Finite Group Databasebeta

← #22572003

Subgroups — #22572003

Order 8576.

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

normal subgroups only

This group has 460 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
1#16864208C212yesmaximal, complemented
2#16864203C212yesmaximal, complemented
3#16863769C211yesmaximal, characteristic, complemented
4#16863660C212yesmaximal, complemented
5#16863659C212yesmaximal, complemented
6#16863580C212yesmaximal, complemented
7#16863565C212yesmaximal, complemented
8#16863514C211yesmaximal, characteristic, complemented
9#443×C67C211yesmaximal, characteristic, complemented
10#14524090C2212yes—
11#14524089C2212yescomplemented
12#14524076—22no—
13#14524076C2212yescomplemented
14#14524075—22no—
15#14524073C2212yescomplemented
16#14524071—22no—
17#14524071C2212yes—
18#14524068C2212yes—
19#14524067—22no—
20#14524065—22no—
21#14524065C2211yescharacteristic
22#14524064—22no—
23C2×#11761617—22no—
24C2×#11761617C2212yescomplemented
25C2×#11761617C2212yescomplemented
26SD16×D67—22no—
27#14524049C2212yescomplemented
28#14524043C2211yescharacteristic, complemented
29#14524040C2211yescharacteristic
30#14523979C2211yescharacteristic, complemented
31#14523971C2212yescomplemented
32#14523961C2212yescomplemented
33#14523959C2212yescomplemented
34#135×C67C2211yescharacteristic
35#131×C67C2212yescomplemented
36#108×C67C2211yescharacteristic, complemented
37#104×C67C2212yescomplemented
38C4×C8×C67C2211yescharacteristic, complemented
39#11761629—22no—
40#11761628—22no—
41#11761628—22no—
42#11761628C2312yes—
43#11761627—22no—
44#11761627—42no—
45C2×Dic134—22no—
46C2×Dic134C2312yes—
47Q8×D67—22no—
48C2×D268—22no—
49C2×D268—41no—
50C2×D268C2312yes—
51D4×D67—22no—
52D4×D67—42no—
53Dic268—22no—
54Dic268D411yescharacteristic, complemented
55#11761618—22no—
56#11761617—22no—
57#11761617—22no—
58#11761617—22no—
59#11761617—22no—
60#11761617—22no—
61#11761617—22no—
62#11761617D412yescomplemented
63#11761616—22no—
64#11761615—22no—
65D536—22no—
66D536D411yescharacteristic, complemented
67#11761613—22no—
68#11761612C2312yes—
69#11761611—22no—
70C8×D67—22no—
71C4⋊Dic67—41no—
72#41×C67C2312yes—
73Q16×C67D411yescharacteristic, complemented
74SD16×C67D412yescomplemented
75D8×C67D411yescharacteristic, complemented
76M16×C67—22no—
77M16×C67C2312yes—
78C2×C8×C67—22no—
79C2×C8×C67C2311yescharacteristic
80C2×C8×C67C2311yescharacteristic
81C42×C67C2311yescharacteristic
82C22×D67—22no—
83C22×D67—42no—
84Q8×C67C2×D412yes—
85D4×C67—22no—
86D4×C67C2×D412yes—
87C2×C4×C67—21no—
88C2×C4×C67—22no—
89C2×C4×C67C2411yescharacteristic
90C22⋊D67—22no—
91C22⋊D67—42no—
92C22⋊D67—42no—
93C2×Dic67—22no—
94C2×Dic67—42no—
95D268—22no—
96D268—22no—
97D268—42no—
98D268—82no—
99D268C2×D412yes—
100C4×D67—22no—
101C4×D67—42no—
102C4×D67—42no—
103Dic134—22no—
104Dic134—22no—
105Dic134—42no—
106Dic134C2×D412yes—
107C8×C67—22no—
108C8×C67C2×D411yescharacteristic
109C8×C67C2×D411yescharacteristic
110C8×C67C2×D411yescharacteristic
111C8×C67C2×D411yescharacteristic
112C67⋊C8—22no—
113C22×C67—22no—
114C22×C67H(5,𝔽2)11yescharacteristic
115C2×D67—22no—
116C2×D67—42no—
117C2×D67—82no—
118C2×D67—162no—
119C4×C67—22no—
120C4×C67—22no—
121C4×C67C22×D411yescharacteristic
122C4×C67C22×D411yescharacteristic
123Dic67—22no—
124Dic67—42no—
125C2×C67—21no—
126C2×C67—42no—
127C2×C67D4211yescharacteristic
128D67—42no—
129D67—82no—
130#3291—671nomaximal, Sylow
131C67#329111yescharacteristic, complemented, Sylow
132#577—672no—
133#577—672no—
134#492—671no—
135#471—672no—
136#471—672no—
137#456—672no—
