Finite Group Databasebeta

← C2×#11720315 #13139974

Subgroups — C2×#11720315 #13139974

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#11720315C214yesmaximal, direct product factor
4C2×#63575—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#63575—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×#818—91no—
25C2×#807—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)D812yes—
40C22×H(3,𝔽3)Q1612yes—
41C22×H(3,𝔽3)C2×D411yescharacteristic
42C4×H(3,𝔽3)C4⋊C412yes—
43C4×H(3,𝔽3)C22×C411yescharacteristic
44C4×H(3,𝔽3)C2×Q811yescharacteristic
45(C3⋊Dic3)⋊C3—44no—
46C22×C3×C8—91no—
47C2×C3×C4⋊C4—92no—
48C2×C4⋊Dic3—32no—
49C2×C4⋊Dic3—92no—
50C22×C3⋊C8—32no—
51C22×C3⋊C8—61no—
52#818—94no—
53#807—98no—
54C23×C32—21no—
55C23×C32#6712yes—
56C2×C32×C4—22no—
57C2×C32×C4—21no—
58C2×C32×C4—22no—
59C2×C32×C4—21no—
60C2×C32×C4Dic614yes—
61C2×C32×C4C4×D314yes—
62C2×C32×C4#6712yes—
63C2×C32×C4C22×D312yes—
64C2×C3⋊Dic3—24no—
65C2×C3⋊Dic3—44no—
66C2×C3×Dic3—64no—
67C2×C3×Dic3—124no—
68C3×C3⋊C8—38no—
69C3×C3⋊C8—64no—
70C2×#107—271noSylow
71C2×H(3,𝔽3)#10714yescomplemented
72C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
73C2×H(3,𝔽3)C2×D811yescharacteristic
74C2×H(3,𝔽3)C2×Q1611yescharacteristic
75C22×C3×C4—32no—
76C22×C3×C4—61no—
77C22×C3×C4—182no—
78C22×C3×C4D3211yescharacteristic
79C22×Dic3—62no—
80C22×Dic3—182no—
81C2×C3×C8—94no—
82C2×C3×C8—92no—
83C3×C4⋊C4—94no—
84C3×C4⋊C4—182no—
85C4⋊Dic3—34no—
86C4⋊Dic3—62no—
87C4⋊Dic3—94no—
88C4⋊Dic3—182no—
89C2×C3⋊C8—34no—
90C2×C3⋊C8—38no—
91C2×C3⋊C8—64no—
92C2×C3⋊C8—62no—
93C22×C32—22no—
94C22×C32—22no—
95C22×C32—22no—
96C22×C32—21no—
97C22×C32Dic3⋊C414yes—
98C22×C32#21314yes—
99C22×C32#21614yes—
100C22×C32C2×#6712yes—
101C32×C4—22no—
102C32×C4—21no—
103C32×C4—21no—
104C32×C4Dic3⋊C414yes—
105C32×C4C2×Dic612yes—
106C32×C4C2×C4×D312yes—
107C3⋊Dic3—44no—
108C3×Dic3—124no—
109C22×C8—91no—
110C2×C4⋊C4—92no—
111#107—274no—
112H(3,𝔽3)C2×#10711yescharacteristic, complemented, Sylow
113C23×C3—32no—
114C23×C3—61no—
115C23×C3#62511yescharacteristic
116C2×C3×C4—34no—
117C2×C3×C4—32no—
118C2×C3×C4—34no—
119C2×C3×C4—32no—
120C2×C3×C4—62no—
121C2×C3×C4—61no—
122C2×C3×C4—62no—
123C2×C3×C4—61no—
124C2×C3×C4—184no—
125C2×C3×C4—364no—
126C2×C3×C4#62412yes—
127C2×C3×C4#62511yescharacteristic
128C2×C3×C4#62712yes—
129C2×C3×C4C2×D3211yescharacteristic
130C2×Dic3—64no—
131C2×Dic3—124no—
132C2×Dic3—184no—
133C2×Dic3—364no—
134C3×C8—94no—
135C3⋊C8—38no—
136C3⋊C8—64no—
137C2×C32—21no—
138C2×C32—24no—
139C2×C32—21no—
140C2×C32—21no—
141C2×C32#80718yescomplemented
142C2×C32C2×Dic3⋊C412yes—
143C2×C32C2×#21312yes—
144C2×C32C2×#21612yes—
145C22×C4—182no—
146C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
147C2×C8—94no—
148C2×C8—92no—
149C4⋊C4—94no—
150C4⋊C4—182no—
151C22×C3—32no—
152C22×C3—34no—
153C22×C3—34no—
154C22×C3—34no—
155C22×C3—61no—
156C22×C3—62no—
157C22×C3—62no—
158C22×C3—62no—
159C22×C3#371312yes—
160C22×C3#371812yes—
161C22×C3#372412yes—
162C22×C3C2×#62511yescharacteristic
163C3×C4—32no—
164C3×C4—32no—
165C3×C4—34no—
166C3×C4—61no—
167C3×C4—61no—
168C3×C4—62no—
169C3×C4—364no—
170C3×C4#372412yes—
171C3×C4C2×#62411yescharacteristic
172C3×C4C2×#62711yescharacteristic
173Dic3—124no—
174Dic3—364no—
175C32—21no—
176C32C2×#80712yescomplemented
177C23#619811yescharacteristic
178C2×C4—184no—
179C2×C4—364no—
180C2×C4#619612yes—
181C2×C4#619811yescharacteristic
182C2×C4#619912yes—
183C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
184C8—94no—
185C2×C3—32no—
186C2×C3—32no—
187C2×C3—32no—
188C2×C3—38no—
189C2×C3—61no—
190C2×C3—61no—
191C2×C3—61no—
192C2×C3—64no—
193C2×C3#6357614yes—
194C2×C3C2×#371311yescharacteristic
195C2×C3C2×#371811yescharacteristic
196C2×C3C2×#372411yescharacteristic
197C22#8843212yes—
198C22#8843312yes—
199C22#8844912yes—
200C22C2×#619811yescharacteristic
201C4—364no—
202C4#8844912yes—
203C4C2×#619611yescharacteristic
204C4C2×#619911yescharacteristic
205C3—32no—
206C3—61no—
207C3C2×#6357611yescharacteristic
208C2#1172031514yesdirect product factor
209C2C2×#8843211yescharacteristic
210C2C2×#8843311yescharacteristic
211C2C2×#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×#107
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. #11720315C2direct (N × H)
7. C2×#6180C4semidirect
23. #6180C2×C4semidirect
71. C2×H(3,𝔽3)#107semidirect
112. H(3,𝔽3)C2×#107semidirect
141. C2×C32#807semidirect
176. C32C2×#807semidirect
208. C2#11720315direct (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
#117203152
C2×#635753