Finite Group Databasebeta

← #13136915

Subgroups — #13136915

Order 1728.

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

normal subgroups only

This group has 329 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×C4×#1088C211yesmaximal, characteristic
2C2×#88626C211yesmaximal, characteristic
3C2×C8×#272C211yesmaximal, characteristic
4C22×#1088C412yes—
5C2×C4×#272C2211yescharacteristic
6C2×#6188C412yescomplemented
7C4×#1088—21no—
8C4×#1088—21no—
9#88626—21no—
10#88626—21no—
11C2×C8×H(3,𝔽3)C411yescharacteristic, complemented
12C2×#6180C411yescharacteristic, complemented
13C8×#272—21no—
14C8×#272—21no—
15C2×C3×D3×C8—62no—
16C22×#272C2×C411yescharacteristic
17C2×#1088—22no—
18C2×#1088—22no—
19C2×#1088C814yescomplemented
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21C2×#1084C2×C411yescharacteristic
22C4×#272D411yescharacteristic
23C4×#272D411yescharacteristic
24C4×#272D411yescharacteristic
25C4×#272Q811yescharacteristic
26#6188—22no—
27C8×H(3,𝔽3)—21no—
28#6180—21no—
29C3×#336—91nomaximal
30C2×C3×C4×D3—62no—
31C2×C32×C8—22no—
32C2×C3×C3⋊C8—62no—
33C3×D3×C8—64no—
34C3×D3×C8—64no—
35C22×H(3,𝔽3)C4211yescharacteristic
36C2×#272C22⋊C411yescharacteristic
37C2×#272C4⋊C411yescharacteristic
38C2×#272C2×C812yes—
39C2×#272M1612yes—
40#1088—24no—
41C4×H(3,𝔽3)C22⋊C411yescharacteristic
42C4×H(3,𝔽3)C4⋊C411yescharacteristic
43#1084C22⋊C411yescharacteristic
44#1084C4⋊C411yescharacteristic
45C22×C3×C8—91no—
46C22×C3×C8—91no—
47C2×C3×C42—91no—
48C2×D3×C8—62no—
49C22×C3×D3—62no—
50C2×C32×C4—22no—
51C2×C3×Dic3—62no—
52C3×C4×D3—62no—
53C3×C4×D3—62no—
54C3×C4×D3—62no—
55C3×C4×D3—62no—
56C32×C8—24no—
57C3×C3⋊C8—64no—
58#336—91noSylow
59C2×H(3,𝔽3)#9511yescharacteristic
60C2×H(3,𝔽3)C4×C811yescharacteristic
61C2×H(3,𝔽3)#9711yescharacteristic
62#272C22⋊C812yes—
63#272C4⋊C812yes—
64C22×C3×C4—91no—
65C22×C3×C4—92no—
66C2×C4×D3—62no—
67C2×C3×C8—62no—
68C2×C3×C8—92no—
69C2×C3×C8—91no—
70C2×C3×C8—181no—
71C2×C3×C8—181no—
72C2×C3×C8—181no—
73C2×C3×C8—181no—
74C2×C3×C8#15711yescharacteristic
75C3×C42—181no—
76C3×C42—181no—
77C2×C3⋊C8—62no—
78D3×C8—64no—
79D3×C8—64no—
80C22×C32—22no—
81C2×C3×D3—62no—
82C2×C3×D3—62no—
83C2×C3×D3—64no—
84C2×C3×D3—64no—
85C32×C4—22no—
86C32×C4—22no—
87C3×Dic3—62no—
88C3×Dic3—62no—
89C22×C8—91no—
90C22×C8—91no—
91C2×C42—91no—
92H(3,𝔽3)#33611yescharacteristic, complemented, Sylow
93C23×C3—91no—
94C22×D3—62no—
95C2×C3×C4—62no—
96C2×C3×C4—91no—
97C2×C3×C4—91no—
98C2×C3×C4—91no—
99C2×C3×C4—91no—
100C2×C3×C4—91no—
101C2×C3×C4—94no—
102C2×C3×C4—182no—
103C2×C3×C4—182no—
104C2×C3×C4C2×#15711yescharacteristic
105C2×Dic3—62no—
106C4×D3—62no—
107C4×D3—62no—
108C4×D3—62no—
109C4×D3—62no—
110C3×C8—21no—
111C3×C8—64no—
112C3×C8—182no—
113C3×C8—181no—
114C3⋊C8—64no—
115C2×C32—22no—
116C2×C32—22no—
117C2×C32—22no—
118C3×D3—64no—
119C3×D3—64no—
120C22×C4—91no—
121C22×C4—92no—
122C2×C8—92no—
123C2×C8—91no—
124C2×C8—181no—
125C2×C8—181no—
126C2×C8—181no—
127C2×C8—181no—
128C2×C8#108811yescharacteristic, complemented
129C42—181no—
130C42—181no—
131C22×C3—62no—
132C22×C3—92no—
133C22×C3—92no—
134C22×C3—91no—
135C22×C3—91no—
136C22×C3C4×#15711yescharacteristic
137C2×D3—62no—
138C2×D3—62no—
139C2×D3—64no—
140C2×D3—64no—
141C3×C4—62no—
142C3×C4—62no—
143C3×C4—91no—
144C3×C4—91no—
145C3×C4—184no—
146C3×C4C4⋊#15711yescharacteristic
147C3×C4C22⋊#15711yescharacteristic
148Dic3—62no—
149Dic3—62no—
150C32—22no—
151C23—91no—
152C2×C4—91no—
153C2×C4—91no—
154C2×C4—91no—
155C2×C4—91no—
156C2×C4—91no—
157C2×C4—94no—
158C2×C4—182no—
159C2×C4—182no—
160C2×C4C2×#108811yescharacteristic
161C8—21no—
162C8—182no—
163C8—181no—
164C2×C3—62no—
165C2×C3—62no—
166C2×C3—62no—
167C2×C3—92no—
168C2×C3—92no—
169C2×C3C8×#15711yescharacteristic
170C2×C3#6376711yescharacteristic
171C2×C3#6378111yescharacteristic
172D3—64no—
173D3—64no—
174C22—92no—
175C22—92no—
176C22—91no—
177C22—91no—
178C22C4×#108811yescharacteristic
179C4—91no—
180C4—91no—
181C4—184no—
182C4C4⋊#108811yescharacteristic
183C4C22⋊#108811yescharacteristic
184C3—62no—
185C3#1059018111yescharacteristic
186C2—92no—
187C2—92no—
188C2C8×#108811yescharacteristic
189C2#1172077311yescharacteristic
190C2#1172078711yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C2×#157
commutatorC2×H(3,𝔽3)C4×C8
FittingC2×C8×H(3,𝔽3)C4
socleC22×C3C4×#157
prime radicalC22×#272C2×C4
cosocleC2×C4×#272C22
hypercenterC2×C3×C8#157
hypocenterH(3,𝔽3)#336
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
6. C2×#6188C4semidirect
11. C2×C8×H(3,𝔽3)C4semidirect
12. C2×#6180C4semidirect
19. C2×#1088C8semidirect
92. H(3,𝔽3)#336semidirect
128. C2×C8#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×#886262
C2×C8×#2722
C3×#3369