Finite Group Databasebeta

← #13146477

Subgroups — #13146477

Order 1728.

283 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 41 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
1#11722296C211yesmaximal, characteristic
2#11720842C212yesmaximal, complemented
3C8×#1088C211yesmaximal, characteristic, complemented
4#11720697C211yesmaximal, characteristic, complemented
5#11720694C212yesmaximal, complemented
6#88650C2212yes—
7C4×#1088C2211yescharacteristic
8#88626C2211yescharacteristic
9#88589C411yescharacteristic, complemented
10#88588C411yescharacteristic, complemented
11#88587C412yescomplemented
12C8×#272C2211yescharacteristic
13#3743⋊C3—22no—
14#88435C2212yes—
15((C3⋊D3)⋊C3)⋊C8—22no—
16#63804—62no—
17C2×#1088—21no—
18C4×#272C2311yescharacteristic
19C4×(C3⋊D3)⋊C3—22no—
20#6200—22no—
21#6198C2×C412yes—
22#6196C2×C412yes—
23#6188C2×C411yescharacteristic
24#6188C2×C411yescharacteristic
25C8×H(3,𝔽3)D411yescharacteristic
26#6180D411yescharacteristic
27(C3⋊(C3⋊C8))⋊C3—22no—
28#4700—91nomaximal
29#3798—62no—
30#3743—22no—
31C3×ΓL(1,𝔽25)—62no—
32C3×D3×C8—62no—
33#3711—62no—
34#3710—62no—
35D3×C3⋊C8—62no—
36C2×#272#4111yescharacteristic
37C2×(C3⋊D3)⋊C3—22no—
38#1088—22no—
39C4×H(3,𝔽3)C2×D411yescharacteristic
40#1084C22×C411yescharacteristic
41(C3⋊Dic3)⋊C3—22no—
42#907—182no—
43#904—91no—
44#901—92no—
45C3×C4×C8—91no—
46#819—91no—
47#805—92no—
48C4×C3⋊D3—22no—
49C3×C4×D3—62no—
50C3×C4×D3—62no—
51#627—62no—
52#626—62no—
53#626—62no—
54#625—62no—
55C32×C8—22no—
56C3⋊(C3⋊C8)—22no—
57C3×C3⋊C8—62no—
58C3×C3⋊C8—62no—
59#443—271noSylow
60C2×H(3,𝔽3)C4×D411yescharacteristic
61#272—21no—
62(C3⋊D3)⋊C3—42no—
63#235—92no—
64#235—182no—
65C3×M16—182no—
66C2×C3×C8—91no—
67C2×C3×C8—91no—
68C3×C42—91no—
69#208—92no—
70C2×C3⋊C8—182no—
71Dic12—91no—
72D24—91no—
73#204—92no—
74ΓL(1,𝔽25)—62no—
75ΓL(1,𝔽25)—182no—
76ΓL(1,𝔽25)—182no—
77D3×C8—62no—
78D3×C8—182no—
79D3×C8—182no—
80C2×C3⋊D3—22no—
81C2×C3×D3—62no—
82C2×C3×D3—62no—
83C32×C4—22no—
84C3⋊Dic3—22no—
85C3×Dic3—62no—
86C3×Dic3—62no—
87#135—271no—
88#131—272no—
89#108—271no—
90#104—272no—
91C4×C8—91no—
92H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
93C2×C3×C4—91no—
94C2×C3×C4—182no—
95C2×C3×C4—181no—
96#67—182no—
97#67—182no—
98#67—182no—
99D12—92no—
100D12—182no—
101C4×D3—62no—
102C4×D3—62no—
103C4×D3—182no—
104Dic6—92no—
105Dic6—182no—
106C3×C8—62no—
107C3×C8—91no—
108C3×C8—91no—
109C3×C8—91no—
110C3×C8—182no—
111C3×C8GO+(4,𝔽2)11yescharacteristic
112C3⋊C8—62no—
113C3⋊C8—62no—
114C3⋊C8—182no—
115C2×C32—22no—
116C3⋊D3—42no—
117C3×D3—122no—
118C3×D3—122no—
119#41—272no—
120Q16—271no—
121SD16—272no—
122D8—271no—
123M16—182no—
124M16—272no—
125C2×C8—91no—
126C2×C8—91no—
127C2×C8—542no—
128C42—91no—
129C22×C3—91no—
130C22×C3—182no—
131C2×D3—62no—
132C2×D3—62no—
133C2×D3—182no—
134C3×C4—62no—
135C3×C4—91no—
136C3×C4—182no—
137C3×C4—182no—
138C3×C4C2×GO+(4,𝔽2)11yescharacteristic
139Dic3—62no—
140Dic3—62no—
141Dic3—182no—
142C32—22no—
143Q8—272no—
144D4—272no—
145D4—542no—
146C2×C4—91no—
147C2×C4—181no—
148C2×C4—182no—
149C8—91no—
150C8—91no—
151C8—91no—
152C8—182no—
153C8#1088⋊C211yescharacteristic
154C2×C3—62no—
155C2×C3—181no—
156C2×C3—362no—
157C2×C3C4×GO+(4,𝔽2)11yescharacteristic
158D3—122no—
159D3—122no—
160D3—362no—
161C22—91no—
162C22—182no—
163C4—91no—
164C4—182no—
165C4—182no—
166C4C2×#1088⋊C211yescharacteristic
167C3—62no—
168C3#1059329911yescharacteristic
169C2—181no—
170C2—362no—
171C2C4×#1088⋊C211yescharacteristic
subgroupG/H
centerC4C2×#1088⋊C2
commutator#1084C22×C4
FrattiniC3×C4C2×GO+(4,𝔽2)
FittingC8×H(3,𝔽3)D4
socleC2×C3C4×GO+(4,𝔽2)
prime radical#11722296C2
cosocleC4×#272C23
hypercenterC8#1088⋊C2
hypocenterH(3,𝔽3)#443
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11720842C2semidirect
3. C8×#1088C2semidirect
4. #11720697C2semidirect
5. #11720694C2semidirect
9. #88589C4semidirect
10. #88588C4semidirect
11. #88587C4semidirect
92. H(3,𝔽3)#443semidirect

Maximal subgroups

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

subgroupindex
#117222962
#117208422
C8×#10882
#117206972
#117206942
#47009