Finite Group Databasebeta

← #22569733

Subgroups — #22569733

Order 8576.

490 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14524073C211yesmaximal, characteristic, complemented
2C22×#11761617C211yesmaximal, characteristic, complemented
3#16863960C211yesmaximal, characteristic
4#16863873C211yesmaximal, characteristic, complemented
5#16863621C214yesmaximal, complemented
6#16863595C214yesmaximal, complemented
7#16863422C211yesmaximal, characteristic, complemented
8#16863400C211yesmaximal, characteristic, complemented
9#416×C67C211yesmaximal, characteristic, complemented
10C22×Dic134C2211yescharacteristic
11C22×D268C2211yescharacteristic
12#14524073—24no—
13C2×#11761618C2212yescomplemented
14C2×#11761617—24no—
15C2×#11761617C2214yes—
16C2×#11761616C2212yescomplemented
17C2×#11761612C2211yescharacteristic
18#14524039C2212yes—
19#14524032C2212yes—
20#14524017—22no—
21#14524011—22no—
22C4×Dic134—22no—
23#14524008C2214yescomplemented
24#14524002C2211yescharacteristic
25#14523999—22no—
26Dic67⋊C8C2214yescomplemented
27#14523963C2214yes—
28C2×#41×C67C2211yescharacteristic
29C22×C8×C67C2211yescharacteristic
30#117×C67C2211yescharacteristic
31#103×C67C2212yescomplemented
32#102×C67C2212yescomplemented
33C23×D67—21no—
34C2×Dic134—21no—
35C2×Dic134—22no—
36C2×Dic134D414yescomplemented
37C2×Dic134C2312yes—
38C2×D268—22no—
39C2×D268—22no—
40C2×D268—21no—
41C2×D268C2312yes—
42C2×C22⋊D67—24no—
43C22×Dic67—21no—
44#11761618—24no—
45#11761617—28no—
46#11761616—24no—
47#11761612—22no—
48C2×C67⋊C8C2312yes—
49C4⋊Dic67D412yes—
50Dic67⋊C4—24no—
51C22⋊Dic67D412yes—
52(C2×D67)⋊C4—24no—
53C4×Dic67—24no—
54C4×Dic67C2312yes—
55#41×C67—22no—
56#41×C67—22no—
57C2×Q8×C67C2311yescharacteristic
58C2×D4×C67—21no—
59C2×D4×C67C2311yescharacteristic
60C22×C4×C67—21no—
61C22×C4×C67C2311yescharacteristic
62C2×C8×C67—22no—
63C2×C8×C67C2312yes—
64C4⋊C4×C67C2312yes—
65C22⋊C4×C67—21no—
66C42×C67—21no—
67C23×C67—21no—
68C23×C67C2×D411yescharacteristic
69C22×D67—22no—
70C22×D67—42no—
71C22×D67—82no—
72Q8×C67—21no—
73Q8×C67—21no—
74D4×C67—21no—
75D4×C67—21no—
76D4×C67—42no—
77C2×C4×C67—22no—
78C2×C4×C67—21no—
79C2×C4×C67—42no—
80C2×C4×C67C2×D411yescharacteristic
81C2×C4×C67C2×D412yes—
82C2×C4×C67#4112yes—
83C2×C4×C67C2411yescharacteristic
84C22⋊D67—84no—
85C2×Dic67—24no—
86C2×Dic67—22no—
87C2×Dic67C2×D414yes—
88D268—22no—
89D268—22no—
90D268—42no—
91Dic134—24no—
92Dic134—22no—
93Dic134C2×D414yes—
94C8×C67—22no—
95C67⋊C8—22no—
96C22×C67—22no—
97C22×C67—41no—
98C22×C67—42no—
99C22×C67#12112yes—
100C22×C67C22×D411yescharacteristic
101C2×D67—42no—
102C2×D67—44no—
103C2×D67—84no—
104C4×C67—41no—
105C4×C67—42no—
106C4×C67#12112yes—
107C4×C67C22×D411yescharacteristic
108C4×C67C2×#4111yescharacteristic
109Dic67—24no—
110Dic67—24no—
111C2×C67—22no—
112C2×C67—81no—
113C2×C67C2×#12111yescharacteristic
114C2×C67#57712yes—
115D67—82no—
116#3043—671nomaximal, Sylow
