Finite Group Databasebeta

← #13135095

Subgroups — #13135095

Order 1728.

215 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 55 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×#3746⋊C3C211yesmaximal, characteristic
2#11720545C211yesmaximal, characteristic
3#63646⋊C3C211yesmaximal, characteristic
4#10589131—31nomaximal
5#10589130—31nomaximal
6C2×#634⋊C3—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#3748⋊C3—21no—
9#3746⋊C3—21no—
10#3746⋊C3C412yes—
11#88386—21no—
12C2×#3746D311yescharacteristic
13C2×C3×Dic12—31no—
14#63646D311yescharacteristic
15C3×#819—31no—
16C3×#812—31no—
17C3×#812—31no—
18C3×#812—61no—
19C2×C4×H(3,𝔽3)D411yescharacteristic
20C2×(C3⋊Dic3)⋊C3—41no—
21#634⋊C3—22no—
22#634⋊C3—41no—
23C8×H(3,𝔽3)C2×C411yescharacteristic
24C8×H(3,𝔽3)D411yescharacteristic
25(C3⋊(C3⋊C8))⋊C3—41no—
26#4565—91no—
27#4517—91no—
28#4496—91no—
29C2×#634—21no—
30C2×C3×Dic6—61no—
31C2×C32×C8—21no—
32C2×C32×C8C2×D311yescharacteristic
33C2×C32×C8C2×D311yescharacteristic
34#3748—21no—
35#3746—21no—
36#3746Dic312yes—
37C3×#208—61no—
38C3×Dic12—32no—
39C3×Dic12—61no—
40C3×C3⋊C16—62no—
41C3×C3⋊C16—61no—
42C3×C3⋊C16—61no—
43C22×H(3,𝔽3)SD1611yescharacteristic
44C4×H(3,𝔽3)C22⋊C411yescharacteristic
45C4×H(3,𝔽3)D811yescharacteristic
46(C3⋊Dic3)⋊C3—42no—
47C2×C3×Q16—91no—
48C2×Dic12—31no—
49C2×Dic12—91no—
50C3×M32—91no—
51C3×#108—91no—
52#819—31no—
53#819—91no—
54#812—31no—
55#812—31no—
56#812—61no—
57C2×C32×C4—21no—
58C2×C32×C4#6711yescharacteristic
59C2×C32×C4#6711yescharacteristic
60C2×C3⋊Dic3—41no—
61#634—22no—
62#634—41no—
63C2×C3×Dic3—121no—
64C3×Dic6—62no—
65C3×Dic6—121no—
66C32×C8—21no—
67C32×C8—21no—
68C32×C8C4×D311yescharacteristic
69C32×C8D1211yescharacteristic
70C32×C8C2×Dic311yescharacteristic
71C32×C8#6711yescharacteristic
72C3⋊(C3⋊C8)—41no—
73C3×C3⋊C8—121no—
74#362—271noSylow
75C2×H(3,𝔽3)—21no—
76C2×H(3,𝔽3)#10211yescharacteristic
77C2×C3×Q8—181no—
78C2×Dic6—61no—
79C2×Dic6—181no—
80C3×Q16—92no—
81C3×Q16—181no—
82C3×M16—181no—
83C2×C3×C8—31no—
84C2×C3×C8—31no—
85C2×C3×C8—61no—
86C2×C3×C8D3211yescharacteristic
87#208—61no—
88#208—181no—
89Dic12—32no—
90Dic12—61no—
91Dic12—92no—
92Dic12—181no—
93C3×C16—181no—
94C3⋊C16—61no—
95C3⋊C16—61no—
96C3⋊C16—62no—
97C22×C32—21no—
98C22×C32#21411yescharacteristic
99C22×C32#21511yescharacteristic
100C32×C4—21no—
101C32×C4—21no—
102C32×C4(C2×D3)⋊C411yescharacteristic
103C32×C4#21311yescharacteristic
104C32×C4#21311yescharacteristic
105C32×C4#21711yescharacteristic
106C3⋊Dic3—42no—
107C3×Dic3—122no—
108C2×Q16—91no—
109M32—91no—
110#108—91no—
111H(3,𝔽3)#36211yescharacteristic, complemented, Sylow
112C3×Q8—182no—
113C3×Q8—361no—
114C2×C3×C4—31no—
115C2×C3×C4—31no—
116C2×C3×C4—61no—
117C2×C3×C4—361no—
118C2×C3×C4#62511yescharacteristic
119C2×Dic3—121no—
120C2×Dic3—361no—
121Dic6—62no—
122Dic6—121no—
123Dic6—182no—
124Dic6—361no—
125C3×C8—31no—
126C3×C8—31no—
127C3×C8—31no—
128C3×C8—31no—
129C3×C8—61no—
130C3×C8—61no—
131C3×C8—361no—
132C3×C8D3×Dic311yescharacteristic
133C3×C8#62611yescharacteristic
134C3⋊C8—121no—
135C3⋊C8—361no—
136C2×C32—21no—
137C2×C32—21no—
138C2×C32—21no—
139C2×C32—41no—
140C2×C32#80911yescharacteristic
141C2×C32#83211yescharacteristic
142C2×Q8—181no—
143Q16—92no—
144Q16—181no—
145M16—181no—
146C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
147C16—181no—
148C22×C3—31no—
149C22×C3—31no—
150C22×C3—61no—
151C22×C3#371511yescharacteristic
152C3×C4—31no—
153C3×C4—31no—
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—61no—
157C3×C4—61no—
158C3×C4—362no—
159C3×C4#371311yescharacteristic
160C3×C4(C2×D3)⋊Dic311yescharacteristic
161Dic3—122no—
162Dic3—362no—
163C32—21no—
164C32#449611yescharacteristic, complemented
165C32#456511yescharacteristic, complemented
166Q8—182no—
167Q8—361no—
168C2×C4—361no—
169C2×C4#619811yescharacteristic
170C8—361no—
171C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
172C8#620011yescharacteristic
173C2×C3—21no—
174C2×C3—31no—
175C2×C3—31no—
176C2×C3—61no—
177C2×C3—61no—
178C2×C3—61no—
179C2×C3—121no—
180C2×C3#6356211yescharacteristic
181C22#8843111yescharacteristic
182C4—362no—
183C4#8843211yescharacteristic
184C4#8845011yescharacteristic
185C3—31no—
186C3—31no—
187C3—61no—
188C3#1058912911yescharacteristic
189C2—21no—
190C2#1172031811yescharacteristic
subgroupG/H
centerC2#11720318
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3#63562
prime radicalC22×H(3,𝔽3)SD16
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#362
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
111. H(3,𝔽3)#362semidirect
164. C32#4496semidirect
165. C32#4565semidirect

Maximal subgroups

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

subgroupindex
C2×#3746⋊C32
#117205452
#63646⋊C32
#105891313
#105891303