Finite Group Databasebeta

← C2×#11720314 #13139973

Subgroups — C2×#11720314 #13139973

Order 1728.

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

normal subgroups only

This group has 531 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
1C22×#6180C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C212yesmaximal
3#11720314C214yesmaximal, direct product factor
4C2×#63573—32nomaximal
5C22×C4×H(3,𝔽3)C2211yescharacteristic
6C22×(C3⋊Dic3)⋊C3—22no—
7C2×#6180C414yescomplemented
8C2×#6180C2212yes—
9C4⋊((C3⋊Dic3)⋊C3)—22no—
10C4⋊((C3⋊Dic3)⋊C3)C2214yes—
11C2×C4⋊(C3⋊Dic3)D312yes—
12C2×C3×C4⋊Dic3—32no—
13C22×C3×C3⋊C8—32no—
14C22×C3×C3⋊C8—61no—
15#63573—38no—
16C23×H(3,𝔽3)D411yescharacteristic
17C2×C4×H(3,𝔽3)C2×C412yes—
18C2×C4×H(3,𝔽3)D411yescharacteristic
19C2×C4×H(3,𝔽3)Q812yes—
20C2×C4×H(3,𝔽3)C2311yescharacteristic
21C2×(C3⋊Dic3)⋊C3—24no—
22C2×(C3⋊Dic3)⋊C3—44no—
23#6180C2×C414yescomplemented
24C2×#817—91no—
25C2×#808—92no—
26C22×C32×C4—21no—
27C22×C32×C4C2×D312yes—
28C22×C3⋊Dic3—22no—
29C22×C3×Dic3—62no—
30C4⋊(C3⋊Dic3)—22no—
31C4⋊(C3⋊Dic3)C2×D314yes—
32C3×C4⋊Dic3—34no—
33C3×C4⋊Dic3—62no—
34C2×C3×C3⋊C8—38no—
35C2×C3×C3⋊C8—34no—
36C2×C3×C3⋊C8—64no—
37C2×C3×C3⋊C8—62no—
38C22×H(3,𝔽3)C4⋊C412yes—
39C22×H(3,𝔽3)SD1614yes—
40C22×H(3,𝔽3)C2×D411yescharacteristic
41C4×H(3,𝔽3)C4⋊C412yes—
42C4×H(3,𝔽3)C22×C411yescharacteristic
43C4×H(3,𝔽3)C2×Q811yescharacteristic
44(C3⋊Dic3)⋊C3—44no—
45C22×C3×C8—91no—
46C2×C3×C4⋊C4—92no—
47C2×C4⋊Dic3—32no—
48C2×C4⋊Dic3—92no—
49C22×C3⋊C8—32no—
50C22×C3⋊C8—61no—
51#817—94no—
52#808—98no—
53C23×C32—21no—
54C23×C32#6712yes—
55C2×C32×C4—22no—
56C2×C32×C4—21no—
57C2×C32×C4—22no—
58C2×C32×C4—21no—
59C2×C32×C4Dic614yes—
60C2×C32×C4C4×D314yes—
61C2×C32×C4#6712yes—
62C2×C32×C4C22×D312yes—
63C2×C3⋊Dic3—24no—
64C2×C3⋊Dic3—44no—
65C2×C3×Dic3—64no—
66C2×C3×Dic3—124no—
67C3×C3⋊C8—38no—
68C3×C3⋊C8—64no—
69C2×#106—271noSylow
70C2×H(3,𝔽3)#10614yescomplemented
71C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
72C2×H(3,𝔽3)C2×SD1612yes—
73C22×C3×C4—32no—
74C22×C3×C4—61no—
75C22×C3×C4—182no—
76C22×C3×C4D3211yescharacteristic
77C22×Dic3—62no—
78C22×Dic3—182no—
79C2×C3×C8—94no—
80C2×C3×C8—92no—
81C3×C4⋊C4—94no—
82C3×C4⋊C4—182no—
83C4⋊Dic3—34no—
84C4⋊Dic3—62no—
85C4⋊Dic3—94no—
86C4⋊Dic3—182no—
87C2×C3⋊C8—34no—
88C2×C3⋊C8—38no—
89C2×C3⋊C8—64no—
90C2×C3⋊C8—62no—
91C22×C32—24no—
92C22×C32—22no—
93C22×C32—21no—
94C22×C32Dic3⋊C414yes—
95C22×C32#21414yes—
96C22×C32#21514yes—
97C22×C32C2×#6712yes—
98C32×C4—22no—
99C32×C4—21no—
100C32×C4—21no—
101C32×C4Dic3⋊C414yes—
102C32×C4C2×Dic612yes—
103C32×C4C2×C4×D312yes—
104C3⋊Dic3—44no—
105C3×Dic3—124no—
