Finite Group Databasebeta

← #11721350⋊C2 #13152206

Subgroups — #11721350⋊C2 #13152206

Order 1728.

342 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 53 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#11721353C211yesmaximal, characteristic, complemented
2#11721350C212yesmaximal, complemented
3C32×#560C311yesmaximal, characteristic
4#560⋊C9C313yesmaximal, complemented
5C22×#1094—41nomaximal
6#88455C2211yescharacteristic, complemented
7C32×#127C2×C311yescharacteristic
8C32×#126C2×C312yes—
9C32×#123—31no—
10C32×#121—32no—
11C32×#120—31no—
12#127⋊C9C2×C313yescomplemented
13#126⋊C9C2×C316yescomplemented
14C2×#1094—42no—
15C2×#1094—41no—
16C3×#560—31no—
17C3×#560C3211yescharacteristic
18C24×C32A411yescharacteristic
19C2×C32×D4—32no—
20C2×C32×D4—31no—
21C2×C32×D4—31no—
22C22×C32×C4—31no—
23C22×C22⋊C9—43no—
24C32×C4⋊C4—32no—
25C32×C22⋊C4—31no—
26C32×C22⋊C4—32no—
27C32×C22⋊C4—31no—
28C32×C42C22×C311yescharacteristic
29C42⋊C9C22×C313yescomplemented
30C22×3−1+2—161no—
31#1094—41no—
32C3×#127—31no—
33C3×#127C2×C3211yescharacteristic
34C3×#126—32no—
35C3×#126C2×C3212yes—
36C3×#123—33no—
37C3×#123—31no—
38C3×#121—32no—
39C3×#121—36no—
40C3×#120—33no—
41C3×#120—31no—
42C23×C32—31no—
43C23×C32—121no—
44C23×C32C2×A412yes—
45C23×C32C2×A411yescharacteristic
46C32×D4—122no—
47C32×D4—121no—
48C32×D4—121no—
49C2×C32×C4—31no—
50C2×C32×C4—32no—
51C2×C32×C4—121no—
52C2×C22⋊C9—46no—
53C2×C22⋊C9—43no—
54#5603−1+211yescharacteristic, complemented, Sylow
55C2×3−1+2—162no—
56C2×3−1+2—161no—
57C24×C3—31no—
58C24×C3C3×A411yescharacteristic
59C2×C3×D4—31no—
60C2×C3×D4—36no—
61C2×C3×D4—33no—
62C2×C3×D4—33no—
63C2×C3×D4—32no—
64C2×C3×D4—31no—
65C22×C3×C4—33no—
66C22×C3×C4—31no—
67C3×C4⋊C4—32no—
68C3×C4⋊C4—36no—
69C3×C22⋊C4—36no—
70C3×C22⋊C4—33no—
71C3×C22⋊C4—33no—
72C3×C22⋊C4—31no—
73C3×C22⋊C4—32no—
74C3×C22⋊C4—31no—
75C3×C42—31no—
76C3×C42C22×C3211yescharacteristic
77C22×C32—41no—
78C22×C32—62no—
79C22×C32—61no—
80C22×C32—61no—
81C22×C32—62no—
82C22×C32—121no—
83C22×C32C22×A411yescharacteristic
84C32×C4—62no—
85C32×C4—61no—
86C22×C9—163no—
87C22⋊C9—43no—
88#127C2×3−1+211yescharacteristic, complemented
89#126C2×3−1+212yescomplemented
90#123—31no—
91#121—32no—
92#120—31no—
933−1+2—161noSylow
94C23×C3—33no—
95C23×C3—31no—
96C23×C3—32no—
97C23×C3—31no—
98C23×C3—121no—
99C23×C3—123no—
100C23×C3C2×C3×A412yes—
101C23×C3C2×C3×A411yescharacteristic
102C3×D4—122no—
103C3×D4—126no—
104C3×D4—121no—
105C3×D4—123no—
106C3×D4—121no—
107C3×D4—123no—
108C2×C3×C4—33no—
109C2×C3×C4—36no—
110C2×C3×C4—31no—
111C2×C3×C4—32no—
112C2×C3×C4—123no—
113C2×C3×C4—121no—
114C2×C32—31no—
115C2×C32—42no—
116C2×C32—41no—
117C2×C32—121no—
118C2×C9—166no—
119C2×C9—163no—
120C24#109411yescharacteristic
121C2×D4—31no—
122C2×D4—32no—
123C2×D4—31no—
124C22×C4—31no—
125C4⋊C4—32no—
126C22⋊C4—31no—
127C22⋊C4—32no—
128C22⋊C4—31no—
129C42C22×3−1+211yescharacteristic, complemented
130C22×C3—31no—
131C22×C3—41no—
132C22×C3—62no—
133C22×C3—61no—
134C22×C3—63no—
135C22×C3—61no—
136C22×C3—66no—
137C22×C3—62no—
138C22×C3—66no—
139C22×C3—63no—
140C22×C3—121no—
141C22×C3—121no—
142C22×C3—123no—
143C22×C3C22×C3×A411yescharacteristic
144C3×C4—66no—
145C3×C4—62no—
146C3×C4—63no—
147C3×C4—61no—
148C32#560⋊C311yescharacteristic
149C9—163no—
150C23—31no—
151C23—121no—
152C23C2×#109412yes—
153C23C2×#109411yescharacteristic
154D4—122no—
155D4—121no—
156D4—121no—
157C2×C4—31no—
158C2×C4—32no—
159C2×C4—121no—
160C2×C3—33no—
161C2×C3—31no—
162C2×C3—42no—
163C2×C3—41no—
164C2×C3—122no—
165C2×C3—121no—
166C2×C3—123no—
167C2×C3—121no—
168C22—41no—
169C22—62no—
170C22—61no—
171C22—61no—
172C22—62no—
173C22—121no—
174C22C22×#109411yescharacteristic
175C4—62no—
176C4—61no—
177C3—31no—
178C3C3×#560⋊C311yescharacteristic
179C2—31no—
180C2—42no—
181C2—41no—
182C2—121no—
subgroupG/H
centerC3C3×#560⋊C3
commutator, cosocleC3×C42C22×C32
Frattini, socleC22×C3C22×C3×A4
Fitting, prime radicalC32×#560C3
hypercenterC32#560⋊C3
hypocenterC42C22×3−1+2
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C3 ◁ C32

Complements

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

NH ≅ G/Ntype
1. #11721353C2semidirect
2. #11721350C2semidirect
4. #560⋊C9C3semidirect
6. #88455C22semidirect
12. #127⋊C9C2×C3semidirect
13. #126⋊C9C2×C3semidirect
29. C42⋊C9C22×C3semidirect
54. #5603−1+2semidirect
88. #127C2×3−1+2semidirect
89. #126C2×3−1+2semidirect
129. C42C22×3−1+2semidirect

Maximal subgroups

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

subgroupindex
#117213532
#117213502
C32×#5603
#560⋊C93
C22×#10944