Finite Group Databasebeta

← #10589735

Subgroups — #10589735

Order 576.

435 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 127 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, complemented
3C9×C22⋊Q8C211yesmaximal, characteristic, complemented
4#63509C211yesmaximal, characteristic, complemented
5C4×#611C211yesmaximal, characteristic, complemented
6#63458C212yesmaximal, complemented
7#63457C211yesmaximal, characteristic, complemented
8#63455C212yesmaximal, complemented
9#63449C211yesmaximal, characteristic, complemented
10#63447C212yesmaximal, complemented
11#63446C212yesmaximal, complemented
12#5633—31nomaximal
13C2×Q8×C9C2211yescharacteristic
14C22×C4×C9C2211yescharacteristic
15C2×#611C2211yescharacteristic
16C2×#611C2212yes—
17Q8×D9—24no—
18#3697—24no—
19C2×D36C2211yescharacteristic
20C2×D36C2212yes—
21C2×C4×D9C2211yescharacteristic
22C2×C4×D9C2212yes—
23C2×C4×D9C2212yes—
24C2×Dic18C2212yes—
25C2×Dic18C2211yescharacteristic
26C9×C4⋊C4C2211yescharacteristic
27C9×C4⋊C4C2212yes—
28C9×C22⋊C4C2212yes—
29#3676C2211yescharacteristic
30(C2×D9)⋊C4C2211yescharacteristic
31(C2×D9)⋊C4C2212yes—
32(C2×D9)⋊C4C2212yes—
33(C2×D9)⋊C4C2212yes—
34Dic9⋊C4C2211yescharacteristic
35Dic9⋊C4C2212yes—
36C4×Dic9C2211yescharacteristic
37C4×Dic9C2212yes—
38C2×D3×Q8—31no—
39C2×#235—31no—
40C3×C22⋊Q8D311yescharacteristic
41#947—31no—
42C4×#67—31no—
43#896—32no—
44#895—31no—
45#893—32no—
46#887—31no—
47#885—32no—
48#884—32no—
49C23×C9C2311yescharacteristic
50C22×D9C2312yes—
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—24no—
60C2×Dic9C2311yescharacteristic
61C2×Dic9C2311yescharacteristic
62C2×Dic9C2311yescharacteristic
63C2×Dic9C2312yes—
64D36—24no—
65D36—22no—
66C4×D9—22no—
67C4×D9—24no—
68C4×D9—24no—
69Dic18—24no—
70Dic18D414yescomplemented
71#548—91noSylow
72C2×C3×Q8C2×D311yescharacteristic
73C22×C3×C4C2×D311yescharacteristic
74C2×#67—31no—
75C2×#67—32no—
76D3×Q8—64no—
77#235—64no—
78C2×D12—32no—
79C2×D12—31no—
80C2×C4×D3—32no—
81C2×C4×D3—32no—
82C2×C4×D3—31no—
83C2×Dic6—31no—
84C2×Dic6—32no—
85C3×C4⋊C4C2×D311yescharacteristic
86C3×C4⋊C4C2×D312yes—
87C3×C22⋊C4C2×D312yes—
88#217—31no—
89(C2×D3)⋊C4—32no—
90(C2×D3)⋊C4—32no—
91(C2×D3)⋊C4—32no—
92(C2×D3)⋊C4—31no—
93Dic3⋊C4—32no—
94Dic3⋊C4—31no—
95C4×Dic3—32no—
96C4×Dic3—31no—
97C22×C9—21no—
98C22×C9—21no—
99C22×C9—21no—
100C22×C9C2411yescharacteristic
101C2×D9—21no—
102C2×D9—21no—
103C2×D9—22no—
104C2×D9—22no—
105C2×D9—22no—
106C2×D9#4112yescomplemented
107C4×C9—22no—
108C4×C9—21no—
109C4×C9—22no—
110C4×C9C2×D412yes—
111Dic9—21no—
112Dic9—21no—
113Dic9—21no—
114Dic9C2×D414yes—
115C2×#41—91no—
116C22×Q8—91no—
117#124—91no—
118#124—92no—
119C22⋊Q8—92no—
120C22⋊Q8D911yescharacteristic, complemented
