Finite Group Databasebeta

← #10589663

Subgroups — #10589663

Order 576.

324 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 117 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, complemented
2#63500C211yesmaximal, characteristic
3#63498C211yesmaximal, characteristic, complemented
4#63491C212yesmaximal, complemented
5#63486C212yesmaximal, complemented
6#63459C211yesmaximal, characteristic, complemented
7#63450C211yesmaximal, characteristic, complemented
8#63447C212yesmaximal, complemented
9#63441C212yesmaximal, complemented
10#63434C211yesmaximal, characteristic, complemented
11C4×Dic18C211yesmaximal, characteristic, complemented
12#5561—31nomaximal
13C2×D4×C9C2211yescharacteristic
14C22×C4×C9C2212yes—
15C2×#611C2212yes—
16C2×C4×D9C2211yescharacteristic
17C2×Dic18C2211yescharacteristic
18C2×Dic18C2212yes—
19C9×C4⋊C4C2211yescharacteristic
20C9×C22⋊C4C2212yes—
21C42×C9C2211yescharacteristic
22#3676C2212yes—
23#3676C2212yes—
24#3676C2212yes—
25(C2×D9)⋊C4C2211yescharacteristic
26(C2×D9)⋊C4C2212yes—
27(C2×D9)⋊C4C2211yescharacteristic
28C4⋊Dic9C2211yescharacteristic
29C4⋊Dic9C2212yes—
30Dic9⋊C4C2212yes—
31Dic9⋊C4C2212yes—
32Dic9⋊C4C2211yescharacteristic
33Dic9⋊C4C2211yescharacteristic
34C4×Dic9C2211yescharacteristic
35C4×Dic9C2212yes—
36C3×C4×D4D311yescharacteristic
37#938—31no—
38#936—31no—
39#929—32no—
40#924—32no—
41#897—31no—
42#888—31no—
43#885—32no—
44#879—32no—
45#872—31no—
46C4×Dic6—31no—
47C23×C9C2312yes—
48C22×D9C2311yescharacteristic
49D4×C9—22no—
50C2×C4×C9—42no—
51C2×C4×C9C2311yescharacteristic
52C2×C4×C9C2311yescharacteristic
53C2×C4×C9C2312yes—
54C2×C4×C9C2311yescharacteristic
55#611—42no—
56C2×Dic9C2311yescharacteristic
57C2×Dic9C2312yes—
58C2×Dic9C2312yes—
59C2×Dic9C2311yescharacteristic
60C2×Dic9C2311yescharacteristic
61C4×D9—22no—
62Dic18—22no—
63Dic18—42no—
64#542—91noSylow
65C2×C3×D4C2×D311yescharacteristic
66C22×C3×C4C2×D312yes—
67C2×#67—32no—
68C2×C4×D3—31no—
69C2×Dic6—31no—
70C2×Dic6—32no—
71C3×C4⋊C4C2×D311yescharacteristic
72C3×C22⋊C4C2×D312yes—
73C3×C42C2×D311yescharacteristic
74#217—32no—
75#217—32no—
76#217—32no—
77(C2×D3)⋊C4—31no—
78(C2×D3)⋊C4—31no—
79(C2×D3)⋊C4—32no—
80C4⋊Dic3—31no—
81C4⋊Dic3—32no—
82Dic3⋊C4—32no—
83Dic3⋊C4—31no—
84Dic3⋊C4—31no—
85Dic3⋊C4—32no—
86C4×Dic3—32no—
87C4×Dic3—31no—
88C22×C9—22no—
89C22×C9—22no—
90C22×C9—22no—
91C22×C9C2411yescharacteristic
92C2×D9—21no—
93C2×D9—21no—
94C2×D9—21no—
95C4×C9—22no—
96C4×C9—21no—
97C4×C9—21no—
98C4×C9#4112yes—
99Dic9—22no—
100Dic9—21no—
101Dic9—21no—
102Dic9—22no—
103Dic9—21no—
104C4⋊Q8—91no—
105#126—92no—
106#124—91no—
107#124—92no—
108#123—92no—
109C22⋊Q8—92no—
110C22⋊Q8—91no—
111#121—91no—
112C4×Q8—91no—
113C4×D4D911yescharacteristic, complemented
