Finite Group Databasebeta

← #13152221

Subgroups — #13152221

Order 1728.

366 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88691C211yesmaximal, characteristic
2#11721359C212yesmaximal, complemented
3C32×#568C311yesmaximal, characteristic
4#568⋊C9C313yesmaximal, complemented
5#88691C411yescharacteristic
6#88691C2211yescharacteristic
7C2×#6204C411yescharacteristic
8M16×3−1+2—41nomaximal
9C2×C32×#41C2×C311yescharacteristic
10C32×#131—32no—
11C32×#131C2×C312yes—
12C2×C32×M16—31no—
13C2×C32×M16—31no—
14C2×#41⋊C9C2×C313yes—
15#131⋊C9C2×C316yescomplemented
16C2×C4×3−1+2—41no—
17#6204C2×C411yescharacteristic
18C8×3−1+2—42no—
19C3×#568—31no—
20C3×#568C3211yescharacteristic
21C32×#41—31no—
22C32×#41—31no—
23C32×#41C3×C411yescharacteristic
24C32×#41C22×C311yescharacteristic
25C2×C32×Q8C3×C411yescharacteristic
26C2×C32×D4—31no—
27C22×C32×C4—31no—
28C32×M16—31no—
29C32×M16—32no—
30C32×M16—32no—
31C32×M16A411yescharacteristic
32C2×C32×C8—32no—
33C2×C32×C8—32no—
34#41⋊C9C3×C413yes—
35#41⋊C9C22×C313yes—
36C2×Q8⋊C9C3×C413yes—
37C9×M16—43no—
38C22×3−1+2—41no—
39C4×3−1+2—41no—
40C4×3−1+2—41no—
41C2×C3×#41—31no—
42C2×C3×#41C2×C3211yescharacteristic
43C3×#131—36no—
44C3×#131—32no—
45C3×#131—32no—
46C3×#131C2×C3212yes—
47C2×C3×M16—33no—
48C2×C3×M16—33no—
49C2×C3×M16—31no—
50C2×C3×M16—31no—
51C23×C32—31no—
52C32×Q8—31no—
53C32×Q8C2×C3×C411yescharacteristic
54C32×D4—31no—
55C32×D4—31no—
56C32×D4—32no—
57C2×C32×C4—31no—
58C2×C32×C4—31no—
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62C2×C32×C4C2×A411yescharacteristic
63C32×C8—32no—
64C32×C8C2×A412yes—
65C2×C4×C9—43no—
66Q8⋊C9C2×C3×C413yes—
67C8×C9—46no—
68#5683−1+211yescharacteristic, complemented, Sylow
69C2×3−1+2—41no—
70C2×3−1+2—81no—
71C3×#41—33no—
72C3×#41—33no—
73C3×#41—31no—
74C3×#41—31no—
75C3×#41—31no—
76C3×#41—31no—
77C3×#41C32×C411yescharacteristic
78C3×#41C22×C3211yescharacteristic
79C2×C3×Q8—31no—
80C2×C3×Q8C32×C411yescharacteristic
81C2×C3×D4—31no—
82C2×C3×D4—33no—
83C22×C3×C4—33no—
84C22×C3×C4—31no—
85C3×M16—33no—
86C3×M16—36no—
87C3×M16—36no—
88C3×M16—31no—
89C3×M16—32no—
90C3×M16—32no—
91C3×M16—31no—
92C3×M16C3×A411yescharacteristic
93C2×C3×C8—36no—
94C2×C3×C8—32no—
95C2×C3×C8—36no—
96C2×C3×C8—32no—
97C22×C32—31no—
98C22×C32—31no—
99C22×C32—121no—
100C22×C32C4×A411yescharacteristic
101C32×C4—31no—
102C32×C4—31no—
103C32×C4C4×A411yescharacteristic
104C32×C4C22×A411yescharacteristic
105C22×C9—43no—
106C4×C9—43no—
107C4×C9—43no—
108C2×#41C2×3−1+211yescharacteristic
109#131—32no—
110#131C2×3−1+212yescomplemented
