Finite Group Databasebeta

← #10589731

Subgroups — #10589731

Order 576.

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

normal subgroups only

This group has 441 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×#3701C211yesmaximal, characteristic, complemented
2C2×#3697C211yesmaximal, characteristic, complemented
3C9×C22⋊Q8C211yesmaximal, characteristic, complemented
4#63508C211yesmaximal, characteristic, complemented
5#63492C211yesmaximal, characteristic, complemented
6#63458C212yesmaximal, complemented
7#63457C212yesmaximal, complemented
8#63457C211yesmaximal, characteristic, complemented
9C4⋊C4×D9C211yesmaximal, characteristic, complemented
10#63446C212yesmaximal, complemented
11#63446C212yesmaximal, complemented
12#5629—31nomaximal
13C2×Q8×C9C2211yescharacteristic
14C22×C4×C9C2211yescharacteristic
15C2×#611C2211yescharacteristic
16C2×#611C2211yescharacteristic
17C2×#611C2212yes—
18#3701—24no—
19#3697—24no—
20C2×D36C2211yescharacteristic
21C2×D36C2212yes—
22C2×D36C2211yescharacteristic
23C2×D36C2212yes—
24C2×C4×D9C2211yescharacteristic
25C2×C4×D9C2212yes—
26C2×C4×D9C2212yes—
27C2×C4×D9C2211yescharacteristic
28C2×Dic18C2211yescharacteristic
29C9×C4⋊C4C2211yescharacteristic
30C9×C4⋊C4C2212yes—
31C9×C22⋊C4C2212yes—
32(C2×D9)⋊C4C2212yes—
33(C2×D9)⋊C4C2212yes—
34(C2×D9)⋊C4C2212yes—
35C4⋊Dic9C2211yescharacteristic
36Dic9⋊C4C2212yes—
37Dic9⋊C4C2212yes—
38C2×#239—31no—
39C2×#235—31no—
40C3×C22⋊Q8D311yescharacteristic
41#946—31no—
42#930—31no—
43#896—32no—
44#895—31no—
45#895—32no—
46D3×C4⋊C4—31no—
47#884—32no—
48#884—32no—
49C23×C9C2311yescharacteristic
50C22×D9C2312yes—
51C22×D9C2311yescharacteristic
52C22×D9C2311yescharacteristic
53Q8×C9—22no—
54C2×C4×C9—22no—
55C2×C4×C9C2312yes—
56C2×C4×C9C2312yes—
57C2×C4×C9C2311yescharacteristic
58C2×C4×C9C2311yescharacteristic
59#611—22no—
60#611—22no—
61#611—42no—
62C2×Dic9C2312yes—
63C2×Dic9C2311yescharacteristic
64C2×Dic9C2311yescharacteristic
65D36—22no—
66D36—24no—
67D36—22no—
68D36—42no—
69C4×D9—22no—
70C4×D9—24no—
71C4×D9—42no—
72C4×D9D414yescomplemented
73Dic18—22no—
74#537—91noSylow
75C2×C3×Q8C2×D311yescharacteristic
76C22×C3×C4C2×D311yescharacteristic
77C2×#67—31no—
78C2×#67—31no—
79C2×#67—32no—
80#239—64no—
81#235—64no—
82C2×D12—32no—
83C2×D12—31no—
84C2×D12—31no—
85C2×D12—32no—
86C2×C4×D3—31no—
87C2×C4×D3—32no—
88C2×C4×D3—31no—
89C2×C4×D3—32no—
90C2×Dic6—31no—
91C3×C4⋊C4C2×D311yescharacteristic
92C3×C4⋊C4C2×D312yes—
93C3×C22⋊C4C2×D312yes—
94(C2×D3)⋊C4—32no—
95(C2×D3)⋊C4—32no—
96(C2×D3)⋊C4—32no—
97C4⋊Dic3—31no—
98Dic3⋊C4—32no—
99Dic3⋊C4—32no—
100C22×C9—21no—
101C22×C9—21no—
102C22×C9—21no—
103C22×C9C2411yescharacteristic
104C2×D9—21no—
105C2×D9—21no—
106C2×D9—22no—
107C2×D9—21no—
108C2×D9—21no—
109C2×D9—22no—
110C2×D9—22no—
111C2×D9—21no—
112C2×D9C2×D412yes—
113C4×C9—21no—
114C4×C9—22no—
115C4×C9—22no—
116C4×C9C2×D412yes—
117Dic9—21no—
118Dic9—22no—
119Dic9C2×D412yes—
120C2×#41—91no—
121C2×#41—91no—
122C22⋊Q8—91no—
123C22⋊Q8—92no—
124C22⋊Q8D911yescharacteristic, complemented
125#121—91no—
126#121—91no—
127#121—92no—
128#121—92no—
129#121—92no—
130C2×C4⋊C4—91no—
131C23×C3C22×D311yescharacteristic
