Finite Group Databasebeta

← #5569⋊C9 #13147203

Subgroups — #5569⋊C9 #13147203

Order 1728.

252 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 61 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⋊C9C211yesmaximal, characteristic
3Dic3×Q8⋊C9C211yesmaximal, characteristic
4C3×#5569C311yesmaximal, characteristic
5#557⋊C9—31nomaximal
6C2×C3×Q8⋊C9C2211yescharacteristic
7C9×Dic3⋊C4—41nomaximal
8C32×C4×Q8C2×C311yescharacteristic
9C3×Q8×Dic3C2×C311yescharacteristic
10C3×Q8×Dic3C2×C311yescharacteristic
11C3×#890—31no—
12C3×#888—31no—
13C3×#870—31no—
14C3×C4×Dic6—31no—
15C4×Q8⋊C9—21no—
16C4×Q8⋊C9—31no—
17C4×Q8⋊C9—31no—
18C4×Q8⋊C9D311yescharacteristic
19C2×C3×C4×C9—41no—
20C2×C9×Dic3—41no—
21C2×C9×Dic3—41no—
22C3×Q8⋊C9Q811yescharacteristic, complemented
23C3×#557—31no—
24#5569C911yescharacteristic, complemented
25C2×C32×Q8C22×C311yescharacteristic
26C2×C3×Dic6—31no—
27C32×C4⋊C4—31no—
28C32×C42—31no—
29C3×C4⋊Dic3—31no—
30C3×C4⋊Dic3—31no—
31C3×C4⋊Dic3—31no—
32C3×Dic3⋊C4—32no—
33C3×Dic3⋊C4—31no—
34C3×Dic3⋊C4A411yescharacteristic
35C3×C4×Dic3—31no—
36C3×C4×Dic3—31no—
37C2×Q8⋊C9—21no—
38C2×Q8⋊C9C2×D311yescharacteristic
39C9×C4⋊C4—121no—
40C22×C3×C9—41no—
41C3×C4×C9—81no—
42C9×Dic3—42no—
43C9×Dic3—81no—
44C3×C4⋊Q8—91no—
45C3×#125—91no—
46C3×#125—91no—
47C3×C4×Q8—21no—
48C3×C4×Q8—31no—
49C3×C4×Q8—31no—
50C3×C4×Q8—91no—
51C3×C4×Q8C2×C911yescharacteristic
52C3×C4×Q8C3×D311yescharacteristic
53Q8×Dic3C2×C911yescharacteristic
54Q8×Dic3C2×C911yescharacteristic
55#890—31no—
56#888—31no—
57#870—31no—
58C4×Dic6—31no—
59C32×Q8—31no—
60C32×Q8C3×Q811yescharacteristic
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C32×C4C2×A411yescharacteristic
64C2×C3×Dic3—31no—
65C2×C3×Dic3—31no—
66C2×C3×Dic3C2×A411yescharacteristic
67C2×C3×Dic3C2×A411yescharacteristic
68C3×Dic6—62no—
69C2×C4×C9—41no—
70C2×C4×C9—81no—
71C2×C4×C9—121no—
72C2×C4×C9—121no—
73Q8⋊C9—21no—
74Q8⋊C9Dic611yescharacteristic, complemented
75#557—31noSylow
76C2×C3×C9—41no—
77C2×C3×C9—41no—
78C2×C3×C9—41no—
79C2×C3×Q8—21no—
80C2×C3×Q8—91no—
81C2×C3×Q8C22×C911yescharacteristic
82C2×C3×Q8C2×C3×D311yescharacteristic
83C2×Dic6—31no—
84C3×C4⋊C4—31no—
85C3×C4⋊C4—31no—
86C3×C4⋊C4—31no—
87C3×C4⋊C4—61no—
88C3×C4⋊C4—91no—
89C3×C4⋊C4—91no—
90C3×C4⋊C4—91no—
91C3×C4⋊C4—92no—
92C3×C4⋊C4—91no—
93C3×C42—31no—
94C3×C42—31no—
95C3×C42—61no—
96C3×C42—91no—
97C3×C42—91no—
98C4⋊Dic3—31no—
99C4⋊Dic3—31no—
100C4⋊Dic3—31no—
101Dic3⋊C4—32no—
102Dic3⋊C4—31no—
103Dic3⋊C4C22⋊C911yescharacteristic
104C4×Dic3—31no—
105C4×Dic3—31no—
106C22×C32C22×A411yescharacteristic
107C32×C4—21no—
108C32×C4—31no—
109C32×C4—31no—
110C32×C4—61no—
111C3×Dic3—21no—
