Finite Group Databasebeta

← #11721801

Subgroups — #11721801

Order 864.

330 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 62 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
1Q8×#272C211yesmaximal, characteristic, complemented
2#88744C211yesmaximal, characteristic
3SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#88550C211yesmaximal, characteristic, complemented
5#88546C211yesmaximal, characteristic, complemented
6#88530C211yesmaximal, characteristic, complemented
7#88527C211yesmaximal, characteristic, complemented
8C3×#915—34nomaximal
9Q8×H(3,𝔽3)C2211yescharacteristic
10D4×H(3,𝔽3)C2211yescharacteristic
11#6235—21no—
12C2×#1084—21no—
13C4×#272—21no—
14C4×#272C2211yescharacteristic
15#6229—21no—
16#6229C2211yescharacteristic
17#6229C2211yescharacteristic
18C8×H(3,𝔽3)C2211yescharacteristic
19#6180C2211yescharacteristic
20C3×D3×Q8—34no—
21C3×#237—34no—
22C32×SD16D314yescomplemented
23C3×#216—34no—
24C3×#214—34no—
25C3×Dic12—34no—
26C3×ΓL(1,𝔽25)—34no—
27C22×H(3,𝔽3)—21no—
28C2×#272D411yescharacteristic
29C4×H(3,𝔽3)—21no—
30C4×H(3,𝔽3)C2311yescharacteristic
31#1084—21no—
32#1084—21no—
33#1084D411yescharacteristic
34C3×#137—91no—
35#915—34no—
36C32×Q8C2×D314yes—
37C32×D4C2×D314yes—
38C3×#67—64no—
39C2×C3×Dic3—64no—
40C3×C4×D3—34no—
41C3×C4×D3—64no—
42C3×Dic6—34no—
43C3×Dic6—34no—
44C3×Dic6—64no—
45C32×C8C2×D314yes—
46C3×C3⋊C8—34no—
47C2×H(3,𝔽3)—41no—
48C2×H(3,𝔽3)C2×D411yescharacteristic
49#272—21no—
50C3×#41—91no—
51C2×C3×Q8—91no—
52D3×Q8—34no—
53#237—34no—
54C3×Q16—91no—
55C3×Q16—91no—
56C3×SD16—34no—
57C3×SD16—91no—
58C3×SD16C3⋊D311yescharacteristic
59C3×M16—91no—
60#216—34no—
61#214—34no—
62Dic12—34no—
63ΓL(1,𝔽25)—34no—
64C22×C32—24no—
65C2×C3×D3—34no—
66C32×C4—24no—
67C32×C4C22×D314yes—
68C3×Dic3—34no—
69C3×Dic3—64no—
70C3×Dic3—64no—
71#137—91noSylow
72H(3,𝔽3)#13711yescharacteristic, complemented, Sylow
73C3×Q8—34no—
74C3×Q8—91no—
75C3×Q8—91no—
76C3×Q8—181no—
77C3×Q8C2×C3⋊D311yescharacteristic
78C3×D4—34no—
79C3×D4—181no—
80C3×D4C2×C3⋊D311yescharacteristic
81C2×C3×C4—91no—
82C2×C3×C4—181no—
83C2×C3×C4—181no—
84#67—64no—
85C2×Dic3—64no—
86C4×D3—34no—
87C4×D3—64no—
88Dic6—34no—
89Dic6—34no—
90Dic6—64no—
91C3×C8—34no—
92C3×C8—91no—
93C3×C8C2×C3⋊D311yescharacteristic
94C3⋊C8—34no—
95C2×C32—44no—
96C2×C32D3×D414yes—
97C3×D3—64no—
98#41—91no—
99C2×Q8—91no—
100Q16—91no—
101Q16—91no—
102SD16—91no—
103SD16#27211yescharacteristic, complemented
104M16—91no—
105C22×C3—21no—
106C22×C3—64no—
107C22×C3—91no—
108C2×D3—34no—
109C3×C4—21no—
110C3×C4—34no—
111C3×C4—64no—
112C3×C4—91no—
113C3×C4—181no—
114C3×C4—181no—
115C3×C4C22×C3⋊D311yescharacteristic
116Dic3—34no—
117Dic3—64no—
118Dic3—64no—
119C32#91514yescomplemented
120Q8—91no—
121Q8—91no—
122Q8—181no—
123Q8C2×#27211yescharacteristic
124D4—181no—
125D4C2×#27211yescharacteristic
126C2×C4—91no—
127C2×C4—181no—
128C2×C4—181no—
129C8—91no—
130C8C2×#27211yescharacteristic
131C2×C3—34no—
132C2×C3—41no—
133C2×C3—124no—
134C2×C3—181no—
135C2×C3D4×C3⋊D311yescharacteristic
136D3—64no—
137C22—21no—
138C22—91no—
139C4—21no—
140C4—91no—
141C4—181no—
142C4—181no—
143C4C22×#27211yescharacteristic
144C3—34no—
145C3#6412411yescharacteristic
146C2—41no—
147C2—181no—
148C2D4×#27211yescharacteristic
subgroupG/H
center, socleC2×C3D4×C3⋊D3
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C22×C3⋊D3
FittingSD16×H(3,𝔽3)C2
prime radical#88744C2
hypercenterC3×SD16C3⋊D3
hypocenterH(3,𝔽3)#137
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×#272C2semidirect
3. SD16×H(3,𝔽3)C2semidirect
4. #88550C2semidirect
5. #88546C2semidirect
6. #88530C2semidirect
7. #88527C2semidirect
22. C32×SD16D3semidirect
72. H(3,𝔽3)#137semidirect
103. SD16#272semidirect
119. C32#915semidirect

Maximal subgroups

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

subgroupindex
Q8×#2722
#887442
SD16×H(3,𝔽3)2
#885502
#885462
#885302
#885272
C3×#9153