Finite Group Databasebeta

← #10589730

Subgroups — #10589730

Order 576.

405 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 125 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×Q8×D9C211yesmaximal, characteristic, complemented
2C2×#3697C211yesmaximal, characteristic
3C9×C22⋊Q8C211yesmaximal, characteristic, complemented
4#63508C211yesmaximal, characteristic, complemented
5#63486C211yesmaximal, characteristic, complemented
6#63459C211yesmaximal, characteristic, complemented
7#63456C212yesmaximal, complemented
8#63454C211yesmaximal, characteristic, complemented
9#63450C212yesmaximal, complemented
10#63447C212yesmaximal, complemented
11#63445C212yesmaximal, complemented
12#5628—31nomaximal
13C2×Q8×C9C2211yescharacteristic
14C22×C4×C9C2211yescharacteristic
15C2×#611C2211yescharacteristic
16C2×#611C2211yescharacteristic
17Q8×D9—24no—
18#3697—24no—
19C2×D36C2211yescharacteristic
20C2×C4×D9C2211yescharacteristic
21C2×C4×D9C2212yes—
22C2×C4×D9C2211yescharacteristic
23C2×Dic18C2212yes—
24C2×Dic18C2211yescharacteristic
25C2×Dic18C2211yescharacteristic
26C9×C4⋊C4C2212yes—
27C9×C4⋊C4C2211yescharacteristic
28C9×C22⋊C4C2212yes—
29#3676C2212yes—
30(C2×D9)⋊C4C2212yes—
31(C2×D9)⋊C4C2212yes—
32(C2×D9)⋊C4C2212yes—
33C4⋊Dic9C2211yescharacteristic
34C4⋊Dic9C2212yes—
35Dic9⋊C4C2212yes—
36Dic9⋊C4C2212yes—
37C4×Dic9C2212yes—
38C2×D3×Q8—31no—
39C2×#235—31no—
40C3×C22⋊Q8D311yescharacteristic
41#946—31no—
42#924—31no—
43#897—31no—
44#894—32no—
45#892—31no—
46#888—32no—
47#885—32no—
48#883—32no—
49C23×C9C2311yescharacteristic
50C22×D9C2311yescharacteristic
51C22×D9C2311yescharacteristic
52Q8×C9—22no—
53C2×C4×C9—22no—
54C2×C4×C9C2312yes—
55C2×C4×C9C2311yescharacteristic
56C2×C4×C9C2312yes—
57C2×C4×C9C2311yescharacteristic
58#611—22no—
59#611—22no—
60C2×Dic9C2312yes—
61C2×Dic9C2311yescharacteristic
62C2×Dic9C2312yes—
63C2×Dic9C2311yescharacteristic
64D36—22no—
65C4×D9—24no—
66C4×D9—22no—
67C4×D9D414yes—
68Dic18—22no—
69Dic18—22no—
70Dic18—24no—
71#536—91noSylow
72C2×C3×Q8C2×D311yescharacteristic
73C22×C3×C4C2×D311yescharacteristic
74C2×#67—31no—
75C2×#67—31no—
76D3×Q8—64no—
77#235—64no—
78C2×D12—31no—
79C2×C4×D3—31no—
80C2×C4×D3—32no—
81C2×C4×D3—31no—
82C2×Dic6—32no—
83C2×Dic6—31no—
84C2×Dic6—31no—
85C3×C4⋊C4C2×D312yes—
86C3×C4⋊C4C2×D311yescharacteristic
87C3×C22⋊C4C2×D312yes—
88#217—32no—
89(C2×D3)⋊C4—32no—
90(C2×D3)⋊C4—32no—
91(C2×D3)⋊C4—32no—
92C4⋊Dic3—32no—
93C4⋊Dic3—31no—
94Dic3⋊C4—32no—
95Dic3⋊C4—32no—
96C4×Dic3—32no—
97C22×C9—21no—
98C22×C9—21no—
99C22×C9—21no—
100C22×C9C2411yescharacteristic
101C2×D9—21no—
102C2×D9—21no—
103C2×D9—21no—
104C2×D9—21no—
105C2×D9—21no—
106C2×D9C2×D412yes—
107C4×C9—22no—
108C4×C9—21no—
109C4×C9—22no—
110C4×C9C2×D412yes—
111Dic9—22no—
112Dic9—21no—
113Dic9—22no—
114Dic9C2×D412yes—
115C2×#41—91no—
116C22×Q8—91no—
117C4⋊Q8—92no—
118#124—92no—
119#123—92no—
120#123—92no—
121C22⋊Q8—91no—
122C22⋊Q8—91no—
123C22⋊Q8—91no—