112C3×Dic3—61no—
113C3×Dic3—61no—
114C3×Dic3#41⋊C312yes—
115C22×C9—41no—
116C22×C9—81no—
117C4×C9—81no—
118C4×C9—82no—
119C4×C9—122no—
120C4×C9—241no—
121C4⋊Q8—91no—
122#125—91no—
123#125—91no—
124C4×Q8—31no—
125C4×Q8—31no—
126C4×Q8—91no—
127C4×Q8D3×C911yescharacteristic
128C3×C9—41noSylow
129C3×Q8—21no—
130C3×Q8—31no—
131C3×Q8—31no—
132C3×Q8—61no—
133C3×Q8—182no—
134C3×Q8Q8×C911yescharacteristic
135C3×Q8C3×Dic611yescharacteristic
136C2×C3×C4—21no—
137C2×C3×C4—31no—
138C2×C3×C4—31no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—61no—
144C2×C3×C4—61no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4C2×C22⋊C911yescharacteristic
148C2×C3×C4D3×A411yescharacteristic
149C2×Dic3—31no—
150C2×Dic3—31no—
151C2×Dic3C2×C22⋊C911yescharacteristic
152C2×Dic3C2×C22⋊C911yescharacteristic
153Dic6—62no—
154C2×C32Q8×A411yescharacteristic
155C2×C32C2×#41⋊C311yescharacteristic
156C2×C322−1+4⋊C311yescharacteristic
157C2×C9—41no—
158C2×C9—41no—
159C2×C9—41no—
160C2×C9—81no—
161C2×C9—81no—
162C2×C9—81no—
163C2×Q8—91no—
164C2×Q8C2×D3×C911yescharacteristic
165C4⋊C4—31no—
166C4⋊C4—31no—
167C4⋊C4—91no—
168C4⋊C4—91no—
169C4⋊C4—91no—
170C4⋊C4—92no—
171C4⋊C4—91no—
172C42—31no—
173C42—91no—
174C42—91no—
175C22×C3—21no—
176C22×C3C22×C22⋊C911yescharacteristic
177C22×C3C2×D3×A411yescharacteristic
178C3×C4—22no—
179C3×C4—21no—
180C3×C4—21no—
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—32no—
184C3×C4—31no—
185C3×C4—31no—
186C3×C4—61no—
187C3×C4—61no—
188C3×C4—61no—
189C3×C4—61no—
190C3×C4—61no—
191C3×C4—121no—
192C3×C4—181no—
193C3×C4—181no—
194Dic3—21no—
195Dic3—61no—
196Dic3—61no—
197Dic3#41⋊C912yes—
198C32#557⋊C311yescharacteristic
199C9—41no—
200C9—81no—
201Q8—31no—
202Q8—182no—
203Q8C9×Dic611yescharacteristic
204C2×C4—31no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4D3×C22⋊C911yescharacteristic
211C2×C3—21no—
212C2×C3—21no—
213C2×C3—21no—
214C2×C3Q8×C22⋊C911yescharacteristic
215C2×C3C2×#41⋊C911yescharacteristic
216C2×C32−1+4⋊C911yescharacteristic
217C2×C3A4×Dic611yescharacteristic
218C2×C3#1007⋊C311yescharacteristic
219C2×C3D3×#41⋊C311yescharacteristic
220C22C2×D3×C22⋊C911yescharacteristic
221C4—21no—
222C4—31no—
223C4—31no—
224C4—32no—
225C4—61no—
226C4—61no—
227C4—181no—
228C4—181no—
229C3—21no—
230C3#557⋊C911yescharacteristic, complemented
231C3#5569⋊C311yescharacteristic
232C2Dic6×C22⋊C911yescharacteristic
233C2#1007⋊C911yescharacteristic
234C2D3×#41⋊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
22. C3×Q8⋊C9Q8semidirect
24. #5569C9semidirect
74. Q8⋊C9Dic6semidirect
230. 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×#55693
#557⋊C93
C9×Dic3⋊C44