C2×C3×C4GO+(4,𝔽2)11yescharacteristic
83#67—368no—
84#67—368no—
85C2×Dic3—62no—
86C2×Dic3—62no—
87D12—124no—
88D12—188no—
89D12—364no—
90D12—364no—
91C3×C8—94no—
92C2×C32—22no—
93C2×C32—22no—
94C2×C32—22no—
95C3⋊D3—84no—
96C3×D3—244no—
97C2×D4—182no—
98C2×D4—274no—
99C2×D4—542no—
100D8—548no—
101C2×C8—92no—
102C42—91no—
103C22×C3—62no—
104C22×C3—364no—
105C22×C3—364no—
106C22×C3—364no—
107C22×C3C2×GO+(4,𝔽2)11yescharacteristic
108C2×D3—124no—
109C2×D3—124no—
110C2×D3—124no—
111C2×D3—364no—
112C2×D3—364no—
113C2×D3—364no—
114C3×C4—64no—
115C3×C4—92no—
116C3×C4—92no—
117C3×C4#377412yes—
118Dic3—64no—
119Dic3—64no—
120C32—22no—
121C23—184no—
122D4—364no—
123D4—548no—
124D4—1084no—
125C2×C4—91no—
126C2×C4—91no—
127C2×C4#1088⋊C211yescharacteristic
128C8—94no—
129C2×C3—62no—
130C2×C3—62no—
131C2×C3—62no—
132C2×C3—724no—
133C2×C3#6422511yescharacteristic
134C2×C3#6423111yescharacteristic
135C2×C3C2×#377411yescharacteristic
136D3—244no—
137D3—724no—
138C22—364no—
139C22—364no—
140C22—364no—
141C22C2×#1088⋊C211yescharacteristic
142C4—92no—
143C4—92no—
144C4#8858812yescomplemented
145C3—62no—
146C3#1059341811yescharacteristic
147C2—724no—
148C2#1172229811yescharacteristic
149C2#1172230511yescharacteristic
150C2C2×#8858811yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#493
LCSG ▷ C2×#1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88588C2semidirect
2. #11721144C2semidirect
3. C4×#6188C2semidirect
18. #6188D4semidirect
29. C4×H(3,𝔽3)D8semidirect
30. #1084D8semidirect
74. H(3,𝔽3)#493semidirect
144. C4#88588semidirect

Maximal subgroups

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

subgroupindex
C2×#885882
#117211442
C4×#61882
#46959