124C22⋊Q8D911yescharacteristic, complemented
125#117—91no—
126C23×C3C22×D311yescharacteristic
127C22×D3—31no—
128C22×D3—31no—
129C3×Q8—22no—
130C2×C3×C4—22no—
131C2×C3×C4C22×D312yes—
132C2×C3×C4C22×D311yescharacteristic
133C2×C3×C4C22×D312yes—
134C2×C3×C4C22×D311yescharacteristic
135#67—62no—
136#67—62no—
137C2×Dic3—31no—
138C2×Dic3—32no—
139C2×Dic3—31no—
140C2×Dic3—32no—
141D12—62no—
142C4×D3—34no—
143C4×D3—64no—
144C4×D3—62no—
145Dic6—62no—
146Dic6—64no—
147Dic6—62no—
148C2×C9—41no—
149C2×C9C22×D411yescharacteristic
150C2×C92−1+411yescharacteristic
151C2×C92−1+411yescharacteristic
152D9—22no—
153D9—41no—
154#41—184no—
155C2×Q8—91no—
156C2×Q8—92no—
157C2×Q8—91no—
158C2×Q8—184no—
159C2×Q8C2×D911yescharacteristic
160C2×D4—91no—
161C2×D4—91no—
162C2×D4—91no—
163C22×C4—91no—
164C22×C4—91no—
165C22×C4—92no—
166C22×C4C2×D911yescharacteristic
167C4⋊C4—91no—
168C4⋊C4—92no—
169C4⋊C4—92no—
170C4⋊C4—92no—
171C4⋊C4C2×D911yescharacteristic
172C4⋊C4C2×D912yes—
173C22⋊C4—92no—
174C22⋊C4—92no—
175C22⋊C4—92no—
176C22⋊C4—92no—
177C22⋊C4C2×D912yes—
178C42—92no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3C23×D311yescharacteristic
183C2×D3—32no—
184C2×D3—61no—
185C2×D3—61no—
186C2×D3—61no—
187C2×D3—61no—
188C2×D3—61no—
189C3×C4—22no—
190C3×C4—21no—
191C3×C4—22no—
192C3×C4D3×D412yes—
193Dic3—32no—
194Dic3—62no—
195Dic3—61no—
196Dic3—62no—
197C9#53611yescharacteristic, complemented, Sylow
198C23—91no—
199C23—91no—
200C23C22×D911yescharacteristic
201Q8—22no—
202Q8—184no—
203Q8—182no—
204Q8—182no—
205D4—182no—
206D4—182no—
207D4—182no—
208C2×C4—22no—
209C2×C4—91no—
210C2×C4—94no—
211C2×C4—92no—
212C2×C4—92no—
213C2×C4—91no—
214C2×C4—184no—
215C2×C4—182no—
216C2×C4C22×D912yes—
217C2×C4C22×D912yes—
218C2×C4C22×D911yescharacteristic
219C2×C4C22×D911yescharacteristic
220C2×C3—41no—
221C2×C3C2×D3×D411yescharacteristic
222C2×C3#100711yescharacteristic
223C2×C3#101011yescharacteristic
224D3—62no—
225D3—121no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—92no—
230C22—181no—
231C22—181no—
232C22—181no—
233C22—181no—
234C22—181no—
235C22C23×D911yescharacteristic
236C4—21no—
237C4—22no—
238C4—22no—
239C4—92no—
240C4—181no—
241C4—182no—
242C4—182no—
243C4D4×D912yes—
244C3#562811yescharacteristic
245C2—41no—
246C2—182no—
247C2—361no—
248C2C2×D4×D911yescharacteristic
249C2#6371311yescharacteristic
250C2#6371611yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×C22⋊Q8C2
prime radicalC2×#3697C2
hypercenterC22⋊Q8D9
hypocenterC9#536
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D9C2semidirect
3. C9×C22⋊Q8C2semidirect
4. #63508C2semidirect
5. #63486C2semidirect
6. #63459C2semidirect
7. #63456C2semidirect
8. #63454C2semidirect
9. #63450C2semidirect
10. #63447C2semidirect
11. #63445C2semidirect
124. C22⋊Q8D9semidirect
197. C9#536semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D92
C2×#36972
C9×C22⋊Q82
#635082
#634862
#634592
#634562
#634542
#634502
#634472
#634452
#56283