Finite Group Databasebeta

← #13138758

Subgroups — #13138758

Order 1728.

387 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721117C212yesmaximal
2C8×#1084C211yesmaximal, characteristic
3#63645⋊C3C212yesmaximal
4#11720315C212yesmaximal
5#10590891—32nomaximal
6C2×#6196—22no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C4×#1084C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10C4⋊((C3⋊Dic3)⋊C3)C2214yes—
11#88449—22no—
12#63918—34no—
13#63645D312yes—
14C3×#818—32no—
15C3×C8×Dic3—32no—
16C3×C8×Dic3—61no—
17#63575—34no—
18C2×C4×H(3,𝔽3)C2311yescharacteristic
19C2×#1084D411yescharacteristic
20C2×#1084D411yescharacteristic
21C2×(C3⋊Dic3)⋊C3—24no—
22#6196—44no—
23C8×H(3,𝔽3)Q812yescomplemented
24#6180Q812yescomplemented
25#4874—92no—
26#4682—91no—
27C2×#627—64no—
28C2×C32×C8—21no—
29C2×C32×C8C2×D312yes—
30C4⋊(C3⋊Dic3)C2×D314yes—
31C3×C4⋊Dic3—34no—
32C3×C4×Dic3—32no—
33C3×C4×Dic3—61no—
34C2×C3×C3⋊C8—32no—
35C2×C3×C3⋊C8—61no—
36Dic3⋊Dic3—64no—
37C22×H(3,𝔽3)C2×D411yescharacteristic
38C4×H(3,𝔽3)C2×Q811yescharacteristic
39C4×H(3,𝔽3)C2×Q811yescharacteristic
40#1084D812yes—
41#1084Q1612yes—
42(C3⋊Dic3)⋊C3—44no—
43#888—94no—
44#869—92no—
45C3×#107—92no—
46C3×C4×C8—91no—
47#818—32no—
48#818—92no—
49#818—92no—
50C8×Dic3—32no—
51C8×Dic3—61no—
52#807—94no—
53C2×C32×C4—21no—
54C2×C32×C4C22×D312yes—
55C2×C3⋊Dic3—24no—
56C2×C3×Dic3—32no—
57C2×C3×Dic3—32no—
58C2×C3×Dic3—61no—
59C2×C3×Dic3—61no—
60C2×C3×Dic3—64no—
61#627—128no—
62C32×C8—22no—
63C32×C8Dic614yescomplemented
64C3×C3⋊C8—34no—
65C3×C3⋊C8—62no—
66#500—271noSylow
67C2×H(3,𝔽3)C4⋊Q811yescharacteristic
68C2×H(3,𝔽3)C2×D811yescharacteristic
69C2×H(3,𝔽3)C2×Q1611yescharacteristic
70C2×Dic6—184no—
71C2×Dic6—182no—
72C2×C3×C8—32no—
73C2×C3×C8—61no—
74C2×C3×C8—91no—
75C2×C3×C8D3211yescharacteristic
76C3×C4⋊C4—94no—
77C3×C42—91no—
78C4⋊Dic3—34no—
79C4⋊Dic3—94no—
80C4⋊Dic3—182no—
81Dic3⋊C4—184no—
82C4×Dic3—32no—
83C4×Dic3—61no—
84C2×C3⋊C8—32no—
85C2×C3⋊C8—61no—
86C22×C32—21no—
87C22×C32D3×D412yes—
88C32×C4—21no—
89C32×C4—21no—
90C32×C4C2×Dic612yes—
91C32×C4D3×Q812yes—
92C3⋊Dic3—44no—
93C3×Dic3—34no—
94C3×Dic3—34no—
95C3×Dic3—62no—
96C3×Dic3—62no—
97C3×Dic3—124no—
98C4⋊Q8—272no—
99#107—92no—
100#107—272no—
101C4×C8—91no—
102H(3,𝔽3)#50011yescharacteristic, complemented, Sylow
103C2×C3×C4—32no—
104C2×C3×C4—61no—
105C2×C3×C4—91no—
106C2×C3×C4—91no—
107C2×C3×C4—184no—
108C2×C3×C4C2×D3211yescharacteristic
109C2×Dic3—32no—
110C2×Dic3—32no—
111C2×Dic3—61no—
112C2×Dic3—61no—
113C2×Dic3—64no—
114C2×Dic3—184no—
115Dic6—368no—
116Dic6—364no—
117C3×C8—34no—
118C3×C8—62no—
119C3×C8—92no—
120C3×C8#62712yes—
121C3⋊C8—34no—
122C3⋊C8—62no—
123C2×C32—21no—
124C2×C32—21no—
125C2×C32—21no—
126C2×C32#88812yes—
127C2×C32D3×D812yes—
128C2×C32D3×Q1612yes—
129C2×Q8—542no—
130C2×C8—91no—
131C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
132C4⋊C4—94no—
133C4⋊C4—542no—
134C42—91no—
135C22×C3—32no—
136C22×C3—61no—
137C22×C3#380211yescharacteristic
138C3×C4—32no—
139C3×C4—32no—
140C3×C4—61no—
141C3×C4—61no—
142C3×C4—92no—
143C3×C4—92no—
144C3×C4—364no—
145C3×C4#379711yescharacteristic
146C3×C4C2×#62711yescharacteristic
147Dic3—34no—
148Dic3—34no—
149Dic3—62no—
150Dic3—62no—
151Dic3—124no—
152Dic3—364no—
153C32—21no—
154C32#487412yescomplemented
155Q8—1084no—
156C2×C4—91no—
157C2×C4—91no—
158C2×C4—184no—
159C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
160C8—92no—
161C8#619612yescomplemented
162C2×C3—32no—
163C2×C3—32no—
164C2×C3—32no—
165C2×C3—61no—
166C2×C3—61no—
167C2×C3—61no—
168C2×C3#6379711yescharacteristic
169C2×C3#6380211yescharacteristic
170C2×C3#6391911yescharacteristic
171C22#8865511yescharacteristic
172C4—92no—
173C4—92no—
174C4—364no—
175C4#8865111yescharacteristic
176C4C2×#619611yescharacteristic
177C3—32no—
178C3—61no—
179C3#1059090211yescharacteristic
180C2#1172083311yescharacteristic
181C2#1172083811yescharacteristic
182C2#1172111711yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#500
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
23. C8×H(3,𝔽3)Q8semidirect
24. #6180Q8semidirect
63. C32×C8Dic6semidirect
102. H(3,𝔽3)#500semidirect
154. C32#4874semidirect
161. C8#6196semidirect

Maximal subgroups

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

subgroupindex
#117211172
C8×#10842
#63645⋊C32
#117203152
#105908913