Finite Group Databasebeta

← #13136897

Subgroups — #13136897

Order 1728.

317 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 47 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×C4×#1088C211yesmaximal, characteristic
2C2×#88627C211yesmaximal, characteristic
3#11721760C211yesmaximal, characteristic
4C22×#1088—21no—
5C2×C4×#272C2211yescharacteristic
6C2×#6188—21no—
7C4×#1088—21no—
8C4×#1088C412yescomplemented
9#88627—21no—
10#88627C412yescomplemented
11C4⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
12#88542—21no—
13C4⋊#1084C411yescharacteristic, complemented
14C4×#1084—21no—
15C3×#892—62no—
16C22×#272D411yescharacteristic
17C2×#1088—22no—
18C2×#1088—22no—
19C2×#1088—22no—
20C2×C4×H(3,𝔽3)—21no—
21C2×C4×H(3,𝔽3)C2×C411yescharacteristic
22C2×#1084—21no—
23C2×#1084D411yescharacteristic
24C4×#272C2×C411yescharacteristic
25C4×#272C2×C411yescharacteristic
26C4×#272D411yescharacteristic
27C4×#272Q811yescharacteristic
28#6188—22no—
29C3×#339—91nomaximal
30C2×C3×C4×D3—62no—
31C32×C4⋊C4—22no—
32C3×(C2×D3)⋊C4—64no—
33C3×C4⋊Dic3—62no—
34C3×C4×Dic3—64no—
35C22×H(3,𝔽3)C22⋊C411yescharacteristic
36C2×#272—21no—
37C2×#272—21no—
38C2×#272C22⋊C411yescharacteristic
39C2×#272C4⋊C411yescharacteristic
40#1088—24no—
41C4×H(3,𝔽3)—41no—
42C4×H(3,𝔽3)C4211yescharacteristic
43C4×H(3,𝔽3)C4⋊C411yescharacteristic
44#1084—41no—
45#1084C22⋊C411yescharacteristic
46#1084C4⋊C411yescharacteristic
47C2×C3×M16—91no—
48C3×#117—91no—
49C2×C3×C42—91no—
50#892—62no—
51C22×C3×D3—62no—
52C2×C32×C4—24no—
53C2×C32×C4—22no—
54C2×C3×Dic3—62no—
55C2×C3×Dic3—64no—
56C3×C4×D3—62no—
57C3×C4×D3—62no—
58C3×C4×D3—62no—
59C3×C4×D3—62no—
60#339—91noSylow
61C2×H(3,𝔽3)#9511yescharacteristic
62C2×H(3,𝔽3)#10411yescharacteristic
63C2×H(3,𝔽3)#10411yescharacteristic
64#272—21no—
65#272—21no—
66C22×C3×C4—91no—
67C22×C3×C4—181no—
68C2×C4×D3—62no—
69C3×M16—92no—
70C3×M16—181no—
71C2×C3×C8—181no—
72C3×C4⋊C4—62no—
73C3×C4⋊C4—91no—
74C3×C4⋊C4#15711yescharacteristic
75C3×C22⋊C4—181no—
76C3×C42—92no—
77C3×C42—181no—
78C3×C42—181no—
79(C2×D3)⋊C4—64no—
80C4⋊Dic3—62no—
81C4×Dic3—64no—
82C22×C32—22no—
83C2×C3×D3—62no—
84C2×C3×D3—62no—
85C2×C3×D3—122no—
86C2×C3×D3—122no—
87C32×C4—22no—
88C32×C4—22no—
89C32×C4—44no—
90C3×Dic3—62no—
91C3×Dic3—62no—
92C3×Dic3—124no—
93C2×M16—91no—
94#117—91no—
95C2×C42—91no—
96H(3,𝔽3)#33911yescharacteristic, complemented, Sylow
97C23×C3—91no—
98C22×D3—62no—
99C2×C3×C4—21no—
100C2×C3×C4—64no—
101C2×C3×C4—62no—
102C2×C3×C4—91no—
103C2×C3×C4—91no—
104C2×C3×C4—91no—
105C2×C3×C4—91no—
106C2×C3×C4—91no—
107C2×C3×C4—182no—
108C2×C3×C4—182no—
109C2×C3×C4—182no—
110C2×C3×C4—181no—
111C2×C3×C4C2×#15711yescharacteristic
112C2×Dic3—62no—
113C2×Dic3—64no—
114C4×D3—62no—
115C4×D3—62no—
116C4×D3—62no—
117C4×D3—62no—
118C3×C8—182no—
119C2×C32—22no—
120C2×C32—22no—
121C2×C32—22no—
122C3×D3—122no—
123C3×D3—122no—
124C22×C4—91no—
125C22×C4—181no—
126M16—92no—
127M16—181no—
128C2×C8—181no—
129C4⋊C4—91no—
130C4⋊C4#108811yescharacteristic, complemented
131C22⋊C4—181no—
132C42—92no—
133C42—181no—
134C42—181no—
135C22×C3—62no—
136C22×C3—91no—
137C22×C3—91no—
138C22×C3—181no—
139C22×C3—181no—
140C22×C3C22⋊#15711yescharacteristic
141C2×D3—62no—
142C2×D3—62no—
143C2×D3—122no—
144C2×D3—122no—
145C3×C4—41no—
146C3×C4—62no—
147C3×C4—62no—
148C3×C4—91no—
149C3×C4—91no—
150C3×C4—124no—
151C3×C4—184no—
152C3×C4—361no—
153C3×C4C4×#15711yescharacteristic
154C3×C4C4⋊#15711yescharacteristic
155Dic3—62no—
156Dic3—62no—
157Dic3—124no—
158C32—22no—
159C23—91no—
160C2×C4—21no—
161C2×C4—91no—
162C2×C4—91no—
163C2×C4—91no—
164C2×C4—91no—
165C2×C4—91no—
166C2×C4—182no—
167C2×C4—182no—
168C2×C4—182no—
169C2×C4—181no—
170C2×C4C2×#108811yescharacteristic
171C8—182no—
172C2×C3—62no—
173C2×C3—62no—
174C2×C3—62no—
175C2×C3—181no—
176C2×C3—181no—
177C2×C3#6378111yescharacteristic
178C2×C3#6378311yescharacteristic
179C2×C3#6378511yescharacteristic
180D3—122no—
181D3—122no—
182C22—91no—
183C22—91no—
184C22—181no—
185C22—181no—
186C22C22⋊#108811yescharacteristic
187C4—41no—
188C4—91no—
189C4—91no—
190C4—184no—
191C4—361no—
192C4C4×#108811yescharacteristic
193C4C4⋊#108811yescharacteristic
194C3—62no—
195C3#1059016411yescharacteristic
196C2—181no—
197C2—181no—
198C2#1172078711yescharacteristic
199C2#1172078911yescharacteristic
200C2#1172079111yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC4×H(3,𝔽3)C42
FrattiniC2×C3×C4C2×#157
FittingC4⋊C4×H(3,𝔽3)C4
prime radicalC22×#272D4
cosocleC2×C4×#272C22
hypercenterC3×C4⋊C4#157
hypocenterH(3,𝔽3)#339
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. C4×#1088C4semidirect
10. #88627C4semidirect
11. C4⋊C4×H(3,𝔽3)C4semidirect
13. C4⋊#1084C4semidirect
96. H(3,𝔽3)#339semidirect
130. C4⋊C4#1088semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10882
C2×#886272
#117217602
C3×#3399