Finite Group Databasebeta

← #13147407

Subgroups — #13147407

Order 1728.

371 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 59 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×#88631C212yesmaximal, complemented
2D4×#1084C211yesmaximal, characteristic
3C22⋊#6188C212yesmaximal, complemented
4C4⋊#6188C211yesmaximal, characteristic, complemented
5C4×#6188C211yesmaximal, characteristic, complemented
6C2×D4×H(3,𝔽3)C411yescharacteristic
7C22×#1084C2212yes—
8#88631—24no—
9C2×#6188C2211yescharacteristic
10C2×#6188C2212yes—
11C2×#6188C2211yescharacteristic
12#88552C412yes—
13C4⋊#1084C411yescharacteristic
14C4×#1084C2211yescharacteristic
15C3×D4×Dic3—62no—
16C23×H(3,𝔽3)C2×C412yes—
17D4×H(3,𝔽3)—22no—
18C2×C4×H(3,𝔽3)C2×C411yescharacteristic
19C2×#1084—22no—
20C2×#1084—22no—
21C2×#1084C2×C412yes—
22C2×#1084C2×C411yescharacteristic
23C2×#1084C2311yescharacteristic
24#6188—22no—
25#6188—21no—
26#6188D412yescomplemented
27C3×#436—91nomaximal
28C2×C32×D4—22no—
29C22×C3×Dic3—64no—
30C3×#217—64no—
31C3×C4⋊Dic3—62no—
32C3×C4×Dic3—62no—
33C22×H(3,𝔽3)—22no—
34C22×H(3,𝔽3)—22no—
35C22×H(3,𝔽3)—22no—
36C22×H(3,𝔽3)C22×C411yescharacteristic
37C4×H(3,𝔽3)M1612yescomplemented
38#1084—22no—
39#1084—21no—
40#1084C2×D411yescharacteristic
41#1084#4111yescharacteristic
42C2×C3×M16—92no—
43C3×C4×D4—91no—
44D4×Dic3—62no—
45C3×C4⋊C8—91no—
46C3×C22⋊C8—92no—
47C3×C4×C8—91no—
48C23×C32—24no—
49C32×D4—28no—
50C2×C32×C4—22no—
51C2×C3×Dic3—62no—
52C2×C3×Dic3—62no—
53C2×C3×Dic3—64no—
54C2×C3×Dic3—124no—
55C2×C3×Dic3—124no—
56#436—91noSylow
57C2×H(3,𝔽3)—42no—
58C2×H(3,𝔽3)C4×D411yescharacteristic
59C2×H(3,𝔽3)C2×M1611yescharacteristic
60C2×H(3,𝔽3)#13111yescharacteristic
61C2×C3×D4—62no—
62C2×C3×D4#15711yescharacteristic
63C22×C3×C4—92no—
64C22×Dic3—64no—
65C3×M16—184no—
66C2×C3×C8—91no—
67C2×C3×C8—91no—
68C2×C3×C8—92no—
69C3×C4⋊C4—91no—
70C3×C22⋊C4—92no—
71C3×C42—91no—
72#217—64no—
73C4⋊Dic3—62no—
74C4×Dic3—62no—
75C22×C32—28no—
76C22×C32—22no—
77C22×C32—44no—
78C22×C32—44no—
79C32×C4—24no—
80C3×Dic3—62no—
81C3×Dic3—62no—
82C3×Dic3—122no—
83C3×Dic3—124no—
84C2×M16—92no—
85C4×D4—91no—
86C4⋊C8—91no—
87C22⋊C8—92no—
88C4×C8—91no—
89H(3,𝔽3)#43611yescharacteristic, complemented, Sylow
90C23×C3—64no—
91C23×C3C2×#15712yes—
92C3×D4—22no—
93C3×D4—68no—
94C2×C3×C4—62no—
95C2×C3×C4—91no—
96C2×C3×C4—91no—
97C2×C3×C4—92no—
98C2×C3×C4—182no—
99C2×C3×C4—182no—
100C2×C3×C4C2×#15711yescharacteristic
101C2×Dic3—62no—
102C2×Dic3—62no—
103C2×Dic3—64no—
104C2×Dic3—124no—
105C2×Dic3—124no—
106C3×C8—92no—
107C3×C8—181no—
108C3×C8—182no—
109C2×C32—22no—
110C2×C32—22no—
111C2×C32—22no—
112C2×C32—48no—
113C2×D4#108811yescharacteristic
114C22×C4—92no—
115M16—184no—
116C2×C8—91no—
117C2×C8—91no—
118C2×C8—92no—
119C4⋊C4—91no—
120C22⋊C4—92no—
121C42—91no—
122C22×C3—22no—
123C22×C3—22no—
124C22×C3—22no—
125C22×C3—68no—
126C22×C3—62no—
127C22×C3—124no—
128C22×C3—124no—
129C22×C3C22×#15711yescharacteristic
130C3×C4—64no—
131C3×C4—91no—
132C3×C4—91no—
133C3×C4—181no—
134C3×C4—182no—
135C3×C4#379212yes—
136Dic3—62no—
137Dic3—62no—
138Dic3—124no—
139Dic3—122no—
140C32—22no—
141C23C2×#108812yes—
142D4—22no—
143C2×C4—91no—
144C2×C4—91no—
145C2×C4—92no—
146C2×C4—182no—
147C2×C4—182no—
148C2×C4C2×#108811yescharacteristic
149C8—92no—
150C8—182no—
151C8—181no—
152C2×C3—42no—
153C2×C3—62no—
154C2×C3—62no—
155C2×C3—62no—
156C2×C3—128no—
157C2×C3#6428711yescharacteristic
158C2×C3D4×#15711yescharacteristic
159C2×C3C2×#379211yescharacteristic
160C22—22no—
161C22—22no—
162C22—22no—
163C22C22×#108811yescharacteristic
164C4—91no—
165C4—91no—
166C4—181no—
167C4—182no—
168C4#8863112yescomplemented
169C3—62no—
170C3#1059382111yescharacteristic
171C2—42no—
172C2#1172244911yescharacteristic
173C2D4×#108811yescharacteristic
174C2C2×#8863111yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
Fitting, prime radicalC2×D4×H(3,𝔽3)C4
cosocleC2×#1084C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#436
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88631C2semidirect
3. C22⋊#6188C2semidirect
4. C4⋊#6188C2semidirect
5. C4×#6188C2semidirect
26. #6188D4semidirect
37. C4×H(3,𝔽3)M16semidirect
89. H(3,𝔽3)#436semidirect
168. C4#88631semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
D4×#10842
C22⋊#61882
C4⋊#61882
C4×#61882
C3×#4369