Finite Group Databasebeta

← #10589643

Subgroups — #10589643

Order 576.

321 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C4×D4×C9C211yesmaximal, characteristic
2#63497C212yesmaximal
3#63488C211yesmaximal, characteristic, complemented
4C2×C4⋊Dic9C211yesmaximal, characteristic, complemented
5#63486C211yesmaximal, characteristic, complemented
6#63486C211yesmaximal, characteristic, complemented
7#63451C212yesmaximal, complemented
8#63441C212yesmaximal, complemented
9#63439C212yesmaximal, complemented
10#63432C211yesmaximal, characteristic, complemented
11C4×Dic18C211yesmaximal, characteristic, complemented
12#5541—31nomaximal
13C2×D4×C9C2211yescharacteristic
14C22×C4×C9C2211yescharacteristic
15C22×C4×C9C2211yescharacteristic
16C22×Dic9C2212yes—
17C2×Dic18C2211yescharacteristic
18C9×C4⋊C4C2211yescharacteristic
19C9×C22⋊C4C2211yescharacteristic
20C9×C22⋊C4C2211yescharacteristic
21C42×C9C2211yescharacteristic
22#3676C2212yes—
23#3676C2212yes—
24#3676C2212yes—
25C4⋊Dic9C2211yescharacteristic
26C4⋊Dic9C2211yescharacteristic
27C4⋊Dic9C2211yescharacteristic
28C4⋊Dic9C2212yes—
29Dic9⋊C4C2212yes—
30Dic9⋊C4C2212yes—
31Dic9⋊C4C2212yes—
32Dic9⋊C4C2212yes—
33Dic9⋊C4C2212yes—
34C4×Dic9C2212yes—
35C4×Dic9C2212yes—
36C3×C4×D4D311yescharacteristic
37#935—32no—
38#926—31no—
39C2×C4⋊Dic3—31no—
40#924—31no—
41#924—31no—
42#889—32no—
43#879—32no—
44#877—32no—
45#870—31no—
46C4×Dic6—31no—
47C23×C9C2311yescharacteristic
48C23×C9C2311yescharacteristic
49D4×C9—22no—
50C2×C4×C9—22no—
51C2×C4×C9—22no—
52C2×C4×C9C2311yescharacteristic
53C2×C4×C9C2311yescharacteristic
54C2×C4×C9C2311yescharacteristic
55C2×C4×C9C2311yescharacteristic
56C2×C4×C9C2311yescharacteristic
57C2×Dic9—24no—
58C2×Dic9C2312yes—
59C2×Dic9C2312yes—
60C2×Dic9C2312yes—
61C2×Dic9C2312yes—
62Dic18—22no—
63#552—91noSylow
64C2×C3×D4C2×D311yescharacteristic
65C22×C3×C4C2×D311yescharacteristic
66C22×C3×C4C2×D311yescharacteristic
67C22×Dic3—32no—
68C2×Dic6—31no—
69C3×C4⋊C4C2×D311yescharacteristic
70C3×C22⋊C4C2×D311yescharacteristic
71C3×C22⋊C4C2×D311yescharacteristic
72C3×C42C2×D311yescharacteristic
73#217—32no—
74#217—32no—
75#217—32no—
76C4⋊Dic3—31no—
77C4⋊Dic3—31no—
78C4⋊Dic3—31no—
79C4⋊Dic3—32no—
80Dic3⋊C4—32no—
81Dic3⋊C4—32no—
82Dic3⋊C4—32no—
83Dic3⋊C4—32no—
84Dic3⋊C4—32no—
85C4×Dic3—32no—
86C4×Dic3—32no—
87C22×C9—21no—
88C22×C9—21no—
89C22×C9—21no—
90C22×C9—21no—
91C22×C9—21no—
92C22×C9#4112yes—
93C22×C9C2411yescharacteristic
94C4×C9—21no—
95C4×C9—21no—
96C4×C9—21no—
97C4×C9—21no—
98C4×C9#4112yes—
99Dic9—22no—
100Dic9—22no—
101Dic9—22no—
102Dic9—22no—
103#126—92no—
104#125—92no—
105#125—91no—
106#123—92no—
107C22⋊Q8—91no—
108C22⋊Q8—91no—
109C4×Q8—91no—
110C4×D4D911yescharacteristic
111#117—91no—
112#117—92no—
113C2×C4⋊C4—91no—
