Finite Group Databasebeta

← #13147388

Subgroups — #13147388

Order 1728.

405 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 71 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
2C2×C4×#1084C211yesmaximal, characteristic
3#11720785C212yesmaximal, complemented
4C4⋊#6188C212yesmaximal, complemented
5C22×C4×H(3,𝔽3)C411yescharacteristic
6C22×#1084C411yescharacteristic
7C22×#1084C2211yescharacteristic
8#88631—24no—
9C2×#6188C2214yes—
10C4×#1084C411yescharacteristic
11C4×#1084C411yescharacteristic
12C4×#1084C2211yescharacteristic
13C4×#1084C2211yescharacteristic
14C2×C3×C4×Dic3—62no—
15C23×H(3,𝔽3)C2×C411yescharacteristic
16C2×C4×H(3,𝔽3)—22no—
17C2×C4×H(3,𝔽3)C2×C411yescharacteristic
18C2×C4×H(3,𝔽3)C2×C411yescharacteristic
19C2×#1084—22no—
20C2×#1084C2×C411yescharacteristic
21C2×#1084C2×C411yescharacteristic
22C2×#1084C2×C411yescharacteristic
23C2×#1084D412yes—
24C2×#1084Q812yes—
25C2×#1084C2311yescharacteristic
26#6188—24no—
27C3×#423—91nomaximal
28C22×C32×C4—22no—
29C22×C3×Dic3—62no—
30C22×C3×Dic3—62no—
31C3×C4×Dic3—62no—
32C3×C4×Dic3—62no—
33C3×C4×Dic3—62no—
34C3×C4×Dic3—62no—
35C22×H(3,𝔽3)—21no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)C4⋊C412yes—
38C22×H(3,𝔽3)C22×C411yescharacteristic
39C4×H(3,𝔽3)—21no—
40C4×H(3,𝔽3)M1612yescomplemented
41#1084—21no—
42#1084C4⋊C412yes—
43#1084M1612yescomplemented
44#1084C2×D411yescharacteristic
45#1084C2×Q811yescharacteristic
46C2×C3×M16—92no—
47C2×C3×C42—91no—
48C2×C4×Dic3—62no—
49C3×C4⋊C8—92no—
50C3×C4⋊C8—92no—
51C23×C32—22no—
52C2×C32×C4—28no—
53C2×C32×C4—22no—
54C2×C32×C4—22no—
55C2×C3×Dic3—62no—
56C2×C3×Dic3—62no—
57C2×C3×Dic3—62no—
58C2×C3×Dic3—64no—
59C2×C3×Dic3—64no—
60C2×C3×Dic3—62no—
61C2×C3×Dic3—68no—
62#423—91noSylow
63C2×H(3,𝔽3)—22no—
64C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
65C2×H(3,𝔽3)C2×M1611yescharacteristic
66C2×H(3,𝔽3)C2×M1611yescharacteristic
67C22×C3×C4—62no—
68C22×C3×C4—91no—
69C22×C3×C4—91no—
70C22×C3×C4#15711yescharacteristic
71C22×Dic3—62no—
72C22×Dic3—62no—
73C3×M16—184no—
74C2×C3×C8—94no—
75C3×C42—91no—
76C3×C42—91no—
77C3×C42—91no—
78C3×C42—91no—
79C4×Dic3—62no—
80C4×Dic3—62no—
81C4×Dic3—62no—
82C4×Dic3—62no—
83C22×C32—24no—
84C22×C32—24no—
85C22×C32—22no—
86C22×C32—24no—
87C32×C4—24no—
88C32×C4—24no—
89C3×Dic3—64no—
90C3×Dic3—64no—
91C3×Dic3—62no—
92C3×Dic3—62no—
93C3×Dic3—64no—
94C2×M16—92no—
95C2×C42—91no—
96C4⋊C8—92no—
97C4⋊C8—92no—
98H(3,𝔽3)#42311yescharacteristic, complemented, Sylow
99C23×C3—62no—
100C23×C3C2×#15711yescharacteristic
101C2×C3×C4—22no—
102C2×C3×C4—68no—
103C2×C3×C4—62no—
104C2×C3×C4—62no—
105C2×C3×C4—91no—
106C2×C3×C4—91no—
107C2×C3×C4—91no—
108C2×C3×C4—92no—
109C2×C3×C4—92no—
110C2×C3×C4—91no—
111C2×C3×C4—182no—
112C2×C3×C4C2×#15711yescharacteristic
113C2×C3×C4C2×#15711yescharacteristic
114C2×Dic3—62no—
115C2×Dic3—62no—
116C2×Dic3—62no—
117C2×Dic3—62no—
118C2×Dic3—64no—
119C2×Dic3—64no—
120C2×Dic3—68no—
121C3×C8—184no—
122C2×C32—28no—
123C2×C32—22no—
124C2×C32—22no—
125C2×C32—22no—
126C22×C4—91no—
127C22×C4—91no—
128C22×C4#108811yescharacteristic
129M16—184no—
130C2×C8—94no—
131C42—91no—
132C42—91no—
133C42—91no—
134C42—91no—
135C22×C3—21no—
136C22×C3—21no—
137C22×C3—64no—
138C22×C3—64no—
139C22×C3—64no—
140C22×C3—62no—
141C22×C3C4⋊#15712yes—
142C22×C3C22×#15711yescharacteristic
143C3×C4—21no—
144C3×C4—64no—
145C3×C4—64no—
146C3×C4—92no—
147C3×C4—91no—
148C3×C4—91no—
149C3×C4—92no—
150C3×C4—181no—
151C3×C4#379212yes—
152Dic3—64no—
153Dic3—62no—
154Dic3—64no—
155Dic3—62no—
156Dic3—64no—
157C32—22no—
158C23C2×#108811yescharacteristic
159C2×C4—22no—
160C2×C4—92no—
161C2×C4—92no—
162C2×C4—91no—
163C2×C4—91no—
164C2×C4—91no—
165C2×C4—91no—
166C2×C4—182no—
167C2×C4C2×#108811yescharacteristic
168C2×C4C2×#108811yescharacteristic
169C8—184no—
170C2×C3—22no—
171C2×C3—68no—
172C2×C3—62no—
173C2×C3—62no—
174C2×C3—62no—
175C2×C3#6428311yescharacteristic
176C2×C3C2×C4⋊#15711yescharacteristic
177C2×C3C2×#379211yescharacteristic
178C22—21no—
179C22—21no—
180C22C4⋊#108812yes—
181C22C22×#108811yescharacteristic
182C4—21no—
183C4—92no—
184C4—91no—
185C4—92no—
186C4—91no—
187C4—181no—
188C4#8863112yescomplemented
189C3—62no—
190C3#1059380511yescharacteristic
191C2—22no—
192C2#1172244511yescharacteristic
193C2C2×C4⋊#108811yescharacteristic
194C2C2×#8863111yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×#1084C23
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)#423
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. #11720785C2semidirect
4. C4⋊#6188C2semidirect
40. C4×H(3,𝔽3)M16semidirect
43. #1084M16semidirect
98. H(3,𝔽3)#423semidirect
188. C4#88631semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
C2×C4×#10842
#117207852
C4⋊#61882
C3×#4239