Finite Group Databasebeta

← C2×#11721350 #13152185

Subgroups — C2×#11721350 #13152185

Order 1728.

336 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 56 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
1#11721350C212yesmaximal, direct product factor
2C2×#88455C211yesmaximal, characteristic, complemented
3C2×C32×#126C311yesmaximal, characteristic
4C2×#126⋊C9C313yesmaximal, complemented
5C22×#1094—41nomaximal
6#88455C2211yescharacteristic, complemented
7C32×#126—32no—
8C32×#126C2×C312yes—
9C2×C32×C4⋊C4—31no—
10C2×C32×C22⋊C4—31no—
11C2×C32×C42C2×C311yescharacteristic
12#126⋊C9C2×C316yescomplemented
13C2×C42⋊C9C2×C313yescomplemented
14C2×#1094—41no—
15C2×#1094—42no—
16C2×C3×#126—31no—
17C2×C3×#126C3211yescharacteristic
18C24×C32A411yescharacteristic
19C22×C32×C4—31no—
20C22×C32×C4—31no—
21C22×C22⋊C9—43no—
22C32×C4⋊C4—32no—
23C32×C4⋊C4—32no—
24C32×C22⋊C4—32no—
25C32×C22⋊C4—32no—
26C32×C42—31no—
27C32×C42C22×C311yescharacteristic
28C42⋊C9C22×C313yescomplemented
29C22×3−1+2—161no—
30#1094—41no—
31C3×#126—36no—
32C3×#126—32no—
33C3×#126—32no—
34C3×#126C2×C3212yes—
35C2×C3×C4⋊C4—31no—
36C2×C3×C4⋊C4—33no—
37C2×C3×C22⋊C4—33no—
38C2×C3×C22⋊C4—31no—
39C2×C3×C42—31no—
40C2×C3×C42C2×C3211yescharacteristic
41C23×C32—61no—
42C23×C32—61no—
43C23×C32C2×A411yescharacteristic
44C23×C32C2×A412yes—
45C2×C32×C4—31no—
46C2×C32×C4—31no—
47C2×C32×C4—32no—
48C2×C32×C4—61no—
49C2×C32×C4—61no—
50C2×C32×C4—61no—
51C2×C32×C4—61no—
52C2×C22⋊C9—43no—
53C2×C22⋊C9—46no—
54C2×#1263−1+211yescharacteristic, complemented, Sylow
55C2×3−1+2—162no—
56C2×3−1+2—161no—
57C24×C3—31no—
58C24×C3C3×A411yescharacteristic
59C22×C3×C4—33no—
60C22×C3×C4—33no—
61C22×C3×C4—31no—
62C22×C3×C4—31no—
63C3×C4⋊C4—32no—
64C3×C4⋊C4—36no—
65C3×C4⋊C4—36no—
66C3×C4⋊C4—32no—
67C3×C22⋊C4—36no—
68C3×C22⋊C4—36no—
69C3×C22⋊C4—32no—
70C3×C22⋊C4—32no—
71C3×C42—31no—
72C3×C42—33no—
73C3×C42—31no—
74C3×C42C22×C3211yescharacteristic
75C22×C32—31no—
76C22×C32—31no—
77C22×C32—41no—
78C22×C32—62no—
79C22×C32—121no—
80C22×C32C22×A411yescharacteristic
81C32×C4—61no—
82C32×C4—61no—
83C32×C4—62no—
84C22×C9—163no—
85C22⋊C9—43no—
86#126—32no—
87#126C2×3−1+212yescomplemented
88C2×C4⋊C4—31no—
89C2×C22⋊C4—31no—
90C2×C42C2×3−1+211yescharacteristic, complemented
913−1+2—161noSylow
92C23×C3—31no—
93C23×C3—32no—
94C23×C3—61no—
95C23×C3—61no—
96C23×C3—63no—
97C23×C3—63no—
98C23×C3C2×C3×A411yescharacteristic
99C23×C3C2×C3×A412yes—
100C2×C3×C4—33no—
101C2×C3×C4—33no—
102C2×C3×C4—36no—
103C2×C3×C4—31no—
104C2×C3×C4—31no—
105C2×C3×C4—32no—
106C2×C3×C4—61no—
107C2×C3×C4—61no—
108C2×C3×C4—63no—
109C2×C3×C4—63no—
110C2×C3×C4—63no—
111C2×C3×C4—63no—
112C2×C3×C4—61no—
113C2×C3×C4—61no—
114C2×C32—31no—
115C2×C32—31no—
116C2×C32—42no—
117C2×C32#126⋊C311yescharacteristic
118C2×C9—166no—
119C2×C9—163no—
120C24#109411yescharacteristic
121C22×C4—31no—
122C22×C4—31no—
123C4⋊C4—32no—
124C4⋊C4—32no—
125C22⋊C4—32no—
126C22⋊C4—32no—
127C42—31no—
128C42C22×3−1+211yescharacteristic, complemented
129C22×C3—31no—
130C22×C3—33no—
131C22×C3—33no—
132C22×C3—31no—
133C22×C3—31no—
134C22×C3—41no—
135C22×C3—62no—
136C22×C3—66no—
137C22×C3—121no—
138C22×C3—121no—
139C22×C3—123no—
140C22×C3C22×C3×A411yescharacteristic
141C3×C4—63no—
142C3×C4—63no—
143C3×C4—66no—
144C3×C4—61no—
145C3×C4—61no—
146C3×C4—62no—
147C32C2×#126⋊C311yescharacteristic
148C9—163no—
149C23—61no—
150C23—61no—
151C23C2×#109411yescharacteristic
152C23C2×#109412yes—
153C2×C4—31no—
154C2×C4—31no—
155C2×C4—32no—
156C2×C4—61no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C3—33no—
161C2×C3—33no—
162C2×C3—31no—
163C2×C3—31no—
164C2×C3—31no—
165C2×C3—42no—
166C2×C3—122no—
167C2×C3C3×#126⋊C311yescharacteristic
168C22—31no—
169C22—31no—
170C22—41no—
171C22—62no—
172C22—121no—
173C22C22×#109411yescharacteristic
174C4—61no—
175C4—61no—
176C4—62no—
177C3—31no—
178C3C2×C3×#126⋊C311yescharacteristic
179C2—31no—
180C2—31no—
181C2—42no—
182C2#1172135011yescharacteristic, direct product factor
subgroupG/H
centerC2×C3C3×#126⋊C3
commutator, cosocleC3×C42C22×C32
FrattiniC22×C3C22×C3×A4
FittingC2×C32×#126C3
socleC23×C3C2×C3×A4
prime radicalC24×C32A4
hypercenterC2×C32#126⋊C3
hypocenterC42C22×3−1+2
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. #11721350C2direct (N × H)
2. C2×#88455C2semidirect
4. C2×#126⋊C9C3semidirect
6. #88455C22semidirect
12. #126⋊C9C2×C3semidirect
13. C2×C42⋊C9C2×C3semidirect
28. C42⋊C9C22×C3semidirect
54. C2×#1263−1+2semidirect
87. #126C2×3−1+2semidirect
90. C2×C42C2×3−1+2semidirect
128. C42C22×3−1+2semidirect
182. C2#11721350direct (N × H)

Maximal subgroups

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

subgroupindex
#117213502
C2×#884552
C2×C32×#1263
C2×#126⋊C93
C22×#10944