Finite Group Databasebeta

← #11721301

Subgroups — #11721301

Order 864.

327 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 60 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
2C22×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3#3757⋊C3C211yesmaximal, characteristic, complemented
4#88451C212yesmaximal, complemented
5#88448C212yesmaximal, complemented
6#63962—31nomaximal
7#63961—31nomaximal
8C23×H(3,𝔽3)C2211yescharacteristic
9C22×#272C2211yescharacteristic
10#6235—22no—
11C2×#1084C2211yescharacteristic
12C2×(C3⋊Dic3)⋊C3—22no—
13C2×(C3⋊Dic3)⋊C3C2212yes—
14C2×(C3⋊Dic3)⋊C3C2212yes—
15C22×C3⋊Dic3D311yescharacteristic, complemented
16C2×C3×#67—31no—
17C2×C3×#67—31no—
18C2×C3×#67—61no—
19C22×C3×Dic3—31no—
20#3757D311yescharacteristic, complemented
21C3×#217—31no—
22#3724—32no—
23Dic3⋊Dic3—32no—
24(C2×D3)⋊Dic3—32no—
25(C2×D3)⋊Dic3—32no—
26C22×H(3,𝔽3)—22no—
27C22×H(3,𝔽3)D412yes—
28C22×H(3,𝔽3)C2311yescharacteristic
29C2×#272—21no—
30C2×#272—22no—
31#1084—21no—
32(C3⋊Dic3)⋊C3—22no—
33(C3⋊Dic3)⋊C3—22no—
34#935—91no—
35#929—91no—
36#886—91no—
37C23×C32—21no—
38C23×C32C2×D311yescharacteristic
39C23×C32C2×D311yescharacteristic
40C22×C3×D3—31no—
41C22×C3×D3—31no—
42C22×C3×D3—61no—
43C2×C3⋊Dic3—22no—
44C2×C3⋊Dic3C2×D312yes—
45C2×C3⋊Dic3C2×D312yes—
46C3×#67—64no—
47C3×#67—62no—
48C3×#67—62no—
49C2×C3×Dic3—32no—
50C2×C3×Dic3—32no—
51C2×C3×Dic3—31no—
52C2×C3×Dic3—31no—
53C2×C3×Dic3—61no—
54C2×C3×Dic3—62no—
55C2×H(3,𝔽3)—22no—
56C2×H(3,𝔽3)C2×D411yescharacteristic
57C2×H(3,𝔽3)#4112yes—
58#272—41no—
59C2×C3×D4—91no—
60C22×C3×C4—91no—
61C2×#67—31no—
62C2×#67—31no—
63C2×#67—61no—
64C22×Dic3—31no—
65C22×Dic3—91no—
66C3×C22⋊C4—91no—
67#217—31no—
68#217—91no—
69#217—92no—
70(C2×D3)⋊C4—92no—
71(C2×D3)⋊C4—92no—
72C4⋊Dic3—92no—
73Dic3⋊C4—92no—
74Dic3⋊C4—92no—
75C22×C32—22no—
76C22×C32—22no—
77C22×C32—22no—
78C22×C32—21no—
79C22×C32—42no—
80C22×C32D1212yes—
81C22×C32#6712yes—
82C22×C32C22×D311yescharacteristic
83C22×C32C22×D311yescharacteristic
84C2×C3×D3—62no—
85C2×C3×D3—61no—
86C2×C3×D3—62no—
87C2×C3×D3—62no—
88C2×C3×D3—61no—
89C2×C3×D3—122no—
90C3⋊Dic3—22no—
91C3⋊Dic3—22no—
92C3×Dic3—62no—
93C3×Dic3—62no—
94C3×Dic3—61no—
95C3×Dic3—62no—
96C3×Dic3—61no—
97#123—271noSylow
98H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
99C23×C3—31no—
100C23×C3—31no—
101C23×C3—61no—
102C23×C3—91no—
103C23×C3D3211yescharacteristic
104C22×D3—31no—
105C22×D3—31no—
106C22×D3—61no—
107C3×D4—182no—
108C2×C3×C4—92no—
109C2×C3×C4—92no—
110C2×C3×C4—91no—
111C2×C3×C4—182no—
112#67—62no—
113#67—62no—
114#67—64no—
115C2×Dic3—31no—
116C2×Dic3—31no—
117C2×Dic3—32no—
118C2×Dic3—32no—
119C2×Dic3—61no—
120C2×Dic3—62no—
121C2×Dic3—92no—
122C2×Dic3—92no—
123C2×Dic3—182no—
124C2×C32—22no—
125C2×C32—22no—
126C2×C32—22no—
127C2×C32—21no—
128C2×C32—42no—
129C2×C32C2×D1211yescharacteristic
130C2×C32#23512yes—
131C2×C32#23712yes—
132C2×C32C2×#6711yescharacteristic
133C3×D3—122no—
134C3×D3—121no—
135C3×D3—121no—
136C2×D4—91no—
137C22×C4—91no—
138C4⋊C4—272no—
139C22⋊C4—91no—
140C22⋊C4—272no—
141C22×C3—22no—
142C22×C3—32no—
143C22×C3—32no—
144C22×C3—31no—
145C22×C3—31no—
146C22×C3—62no—
147C22×C3—62no—
148C22×C3—62no—
149C22×C3—61no—
150C22×C3—122no—
151C22×C3—182no—
152C22×C3—181no—
153C22×C3#62612yes—
154C22×C3C2×D3211yescharacteristic
155C2×D3—61no—
156C2×D3—61no—
157C2×D3—62no—
158C2×D3—62no—
159C2×D3—62no—
160C2×D3—122no—
161C3×C4—182no—
162C3×C4—182no—
163C3×C4—181no—
164Dic3—61no—
165Dic3—61no—
166Dic3—62no—
167Dic3—62no—
168Dic3—62no—
169Dic3—182no—
170Dic3—182no—
171C32—21no—
172C32#88611yescharacteristic, complemented
173C32#92911yescharacteristic, complemented
174C23—91no—
175C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
176D4—182no—
177C2×C4—92no—
178C2×C4—92no—
179C2×C4—91no—
180C2×C4—182no—
181C2×C3—22no—
182C2×C3—32no—
183C2×C3—32no—
184C2×C3—31no—
185C2×C3—31no—
186C2×C3—62no—
187C2×C3—62no—
188C2×C3—62no—
189C2×C3—61no—
190C2×C3—122no—
191C2×C3—361no—
192C2×C3#380412yes—
193C2×C3C2×#62611yescharacteristic
194D3—121no—
195D3—121no—
196D3—122no—
197C22—22no—
198C22—182no—
199C22—181no—
200C22#620012yes—
201C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
202C4—182no—
203C4—182no—
204C4—181no—
205C3—31no—
206C3—31no—
207C3—61no—
208C3#6396611yescharacteristic
209C2—22no—
210C2—361no—
211C2#8867112yes—
212C2C2×#620011yescharacteristic
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. C22×(C3⋊Dic3)⋊C3C2semidirect
3. #3757⋊C3C2semidirect
4. #88451C2semidirect
5. #88448C2semidirect
15. C22×C3⋊Dic3D3semidirect
20. #3757D3semidirect
98. H(3,𝔽3)#123semidirect
172. C32#886semidirect
173. C32#929semidirect

Maximal subgroups

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

subgroupindex
C2×#62352
C22×(C3⋊Dic3)⋊C32
#3757⋊C32
#884512
#884482
#639623
#639613