Finite Group Databasebeta

← C2×#11721095 #13150622

Subgroups — C2×#11721095 #13150622

Order 1728.

899 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 899 conjugacy classes of subgroups — too many to draw. Only its 219 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
3C2×#88448C214yesmaximal
4#11721095C218yesmaximal, direct product factor
5C2×#63869—32nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084C2212yes—
8C22×(C3⋊Dic3)⋊C3C2214yes—
9C4×#1084C2214yes—
10C4⋊((C3⋊Dic3)⋊C3)C2218yes—
11#88448C22116yes—
12C2×C4⋊(C3⋊Dic3)D312yes—
13C2×C3×C4⋊Dic3—32no—
14C2×C3×C4×Dic3—32no—
15C2×C3×C4×Dic3—61no—
16C2×#3724—34no—
17C2×Dic3⋊Dic3—34no—
18#63869—316no—
19C23×H(3,𝔽3)C2311yescharacteristic
20C2×C4×H(3,𝔽3)Q814yes—
21C2×C4×H(3,𝔽3)C2312yes—
22C2×#1084—24no—
23C2×#1084C2314yes—
24C2×(C3⋊Dic3)⋊C3—28no—
25C2×(C3⋊Dic3)⋊C3C2318yes—
26C2×#890—92no—
27C2×#870—91no—
28C22×C32×C4—21no—
29C22×C32×C4C2×D312yes—
30C22×C3⋊Dic3C2×D314yes—
31C22×C3×Dic3—32no—
32C22×C3×Dic3—32no—
33C22×C3×Dic3—34no—
34C22×C3×Dic3—62no—
35C4⋊(C3⋊Dic3)C2×D318yes—
36C3×C4⋊Dic3—38no—
37C3×C4×Dic3—38no—
38C3×C4×Dic3—64no—
39#3724—316no—
40Dic3⋊Dic3—316no—
41C22×H(3,𝔽3)C2×Q812yes—
42C22×H(3,𝔽3)#4114yes—
43C22×H(3,𝔽3)C2411yescharacteristic
44C4×H(3,𝔽3)C2×Q814yes—
45#1084—24no—
46(C3⋊Dic3)⋊C3—28no—
47C2×C3×C4⋊C4—92no—
48C2×C3×C42—91no—
49C2×C4⋊Dic3—32no—
50C2×C4⋊Dic3—92no—
51C2×C4⋊Dic3—94no—
52C2×Dic3⋊C4—94no—
53C2×Dic3⋊C4—94no—
54C2×C4×Dic3—32no—
55C2×C4×Dic3—61no—
56#890—916no—
57#870—98no—
58C23×C32—21no—
59C23×C32C22×D312yes—
60C2×C32×C4—24no—
61C2×C32×C4—22no—
62C2×C32×C4Dic618yes—
63C2×C32×C4C22×D314yes—
64C2×C3⋊Dic3—28no—
65C2×C3⋊Dic3C22×D318yes—
66C2×C3×Dic3—38no—
67C2×C3×Dic3—34no—
68C2×C3×Dic3—34no—
69C2×C3×Dic3—64no—
70C2×C3×Dic3—68no—
71C2×C3×Dic3—64no—
72C2×C3×Dic3—68no—
73C2×C3×Dic3—64no—
74C2×#125—271noSylow
75C2×H(3,𝔽3)#12514yescomplemented
76C2×H(3,𝔽3)C22×Q811yescharacteristic
77C2×H(3,𝔽3)C2×#4112yes—
78C22×C3×C4—32no—
79C22×C3×C4—61no—
80C22×C3×C4—94no—
81C22×C3×C4—92no—
82C22×C3×C4D3211yescharacteristic
83C22×Dic3—32no—
84C22×Dic3—32no—
85C22×Dic3—34no—
86C22×Dic3—62no—
87C22×Dic3—94no—
88C3×C4⋊C4—98no—
89C3×C42—94no—
90C4⋊Dic3—38no—
91C4⋊Dic3—98no—
92C4⋊Dic3—916no—
93Dic3⋊C4—916no—
94Dic3⋊C4—916no—
95C4×Dic3—38no—
96C4×Dic3—64no—
97C22×C32—21no—
98C22×C32—24no—
99C22×C32—22no—
100C22×C32C2×Dic614yes—
101C22×C32#23714yes—
102C22×C32#23914yes—
103C22×C32C23×D312yes—
