Finite Group Databasebeta

← #11721107

Subgroups — #11721107

Order 864.

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

normal subgroups only

This group has 297 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×#6229C211yesmaximal, characteristic, complemented
2C2×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4#88450C212yesmaximal
5#88448C212yesmaximal, complemented
6#63871—31nomaximal
7#63856—31nomaximal
8C22×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#1084C2212yes—
11#6229—22no—
12C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×(C3⋊Dic3)⋊C3C2212yes—
14C4×(C3⋊D3)⋊C3—22no—
15C2×C4×C3⋊D3D311yescharacteristic
16C2×C3×C4×D3—31no—
17C2×C3×Dic6—31no—
18C2×C3×Dic6—31no—
19C2×C3×Dic6—61no—
20C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
21C3×C4⋊Dic3—31no—
22#3724—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—21no—
33(C3⋊Dic3)⋊C3—22no—
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—
45C2×C3×Dic3—32no—
46C2×C3×Dic3—31no—
47C2×C3×Dic3—32no—
48C2×C3×Dic3—32no—
49C2×C3×Dic3—62no—
50C3×C4×D3—62no—
51C3×Dic6—64no—
52C3×Dic6—62no—
53C3×Dic6—62no—
54C2×H(3,𝔽3)C2×D411yescharacteristic
55C2×H(3,𝔽3)C2×Q811yescharacteristic
56C2×H(3,𝔽3)#4111yescharacteristic
57(C3⋊D3)⋊C3—22no—
58C2×C3×Q8—91no—
59C22×C3×C4—91no—
60C2×C4×D3—31no—
61C2×C4×D3—181no—
62C2×Dic6—31no—
63C2×Dic6—31no—
64C2×Dic6—61no—
65C3×C4⋊C4—91no—
66#217—92no—
67(C2×D3)⋊C4—92no—
68(C2×D3)⋊C4—92no—
69C4⋊Dic3—31no—
70C4⋊Dic3—91no—
71C4⋊Dic3—92no—
72Dic3⋊C4—92no—
73Dic3⋊C4—92no—
74C22×C32—21no—
75C22×C32C22×D311yescharacteristic
76C22×C32C22×D311yescharacteristic
77C2×C3⋊D3—21no—
78C2×C3⋊D3—21no—
79C2×C3⋊D3Dic612yescomplemented
80C2×C3×D3—32no—
81C2×C3×D3—61no—
82C2×C3×D3—61no—
83C32×C4—22no—
84C32×C4D1212yes—
85C32×C4#6712yes—
86C3⋊Dic3—22no—
87C3⋊Dic3—21no—
88C3×Dic3—64no—
89C3×Dic3—61no—
90C3×Dic3—62no—
91C3×Dic3—62no—
92C3×Dic3—62no—
93C22⋊Q8—271noSylow
94H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
95C23×C3—91no—
96C22×D3—31no—
97C22×D3—181no—
98C3×Q8—182no—
99C2×C3×C4—31no—
100C2×C3×C4—61no—
101C2×C3×C4—91no—
102C2×C3×C4—92no—
103C2×C3×C4—92no—
104C2×C3×C4—121no—
105C2×C3×C4—182no—
106C2×C3×C4D3211yescharacteristic
107C2×Dic3—32no—
108C2×Dic3—32no—
109C2×Dic3—31no—
110C2×Dic3—32no—
111C2×Dic3—62no—
112C2×Dic3—92no—
113C2×Dic3—181no—
114C4×D3—62no—
115C4×D3—182no—
116Dic6—62no—
117Dic6—62no—
118Dic6—64no—
119C2×C32—21no—
120C2×C32—21no—
121C2×C32—21no—
122C2×C32C2×Dic611yescharacteristic
123C2×C32C2×D1211yescharacteristic
124C2×C32#23511yescharacteristic
125C2×C32#23711yescharacteristic
126C2×C32D3×Q811yescharacteristic
127C2×C32C2×#6711yescharacteristic
128C3⋊D3—22no—
129C3×D3—62no—
130C2×Q8—91no—
131C22×C4—91no—
132C4⋊C4—91no—
133C4⋊C4—272no—
134C22⋊C4—272no—
135C22×C3—31no—
136C22×C3—61no—
137C22×C3—92no—
138C22×C3—121no—
139C22×C3—181no—
140C22×C3—181no—
141C22×C3C2×D3211yescharacteristic
142C2×D3—32no—
143C2×D3—61no—
144C2×D3—61no—
145C2×D3—182no—
146C2×D3—181no—
147C2×D3—181no—
148C3×C4—32no—
149C3×C4—62no—
150C3×C4—122no—
151C3×C4—181no—
152C3×C4—182no—
153C3×C4—182no—
154C3×C4#62612yes—
155Dic3—62no—
156Dic3—62no—
157Dic3—64no—
158Dic3—62no—
159Dic3—61no—
160Dic3—182no—
161Dic3—181no—
162C32—21no—
163C32#89711yescharacteristic, complemented
164C32#92411yescharacteristic, complemented
165C23—91no—
166Q8—182no—
167C2×C4—92no—
168C2×C4—92no—
169C2×C4—91no—
170C2×C4—182no—
171C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
172C2×C3—31no—
173C2×C3—31no—
174C2×C3—31no—
175C2×C3—61no—
176C2×C3—61no—
177C2×C3—61no—
178C2×C3—121no—
179C2×C3—121no—
180C2×C3—121no—
181C2×C3—182no—
182C2×C3D3×Dic611yescharacteristic
183C2×C3#379511yescharacteristic
184C2×C3C2×#62611yescharacteristic
185D3—62no—
186D3—182no—
187C22—92no—
188C22—181no—
189C22—181no—
190C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
191C4—182no—
192C4—181no—
193C4—182no—
194C4#620012yes—
195C3—31no—
196C3—61no—
197C3—121no—
198C3#6385711yescharacteristic
199C2—182no—
200C2((C3⋊D3)⋊C3)⋊Q811yescharacteristic
201C2#8864811yescharacteristic
202C2C2×#620011yescharacteristic
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
1. C2×#6229C2semidirect
3. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #88448C2semidirect
20. C4⋊(C3⋊Dic3)D3semidirect
29. C2×(C3⋊D3)⋊C3Q8semidirect
79. C2×C3⋊D3Dic6semidirect
94. H(3,𝔽3)C22⋊Q8semidirect
163. C32#897semidirect
164. C32#924semidirect

Maximal subgroups

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

subgroupindex
C2×#62292
C2×C4×(C3⋊D3)⋊C32
C4⋊((C3⋊Dic3)⋊C3)2
#884502
#884482
#638713
#638563