Finite Group Databasebeta

← #11721870

Subgroups — #11721870

Order 864.

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

normal subgroups only

This group has 374 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
2C2×C4⋊#272C211yesmaximal, characteristic, complemented
3#88549C212yesmaximal, complemented
4#88545C212yesmaximal, complemented
5#88538C211yesmaximal, characteristic, complemented
6C3×#949—33nomaximal
7C3×#949—31nomaximal
8C22×#272—21no—
9Q8×H(3,𝔽3)C2211yescharacteristic, complemented
10D4×H(3,𝔽3)—21no—
11D4×H(3,𝔽3)C2211yescharacteristic, complemented
12C2×C4×H(3,𝔽3)—21no—
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#272—21no—
15C4⋊#272C2212yes—
16#6180C2212yescomplemented
17C32×#41D313yescomplemented
18C32×#41D311yescharacteristic, complemented
19C2×C3×D12—33no—
20C2×C3×D12—31no—
21C3×#215—36no—
22C3×#215—32no—
23C3×#213—36no—
24C3×#213—32no—
25C3×#208—33no—
26C3×#208—31no—
27C22×H(3,𝔽3)—21no—
28C22×H(3,𝔽3)D411yescharacteristic
29C2×#272—22no—
30C2×#272—42no—
31C4×H(3,𝔽3)—21no—
32C4×H(3,𝔽3)D411yescharacteristic
33C4×H(3,𝔽3)C2311yescharacteristic
34C3×AGL(1,ℤ/8)—91no—
35#949—33no—
36#949—31no—
37C22×C3×D3—63no—
38C22×C3×D3—61no—
39C32×Q8C2×D313yescomplemented
40C32×Q8C2×D311yescharacteristic, complemented
41C32×D4—23no—
42C32×D4—21no—
43C32×D4C2×D313yescomplemented
44C32×D4C2×D311yescharacteristic, complemented
45C2×C32×C4—23no—
46C2×C32×C4—21no—
47C2×C32×C4C2×D313yes—
48C2×C32×C4C2×D311yescharacteristic
49C3×D12—36no—
50C3×D12—32no—
51C3×D12—63no—
52C3×D12—61no—
53C3×C3⋊C8—36no—
54C3×C3⋊C8—32no—
55C2×H(3,𝔽3)—21no—
56C2×H(3,𝔽3)—41no—
57C2×H(3,𝔽3)C2×D411yescharacteristic
58#272—42no—
59C3×#41—33no—
60C3×#41—61no—
61C3×#41C3⋊D311yescharacteristic
62C2×C3×D4—91no—
63C2×D12—33no—
64C2×D12—31no—
65C3×SD16—92no—
66C3×D8—92no—
67C3×M16—91no—
68#215—36no—
69#215—32no—
70#213—36no—
71#213—32no—
72#208—33no—
73#208—31no—
74C22×C32—23no—
75C22×C32—21no—
76C22×C32#6713yes—
77C22×C32#6711yescharacteristic
78C2×C3×D3—66no—
79C2×C3×D3—62no—
80C2×C3×D3—126no—
81C2×C3×D3—122no—
82C32×C4—23no—
83C32×C4—21no—
84C32×C4#6713yes—
85C32×C4#6711yescharacteristic
86C32×C4C22×D313yes—
87C32×C4C22×D311yescharacteristic
88AGL(1,ℤ/8)—91noSylow
89H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
90C23×C3—181no—
91C22×D3—63no—
92C22×D3—61no—
93C3×Q8—33no—
94C3×Q8—61no—
95C3×Q8C2×C3⋊D311yescharacteristic
96C3×D4—21no—
97C3×D4—33no—
98C3×D4—63no—
99C3×D4—61no—
100C3×D4—61no—
101C3×D4—92no—
102C3×D4—181no—
103C3×D4C2×C3⋊D311yescharacteristic
104C2×C3×C4—21no—
105C2×C3×C4—33no—
106C2×C3×C4—63no—
107C2×C3×C4—61no—
108C2×C3×C4—61no—
109C2×C3×C4C2×C3⋊D311yescharacteristic
110D12—36no—
111D12—32no—
112D12—63no—
113D12—61no—
114C3×C8—92no—
115C3⋊C8—36no—
116C3⋊C8—32no—
117C2×C32—23no—
118C2×C32—21no—
119C2×C32—43no—
120C2×C32—41no—
121C2×C32C2×#6713yes—
122C2×C32C2×#6711yescharacteristic
123C3×D3—126no—
124C3×D3—122no—
125#41#27211yescharacteristic, complemented
126C2×D4—91no—
127SD16—92no—
128D8—92no—
129M16—91no—
130C22×C3—21no—
131C22×C3—33no—
132C22×C3—63no—
133C22×C3—61no—
134C22×C3—61no—
135C22×C3—182no—
136C22×C3—362no—
137C22×C3#63811yescharacteristic
138C2×D3—66no—
139C2×D3—62no—
140C2×D3—126no—
141C2×D3—122no—
142C3×C4—21no—
143C3×C4—33no—
144C3×C4—33no—
145C3×C4—63no—
146C3×C4—61no—
147C3×C4—61no—
148C3×C4—61no—
149C3×C4#63811yescharacteristic
150C3×C4C22×C3⋊D311yescharacteristic
151C32#94913yescomplemented
152C32#94911yescharacteristic, complemented
153C23—181no—
154Q8C2×#27211yescharacteristic, complemented
155D4—21no—
156D4—92no—
157D4—181no—
158D4C2×#27211yescharacteristic, complemented
159C2×C4—21no—
160C2×C4C2×#27211yescharacteristic
161C8—92no—
162C2×C3—21no—
163C2×C3—33no—
164C2×C3—41no—
165C2×C3—63no—
166C2×C3—61no—
167C2×C3—121no—
168C2×C3—123no—
169C2×C3—121no—
170C2×C3—362no—
171C2×C3C2×#63811yescharacteristic
172D3—126no—
173D3—122no—
174C22—21no—
175C22—182no—
176C22—362no—
177C22#623511yescharacteristic
178C4—21no—
179C4#623511yescharacteristic
180C4C22×#27211yescharacteristic
181C3—33no—
182C3—61no—
183C3#6415811yescharacteristic
184C2—21no—
185C2—41no—
186C2—362no—
187C2C2×#623511yescharacteristic
subgroupG/H
center, socleC2×C3C2×#638
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C22×C3⋊D3
Fitting#41×H(3,𝔽3)C2
hypercenterC3×#41C3⋊D3
hypocenterH(3,𝔽3)AGL(1,ℤ/8)
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ 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. C2×C4⋊#272C2semidirect
3. #88549C2semidirect
4. #88545C2semidirect
5. #88538C2semidirect
9. Q8×H(3,𝔽3)C22semidirect
11. D4×H(3,𝔽3)C22semidirect
16. #6180C22semidirect
17. C32×#41D3semidirect
18. C32×#41D3semidirect
39. C32×Q8C2×D3semidirect
40. C32×Q8C2×D3semidirect
43. C32×D4C2×D3semidirect
44. C32×D4C2×D3semidirect
89. H(3,𝔽3)AGL(1,ℤ/8)semidirect
125. #41#272semidirect
151. C32#949semidirect
152. C32#949semidirect
154. Q8C2×#272semidirect
158. D4C2×#272semidirect

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
C2×C4⋊#2722
#885492
#885452
#885382
C3×#9493