111C2×M16—31no—
112C2×M16—31no—
1133−1+2—41noSylow
114C23×C3—33no—
115C23×C3—31no—
116C3×Q8—31no—
117C3×Q8—33no—
118C3×Q8—31no—
119C3×Q8C2×C32×C411yescharacteristic
120C3×D4—31no—
121C3×D4—31no—
122C3×D4—33no—
123C3×D4—33no—
124C3×D4—36no—
125C3×D4—32no—
126C2×C3×C4—33no—
127C2×C3×C4—33no—
128C2×C3×C4—33no—
129C2×C3×C4—33no—
130C2×C3×C4—33no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—31no—
134C2×C3×C4—31no—
135C2×C3×C4—31no—
136C2×C3×C4—31no—
137C2×C3×C4C2×C3×A411yescharacteristic
138C3×C8—36no—
139C3×C8—32no—
140C3×C8—32no—
141C3×C8C2×C3×A412yes—
142C2×C32—21no—
143C2×C32—61no—
144C2×C32—61no—
145C2×C32C2×C4×A411yescharacteristic
146C2×C9—43no—
147C2×C9—83no—
148#41—31no—
149#41—31no—
150#41C4×3−1+211yescharacteristic
151#41C22×3−1+211yescharacteristic
152C2×Q8C4×3−1+211yescharacteristic
153C2×D4—31no—
154C22×C4—31no—
155M16—31no—
156M16—32no—
157M16—32no—
158M16#109411yescharacteristic
159C2×C8—32no—
160C2×C8—32no—
161C22×C3—33no—
162C22×C3—33no—
163C22×C3—31no—
164C22×C3—31no—
165C22×C3—31no—
166C22×C3—121no—
167C22×C3—123no—
168C22×C3C3×C4×A411yescharacteristic
169C3×C4—33no—
170C3×C4—33no—
171C3×C4—31no—
172C3×C4—31no—
173C3×C4—31no—
174C3×C4—31no—
175C3×C4C3×C4×A411yescharacteristic
176C3×C4C22×C3×A411yescharacteristic
177C32#568⋊C311yescharacteristic
178C9—43no—
179C23—31no—
180Q8—31no—
181Q8C2×C4×3−1+211yescharacteristic
182D4—31no—
183D4—31no—
184D4—32no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4C2×#109411yescharacteristic
191C8—32no—
192C8C2×#109412yes—
193C2×C3—21no—
194C2×C3—31no—
195C2×C3—61no—
196C2×C3—63no—
197C2×C3—63no—
198C2×C3—61no—
199C2×C3—61no—
200C2×C3C2×C3×C4×A411yescharacteristic
201C22—31no—
202C22—31no—
203C22—121no—
204C22C4×#109411yescharacteristic
205C4—31no—
206C4—31no—
207C4C4×#109411yescharacteristic
208C4C22×#109411yescharacteristic
209C3—31no—
210C3C3×#568⋊C311yescharacteristic
211C2—21no—
212C2—61no—
213C2—61no—
214C2C2×C4×#109411yescharacteristic
subgroupG/H
center, FrattiniC3×C4C22×C3×A4
commutatorC3×Q8C2×C32×C4
FittingC32×#568C3
socleC2×C3C2×C3×C4×A4
prime radicalC2×C32×#41C2×C3
cosocleC3×#41C22×C32
hypercenterC32×M16A4
hypocenterQ8C2×C4×3−1+2
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C3×C4 ◁ C32×M16

Complements

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

NH ≅ G/Ntype
2. #11721359C2semidirect
4. #568⋊C9C3semidirect
15. #131⋊C9C2×C3semidirect
68. #5683−1+2semidirect
110. #131C2×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C2×#886912
#117213592
C32×#5683
#568⋊C93
M16×3−1+24