Finite Group Databasebeta

← #11722432

Subgroups — #11722432

Order 864.

225 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 28 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
1C2×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#88623C211yesmaximal, characteristic, complemented
3#88622C211yesmaximal, characteristic, complemented
4#64364C311yesmaximal, characteristic, complemented
5#1007⋊C3—31nomaximal
6#638⋊C3—41nomaximal
7Q8⋊H(3,𝔽3)C2211yescharacteristic
8C2×C32×Q8C2×C311yescharacteristic, complemented
9#3832—31no—
10#3832C2×C311yescharacteristic, complemented
11Q8×C3⋊D3—31no—
12Q8×C3⋊D3C2×C311yescharacteristic, complemented
13#3828—31no—
14#3828—31no—
15C2×C3×SL(2,𝔽3)—21no—
16C2×C3×SL(2,𝔽3)D311yescharacteristic, complemented
17D3×SL(2,𝔽3)—31no—
18C3⋊(#41⋊C3)—31no—
19C22×H(3,𝔽3)—41no—
20C2×(C3⋊D3)⋊C3—41no—
21(C3⋊Dic3)⋊C3—41no—
22#1007—31no—
23#1007—91no—
242−1+4⋊C3—91no—
25C32×Q8—31no—
26C32×Q8C22×C311yescharacteristic
27C2×C32×C4—31no—
28#638—31no—
29#638A411yescharacteristic
30C4⋊(C3⋊D3)—31no—
31C4⋊(C3⋊D3)—31no—
32C4×C3⋊D3—32no—
33C4×C3⋊D3—31no—
34C4×C3⋊D3—31no—
35#634—31no—
36#634—31no—
37C3×#67—121no—
38C3×SL(2,𝔽3)—21no—
39C3×SL(2,𝔽3)C2×D311yescharacteristic
40C2×H(3,𝔽3)—41no—
41C2×H(3,𝔽3)—81no—
42(C3⋊D3)⋊C3—81no—
43C2×C3×Q8—31no—
44C2×C3×Q8C3×D311yescharacteristic
45#239—31no—
46#239—91no—
47#239—93no—
48#239—91no—
49D3×Q8—31no—
50D3×Q8—91no—
51D3×Q8—93no—
52D3×Q8—91no—
53#235—91no—
54#235—93no—
55#235—93no—
56#235—91no—
57#41⋊C3—91no—
58C2×SL(2,𝔽3)—31no—
59C2×SL(2,𝔽3)—61no—
60C2×SL(2,𝔽3)—91no—
61C22×C32—41no—
62C22×C32—81no—
63C22×C32C2×A411yescharacteristic
64C2×C3⋊D3—31no—
65C2×C3⋊D3C2×A411yescharacteristic
66C2×C3×D3—121no—
67C32×C4—31no—
68C32×C4—31no—
69C3⋊Dic3—31no—
70C3⋊Dic3C2×A411yescharacteristic
71C3×Dic3—121no—
722−1+4—91noSylow
73H(3,𝔽3)—41noSylow
74C3×Q8—31no—
75C3×Q8—33no—
76C3×Q8—31no—
77C3×Q8C2×C3×D311yescharacteristic
78C3×D4—361no—
79C2×C3×C4—33no—
80C2×C3×C4—31no—
81#67—31no—
82#67—91no—
83#67—93no—
84#67—91no—
85D12—91no—
86D12—91no—
87D12—93no—
88D12—93no—
89C4×D3—91no—
90C4×D3—91no—
91C4×D3—93no—
92C4×D3—93no—
93C4×D3—96no—
94C4×D3—92no—
95Dic6—91no—
96Dic6—93no—
97Dic6—93no—
98Dic6—91no—
99SL(2,𝔽3)—31no—
100SL(2,𝔽3)—61no—
101C2×C32—21no—
102C2×C32—41no—
103C2×C32—81no—
104C2×C32—82no—
105C2×C32—81no—
106C2×C32C22×A411yescharacteristic
107C3⋊D3—21no—
108C3⋊D3—61no—
109C3×D3—241no—
110#41—91no—
111#41—271no—
112#41—271no—
113#41—271no—
114C2×Q8—91no—
115C2×Q8—271no—
116C2×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
117C22×C3—31no—
118C22×C3—121no—
119C22×C3—241no—
120C22×C3—361no—
121C22×C3D3×A411yescharacteristic
122C2×D3—31no—
123C2×D3—91no—
124C2×D3—93no—
125C2×D3—91no—
126C3×C4—33no—
127C3×C4—33no—
128C3×C4—31no—
129C3×C4—31no—
130C3×C4—361no—
131Dic3—31no—
132Dic3—91no—
133Dic3—93no—
134Dic3—91no—
135C32—41no—
136C32—81no—
137C322−1+4⋊C311yescharacteristic, complemented
138Q8—31no—
139Q8—271no—
140Q8—271no—
141Q8C2×(C3⋊D3)⋊C311yescharacteristic
142D4—91no—
143D4—271no—
144D4—271no—
145D4—271no—
146C2×C4—31no—
147C2×C4—271no—
148C2×C4—271no—
149C2×C4—272no—
150C2×C3—21no—
151C2×C3—31no—
152C2×C3—61no—
153C2×C3—121no—
154C2×C3—241no—
155C2×C3—242no—
156C2×C3—241no—
157C2×C3—721no—
158C2×C3C2×D3×A411yescharacteristic
159D3—61no—
160D3—181no—
161D3—183no—
162D3—181no—
163C22—91no—
164C22—271no—
165C22(C3⋊D3)⋊A411yescharacteristic
166C4—31no—
167C4—31no—
168C4—91no—
169C4—271no—
170C3—31no—
171C3—121no—
172C3—241no—
173C3#1007⋊C311yescharacteristic
174C2—21no—
175C2—181no—
176C2—541no—
177C2C2×(C3⋊D3)⋊A411yescharacteristic
subgroupG/H
centerC2C2×(C3⋊D3)⋊A4
commutator, cosocle, hypocenterC32×Q8C22×C3
Frattini, socleC2×C3C2×D3×A4
FittingC2×C32×Q8C2×C3
hypercenterC22(C3⋊D3)⋊A4
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×Q8⋊H(3,𝔽3)C2semidirect
2. #88623C2semidirect
3. #88622C2semidirect
4. #64364C3semidirect
8. C2×C32×Q8C2×C3semidirect
10. #3832C2×C3semidirect
12. Q8×C3⋊D3C2×C3semidirect
16. C2×C3×SL(2,𝔽3)D3semidirect
116. C2×Q8(C3⋊D3)⋊C3semidirect
137. C322−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×Q8⋊H(3,𝔽3)2
#886232
#886222
#643643
#1007⋊C33
#638⋊C34