Finite Group Databasebeta

← #13140009

Subgroups — #13140009

Order 1728.

647 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 163 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×#1084C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C212yesmaximal
3#11720334C214yesmaximal
4#10591224—32nomaximal
5C22×C4×H(3,𝔽3)C2211yescharacteristic
6C22×#1084C2212yes—
7C22×(C3⋊Dic3)⋊C3C2214yes—
8C4×#1084C414yescomplemented
9C4⋊((C3⋊Dic3)⋊C3)—24no—
10C2×C4⋊(C3⋊Dic3)D312yes—
11C2×C3×C4⋊Dic3—32no—
12C2×C3×C4×Dic3—32no—
13C2×C3×C4×Dic3—61no—
14#63579—38no—
15C23×H(3,𝔽3)C2311yescharacteristic
16C2×C4×H(3,𝔽3)C2×C412yes—
17C2×C4×H(3,𝔽3)D412yes—
18C2×C4×H(3,𝔽3)Q812yes—
19C2×#1084—24no—
20C2×#1084C2×C414yes—
21C2×(C3⋊Dic3)⋊C3—24no—
22C2×(C3⋊Dic3)⋊C3—28no—
23#4977—92no—
24#4936—91no—
25C22×C32×C4—21no—
26C22×C32×C4C2×D312yes—
27C22×C3⋊Dic3C2×D314yes—
28C22×C3×Dic3—34no—
29C22×C3×Dic3—34no—
30C22×C3×Dic3—62no—
31C4⋊(C3⋊Dic3)—24no—
32C3×C4⋊Dic3—64no—
33C3×C4×Dic3—38no—
34C3×C4×Dic3—64no—
35C22×H(3,𝔽3)C22×C411yescharacteristic
36C22×H(3,𝔽3)C2×D411yescharacteristic
37C22×H(3,𝔽3)C2×Q811yescharacteristic
38C22×H(3,𝔽3)#4114yes—
39C4×H(3,𝔽3)C4⋊C414yes—
40#1084—24no—
41(C3⋊Dic3)⋊C3—44no—
42C2×C3×C4⋊C4—92no—
43C2×C3×C42—91no—
44C2×C4⋊Dic3—32no—
45C2×C4⋊Dic3—92no—
46C2×C4×Dic3—32no—
47C2×C4×Dic3—61no—
48C3⋊#95—94no—
49C3⋊#95—98no—
50C23×C32—21no—
51C23×C32C22×D312yes—
52C2×C32×C4—22no—
53C2×C32×C4—22no—
54C2×C32×C4—22no—
55C2×C32×C4Dic614yes—
56C2×C32×C4C4×D314yes—
57C2×C32×C4#6714yes—
58C2×C3⋊Dic3—28no—
59C2×C3⋊Dic3—24no—
60C2×C3×Dic3—38no—
61C2×C3×Dic3—64no—
62C2×C3×Dic3—68no—
63C2×C3×Dic3—64no—
64C2×C3×Dic3—64no—
65C2×C3×Dic3—68no—
66C2×C3×Dic3—64no—
67#382—271noSylow
68C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
69C2×H(3,𝔽3)#11712yes—
70C2×H(3,𝔽3)#12412yes—
71C2×H(3,𝔽3)#12512yes—
72C22×C3×C4—32no—
73C22×C3×C4—61no—
74C22×C3×C4—92no—
75C22×C3×C4—94no—
76C22×C3×C4D3211yescharacteristic
77C22×Dic3—34no—
78C22×Dic3—34no—
79C22×Dic3—62no—
80C22×Dic3—94no—
81C3×C4⋊C4—184no—
82C3×C42—94no—
83C4⋊Dic3—64no—
84C4⋊Dic3—184no—
85C4×Dic3—38no—
86C4×Dic3—64no—
87C22×C32—24no—
88C22×C32—21no—
89C22×C32—21no—
90C22×C32—21no—
91C22×C32C2×Dic612yes—
92C22×C32C2×C4×D312yes—
93C22×C32#23714yes—
94C22×C32#23914yes—
95C22×C32C2×#6712yes—
96C32×C4—24no—
97C32×C4Dic3⋊C418yes—
98C3⋊Dic3—44no—
99C3×Dic3—68no—
100C3×Dic3—68no—
