Finite Group Databasebeta

← #11721106

Subgroups — #11721106

Order 864.

285 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 64 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×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#634⋊C3C211yesmaximal, characteristic, complemented
3C4⋊#1084C211yesmaximal, characteristic, complemented
4#88450C212yesmaximal
5#88449C212yesmaximal, complemented
6#63871—31nomaximal
7#63857—31nomaximal
8C22×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#1084C2212yes—
11C2×(C3⋊Dic3)⋊C3C2212yes—
12C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4×(C3⋊D3)⋊C3—22no—
14#634⋊C3—22no—
15C2×C4×C3⋊D3D311yescharacteristic
16C2×#634D311yescharacteristic, complemented
17C2×C3×C4×D3—31no—
18C2×C3×Dic6—31no—
19C3×C4⋊Dic3—31no—
20C3×C4⋊Dic3—31no—
21C3×C4⋊Dic3—61no—
22Dic3⋊Dic3—32no—
23Dic3⋊Dic3—32no—
24#3722—32no—
25(C2×D3)⋊Dic3—32no—
26C22×H(3,𝔽3)C2311yescharacteristic
27C2×(C3⋊D3)⋊C3—21no—
28C2×(C3⋊D3)⋊C3—21no—
29C2×(C3⋊D3)⋊C3Q812yescomplemented
30C4×H(3,𝔽3)D412yes—
31#1084—22no—
32(C3⋊Dic3)⋊C3—22no—
33(C3⋊Dic3)⋊C3—21no—
34#946—91no—
35#924—91no—
36#897—91no—
37C22×C3⋊D3C2×D311yescharacteristic
38C22×C3×D3—31no—
39C2×C32×C4—21no—
40C2×C32×C4C2×D311yescharacteristic
41C2×C32×C4C2×D311yescharacteristic
42C2×C3⋊Dic3C2×D312yes—
43C2×C3⋊Dic3C2×D311yescharacteristic
44C4×C3⋊D3—22no—
45#634—22no—
46C2×C3×Dic3—32no—
47C2×C3×Dic3—31no—
48C2×C3×Dic3—32no—
49C2×C3×Dic3—32no—
50C2×C3×Dic3—62no—
51C3×C4×D3—62no—
52C3×Dic6—62no—
53C2×H(3,𝔽3)C2×D411yescharacteristic
54C2×H(3,𝔽3)C2×Q811yescharacteristic
55C2×H(3,𝔽3)#4111yescharacteristic
56(C3⋊D3)⋊C3—22no—
57C2×C3×Q8—91no—
58C22×C3×C4—91no—
59C2×C4×D3—31no—
60C2×C4×D3—181no—
61C2×Dic6—31no—
62C2×Dic6—181no—
63C3×C4⋊C4—91no—
64#217—92no—
65(C2×D3)⋊C4—92no—
66(C2×D3)⋊C4—92no—
67C4⋊Dic3—31no—
68C4⋊Dic3—31no—
69C4⋊Dic3—61no—
70C4⋊Dic3—92no—
71Dic3⋊C4—92no—
72Dic3⋊C4—92no—
73C22×C32—21no—
74C22×C32C22×D311yescharacteristic
75C22×C32C22×D311yescharacteristic
76C2×C3⋊D3—21no—
77C2×C3⋊D3—21no—
78C2×C3⋊D3Dic612yescomplemented
79C2×C3×D3—32no—
80C2×C3×D3—61no—
81C2×C3×D3—61no—
82C32×C4—22no—
83C32×C4#6712yes—
84C32×C4#6712yes—
85C3⋊Dic3—22no—
86C3⋊Dic3—21no—
87C3×Dic3—62no—
88C3×Dic3—62no—
89C3×Dic3—62no—
90C3×Dic3—61no—
91C3×Dic3—122no—
92C22⋊Q8—271noSylow
93H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
94C23×C3—91no—
95C22×D3—31no—
96C22×D3—181no—
97C3×Q8—182no—
98C2×C3×C4—61no—
99C2×C3×C4—61no—
100C2×C3×C4—91no—
101C2×C3×C4—92no—
102C2×C3×C4—92no—
103C2×C3×C4—121no—
104C2×C3×C4—182no—
105C2×C3×C4D3211yescharacteristic
106C2×Dic3—32no—
107C2×Dic3—32no—
108C2×Dic3—31no—
109C2×Dic3—32no—
110C2×Dic3—62no—
111C2×Dic3—181no—
112C2×Dic3—182no—
113C4×D3—62no—
114C4×D3—182no—
115Dic6—62no—
116Dic6—182no—
117C2×C32—21no—
118C2×C32—21no—
119C2×C32—21no—
120C2×C32C2×Dic611yescharacteristic
121C2×C32#23511yescharacteristic
122C2×C32D3×Q811yescharacteristic
123C2×C32#23911yescharacteristic
124C2×C32C2×#6711yescharacteristic
125C2×C32C2×#6711yescharacteristic
126C3⋊D3—22no—
127C3×D3—62no—
128C2×Q8—91no—
129C22×C4—91no—
130C4⋊C4—91no—
131C4⋊C4—272no—
132C22⋊C4—272no—
133C22×C3—61no—
134C22×C3—61no—
135C22×C3—92no—
136C22×C3—121no—
137C22×C3—181no—
138C22×C3—181no—
139C22×C3C2×D3211yescharacteristic
140C2×D3—32no—
141C2×D3—61no—
142C2×D3—61no—
143C2×D3—182no—
144C2×D3—181no—
145C2×D3—181no—
146C3×C4—62no—
147C3×C4—62no—
148C3×C4—122no—
149C3×C4—181no—
150C3×C4—182no—
151C3×C4—182no—
152C3×C4#62512yes—
153Dic3—62no—
154Dic3—62no—
155Dic3—62no—
156Dic3—61no—
157Dic3—122no—
158Dic3—182no—
159Dic3—181no—
160C32—21no—
161C32#92411yescharacteristic, complemented
162C32#94611yescharacteristic, complemented
163C23—91no—
164Q8—182no—
165C2×C4—92no—
166C2×C4—92no—
167C2×C4—91no—
168C2×C4—182no—
169C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
170C2×C3—61no—
171C2×C3—61no—
172C2×C3—61no—
173C2×C3—61no—
174C2×C3—61no—
175C2×C3—61no—
176C2×C3—121no—
177C2×C3—121no—
178C2×C3—121no—
179C2×C3—182no—
180C2×C3D3×Dic611yescharacteristic
181C2×C3#379911yescharacteristic
182C2×C3C2×#62511yescharacteristic
183D3—62no—
184D3—182no—
185C22—92no—
186C22—181no—
187C22—181no—
188C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
189C4—182no—
190C4—181no—
191C4—182no—
192C4#619812yes—
193C3—61no—
194C3—61no—
195C3—121no—
196C3#6385611yescharacteristic
197C2—182no—
198C2((C3⋊D3)⋊C3)⋊Q811yescharacteristic
199C2#8865211yescharacteristic
200C2C2×#619811yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#634⋊C3C2semidirect
3. C4⋊#1084C2semidirect
5. #88449C2semidirect
16. C2×#634D3semidirect
29. C2×(C3⋊D3)⋊C3Q8semidirect
78. C2×C3⋊D3Dic6semidirect
93. H(3,𝔽3)C22⋊Q8semidirect
161. C32#924semidirect
162. C32#946semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C3⋊D3)⋊C32
C2×#634⋊C32
C4⋊#10842
#884502
#884492
#638713
#638573