Finite Group Databasebeta

← #11721309

Subgroups — #11721309

Order 864.

313 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C22×#1084C211yesmaximal, characteristic
2C2×#6196C211yesmaximal, characteristic, complemented
3#3757⋊C3C212yesmaximal
4#88449C211yesmaximal, characteristic, complemented
5#88448C212yesmaximal, complemented
6#63964—32nomaximal
7C23×H(3,𝔽3)C2211yescharacteristic
8C2×#1084—22no—
9C2×#1084C2211yescharacteristic
10C2×#1084C2211yescharacteristic
11C2×(C3⋊Dic3)⋊C3C2212yes—
12C2×(C3⋊Dic3)⋊C3C2212yes—
13#6196—22no—
14C22×C3×Dic3—32no—
15C22×C3×Dic3—61no—
16C2×#627—32no—
17#3757D312yes—
18C3×#217—32no—
19#3724—32no—
20Dic3⋊Dic3—32no—
21Dic3⋊Dic3—32no—
22C22×H(3,𝔽3)—21no—
23C22×H(3,𝔽3)—21no—
24C22×H(3,𝔽3)Q812yescomplemented
25C22×H(3,𝔽3)C2311yescharacteristic
26#1084—21no—
27#1084D412yes—
28(C3⋊Dic3)⋊C3—22no—
29(C3⋊Dic3)⋊C3—22no—
30#924—91no—
31#878—92no—
32C23×C32—21no—
33C23×C32C2×D312yes—
34C2×C3⋊Dic3C2×D312yes—
35C2×C3⋊Dic3C2×D312yes—
36C2×C3×Dic3—32no—
37C2×C3×Dic3—32no—
38C2×C3×Dic3—32no—
39C2×C3×Dic3—32no—
40C2×C3×Dic3—61no—
41C2×C3×Dic3—61no—
42C2×C3×Dic3—64no—
43C2×C3×Dic3—64no—
44#627—64no—
45C2×H(3,𝔽3)—22no—
46C2×H(3,𝔽3)C2×D411yescharacteristic
47C2×H(3,𝔽3)C2×Q811yescharacteristic
48C2×H(3,𝔽3)#4111yescharacteristic
49C22×C3×C4—91no—
50C22×Dic3—32no—
51C22×Dic3—61no—
52C2×Dic6—91no—
53C2×Dic6—92no—
54C3×C22⋊C4—92no—
55#217—32no—
56#217—92no—
57C4⋊Dic3—91no—
58C4⋊Dic3—92no—
59Dic3⋊C4—92no—
60Dic3⋊C4—92no—
61Dic3⋊C4—92no—
62C22×C32—22no—
63C22×C32—22no—
64C22×C32—22no—
65C22×C32—22no—
66C22×C32—22no—
67C22×C32—21no—
68C22×C32Dic614yescomplemented
69C22×C32C22×D312yes—
70C3⋊Dic3—22no—
71C3⋊Dic3—22no—
72C3×Dic3—34no—
73C3×Dic3—62no—
74C3×Dic3—62no—
75C3×Dic3—62no—
76C3×Dic3—62no—
77C3×Dic3—62no—
78C22⋊Q8—271noSylow
79H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
80C23×C3—32no—
81C23×C3—61no—
82C23×C3D3211yescharacteristic
83C2×C3×C4—91no—
84C2×C3×C4—92no—
85C2×C3×C4—92no—
86C2×C3×C4—91no—
87C2×C3×C4—182no—
88C2×Dic3—32no—
89C2×Dic3—32no—
90C2×Dic3—32no—
91C2×Dic3—32no—
92C2×Dic3—61no—
93C2×Dic3—61no—
94C2×Dic3—64no—
95C2×Dic3—64no—
96C2×Dic3—92no—
97C2×Dic3—92no—
98Dic6—184no—
99Dic6—182no—
100C2×C32—24no—
101C2×C32—24no—
102C2×C32—21no—
103C2×C32—21no—
104C2×C32—21no—
105C2×C32C2×Dic612yes—
106C2×C32D3×D412yes—
107C2×C32#23712yes—
108C2×Q8—271no—
109C22×C4—91no—
110C4⋊C4—271no—
111C4⋊C4—272no—
112C22⋊C4—92no—
113C22×C3—21no—
114C22×C3—21no—
115C22×C3—34no—
116C22×C3—32no—
117C22×C3—62no—
118C22×C3—62no—
119C22×C3—62no—
120C22×C3—62no—
121C22×C3—62no—
122C22×C3—61no—
123C22×C3#62712yes—
124C22×C3C2×D3211yescharacteristic
125C3×C4—92no—
126C3×C4—182no—
127C3×C4—182no—
128C3×C4—181no—
129Dic3—34no—
130Dic3—62no—
131Dic3—62no—
132Dic3—62no—
133Dic3—62no—
134Dic3—62no—
135Dic3—182no—
136Dic3—182no—
137C32—21no—
138C32#87812yescomplemented
139C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
140Q8—542no—
141C2×C4—91no—
142C2×C4—92no—
143C2×C4—92no—
144C2×C4—91no—
145C2×C4—182no—
146C2×C3—22no—
147C2×C3—32no—
148C2×C3—32no—
149C2×C3—32no—
150C2×C3—64no—
151C2×C3—64no—
152C2×C3—61no—
153C2×C3—61no—
154C2×C3—61no—
155C2×C3#380511yescharacteristic
156C2×C3C2×#62711yescharacteristic
157C2×C3#381111yescharacteristic
158C22—21no—
159C22—21no—
160C22#619612yescomplemented
161C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
162C4—92no—
163C4—182no—
164C4—181no—
165C4—182no—
166C3—32no—
167C3—61no—
168C3#6398211yescharacteristic
169C2—22no—
170C2C2×#619611yescharacteristic
171C2#8867211yescharacteristic
172C2#8867711yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
Fitting, prime radicalC23×H(3,𝔽3)C22
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22⋊Q8
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×#6196C2semidirect
4. #88449C2semidirect
5. #88448C2semidirect
24. C22×H(3,𝔽3)Q8semidirect
68. C22×C32Dic6semidirect
79. H(3,𝔽3)C22⋊Q8semidirect
138. C32#878semidirect
160. C22#6196semidirect

Maximal subgroups

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

subgroupindex
C22×#10842
C2×#61962
#3757⋊C32
#884492
#884482
#639643