138#453—672no—
139#443—671no—
140#443D6711yescharacteristic, complemented
1412−1+4—672no—
142H(5,𝔽2)—672no—
143#137—672no—
144#137—672no—
145#137—1342no—
146#137—1342no—
147AGL(1,ℤ/8)—672no—
148AGL(1,ℤ/8)—672no—
149AGL(1,ℤ/8)—1342no—
150AGL(1,ℤ/8)—1342no—
151#135—671no—
152#135—1342no—
153#135—1342no—
154#135C2×D6711yescharacteristic
155C2×SD16—672no—
156C2×SD16—672no—
157C2×SD16—1342no—
158C2×SD16—1342no—
159#131—672no—
160#131C2×D6712yescomplemented
161C4⋊Q8—671no—
162#127—671no—
163#108—671no—
164#108C2×D6711yescharacteristic, complemented
165#104—672no—
166#104C2×D6712yescomplemented
167#101—672no—
168#100—672no—
169C4×C8C2×D6711yescharacteristic, complemented
170#41—2682no—
171#41—672no—
172#41—1342no—
173#41—1342no—
174#41—1342no—
175#41—1342no—
176#41C22×D6712yes—
177C2×Q8—672no—
178C2×Q8—1342no—
179C2×Q8—1342no—
180C2×D4—2681no—
181C2×D4—2682no—
182C2×D4—672no—
183C2×D4—1342no—
184C2×D4—1342no—
185Q16—671no—
186Q16—1342no—
187Q16—1342no—
188Q16D26811yescharacteristic, complemented
189SD16—672no—
190SD16—1342no—
191SD16—1342no—
192SD16—1342no—
193SD16—1342no—
194SD16—1342no—
195SD16—1342no—
196SD16—1342no—
197SD16—1342no—
198SD16D26812yescomplemented
199D8—671no—
200D8—1342no—
201D8—1342no—
202D8D26811yescharacteristic, complemented
203M16—22no—
204M16—672no—
205M16—1342no—
206M16C22×D6712yes—
207C2×C8—22no—
208C2×C8—1342no—
209C2×C8C22×D6711yescharacteristic
210C2×C8C22×D6711yescharacteristic
211C4⋊C4—2681no—
212C42C22×D6711yescharacteristic
213C23—2682no—
214C23—1342no—
215Q8—2682no—
216Q8—672no—
217Q8—1342no—
218Q8—1342no—
219Q8C2×D26812yes—
220D4—22no—
221D4—2682no—
222D4—2682no—
223D4—2682no—
224D4—5362no—
225D4—672no—
226D4—1342no—
227D4—1342no—
228D4—1342no—
229D4C2×D26812yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—2682no—
233C2×C4—2682no—
234C2×C4—2682no—
235C2×C4—1342no—
236C2×C4—1342no—
237C2×C4C23×D6711yescharacteristic
238C8—22no—
239C8—1342no—
240C8D4×D6711yescharacteristic
241C8D4×D6711yescharacteristic
242C8C2×D26811yescharacteristic
243C8C2×D26811yescharacteristic
244C22—22no—
245C22—2682no—
246C22—5362no—
247C22—10722no—
248C22—1342no—
249C22#1452408911yescharacteristic
250C4—22no—
251C4—22no—
252C4—2682no—
253C4—1342no—
254C4C2×D4×D6711yescharacteristic
255C4C22×D26811yescharacteristic
256C2—21no—
257C2—42no—
258C2—2682no—
259C2—5362no—
260C2D4×D26811yescharacteristic
subgroupG/H
centerC2D4×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#443×C67C2
socleC2×C67D42
hypercenter#443D67
hypocenterC67#3291
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #16864208C2semidirect
2. #16864203C2semidirect
3. #16863769C2semidirect
4. #16863660C2semidirect
5. #16863659C2semidirect
6. #16863580C2semidirect
7. #16863565C2semidirect
8. #16863514C2semidirect
9. #443×C67C2semidirect
11. #14524089C22semidirect
13. #14524076C22semidirect
15. #14524073C22semidirect
24. C2×#11761617C22semidirect
25. C2×#11761617C22semidirect
27. #14524049C22semidirect
28. #14524043C22semidirect
30. #14523979C22semidirect
31. #14523971C22semidirect
32. #14523961C22semidirect
33. #14523959C22semidirect
35. #131×C67C22semidirect
36. #108×C67C22semidirect
37. #104×C67C22semidirect
38. C4×C8×C67C22semidirect
54. Dic268D4semidirect
62. #11761617D4semidirect
66. D536D4semidirect
73. Q16×C67D4semidirect
74. SD16×C67D4semidirect
75. D8×C67D4semidirect
131. C67#3291semidirect
140. #443D67semidirect
160. #131C2×D67semidirect
164. #108C2×D67semidirect
166. #104C2×D67semidirect
169. C4×C8C2×D67semidirect
188. Q16D268semidirect
198. SD16D268semidirect
202. D8D268semidirect