Finite Group Databasebeta

← #22569729

Subgroups — #22569729

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×#14524076C211yesmaximal, characteristic, complemented
2C22×#11761616C211yesmaximal, characteristic, complemented
3#16863963C211yesmaximal, characteristic
4#16863873C211yesmaximal, characteristic, complemented
5#16863617C214yesmaximal, complemented
6#16863595C214yesmaximal, complemented
7#16863424C211yesmaximal, characteristic, complemented
8#16863390C211yesmaximal, characteristic, complemented
9#418×C67C211yesmaximal, characteristic, complemented
10C2×#11761628C2211yescharacteristic
11C22×Dic134C2211yescharacteristic
12#14524076—24no—
13C2×Dic268C2212yescomplemented
14C2×#11761617C2212yescomplemented
15C2×#11761616—24no—
16C2×#11761616C2214yes—
17C22×C67⋊C8C2211yescharacteristic
18#14524039C2212yes—
19#14524030C2212yes—
20#14524014—22no—
21#14524012—22no—
22C4×Dic134—22no—
23#14524008C2214yescomplemented
24#14524002C2211yescharacteristic
25#14523999—22no—
26Dic67⋊C8C2214yescomplemented
27#14523966C2212yescomplemented
28#14523963C2212yescomplemented
29C22×D4×C67C2211yescharacteristic
30C2×M16×C67C2211yescharacteristic
31#117×C67C2211yescharacteristic
32#102×C67C2214yes—
33#11761628—22no—
34#11761628—22no—
35C2×Dic134—21no—
36C2×Dic134—22no—
37C2×Dic134D414yescomplemented
38C2×Dic134C2311yescharacteristic
39C2×Dic134C2312yes—
40C2×D268C2311yescharacteristic
41C2×C22⋊D67—21no—
42C2×C22⋊D67—24no—
43C22×Dic67—21no—
44C2×C4×D67—21no—
45Dic268—24no—
46#11761617—24no—
47#11761616—28no—
48C2×C67⋊C8—22no—
49C2×C67⋊C8C2312yes—
50C4⋊Dic67D412yes—
51Dic67⋊C4—24no—
52C22⋊Dic67—24no—
53C22⋊Dic67D412yes—
54C4×Dic67—24no—
55C4×Dic67C2312yes—
56C24×C67—21no—
57C2×D4×C67—22no—
58C2×D4×C67—21no—
59C2×D4×C67—22no—
60C2×D4×C67C2312yes—
61C22×C4×C67C2311yescharacteristic
62M16×C67—22no—
63C2×C8×C67C2312yes—
64C4⋊C4×C67C2312yes—
65C22⋊C4×C67—21no—
66C42×C67—21no—
67C23×C67—22no—
68C23×C67—42no—
69C23×C67—82no—
70C23×C67C2×D411yescharacteristic
71C22×D67—21no—
72D4×C67—22no—
73D4×C67—22no—
74D4×C67—42no—
75C2×C4×C67—22no—
76C2×C4×C67C2×D411yescharacteristic
77C2×C4×C67C2×D412yes—
78C2×C4×C67#4112yes—
79C2×C4×C67C2411yescharacteristic
80C22⋊D67—42no—
81C22⋊D67—84no—
82C2×Dic67—24no—
83C2×Dic67—22no—
84C2×Dic67—21no—
85C2×Dic67C2×D414yes—
86D268—21no—
87D268—21no—
88C4×D67—42no—
89Dic134—24no—
90Dic134—22no—
91Dic134—21no—
92Dic134—21no—
93Dic134C2×D414yes—
94C8×C67—22no—
95C67⋊C8—22no—
96C22×C67—22no—
97C22×C67—42no—
98C22×C67—44no—
99C22×C67—84no—
100C22×C67#12112yes—
101C22×C67C22×D411yescharacteristic
102C2×D67—41no—
103C2×D67—42no—
104C4×C67—42no—
105C4×C67#12112yes—
106C4×C67C22×D411yescharacteristic
107C4×C67C2×#4111yescharacteristic
108Dic67—24no—
109Dic67—24no—
110Dic67—41no—
111C2×C67—22no—
112C2×C67—82no—
113C2×C67C2×#12111yescharacteristic
114C2×C67#57712yes—
115D67—81no—
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#418D6711yescharacteristic, complemented
126#416—671no—
127C2×#41—671no—
128C22×Q8—671no—
129C22×D4C2×D6711yescharacteristic
130#137—1344no—
131C2×Q16—672no—
132C2×SD16—674no—
133C2×SD16—672no—
134C2×SD16—1344no—
135C2×M16C2×D6711yescharacteristic
136C22×C8—671no—
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#103—672no—
148#102—672no—
149#102C2×D6714yes—
150C24—21no—
151#41—1342no—
152#41—1342no—
153C2×Q8—674no—
154C2×Q8—672no—
155C2×Q8—671no—
156C2×Q8—1341no—
157C2×Q8—1342no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—671no—
162C2×D4—1344no—
163C2×D4—1341no—
164C2×D4C22×D6712yes—
165C22×C4—1341no—
166C22×C4—1341no—
167C22×C4C22×D6711yescharacteristic
168Q16—1344no—
169SD16—1344no—
170SD16—1348no—
171M16—22no—
172C2×C8—672no—
173C2×C8—1342no—
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—22no—
185C23—42no—
186C23—82no—
187C23—1341no—
188C23D4×D6711yescharacteristic
189Q8—674no—
190Q8—1341no—
191Q8—1341no—
192Q8—1344no—
193Q8—1342no—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—2682no—
198D4—5364no—
199D4—1341no—
200D4—1341no—
201C2×C4—22no—
202C2×C4—2682no—
203C2×C4—674no—
204C2×C4—1341no—
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—42no—
215C22—44no—
216C22—84no—
217C22—2681no—
218C22—2682no—
219C22#1452401612yes—
220C22C2×D4×D6711yescharacteristic
221C4—42no—
222C4—2681no—
223C4—1344no—
224C4—1344no—
225C4#1452401612yes—
226C4C2×D4×D6711yescharacteristic
227C4C2×#1176162811yescharacteristic
228C2—22no—
229C2—82no—
230C2—5361no—
231C2C2×#1452401611yescharacteristic
232C2#1686420612yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#418×C67C2
socleC22×C67C22×D4
hypercenter#418D67
hypocenterC67#3043
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#14524076C2semidirect
2. C22×#11761616C2semidirect
4. #16863873C2semidirect
5. #16863617C2semidirect
6. #16863595C2semidirect
7. #16863424C2semidirect
8. #16863390C2semidirect
9. #418×C67C2semidirect
13. C2×Dic268C22semidirect
14. C2×#11761617C22semidirect
23. #14524008C22semidirect
26. Dic67⋊C8C22semidirect
27. #14523966C22semidirect
28. #14523963C22semidirect
37. C2×Dic134D4semidirect
117. C67#3043semidirect
125. #418D67semidirect