Finite Group Databasebeta

← #13141236

Subgroups — #13141236

Order 1728.

370 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 29 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#11720339C211yesmaximal, characteristic, complemented
2C32×#392C311yesmaximal, characteristic
3#392⋊C9C313yesmaximal, complemented
4C4×#1094—41nomaximal
5C22×C32×D4—31no—
6C2×C32×C22⋊C4—31no—
7C32×#95C2×C311yescharacteristic
8#95⋊C9C2×C313yescomplemented
9C2×#1094—41no—
10C3×#392—31no—
11C3×#392C3211yescharacteristic
12C24×C32—31no—
13C2×C32×D4—62no—
14C2×C32×D4—64no—
15C22×C32×C4—31no—
16C22×C32×C4A411yescharacteristic
17C32×C22⋊C4—62no—
18C4×C22⋊C9—43no—
19#1094—41no—
20C4×3−1+2—161no—
21C22×C3×D4—31no—
22C22×C3×D4—33no—
23C2×C3×C22⋊C4—33no—
24C2×C3×C22⋊C4—31no—
25C3×#95—31no—
26C3×#95C2×C3211yescharacteristic
27C23×C32—32no—
28C23×C32—61no—
29C23×C32—62no—
30C23×C32—62no—
31C23×C32—61no—
32C23×C32C2×A411yescharacteristic
33C32×D4—68no—
34C2×C32×C4—32no—
35C2×C32×C4—61no—
36C2×C32×C4—62no—
37C2×C22⋊C9—43no—
38#3923−1+211yescharacteristic, complemented, Sylow
39C2×3−1+2—161no—
40C24×C3—33no—
41C24×C3—31no—
42C2×C3×D4—62no—
43C2×C3×D4—66no—
44C2×C3×D4—64no—
45C2×C3×D4—612no—
46C22×C3×C4—33no—
47C22×C3×C4—31no—
48C22×C3×C4—31no—
49C22×C3×C4C3×A411yescharacteristic
50C3×C22⋊C4—62no—
51C3×C22⋊C4—66no—
52C22×C32—31no—
53C22×C32—31no—
54C22×C32—64no—
55C22×C32—62no—
56C22×C32—121no—
57C22×C32—122no—
58C22×C32—121no—
59C22×C32#41⋊C311yescharacteristic
60C32×C4—41no—
61C32×C4—62no—
62C22⋊C9—43no—
63C4×C9—163no—
64C22×D4—31no—
65C2×C22⋊C4—31no—
66#95C2×3−1+211yescharacteristic
673−1+2—161noSylow
68C23×C3—32no—
69C23×C3—36no—
70C23×C3—31no—
71C23×C3—63no—
72C23×C3—61no—
73C23×C3—62no—
74C23×C3—66no—
75C23×C3—66no—
76C23×C3—62no—
77C23×C3—63no—
78C23×C3—61no—
79C23×C3C2×C3×A411yescharacteristic
80C3×D4—624no—
81C3×D4—68no—
82C2×C3×C4—36no—
83C2×C3×C4—32no—
84C2×C3×C4—61no—
85C2×C3×C4—62no—
86C2×C3×C4—66no—
87C2×C3×C4—63no—
88C2×C32—31no—
89C2×C32—31no—
90C2×C32—122no—
91C2×C32#127⋊C311yescharacteristic
92C2×C9—163no—
93C24—31no—
94C2×D4—62no—
95C2×D4—64no—
96C22×C4—31no—
97C22×C4#109411yescharacteristic
98C22⋊C4—62no—
99C22×C3—31no—
100C22×C3—33no—
101C22×C3—33no—
102C22×C3—31no—
103C22×C3—31no—
104C22×C3—612no—
105C22×C3—64no—
106C22×C3—66no—
107C22×C3—62no—
108C22×C3—123no—
109C22×C3—126no—
110C22×C3—121no—
111C22×C3—122no—
112C22×C3—123no—
113C22×C3—121no—
114C22×C3C3×#41⋊C311yescharacteristic
115C3×C4—41no—
116C3×C4—66no—
117C3×C4—62no—
118C3×C4—121no—
119C32#392⋊C311yescharacteristic
120C9—163no—
121C23—32no—
122C23—61no—
123C23—62no—
124C23—62no—
125C23—61no—
126C23C2×#109411yescharacteristic
127D4—68no—
128C2×C4—32no—
129C2×C4—61no—
130C2×C4—62no—
131C2×C3—33no—
132C2×C3—33no—
133C2×C3—31no—
134C2×C3—31no—
135C2×C3—31no—
136C2×C3—126no—
137C2×C3—122no—
138C2×C3C3×#127⋊C311yescharacteristic
139C22—31no—
140C22—31no—
141C22—64no—
142C22—62no—
143C22—121no—
144C22—122no—
145C22—121no—
146C22#8869111yescharacteristic
147C4—41no—
148C4—62no—
149C3—31no—
150C3C3×#392⋊C311yescharacteristic
151C2—31no—
152C2—31no—
153C2—122no—
154C2#1172135311yescharacteristic
subgroupG/H
centerC2×C3C3×#127⋊C3
commutator, cosocleC3×#95C2×C32
Frattini, socleC23×C3C2×C3×A4
Fitting, prime radicalC32×#392C3
hypercenterC2×C32#127⋊C3
hypocenter#95C2×3−1+2
LCSG ▷ C3×#95 ▷ #95
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. #11720339C2semidirect
3. #392⋊C9C3semidirect
8. #95⋊C9C2×C3semidirect
38. #3923−1+2semidirect

Maximal subgroups

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

subgroupindex
#117203392
C32×#3923
#392⋊C93
C4×#10944