Finite Group Databasebeta

← #11721319

Subgroups — #11721319

Order 864.

327 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 52 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×#6235C211yesmaximal, characteristic
2C2×#6199C211yesmaximal, characteristic, complemented
3#3757⋊C3C212yesmaximal, complemented
4#88451C211yesmaximal, characteristic, complemented
5#88448C212yesmaximal, complemented
6#63969—32nomaximal
7C23×H(3,𝔽3)C2211yescharacteristic
8C22×#272C2211yescharacteristic
9#6235—22no—
10C2×#1084C2211yescharacteristic
11C2×(C3⋊Dic3)⋊C3C2212yes—
12C2×(C3⋊Dic3)⋊C3C2212yes—
13#6199—22no—
14C2×C3×#67—32no—
15C2×C3×#67—61no—
16C2×D3×Dic3—32no—
17#3757D312yescomplemented
18C3×#217—32no—
19#3724—32no—
20Dic3⋊Dic3—32no—
21(C2×D3)⋊Dic3—32no—
22C22×H(3,𝔽3)—21no—
23C22×H(3,𝔽3)—22no—
24C22×H(3,𝔽3)C2311yescharacteristic
25C2×#272—22no—
26C2×#272D412yes—
27#1084—21no—
28(C3⋊Dic3)⋊C3—22no—
29(C3⋊Dic3)⋊C3—22no—
30#935—91no—
31#883—92no—
32C23×C32—21no—
33C23×C32C2×D312yes—
34C22×C3×D3—32no—
35C22×C3×D3—61no—
36C2×C3⋊Dic3C2×D312yes—
37C2×C3⋊Dic3C2×D312yes—
38C3×#67—64no—
39C3×#67—64no—
40C2×C3×Dic3—32no—
41C2×C3×Dic3—32no—
42C2×C3×Dic3—32no—
43C2×C3×Dic3—61no—
44D3×Dic3—64no—
45C2×H(3,𝔽3)—41no—
46C2×H(3,𝔽3)C2×D411yescharacteristic
47C2×H(3,𝔽3)#4112yes—
48#272—22no—
49C2×C3×D4—91no—
50C2×#67—32no—
51C2×#67—61no—
52C22×Dic3—91no—
53C2×C4×D3—92no—
54C3×C22⋊C4—92no—
55#217—32no—
56#217—92no—
57#217—91no—
58(C2×D3)⋊C4—92no—
59C4⋊Dic3—92no—
60Dic3⋊C4—92no—
61Dic3⋊C4—92no—
62C22×C32—22no—
63C22×C32—22no—
64C22×C32—22no—
65C22×C32—22no—
66C22×C32—21no—
67C22×C32—42no—
68C22×C32C22×D312yes—
69C2×C3×D3—34no—
70C2×C3×D3—62no—
71C2×C3×D3—62no—
72C2×C3×D3—62no—
73C2×C3×D3—122no—
74C3⋊Dic3—22no—
75C3⋊Dic3—22no—
76C3×Dic3—62no—
77C3×Dic3—62no—
78C3×Dic3—62no—
79C3×Dic3—62no—
80#123—271noSylow
81H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
82C23×C3—32no—
83C23×C3—61no—
84C23×C3—91no—
85C23×C3D3211yescharacteristic
86C22×D3—32no—
87C22×D3—61no—
88C3×D4—182no—
89C2×C3×C4—92no—
90C2×C3×C4—92no—
91C2×C3×C4—91no—
92#67—64no—
93#67—64no—
94C2×Dic3—32no—
95C2×Dic3—32no—
96C2×Dic3—32no—
97C2×Dic3—61no—
98C2×Dic3—92no—
99C2×Dic3—92no—
100C2×Dic3—182no—
101C4×D3—184no—
102C2×C32—22no—
103C2×C32—21no—
104C2×C32—42no—
105C2×C32—42no—
106C2×C32#23512yes—
107C2×C32D3×D412yes—
108C2×C32#23712yes—
109C3×D3—64no—
110C3×D3—122no—
111C2×D4—91no—
112C22×C4—271no—
113C4⋊C4—272no—
114C22⋊C4—92no—
115C22⋊C4—271no—
116C22×C3—22no—
117C22×C3—21no—
118C22×C3—32no—
119C22×C3—62no—
120C22×C3—62no—
121C22×C3—62no—
122C22×C3—62no—
123C22×C3—61no—
124C22×C3—92no—
125C22×C3—122no—
126C22×C3—182no—
127C22×C3C2×D3211yescharacteristic
128C2×D3—34no—
129C2×D3—62no—
130C2×D3—62no—
131C2×D3—62no—
132C2×D3—122no—
133C3×C4—182no—
134C3×C4—182no—
135C3×C4—181no—
136Dic3—62no—
137Dic3—62no—
138Dic3—62no—
139Dic3—62no—
140Dic3—182no—
141Dic3—182no—
142C32—21no—
143C32#88312yescomplemented
144C23—91no—
145C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
146D4—182no—
147C2×C4—92no—
148C2×C4—92no—
149C2×C4—91no—
150C2×C4—542no—
151C2×C3—32no—
152C2×C3—32no—
153C2×C3—32no—
154C2×C3—41no—
155C2×C3—62no—
156C2×C3—61no—
157C2×C3—122no—
158C2×C3—122no—
159C2×C3—182no—
160C2×C3#380412yes—
161C2×C3#381111yescharacteristic
162D3—64no—
163D3—122no—
164C22—22no—
165C22—21no—
166C22—92no—
167C22—182no—
168C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
169C4—182no—
170C4—182no—
171C4—181no—
172C3—32no—
173C3—61no—
174C3#6397511yescharacteristic
175C2—41no—
176C2—182no—
177C2#8867112yes—
178C2#8867711yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC23×H(3,𝔽3)C22
prime radicalC2×#6235C2
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#6199C2semidirect
3. #3757⋊C3C2semidirect
4. #88451C2semidirect
5. #88448C2semidirect
17. #3757D3semidirect
81. H(3,𝔽3)#123semidirect
143. C32#883semidirect

Maximal subgroups

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

subgroupindex
C2×#62352
C2×#61992
#3757⋊C32
#884512
#884482
#639693