106C22×C8—91no—
107C2×C4⋊C4—92no—
108#106—274no—
109H(3,𝔽3)C2×#10611yescharacteristic, complemented, Sylow
110C23×C3—32no—
111C23×C3—61no—
112C23×C3#62511yescharacteristic
113C2×C3×C4—34no—
114C2×C3×C4—32no—
115C2×C3×C4—34no—
116C2×C3×C4—32no—
117C2×C3×C4—62no—
118C2×C3×C4—61no—
119C2×C3×C4—62no—
120C2×C3×C4—61no—
121C2×C3×C4—184no—
122C2×C3×C4—364no—
123C2×C3×C4#62412yes—
124C2×C3×C4#62511yescharacteristic
125C2×C3×C4#62712yes—
126C2×C3×C4C2×D3211yescharacteristic
127C2×Dic3—64no—
128C2×Dic3—124no—
129C2×Dic3—184no—
130C2×Dic3—364no—
131C3×C8—94no—
132C3⋊C8—38no—
133C3⋊C8—64no—
134C2×C32—22no—
135C2×C32—24no—
136C2×C32—21no—
137C2×C32#80818yescomplemented
138C2×C32C2×Dic3⋊C412yes—
139C2×C32C2×#21412yes—
140C2×C32C2×#21512yes—
141C22×C4—182no—
142C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
143C2×C8—94no—
144C2×C8—92no—
145C4⋊C4—94no—
146C4⋊C4—182no—
147C22×C3—32no—
148C22×C3—34no—
149C22×C3—34no—
150C22×C3—34no—
151C22×C3—61no—
152C22×C3—64no—
153C22×C3—62no—
154C22×C3#371514yes—
155C22×C3#372412yes—
156C22×C3C2×#62511yescharacteristic
157C3×C4—32no—
158C3×C4—32no—
159C3×C4—34no—
160C3×C4—61no—
161C3×C4—61no—
162C3×C4—62no—
163C3×C4—364no—
164C3×C4#372412yes—
165C3×C4C2×#62411yescharacteristic
166C3×C4C2×#62711yescharacteristic
167Dic3—124no—
168Dic3—364no—
169C32—21no—
170C32C2×#80812yescomplemented
171C23#619811yescharacteristic
172C2×C4—184no—
173C2×C4—364no—
174C2×C4#619612yes—
175C2×C4#619811yescharacteristic
176C2×C4#619912yes—
177C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
178C8—94no—
179C2×C3—32no—
180C2×C3—32no—
181C2×C3—32no—
182C2×C3—38no—
183C2×C3—61no—
184C2×C3—62no—
185C2×C3—64no—
186C2×C3#6357414yes—
187C2×C3C2×#371512yes—
188C2×C3C2×#372411yescharacteristic
189C22#8843114yes—
190C22#8844912yes—
191C22C2×#619811yescharacteristic
192C4—364no—
193C4#8844912yes—
194C4C2×#619611yescharacteristic
195C4C2×#619911yescharacteristic
196C3—32no—
197C3—61no—
198C3C2×#6357411yescharacteristic
199C2#1172031414yesdirect product factor
200C2C2×#8843112yes—
201C2C2×#8844911yescharacteristic
subgroupG/H
centerC23#6198
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3#625
prime radicalC23×H(3,𝔽3)D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#106
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11720314C2direct (N × H)
7. C2×#6180C4semidirect
23. #6180C2×C4semidirect
70. C2×H(3,𝔽3)#106semidirect
109. H(3,𝔽3)C2×#106semidirect
137. C2×C32#808semidirect
170. C32C2×#808semidirect
199. C2#11720314direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#61802
C2×C4⋊((C3⋊Dic3)⋊C3)2
#117203142
C2×#635733