114C23×C3C22×D311yescharacteristic
115C23×C3C22×D311yescharacteristic
116C3×D4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4C22×D311yescharacteristic
120C2×C3×C4C22×D311yescharacteristic
121C2×C3×C4C22×D311yescharacteristic
122C2×C3×C4C22×D311yescharacteristic
123C2×C3×C4C22×D311yescharacteristic
124C2×Dic3—32no—
125C2×Dic3—32no—
126C2×Dic3—32no—
127C2×Dic3—32no—
128C2×Dic3—64no—
129Dic6—62no—
130C2×C9—22no—
131C2×C9—41no—
132C2×C9C2×#4111yescharacteristic
133C2×C9C2×#4111yescharacteristic
134C2×C92−1+411yescharacteristic
135C2×Q8—91no—
136C2×D4C2×D911yescharacteristic
137C22×C4—92no—
138C22×C4C2×D911yescharacteristic
139C22×C4C2×D911yescharacteristic
140C4⋊C4—91no—
141C4⋊C4—91no—
142C4⋊C4—92no—
143C4⋊C4—92no—
144C4⋊C4—91no—
145C4⋊C4—92no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4—92no—
149C4⋊C4C2×D911yescharacteristic
150C22⋊C4—92no—
151C22⋊C4—92no—
152C22⋊C4—92no—
153C22⋊C4C2×D911yescharacteristic
154C22⋊C4C2×D911yescharacteristic
155C42—92no—
156C42—92no—
157C42C2×D911yescharacteristic
158C22×C3—21no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3#23512yes—
164C22×C3C23×D311yescharacteristic
165C3×C4—21no—
166C3×C4—21no—
167C3×C4—21no—
168C3×C4—21no—
169C3×C4#23712yes—
170Dic3—62no—
171Dic3—62no—
172Dic3—62no—
173Dic3—62no—
174C9#55211yescharacteristic, complemented, Sylow
175C23C22×D911yescharacteristic
176C23C22×D911yescharacteristic
177Q8—182no—
178D4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—92no—
182C2×C4—92no—
183C2×C4—92no—
184C2×C4—92no—
185C2×C4—184no—
186C2×C4C22×D911yescharacteristic
187C2×C4C22×D911yescharacteristic
188C2×C4C22×D911yescharacteristic
189C2×C4C22×D911yescharacteristic
190C2×C4C22×D911yescharacteristic
191C2×C3—22no—
192C2×C3—41no—
193C2×C3C2×#23511yescharacteristic
194C2×C3C2×#23711yescharacteristic
195C2×C3#101011yescharacteristic
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22#369712yes—
202C22C23×D911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—182no—
208C4—182no—
209C4—182no—
210C4—182no—
211C4#369912yes—
212C3#554111yescharacteristic
213C2—22no—
214C2—41no—
215C2C2×#369711yescharacteristic
216C2C2×#369911yescharacteristic
217C2#6371611yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC4×D4×C9C2
prime radicalC2×C3×D4C2×D3
hypercenterC4×D4D9
hypocenterC9#552
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
3. #63488C2semidirect
4. C2×C4⋊Dic9C2semidirect
5. #63486C2semidirect
6. #63486C2semidirect
7. #63451C2semidirect
8. #63441C2semidirect
9. #63439C2semidirect
10. #63432C2semidirect
11. C4×Dic18C2semidirect
174. C9#552semidirect

Maximal subgroups

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

subgroupindex
C4×D4×C92
#634972
#634882
C2×C4⋊Dic92
#634862
#634512
#634412
#634392
#634322
C4×Dic182
#55413