Finite Group Databasebeta

← #11721871

Subgroups — #11721871

Order 864.

341 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 74 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#41×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#88740C211yesmaximal, characteristic, complemented
3#88550C211yesmaximal, characteristic, complemented
4#88549C211yesmaximal, characteristic, complemented
5#88546C211yesmaximal, characteristic, complemented
6#88545C211yesmaximal, characteristic, complemented
7C2×#6180C211yesmaximal, characteristic, complemented
8C3×#950—34nomaximal
9Q8×H(3,𝔽3)C2211yescharacteristic
10D4×H(3,𝔽3)—21no—
11D4×H(3,𝔽3)C2211yescharacteristic
12C2×C4×H(3,𝔽3)—21no—
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14#6235—21no—
15C4⋊#272C2211yescharacteristic
16C4×#272—21no—
17#6229C2211yescharacteristic
18#6180C2211yescharacteristic
19#6180C2211yescharacteristic
20C32×#41D314yescomplemented
21C3×#235—34no—
22C3×#216—34no—
23C3×#215—34no—
24C3×#214—34no—
25C3×#213—34no—
26C2×C3×C3⋊C8—34no—
27C22×H(3,𝔽3)—21no—
28C22×H(3,𝔽3)D411yescharacteristic
29C2×#272—21no—
30C4×H(3,𝔽3)—21no—
31C4×H(3,𝔽3)D411yescharacteristic
32C4×H(3,𝔽3)C2311yescharacteristic
33#1084—21no—
34C3×#135—91no—
35#950—34no—
36C32×Q8C2×D314yes—
37C32×D4—24no—
38C32×D4C2×D314yes—
39C2×C32×C4—24no—
40C2×C32×C4C2×D314yes—
41C3×#67—64no—
42C3×D12—34no—
43C3×C4×D3—64no—
44C3×Dic6—34no—
45C3×C3⋊C8—34no—
46C3×C3⋊C8—34no—
47C2×H(3,𝔽3)—21no—
48C2×H(3,𝔽3)—41no—
49C2×H(3,𝔽3)C2×D411yescharacteristic
50#272—41no—
51C3×#41—64no—
52C3×#41—91no—
53C3×#41C3⋊D311yescharacteristic
54#235—34no—
55C3×Q16—91no—
56C3×SD16—91no—
57C3×SD16—91no—
58C3×D8—91no—
59C2×C3×C8—91no—
60#216—34no—
61#215—34no—
62#214—34no—
63#213—34no—
64C2×C3⋊C8—34no—
65C22×C32—24no—
66C22×C32#6714yes—
67C2×C3×D3—64no—
68C32×C4—24no—
69C32×C4#6714yes—
70C32×C4C22×D314yes—
71C3×Dic3—64no—
72#135—91noSylow
73H(3,𝔽3)#13511yescharacteristic, complemented, Sylow
74C3×Q8—64no—
75C3×Q8—91no—
76C3×Q8C2×C3⋊D311yescharacteristic
77C3×D4—21no—
78C3×D4—64no—
79C3×D4—64no—
80C3×D4—91no—
81C3×D4—181no—
82C3×D4C2×C3⋊D311yescharacteristic
83C2×C3×C4—21no—
84C2×C3×C4—64no—
85C2×C3×C4—64no—
86C2×C3×C4—181no—
87C2×C3×C4C2×C3⋊D311yescharacteristic
88#67—64no—
89D12—34no—
90C4×D3—64no—
91Dic6—34no—
92C3×C8—91no—
93C3×C8—91no—
94C3⋊C8—34no—
95C3⋊C8—34no—
96C2×C32—24no—
97C2×C32—44no—
98C2×C32C2×#6714yes—
99C3×D3—124no—
100#41—91no—
101#41#27211yescharacteristic, complemented
102Q16—91no—
103SD16—91no—
104SD16—91no—
105D8—91no—
106C2×C8—91no—
107C22×C3—21no—
108C22×C3—64no—
109C22×C3—64no—
110C22×C3—181no—
111C22×C3#63811yescharacteristic
112C2×D3—64no—
113C3×C4—21no—
114C3×C4—64no—
115C3×C4—64no—
116C3×C4—64no—
117C3×C4—181no—
118C3×C4#63811yescharacteristic
119C3×C4C22×C3⋊D311yescharacteristic
120Dic3—64no—
121C32#95014yescomplemented
122Q8—91no—
123Q8C2×#27211yescharacteristic
124D4—21no—
125D4—91no—
126D4—181no—
127D4C2×#27211yescharacteristic
128C2×C4—21no—
129C2×C4—181no—
130C2×C4C2×#27211yescharacteristic
131C8—91no—
132C8—91no—
133C2×C3—21no—
134C2×C3—41no—
135C2×C3—64no—
136C2×C3—124no—
137C2×C3—124no—
138C2×C3—361no—
139C2×C3C2×#63811yescharacteristic
140D3—124no—
141C22—21no—
142C22—181no—
143C22#623511yescharacteristic
144C4—21no—
145C4—181no—
146C4#623511yescharacteristic
147C4C22×#27211yescharacteristic
148C3—64no—
149C3#6415911yescharacteristic
150C2—21no—
151C2—41no—
152C2—361no—
153C2C2×#623511yescharacteristic
subgroupG/H
centerC3×C4#638
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C22×C3⋊D3
Fitting#41×H(3,𝔽3)C2
socleC2×C3C2×#638
hypercenterC3×#41C3⋊D3
hypocenterH(3,𝔽3)#135
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
1. #41×H(3,𝔽3)C2semidirect
2. #88740C2semidirect
3. #88550C2semidirect
4. #88549C2semidirect
5. #88546C2semidirect
6. #88545C2semidirect
7. C2×#6180C2semidirect
20. C32×#41D3semidirect
73. H(3,𝔽3)#135semidirect
101. #41#272semidirect
121. C32#950semidirect

Maximal subgroups

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

subgroupindex
#41×H(3,𝔽3)2
#887402
#885502
#885492
#885462
#885452
C2×#61802
C3×#9503