121#121—91no—
122#121—92no—
123C4×Q8—91no—
124C4×D4—91no—
125C4×D4—92no—
126C23×C3C22×D311yescharacteristic
127C22×D3—31no—
128C22×D3—32no—
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—64no—
136#67—62no—
137C2×Dic3—32no—
138C2×Dic3—31no—
139C2×Dic3—31no—
140C2×Dic3—31no—
141D12—64no—
142D12—62no—
143C4×D3—64no—
144C4×D3—64no—
145C4×D3—62no—
146Dic6—34no—
147Dic6—64no—
148C2×C9—41no—
149C2×C9C22×D411yescharacteristic
150C2×C9C2×#4111yescharacteristic
151C2×C92−1+411yescharacteristic
152D9—22no—
153D9—42no—
154#41—184no—
155C2×Q8—91no—
156C2×Q8—92no—
157C2×Q8—184no—
158C2×Q8C2×D911yescharacteristic
159C2×D4—91no—
160C2×D4—92no—
161C2×D4—92no—
162C2×D4—91no—
163C22×C4—92no—
164C22×C4—91no—
165C22×C4—92no—
166C22×C4C2×D911yescharacteristic
167C4⋊C4—91no—
168C4⋊C4—92no—
169C4⋊C4C2×D912yes—
170C4⋊C4C2×D911yescharacteristic
171C22⋊C4—91no—
172C22⋊C4—91no—
173C22⋊C4—92no—
174C22⋊C4—92no—
175C22⋊C4—92no—
176C22⋊C4C2×D912yes—
177C42—92no—
178C42—91no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3C23×D311yescharacteristic
183C2×D3—32no—
184C2×D3—61no—
185C2×D3—61no—
186C2×D3—62no—
187C2×D3—62no—
188C2×D3—62no—
189C3×C4—22no—
190C3×C4—21no—
191C3×C4—22no—
192C3×C4D3×D412yes—
193Dic3—34no—
194Dic3—61no—
195Dic3—61no—
196Dic3—61no—
197C9#54811yescharacteristic, complemented, Sylow
198C23—91no—
199C23—92no—
200C23C22×D911yescharacteristic
201Q8—22no—
202Q8—94no—
203Q8—184no—
204D4—184no—
205D4—184no—
206D4—182no—
207D4—182no—
208C2×C4—22no—
209C2×C4—92no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—184no—
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×C3D3×#4111yescharacteristic
224D3—62no—
225D3—122no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—92no—
230C22—181no—
231C22—181no—
232C22—182no—
233C22—182no—
234C22—182no—
235C22C23×D911yescharacteristic
236C4—21no—
237C4—22no—
238C4—22no—
239C4—94no—
240C4—181no—
241C4—181no—
242C4—181no—
243C4D4×D912yes—
244C3#563311yescharacteristic
245C2—41no—
246C2—182no—
247C2—362no—
248C2C2×D4×D911yescharacteristic
249C2#6371311yescharacteristic
250C2#41×D911yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×C22⋊Q8C2
hypercenterC22⋊Q8D9
hypocenterC9#548
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
2. C2×#3697C2semidirect
3. C9×C22⋊Q8C2semidirect
4. #63509C2semidirect
5. C4×#611C2semidirect
6. #63458C2semidirect
7. #63457C2semidirect
8. #63455C2semidirect
9. #63449C2semidirect
10. #63447C2semidirect
11. #63446C2semidirect
70. Dic18D4semidirect
106. C2×D9#41semidirect
120. C22⋊Q8D9semidirect
197. C9#548semidirect

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
#635092
C4×#6112
#634582
#634572
#634552
#634492
#634472
#634462
#56333