Finite Group Databasebeta

← #5570⋊C9 #13147205

Subgroups — #5570⋊C9 #13147205

Order 1728.

252 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C3×C4×Q8⋊C9C211yesmaximal, characteristic
2Dic3×Q8⋊C9C212yesmaximal
3C3×#5570C311yesmaximal, characteristic
4#557⋊C9—31nomaximal
5C2×C3×Q8⋊C9C2211yescharacteristic
6C9×C4⋊Dic3—41nomaximal
7C32×C4×Q8C2×C311yescharacteristic
8C3×Q8×Dic3C2×C312yes—
9C3×#890—32no—
10C3×#869—31no—
11C3×C4×Dic6—31no—
12C4×Q8⋊C9—21no—
13C4×Q8⋊C9—32no—
14C4×Q8⋊C9D311yescharacteristic
15C2×C3×C4×C9—41no—
16C2×C9×Dic3—42no—
17C3×Q8⋊C9Q811yescharacteristic, complemented
18C3×#557—31no—
19#5570C911yescharacteristic, complemented
20C2×C32×Q8C22×C311yescharacteristic
21C2×C3×Dic6—31no—
22C32×C4⋊C4—31no—
23C32×C42—31no—
24C3×C4⋊Dic3—32no—
25C3×C4⋊Dic3—32no—
26C3×C4⋊Dic3A411yescharacteristic
27C3×Dic3⋊C4—32no—
28C3×C4×Dic3—32no—
29C2×Q8⋊C9—21no—
30C2×Q8⋊C9C2×D311yescharacteristic
31C9×C4⋊C4—121no—
32C22×C3×C9—41no—
33C3×C4×C9—42no—
34C9×Dic3—82no—
35C3×C4⋊Q8—91no—
36C3×#125—92no—
37C3×C4×Q8—21no—
38C3×C4×Q8—32no—
39C3×C4×Q8—91no—
40C3×C4×Q8C2×C911yescharacteristic
41C3×C4×Q8C3×D311yescharacteristic
42Q8×Dic3C2×C912yes—
43#890—32no—
44#869—31no—
45C4×Dic6—31no—
46C32×Q8—31no—
47C32×Q8C3×Q811yescharacteristic
48C2×C32×C4—31no—
49C2×C32×C4—31no—
50C2×C32×C4C2×A411yescharacteristic
51C2×C3×Dic3—32no—
52C2×C3×Dic3C2×A412yes—
53C3×Dic6—62no—
54C2×C4×C9—41no—
55C2×C4×C9—81no—
56C2×C4×C9—122no—
57Q8⋊C9—21no—
58Q8⋊C9Dic611yescharacteristic, complemented
59#557—31noSylow
60C2×C3×C9—41no—
61C2×C3×C9—41no—
62C2×C3×C9—41no—
63C2×C3×Q8—21no—
64C2×C3×Q8—91no—
65C2×C3×Q8C22×C911yescharacteristic
66C2×C3×Q8C2×C3×D311yescharacteristic
67C2×Dic6—31no—
68C3×C4⋊C4—31no—
69C3×C4⋊C4—31no—
70C3×C4⋊C4—31no—
71C3×C4⋊C4—61no—
72C3×C4⋊C4—92no—
73C3×C4⋊C4—92no—
74C3×C4⋊C4—92no—
75C3×C42—31no—
76C3×C42—31no—
77C3×C42—61no—
78C3×C42—92no—
79C4⋊Dic3—32no—
80C4⋊Dic3—32no—
81C4⋊Dic3C22⋊C911yescharacteristic
82Dic3⋊C4—32no—
83C4×Dic3—32no—
84C22×C32C22×A411yescharacteristic
85C32×C4—31no—
86C32×C4—31no—
87C32×C4—61no—
88C32×C4#41⋊C312yes—
89C3×Dic3—22no—
90C3×Dic3—62no—
91C22×C9—41no—
92C22×C9—81no—
93C4×C9—42no—
94C4×C9—82no—
95C4×C9—242no—
96C4⋊Q8—91no—
97#125—92no—
98C4×Q8—32no—
99C4×Q8—91no—
100C4×Q8D3×C911yescharacteristic
101C3×C9—41noSylow
102C3×Q8—21no—
103C3×Q8—31no—
104C3×Q8—31no—
105C3×Q8—61no—
106C3×Q8—182no—