104C32×C4—24no—
105C32×C4C2×Dic618yes—
106C3⋊Dic3—28no—
107C3×Dic3—68no—
108C3×Dic3—64no—
109C3×Dic3—68no—
110C3×Dic3—64no—
111#125—278no—
112C2×C4⋊C4—92no—
113C2×C4⋊C4—274no—
114C2×C42—91no—
115H(3,𝔽3)C2×#12511yescharacteristic, complemented, Sylow
116C23×C3—32no—
117C23×C3—61no—
118C23×C3C2×D3211yescharacteristic
119C2×C3×C4—38no—
120C2×C3×C4—34no—
121C2×C3×C4—64no—
122C2×C3×C4—62no—
123C2×C3×C4—98no—
124C2×C3×C4—94no—
125C2×C3×C4—188no—
126C2×C3×C4—184no—
127C2×C3×C4#62714yes—
128C2×C3×C4C2×D3212yes—
129C2×Dic3—34no—
130C2×Dic3—34no—
131C2×Dic3—38no—
132C2×Dic3—64no—
133C2×Dic3—64no—
134C2×Dic3—64no—
135C2×Dic3—68no—
136C2×Dic3—68no—
137C2×Dic3—98no—
138C2×Dic3—188no—
139C2×C32—22no—
140C2×C32—21no—
141C2×C32—24no—
142C2×C32#89018yescomplemented
143C2×C32C22×Dic612yes—
144C2×C32C2×#23712yes—
145C2×C32C2×#23912yes—
146C22×C4—92no—
147C22×C4—94no—
148C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
149C4⋊C4—98no—
150C4⋊C4—2716no—
151C42—94no—
152C22×C3—32no—
153C22×C3—34no—
154C22×C3—34no—
155C22×C3—34no—
156C22×C3—61no—
157C22×C3—64no—
158C22×C3—62no—
159C22×C3#379614yes—
160C22×C3C2×#62712yes—
161C22×C3C22×D3211yescharacteristic
162C3×C4—38no—
163C3×C4—64no—
164C3×C4—188no—
165C3×C4—184no—
166C3×C4C2×#62714yes—
167Dic3—64no—
168Dic3—64no—
169Dic3—68no—
170Dic3—68no—
171Dic3—188no—
172C32—21no—
173C32C2×#89012yescomplemented
174C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
175C2×C4—94no—
176C2×C4—98no—
177C2×C4—184no—
178C2×C4—188no—
179C2×C4#619614yes—
180C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
181C2×C3—32no—
182C2×C3—38no—
183C2×C3—32no—
184C2×C3—32no—
185C2×C3—61no—
186C2×C3—64no—
187C2×C3—62no—
188C2×C3#6387214yes—
189C2×C3C2×#379612yes—
190C2×C3C22×#62711yescharacteristic
191C22#8864914yes—
192C22C2×#619612yes—
193C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
194C4—184no—
195C4—188no—
196C4C2×#619614yes—
197C3—32no—
198C3—61no—
199C3C2×#6387211yescharacteristic
200C2#1172109514yesdirect product factor
201C2C2×#8864912yes—
202C2C22×#619611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#125
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
4. #11721095C2direct (N × H)
75. C2×H(3,𝔽3)#125semidirect
115. H(3,𝔽3)C2×#125semidirect
142. C2×C32#890semidirect
173. C32C2×#890semidirect
200. C2#11721095direct (N × H)

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
C2×#884482
#117210952
C2×#638693