117C67#304311yescharacteristic, complemented, Sylow
118C2×#137—671no—
119C22×SD16—671no—
120#535—671no—
121#519—671no—
122#464—674no—
123#460—674no—
124#424—671no—
125#418—671no—
126#416D6711yescharacteristic, complemented
127C2×#41C2×D6711yescharacteristic
128C22×Q8—671no—
129C22×D4—671no—
130#137—1344no—
131C2×Q16—672no—
132C2×SD16—672no—
133C2×SD16—674no—
134C2×SD16—1344no—
135C2×M16—671no—
136C22×C8C2×D6711yescharacteristic
137#127—672no—
138#124—672no—
139#121—1342no—
140#120—1342no—
141C4×Q8—674no—
142C4×Q8—1342no—
143#117—671no—
144#117—1342no—
145#117C2×D6711yescharacteristic
146C4⋊C8—674no—
147#103C2×D6712yescomplemented
148#102—674no—
149#102C2×D6712yescomplemented
150C24—1341no—
151#41—22no—
152#41—22no—
153C2×Q8—674no—
154C2×Q8—672no—
155C2×Q8—1341no—
156C2×Q8—1342no—
157C2×Q8C22×D6711yescharacteristic
158C2×D4—21no—
159C2×D4—672no—
160C2×D4—1344no—
161C2×D4—1341no—
162C2×D4—1342no—
163C2×D4—1342no—
164C2×D4C22×D6711yescharacteristic
165C22×C4—21no—
166C22×C4—1341no—
167C22×C4C22×D6711yescharacteristic
168Q16—1344no—
169SD16—1348no—
170SD16—1344no—
171M16—1342no—
172C2×C8—22no—
173C2×C8—672no—
174C2×C8C22×D6712yes—
175C4⋊C4—672no—
176C4⋊C4—1344no—
177C4⋊C4C22×D6712yes—
178C22⋊C4—21no—
179C22⋊C4—672no—
180C22⋊C4—1344no—
181C42—21no—
182C42—672no—
183C42—1344no—
184C23—21no—
185C23—2682no—
186C23—5362no—
187C23—1342no—
188C23D4×D6711yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—674no—
192Q8—1344no—
193Q8—1342no—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—2682no—
198D4—5364no—
199D4—1342no—
200D4—1342no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—674no—
205C2×C4—1344no—
206C2×C4—1342no—
207C2×C4D4×D6711yescharacteristic
208C2×C4D4×D6712yes—
209C2×C4#1176162812yes—
210C2×C4C23×D6711yescharacteristic
211C8—22no—
212C8—1342no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—2682no—
217C22—2684no—
218C22—5364no—
219C22#1452401612yes—
220C22C2×D4×D6711yescharacteristic
221C4—41no—
222C4—42no—
223C4—1344no—
224C4—1344no—
225C4#1452401612yes—
226C4C2×D4×D6711yescharacteristic
227C4C2×#1176162811yescharacteristic
228C2—22no—
229C2—81no—
230C2—5362no—
231C2C2×#1452401611yescharacteristic
232C2#1686420312yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#416×C67C2
socleC22×C67C22×D4
hypercenter#416D67
hypocenterC67#3043
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#14524073C2semidirect
2. C22×#11761617C2semidirect
4. #16863873C2semidirect
5. #16863621C2semidirect
6. #16863595C2semidirect
7. #16863422C2semidirect
8. #16863400C2semidirect
9. #416×C67C2semidirect
13. C2×#11761618C22semidirect
16. C2×#11761616C22semidirect
23. #14524008C22semidirect
26. Dic67⋊C8C22semidirect
31. #103×C67C22semidirect
32. #102×C67C22semidirect
36. C2×Dic134D4semidirect
117. C67#3043semidirect
126. #416D67semidirect
147. #103C2×D67semidirect
149. #102C2×D67semidirect