107C3×Q8Q8×C911yescharacteristic
108C3×Q8C3×Dic611yescharacteristic
109C2×C3×C4—21no—
110C2×C3×C4—31no—
111C2×C3×C4—31no—
112C2×C3×C4—32no—
113C2×C3×C4—31no—
114C2×C3×C4—31no—
115C2×C3×C4—61no—
116C2×C3×C4—61no—
117C2×C3×C4—92no—
118C2×C3×C4C2×C22⋊C911yescharacteristic
119C2×C3×C4D3×A411yescharacteristic
120C2×Dic3—32no—
121C2×Dic3C2×C22⋊C912yes—
122Dic6—62no—
123C2×C32Q8×A411yescharacteristic
124C2×C32C2×#41⋊C311yescharacteristic
125C2×C322−1+4⋊C311yescharacteristic
126C2×C9—41no—
127C2×C9—41no—
128C2×C9—41no—
129C2×C9—81no—
130C2×C9—81no—
131C2×C9—81no—
132C2×Q8—91no—
133C2×Q8C2×D3×C911yescharacteristic
134C4⋊C4—31no—
135C4⋊C4—31no—
136C4⋊C4—92no—
137C4⋊C4—92no—
138C4⋊C4—92no—
139C42—31no—
140C42—92no—
141C22×C3—21no—
142C22×C3C22×C22⋊C911yescharacteristic
143C22×C3C2×D3×A411yescharacteristic
144C3×C4—22no—
145C3×C4—31no—
146C3×C4—31no—
147C3×C4—31no—
148C3×C4—31no—
149C3×C4—61no—
150C3×C4—61no—
151C3×C4—62no—
152C3×C4—61no—
153C3×C4—61no—
154C3×C4—121no—
155C3×C4—182no—
156C3×C4#41⋊C912yes—
157C3×C4C3⋊(#41⋊C3)12yes—
158Dic3—22no—
159Dic3—62no—
160C32#557⋊C311yescharacteristic
161C9—41no—
162C9—81no—
163Q8—31no—
164Q8—182no—
165Q8C9×Dic611yescharacteristic
166C2×C4—31no—
167C2×C4—31no—
168C2×C4—32no—
169C2×C4—92no—
170C2×C4D3×C22⋊C911yescharacteristic
171C2×C3—21no—
172C2×C3—21no—
173C2×C3—21no—
174C2×C3Q8×C22⋊C911yescharacteristic
175C2×C3C2×#41⋊C911yescharacteristic
176C2×C32−1+4⋊C911yescharacteristic
177C2×C3A4×Dic611yescharacteristic
178C2×C3C2×C3⋊(#41⋊C3)11yescharacteristic
179C2×C3#1010⋊C311yescharacteristic
180C22C2×D3×C22⋊C911yescharacteristic
181C4—31no—
182C4—31no—
183C4—62no—
184C4—61no—
185C4—182no—
186C4C3⋊(#41⋊C9)12yes—
187C3—21no—
188C3#557⋊C911yescharacteristic, complemented
189C3#5570⋊C311yescharacteristic
190C2Dic6×C22⋊C911yescharacteristic
191C2C2×C3⋊(#41⋊C9)11yescharacteristic
192C2#1010⋊C911yescharacteristic
subgroupG/H
center, FrattiniC22×C3C2×D3×A4
commutatorC2×C3×Q8C22×C9
FittingC32×C4×Q8C2×C3
socle, prime radicalC22×C32C22×A4
cosocleC2×C32×Q8C22×C3
hypercenterC2×C3×C4D3×A4
hypocenterC3×Q8Q8×C9
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
17. C3×Q8⋊C9Q8semidirect
19. #5570C9semidirect
58. Q8⋊C9Dic6semidirect
188. C3#557⋊C9semidirect

Maximal subgroups

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

subgroupindex
C3×C4×Q8⋊C92
Dic3×Q8⋊C92
C3×#55703
#557⋊C93
C9×C4⋊Dic34