Finite Group Databasebeta

← #11721074

Subgroups — #11721074

Order 864.

248 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 50 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
2#3828⋊C3C211yesmaximal, characteristic
3#3748⋊C3C211yesmaximal, characteristic, complemented
4#88439C212yesmaximal, complemented
5#88438C212yesmaximal, complemented
6#63833—31nomaximal
7#63827—31nomaximal
8C2×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#1084—21no—
10#6229—21no—
11#6229C2212yes—
12#638⋊C3—21no—
13C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C4×(C3⋊D3)⋊C3—21no—
15#634⋊C3C2211yescharacteristic
16(C3⋊(C3⋊C8))⋊C3C2212yes—
17#3828D311yescharacteristic
18C3×#235—31no—
19C2×C3×Dic6—31no—
20C2×C3×Dic6—31no—
21C2×C3×Dic6—61no—
22#3748D311yescharacteristic, complemented
23C3×#208—31no—
24#3719—32no—
25#3718—32no—
26#3717—32no—
27#3715—32no—
28C22×H(3,𝔽3)D411yescharacteristic
29C2×(C3⋊D3)⋊C3—21no—
30C4×H(3,𝔽3)D411yescharacteristic
31C4×H(3,𝔽3)C2311yescharacteristic
32#1084—22no—
33(C3⋊Dic3)⋊C3—21no—
34#951—91no—
35#942—91no—
36#909—91no—
37C2×C32×C4—21no—
38C2×C32×C4C2×D311yescharacteristic
39C2×C32×C4C2×D311yescharacteristic
40#638—21no—
41C4⋊(C3⋊D3)C2×D311yescharacteristic
42C4×C3⋊D3—21no—
43#634C2×D311yescharacteristic
44C3×#67—61no—
45C2×C3×Dic3—62no—
46C2×C3×Dic3—61no—
47C2×C3×Dic3—61no—
48C3×D12—31no—
49C3×C4×D3—61no—
50C3×Dic6—31no—
51C3×Dic6—32no—
52C3×Dic6—32no—
53C3×Dic6—62no—
54C3×Dic6—62no—
55C3×Dic6—61no—
56C3×Dic6—61no—
57C3⋊(C3⋊C8)C2×D312yes—
58C3×C3⋊C8—32no—
59C2×H(3,𝔽3)—21no—
60C2×H(3,𝔽3)C2×D411yescharacteristic
61(C3⋊D3)⋊C3—41no—
62C3×#41—91no—
63C2×C3×Q8—91no—
64#235—31no—
65#235—181no—
66C2×Dic6—31no—
67C2×Dic6—31no—
68C2×Dic6—61no—
69C3×M16—91no—
70#216—92no—
71#216—92no—
72#215—92no—
73#214—92no—
74#208—31no—
75#208—91no—
76Dic12—92no—
77#204—92no—
78C22×C32—21no—
79C22×C32D1211yescharacteristic
80C22×C32#6711yescharacteristic
81C2×C3⋊D3—21no—
82C2×C3×D3—61no—
83C32×C4—21no—
84C32×C4—21no—
85C32×C4D1211yescharacteristic
86C32×C4#6711yescharacteristic
87C32×C4C22×D311yescharacteristic
88C32×C4C22×D311yescharacteristic
89C3⋊Dic3—21no—
90C3×Dic3—64no—
91C3×Dic3—61no—
92C3×Dic3—62no—
93C3×Dic3—62no—
94#137—271noSylow
95H(3,𝔽3)#13711yescharacteristic, complemented, Sylow
96C3×Q8—91no—
97C3×Q8—92no—
98C3×Q8—181no—
99C3×D4—91no—
100C3×D4—181no—
101C2×C3×C4—31no—
102C2×C3×C4—61no—
103C2×C3×C4—61no—
104C2×C3×C4—181no—
105C2×C3×C4—181no—
106C2×C3×C4D3211yescharacteristic
107#67—61no—
108#67—181no—
109C2×Dic3—61no—
110C2×Dic3—61no—
111C2×Dic3—62no—
112D12—31no—
113D12—181no—
114C4×D3—61no—
115C4×D3—181no—
116Dic6—32no—
117Dic6—32no—
118Dic6—31no—
119Dic6—62no—
120Dic6—61no—
121Dic6—61no—
122Dic6—62no—
123Dic6—181no—
124C3×C8—92no—
125C3⋊C8—32no—
126C3⋊C8—92no—
127C2×C32—22no—
128C2×C32—21no—
129C2×C32—21no—
130C2×C32—21no—
131C2×C32C2×D1211yescharacteristic
132C2×C32C2×#6711yescharacteristic
133C3⋊D3—41no—
134C3×D3—121no—
135#41—91no—
136C2×Q8—91no—
137Q16—272no—
138SD16—272no—
139M16—91no—
140C22×C3—31no—
141C22×C3—61no—
142C22×C3—61no—
143C22×C3—181no—
144C22×C3#62611yescharacteristic
145C2×D3—61no—
146C2×D3—181no—
147C3×C4—31no—
148C3×C4—31no—
149C3×C4—61no—
150C3×C4—61no—
151C3×C4—61no—
152C3×C4—61no—
153C3×C4—181no—
154C3×C4—182no—
155C3×C4#62611yescharacteristic
156C3×C4C2×D3211yescharacteristic
157Dic3—62no—
158Dic3—62no—
159Dic3—64no—
160Dic3—61no—
161Dic3—181no—
162C32—21no—
163C32#90911yescharacteristic, complemented
164C32#95111yescharacteristic, complemented
165Q8—92no—
166Q8—91no—
167Q8—181no—
168D4—91no—
169D4—181no—
170C2×C4—181no—
171C2×C4—181no—
172C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
173C8—92no—
174C2×C3—21no—
175C2×C3—31no—
176C2×C3—61no—
177C2×C3—62no—
178C2×C3—61no—
179C2×C3—61no—
180C2×C3—121no—
181C2×C3—361no—
182C2×C3C2×#62611yescharacteristic
183D3—121no—
184D3—361no—
185C22—181no—
186C22#620011yescharacteristic
187C4—181no—
188C4—182no—
189C4#620011yescharacteristic
190C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
191C3—31no—
192C3—61no—
193C3—61no—
194C3#6383111yescharacteristic
195C2—21no—
196C2—361no—
197C2C2×#620011yescharacteristic
subgroupG/H
centerC2C2×#6200
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C2×D32
FittingC2×C4×H(3,𝔽3)C22
socleC2×C3C2×#626
prime radical#3828⋊C3C2
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#137
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#6229C2semidirect
3. #3748⋊C3C2semidirect
4. #88439C2semidirect
5. #88438C2semidirect
22. #3748D3semidirect
95. H(3,𝔽3)#137semidirect
163. C32#909semidirect
164. C32#951semidirect

Maximal subgroups

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

subgroupindex
C2×#62292
#3828⋊C32
#3748⋊C32
#884392
#884382
#638333
#638273