114#117—91no—
115C23×C3C22×D312yes—
116C22×D3—31no—
117C3×D4—22no—
118C2×C3×C4—42no—
119C2×C3×C4C22×D311yescharacteristic
120C2×C3×C4C22×D311yescharacteristic
121C2×C3×C4C22×D312yes—
122C2×C3×C4C22×D311yescharacteristic
123#67—122no—
124C2×Dic3—31no—
125C2×Dic3—32no—
126C2×Dic3—32no—
127C2×Dic3—31no—
128C2×Dic3—31no—
129C4×D3—62no—
130Dic6—62no—
131Dic6—122no—
132C2×C9—42no—
133C2×C9C2×#4111yescharacteristic
134C2×C9H(5,𝔽2)11yescharacteristic
135C2×C92−1+411yescharacteristic
136D9—41no—
137C2×Q8—91no—
138C2×Q8—92no—
139C2×D4—92no—
140C2×D4C2×D911yescharacteristic
141C22×C4—91no—
142C22×C4C2×D912yes—
143C4⋊C4—91no—
144C4⋊C4—91no—
145C4⋊C4—91no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4—92no—
149C4⋊C4C2×D911yescharacteristic
150C22⋊C4—92no—
151C22⋊C4—92no—
152C22⋊C4—92no—
153C22⋊C4—91no—
154C22⋊C4—91no—
155C22⋊C4—92no—
156C22⋊C4C2×D912yes—
157C42—91no—
158C42—92no—
159C42C2×D911yescharacteristic
160C22×C3—22no—
161C22×C3—22no—
162C22×C3—22no—
163C22×C3C23×D311yescharacteristic
164C2×D3—61no—
165C2×D3—61no—
166C2×D3—61no—
167C3×C4—22no—
168C3×C4—21no—
169C3×C4—21no—
170C3×C4#23512yes—
171Dic3—62no—
172Dic3—61no—
173Dic3—61no—
174Dic3—62no—
175Dic3—61no—
176C9#54211yescharacteristic, complemented, Sylow
177C23—91no—
178C23C22×D912yes—
179Q8—182no—
180Q8—362no—
181D4—22no—
182D4—362no—
183C2×C4—42no—
184C2×C4—91no—
185C2×C4—92no—
186C2×C4—91no—
187C2×C4—92no—
188C2×C4—91no—
189C2×C4—182no—
190C2×C4C22×D911yescharacteristic
191C2×C4C22×D911yescharacteristic
192C2×C4C22×D912yes—
193C2×C4C22×D911yescharacteristic
194C2×C3—42no—
195C2×C3C2×#23511yescharacteristic
196C2×C3#100411yescharacteristic
197C2×C3#101011yescharacteristic
198D3—121no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—181no—
203C22—181no—
204C22—181no—
205C22C23×D911yescharacteristic
206C4—21no—
207C4—22no—
208C4—21no—
209C4—181no—
210C4—182no—
211C4—182no—
212C4—181no—
213C4—181no—
214C4#369712yes—
215C3#556111yescharacteristic
216C2—42no—
217C2—361no—
218C2C2×#369711yescharacteristic
219C2#6371011yescharacteristic
220C2#6371611yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC4×D4×C9C2
prime radical#63500C2
hypercenterC4×D4D9
hypocenterC9#542
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×D4×C9C2semidirect
3. #63498C2semidirect
4. #63491C2semidirect
5. #63486C2semidirect
6. #63459C2semidirect
7. #63450C2semidirect
8. #63447C2semidirect
9. #63441C2semidirect
10. #63434C2semidirect
11. C4×Dic18C2semidirect
113. C4×D4D9semidirect
176. C9#542semidirect

Maximal subgroups

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

subgroupindex
C4×D4×C92
#635002
#634982
#634912
#634862
#634592
#634502
#634472
#634412
#634342
C4×Dic182
#55613