Finite Group Databasebeta

← #13144766

Subgroups — #13144766

Order 1728.

627 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 113 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×Q16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721863C212yesmaximal
3C2×C8×#272C211yesmaximal, characteristic
4#11720583C212yesmaximal, complemented
5#11720551C211yesmaximal, characteristic, complemented
6C3×#5188—34nomaximal
7C2×Q8×H(3,𝔽3)C2212yes—
8C2×C4×#272C2211yescharacteristic
9Q16×H(3,𝔽3)—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88542—22no—
12C4⋊#1084C2212yes—
13#88540—22no—
14C2×#6180C2211yescharacteristic
15C8×#272—22no—
16C2×C32×Q16D314yescomplemented
17C3×#946—38no—
18C2×C3×D3×C8—34no—
19C3×#835—38no—
20C3×#818—34no—
21C22×#272D411yescharacteristic
22Q8×H(3,𝔽3)—22no—
23Q8×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—22no—
27C2×#1084D411yescharacteristic
28C4×#272—21no—
29C4×#272—21no—
30C8×H(3,𝔽3)D412yes—
31#6180—21no—
32C3×#467—91no—
33#5188—34no—
34C2×C32×Q8C2×D318yes—
35C2×C3×C4×D3—34no—
36C32×Q16—28no—
37C2×C32×C8C2×D314yes—
38C3×(C2×D3)⋊C4—68no—
39C3×C4⋊Dic3—38no—
40C3×Dic3⋊C4—68no—
41C2×C3×C3⋊C8—34no—
42C3×D3×C8—68no—
43C22×H(3,𝔽3)C2×D411yescharacteristic
44C2×#272—21no—
45C2×#272—21no—
46C2×#272Q1612yescomplemented
47C4×H(3,𝔽3)—42no—
48C4×H(3,𝔽3)C2×D411yescharacteristic
49C4×H(3,𝔽3)#4111yescharacteristic
50#1084—21no—
51#1084—42no—
52C2×C3×Q16—34no—
53C2×C3×Q16C3⋊D311yescharacteristic
54C22×C3×C8—91no—
55C3×C22⋊Q8—92no—
56#946—38no—
57C2×D3×C8—34no—
58C3×#107—91no—
59C3×#103—92no—
60#835—38no—
61#818—34no—
62C22×C3×D3—34no—
63C32×Q8—28no—
64C32×Q8—28no—
65C2×C32×C4—28no—
66C2×C32×C4C22×D314yes—
67C2×C3×Dic3—34no—
68C2×C3×Dic3—68no—
69C3×C4×D3—64no—
70C3×C4×D3—64no—
71C32×C8#6718yes—
72C3×C3⋊C8—64no—
73#467—91noSylow
74C2×H(3,𝔽3)#12111yescharacteristic
75C2×H(3,𝔽3)C2×Q1611yescharacteristic
76C2×H(3,𝔽3)#13511yescharacteristic
77#272—22no—
78C2×C3×Q8—38no—
79C2×C3×Q8C2×C3⋊D312yes—
80C22×C3×C4—91no—
81C2×C4×D3—34no—
82C3×Q16—22no—
83C3×Q16—68no—
84C2×C3×C8—34no—
85C2×C3×C8—91no—
86C2×C3×C8—182no—
87C2×C3×C8C2×C3⋊D311yescharacteristic
88C3×C4⋊C4—92no—
89C3×C4⋊C4—182no—
90C3×C22⋊C4—182no—
91(C2×D3)⋊C4—68no—
92C4⋊Dic3—38no—
93Dic3⋊C4—68no—
94C2×C3⋊C8—34no—
95D3×C8—68no—
96C22×C32D3×D414yes—
97C2×C3×D3—38no—
98C2×C3×D3—64no—
99C2×C3×D3—64no—
100C32×C4—48no—
101C32×C4#23714yes—
102C32×C4C2×#6714yes—
103C3×Dic3—64no—
104C3×Dic3—128no—
105C2×Q16#27211yescharacteristic, complemented
106C22×C8—91no—
107C22⋊Q8—92no—
108#107—91no—
109#103—92no—
110H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
111C23×C3—91no—
112C22×D3—34no—
113C3×Q8—22no—
114C3×Q8—22no—
115C3×Q8—68no—
116C3×Q8—68no—
117C2×C3×C4—22no—
118C2×C3×C4—34no—
119C2×C3×C4—68no—
120C2×C3×C4—91no—
121C2×C3×C4—181no—
122C2×C3×C4—181no—
123C2×C3×C4—182no—
124C2×C3×C4C22×C3⋊D311yescharacteristic
125C2×Dic3—34no—
126C2×Dic3—68no—
127C4×D3—64no—
128C4×D3—64no—
129C3×C8—38no—
130C3×C8—181no—
131C3×C8#63812yes—
132C3⋊C8—64no—
133C2×C32D3×Q1614yes—
134C2×C32#91914yes—
135C2×C32#93814yes—
136C3×D3—68no—
137C2×Q8C2×#27212yes—
138C22×C4—91no—
139Q16—22no—
140C2×C8—91no—
141C2×C8—182no—
142C2×C8C2×#27211yescharacteristic
143C4⋊C4—92no—
144C4⋊C4—182no—
145C22⋊C4—182no—
146C22×C3—34no—
147C22×C3—92no—
148C22×C3—181no—
149C22×C3—181no—
150C22×C3D4×C3⋊D311yescharacteristic
151C2×D3—38no—
152C2×D3—64no—
153C2×D3—64no—
154C3×C4—34no—
155C3×C4—34no—
156C3×C4—42no—
157C3×C4—128no—
158C3×C4—181no—
159C3×C4—362no—
160C3×C4#383011yescharacteristic
161C3×C4C2×#63811yescharacteristic
162Dic3—64no—
163Dic3—128no—
164C32#518814yescomplemented
165C23—91no—
166Q8—22no—
167Q8—22no—
168C2×C4—22no—
169C2×C4—91no—
170C2×C4—181no—
171C2×C4—181no—
172C2×C4—182no—
173C2×C4C22×#27211yescharacteristic
174C8—181no—
175C8#623512yes—
176C2×C3—34no—
177C2×C3—34no—
178C2×C3—34no—
179C2×C3—182no—
180C2×C3Q16×C3⋊D311yescharacteristic
181C2×C3#6412811yescharacteristic
182C2×C3#6414711yescharacteristic
183D3—68no—
184C22—92no—
185C22—181no—
186C22—181no—
187C22D4×#27211yescharacteristic
188C4—42no—
189C4—181no—
190C4—362no—
191C4#8874411yescharacteristic
192C4C2×#623511yescharacteristic
193C3—34no—
194C3#1059288711yescharacteristic
195C2—182no—
196C2Q16×#27211yescharacteristic
197C2#1172180811yescharacteristic
198C2#1172184811yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×Q16×H(3,𝔽3)C2
prime radicalC22×#272D4
hypercenterC2×C3×Q16C3⋊D3
hypocenterH(3,𝔽3)#467
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
1. C2×Q16×H(3,𝔽3)C2semidirect
4. #11720583C2semidirect
5. #11720551C2semidirect
16. C2×C32×Q16D3semidirect
46. C2×#272Q16semidirect
105. C2×Q16#272semidirect
110. H(3,𝔽3)#467semidirect
164. C32#5188semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×H(3,𝔽3)2
#117218632
C2×C8×#2722
#117205832
#117205512
C3×#51883