Finite Group Databasebeta

← C2×#11724469 #13181404

Subgroups — C2×#11724469 #13181404

Order 1728.

1406 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1406 conjugacy classes of subgroups — too many to draw. Only its 199 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×C32×#41⋊C3C211yesmaximal, characteristic
2#11724469C214yesmaximal, direct product factor
3C2×C3⋊(C3⋊#226)C212yesmaximal, complemented
4C2×#64265—312nomaximal
5C2×#64160—31nomaximal
6C2×#6308—41nomaximal
7C32×#41⋊C3C2212yes—
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C3⋊(C3⋊#226)C2214yescomplemented
10C2×C32×#41D311yescharacteristic
11C22×#634—31no—
12C2×C3×#41⋊C3D3112yes—
13#64265—348no—
14C2×C3⋊#226—324no—
15#64160—34no—
16#64160—34no—
17C2×#3756—32no—
18C2×#3754—32no—
19C2×#3748—31no—
20C2×C33×C4—41no—
21C2×C3⋊(C3⋊Dic3)—42no—
22#6308—44no—
23C32×SL(2,𝔽3)C2311yescharacteristic
24C2×#984—99no—
25C2×#951—94no—
26C32×#41—32no—
27C32×#41C2×D312yes—
28C2×C32×Q8C2×D311yescharacteristic
29C2×C32×D4—31no—
30C22×C32×C4—31no—
31C22×C3⋊Dic3—61no—
32C2×#634—32no—
33C2×#634—34no—
34C2×#634—62no—
35C2×#634—61no—
36C2×#634—1212no—
37C3×#41⋊C3C2×D3124yes—
38C2×C3×SL(2,𝔽3)C2×D3112yes—
39C3⋊#226—348no—
40#3756—34no—
41#3756—34no—
42#3754—38no—
43#3748—34no—
44C2×C3⋊(C3⋊C8)—32no—
45C22×C33—41no—
46C33×C4—42no—
47C3⋊(C3⋊Dic3)—44no—
48C2×C3×#41C3⋊D314yes—
49C22×Dic6—94no—
50C2×#41⋊C3C3⋊D319yes—
51#984—936no—
52C2×#226—918no—
53#951—916no—
54#951—916no—
55C2×#216—98no—
56C2×#214—98no—
57C2×#208—94no—
58C23×C32—31no—
59C32×Q8—31no—
60C32×Q8C22×D311yescharacteristic
61C32×D4—32no—
62C32×D4—61no—
63C2×C32×C4—31no—
64C2×C32×C4—32no—
65C2×C32×C4—32no—
66C2×C32×C4—412no—
67C2×C32×C4S411yescharacteristic
68C2×C3⋊Dic3—64no—
69C2×C3⋊Dic3—62no—
70C2×C3⋊Dic3—1224no—
71#634—34no—
72#634—62no—
73#634—64no—
74#634—1248no—
75C3×SL(2,𝔽3)C22×D3112yes—
76C3⋊(C3⋊C8)—34no—
77C2×#137—271noSylow
78C2×C33—42no—
79C2×C33—41no—
80C3×#41—38no—
81C3×#41C2×C3⋊D318yes—
82C2×C3×Q8C2×C3⋊D314yes—
83C2×C3×D4—34no—
84C22×C3×C4—34no—
85C22×Dic3—184no—
86C2×Dic6—98no—
87C2×Dic6—916no—
88C2×Dic6—188no—
89C2×Dic6—184no—
90C2×Dic6—369no—
91#41⋊C3C2×C3⋊D3118yes—
92C2×SL(2,𝔽3)C2×C3⋊D319yes—
93#226—936no—
94#216—916no—
95#216—916no—
96#214—932no—
97#208—916no—
98C2×C3⋊C8—98no—
99C22×C32—32no—
100C22×C32—412no—
101C22×C32—62no—
102C22×C32C2×S411yescharacteristic
103C32×C4—31no—
104C32×C4—31no—
105C32×C4—424no—
106C32×C4C2×S412yes—
107C3⋊Dic3—64no—
108C3⋊Dic3—1248no—
109C2×#41C3⋊(C3⋊D3)11yescharacteristic
110C22×Q8—271no—
111#137—274no—
112#137—274no—
113C2×Q16—272no—
114C2×SD16—272no—
115C2×M16—271no—
116C33—41noSylow
117C23×C3—34no—
118C3×Q8—34no—
119C3×Q8C22×C3⋊D314yes—
120C3×D4—38no—
121C3×D4—64no—
122C2×C3×C4—34no—
123C2×C3×C4—38no—
124C2×C3×C4—38no—
125C2×C3×C4—49no—
126C2×C3×C4C3⋊S414yes—
127C2×Dic3—1816no—
128C2×Dic3—188no—
129C2×Dic3—3618no—
130Dic6—916no—
131Dic6—188no—
132Dic6—1816no—
133Dic6—3636no—
134SL(2,𝔽3)C22×C3⋊D319yes—
135C3⋊C8—916no—
136C2×C32—424no—
137C2×C32—412no—
138C2×C32—62no—
139C2×C32#98412yescomplemented
140C2×C32C22×S411yescharacteristic
141#41—32no—
142#41C2×C3⋊(C3⋊D3)12yes—
143C2×Q8—272no—
144C2×Q8—274no—
145C2×Q8—542no—
146C2×Q8—541no—
147C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
148C2×D4—31no—
149C22×C4—31no—
150C22×C4—541no—
151Q16—274no—
152Q16—274no—
153SD16—278no—
154M16—274no—
155C2×C8—272no—
156C22×C3—38no—
157C22×C3—49no—
158C22×C3—68no—
159C22×C3C2×C3⋊S414yes—
160C3×C4—34no—
161C3×C4—34no—
162C3×C4—418no—
163C3×C4C2×C3⋊S418yes—
164Dic3—1816no—
165Dic3—3636no—
166C32—412no—
167C32C2×#98411yescharacteristic, complemented
168C23—31no—
169Q8—31no—
170Q8—274no—
171Q8—542no—
172Q8—544no—
173Q8C22×C3⋊(C3⋊D3)11yescharacteristic
174D4—32no—
175D4—61no—
176C2×C4—31no—
177C2×C4—32no—
178C2×C4—32no—
179C2×C4—544no—
180C2×C4—542no—
181C2×C4C3⋊(C3⋊S4)11yescharacteristic
182C8—274no—
183C2×C3—418no—
184C2×C3—49no—
185C2×C3—68no—
186C2×C3#6426518yescomplemented
187C2×C3C22×C3⋊S414yes—
188C22—32no—
189C22—62no—
190C22C2×C3⋊(C3⋊S4)11yescharacteristic
191C4—31no—
192C4—31no—
193C4—544no—
194C4C2×C3⋊(C3⋊S4)12yes—
195C3—49no—
196C3C2×#6426514yescomplemented
197C2—62no—
198C2#1172446912yesdirect product factor
199C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C22×C3⋊(C3⋊S4)
FittingC2×C32×#41D3
socleC22×C32C2×S4
prime radicalC2×C32×#41⋊C3C2
hypercenterC2×C4C3⋊(C3⋊S4)
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724469C2direct (N × H)
3. C2×C3⋊(C3⋊#226)C2semidirect
9. C3⋊(C3⋊#226)C22semidirect
139. C2×C32#984semidirect
167. C32C2×#984semidirect
186. C2×C3#64265semidirect
196. C3C2×#64265semidirect
198. C2#11724469direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
#117244692
C2×C3⋊(C3⋊#226)2
C2×#642653
C2×#641603
C2×#63084