132C22×D3—31no—
133C22×D3—32no—
134C22×D3—31no—
135C3×Q8—22no—
136C2×C3×C4—22no—
137C2×C3×C4C22×D312yes—
138C2×C3×C4C22×D312yes—
139C2×C3×C4C22×D311yescharacteristic
140C2×C3×C4C22×D311yescharacteristic
141#67—62no—
142#67—62no—
143#67—122no—
144C2×Dic3—31no—
145C2×Dic3—32no—
146C2×Dic3—31no—
147D12—62no—
148D12—64no—
149D12—62no—
150D12—122no—
151C4×D3—34no—
152C4×D3—64no—
153C4×D3—62no—
154C4×D3—122no—
155Dic6—62no—
156C2×C9—41no—
157C2×C9C22×D411yescharacteristic
158C2×C9H(5,𝔽2)11yescharacteristic
159C2×C92−1+411yescharacteristic
160D9—22no—
161D9—42no—
162D9—41no—
163#41—184no—
164#41—184no—
165C2×Q8—91no—
166C2×Q8C2×D911yescharacteristic
167C2×D4—91no—
168C2×D4—91no—
169C2×D4—92no—
170C2×D4—91no—
171C2×D4—92no—
172C2×D4—91no—
173C2×D4—92no—
174C22×C4—91no—
175C22×C4—92no—
176C22×C4—91no—
177C22×C4—92no—
178C22×C4C2×D911yescharacteristic
179C4⋊C4—92no—
180C4⋊C4—91no—
181C4⋊C4—92no—
182C4⋊C4C2×D911yescharacteristic
183C4⋊C4C2×D912yes—
184C22⋊C4—92no—
185C22⋊C4—92no—
186C22⋊C4—92no—
187C22⋊C4C2×D912yes—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—21no—
191C22×C3C23×D311yescharacteristic
192C2×D3—32no—
193C2×D3—61no—
194C2×D3—61no—
195C2×D3—62no—
196C2×D3—61no—
197C2×D3—62no—
198C2×D3—61no—
199C2×D3—61no—
200C2×D3—62no—
201C3×C4—21no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4D3×D412yes—
205Dic3—32no—
206Dic3—61no—
207Dic3—62no—
208C9#53711yescharacteristic, complemented, Sylow
209C23—91no—
210C23—91no—
211C23—92no—
212C23C22×D911yescharacteristic
213Q8—22no—
214Q8—182no—
215D4—182no—
216D4—184no—
217D4—182no—
218D4—182no—
219D4—182no—
220D4—362no—
221D4—362no—
222C2×C4—22no—
223C2×C4—91no—
224C2×C4—94no—
225C2×C4—92no—
226C2×C4—91no—
227C2×C4—184no—
228C2×C4—182no—
229C2×C4—362no—
230C2×C4C22×D911yescharacteristic
231C2×C4C22×D912yes—
232C2×C4C22×D912yes—
233C2×C4C22×D911yescharacteristic
234C2×C3—41no—
235C2×C3C2×D3×D411yescharacteristic
236C2×C3#100711yescharacteristic
237C2×C3#100911yescharacteristic
238D3—62no—
239D3—122no—
240D3—121no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—92no—
245C22—181no—
246C22—181no—
247C22—182no—
248C22—181no—
249C22—181no—
250C22—182no—
251C22—181no—
252C22—182no—
253C22C23×D911yescharacteristic
254C4—21no—
255C4—22no—
256C4—22no—
257C4—92no—
258C4—181no—
259C4—182no—
260C4D4×D912yes—
261C3#562911yescharacteristic
262C2—41no—
263C2—182no—
264C2—362no—
265C2—361no—
266C2C2×D4×D911yescharacteristic
267C2#6371311yescharacteristic
268C2#6371511yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×C22⋊Q8C2
hypercenterC22⋊Q8D9
hypocenterC9#537
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#3701C2semidirect
2. C2×#3697C2semidirect
3. C9×C22⋊Q8C2semidirect
4. #63508C2semidirect
5. #63492C2semidirect
6. #63458C2semidirect
7. #63457C2semidirect
8. #63457C2semidirect
9. C4⋊C4×D9C2semidirect
10. #63446C2semidirect
11. #63446C2semidirect
72. C4×D9D4semidirect
124. C22⋊Q8D9semidirect
208. C9#537semidirect

Maximal subgroups

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

subgroupindex
C2×#37012
C2×#36972
C9×C22⋊Q82
#635082
#634922
#634582
#634572
C4⋊C4×D92
#634462
#56293