101C3×Dic3—124no—
102C2×C4⋊C4—92no—
103C2×C42—91no—
104#95—274no—
105H(3,𝔽3)#38211yescharacteristic, complemented, Sylow
106C23×C3—32no—
107C23×C3—61no—
108C23×C3C2×D3211yescharacteristic
109C2×C3×C4—34no—
110C2×C3×C4—34no—
111C2×C3×C4—34no—
112C2×C3×C4—62no—
113C2×C3×C4—62no—
114C2×C3×C4—62no—
115C2×C3×C4—94no—
116C2×C3×C4—184no—
117C2×C3×C4—184no—
118C2×C3×C4—188no—
119C2×C3×C4#62412yes—
120C2×C3×C4#62512yes—
121C2×C3×C4#62712yes—
122C2×Dic3—38no—
123C2×Dic3—64no—
124C2×Dic3—64no—
125C2×Dic3—64no—
126C2×Dic3—68no—
127C2×Dic3—68no—
128C2×Dic3—64no—
129C2×Dic3—184no—
130C2×Dic3—188no—
131C2×C32—22no—
132C2×C32—22no—
133C2×C32—22no—
134C2×C32—21no—
135C2×C32#89014yes—
136C2×C32#89214yes—
137C2×C32C2×Dic3⋊C412yes—
138C2×C32#93612yes—
139C2×C32#94712yes—
140C22×C4—92no—
141C22×C4—94no—
142C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
143C4⋊C4—184no—
144C42—94no—
145C22×C3—34no—
146C22×C3—34no—
147C22×C3—32no—
148C22×C3—32no—
149C22×C3—32no—
150C22×C3—64no—
151C22×C3—61no—
152C22×C3—61no—
153C22×C3—61no—
154C22×C3#379614yes—
155C22×C3C2×#62411yescharacteristic
156C22×C3C2×#62511yescharacteristic
157C22×C3C2×#62711yescharacteristic
158C3×C4—38no—
159C3×C4—64no—
160C3×C4—184no—
161C3×C4—364no—
162C3×C4#372414yes—
163Dic3—68no—
164Dic3—68no—
165Dic3—124no—
166Dic3—364no—
167C32—21no—
168C32#497712yescomplemented
169C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
170C2×C4—94no—
171C2×C4—184no—
172C2×C4—188no—
173C2×C4—184no—
174C2×C4#619612yes—
175C2×C4#619812yes—
176C2×C4#619912yes—
177C2×C3—32no—
178C2×C3—34no—
179C2×C3—34no—
180C2×C3—32no—
181C2×C3—32no—
182C2×C3—61no—
183C2×C3—62no—
184C2×C3—62no—
185C2×C3—62no—
186C2×C3#6384912yes—
187C2×C3#6386312yes—
188C2×C3#6387212yes—
189C2×C3C2×#372411yescharacteristic
190C22#8864914yes—
191C22C2×#619611yescharacteristic
192C22C2×#619811yescharacteristic
193C22C2×#619911yescharacteristic
194C4—184no—
195C4—364no—
196C4#8844914yes—
197C3—32no—
198C3—61no—
199C3#1059123011yescharacteristic
200C2#1172108912yes—
201C2#1172109512yes—
202C2#1172111312yes—
203C2C2×#8844911yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#382
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
8. C4×#1084C4semidirect
105. H(3,𝔽3)#382semidirect
168. C32#4977semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
C2×C4⋊((C3⋊Dic3)⋊C3)2
#117203342
#105912243