Finite Group Databasebeta

← #22569718

Subgroups — #22569718

Order 8576.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14524069C211yesmaximal, characteristic
2#16863970C211yesmaximal, characteristic, complemented
3#16863871C211yesmaximal, characteristic, complemented
4#16863613C214yesmaximal, complemented
5#16863599C214yesmaximal, complemented
6#16863422C211yesmaximal, characteristic, complemented
7#16863405C211yesmaximal, characteristic, complemented
8#16863393C211yesmaximal, characteristic, complemented
9C22×SD16×C67C211yesmaximal, characteristic, complemented
10C2×#11761628C2211yescharacteristic
11#14524069—24no—
12C2×#11761616C2212yes—
13C2×#11761613C2212yes—
14C2×#11761612C2211yescharacteristic
15#14524041C2212yescomplemented
16#14524032C2212yescomplemented
17#14524026—22no—
18#14524019—22no—
19D4×Dic67—22no—
20D4×Dic67C2214yes—
21#14524002C2211yescharacteristic
22#14524002C2211yescharacteristic
23C2×C22⋊Dic67—22no—
24Dic67⋊C8C2214yescomplemented
25#14523968C2212yescomplemented
26#14523967C2214yescomplemented
27#14523965C2212yescomplemented
28C22×Q8×C67C2211yescharacteristic
29C22×D4×C67C2211yescharacteristic
30C2×SD16×C67—24no—
31C2×SD16×C67C2214yes—
32C22×C8×C67C2211yescharacteristic
33#11761628—22no—
34#11761628—22no—
35C2×Dic134C2311yescharacteristic
36C2×D268C2311yescharacteristic
37C2×C22⋊D67—21no—
38C22×Dic67—24no—
39C2×C4×D67—21no—
40#11761616—24no—
41#11761613—24no—
42#11761612—22no—
43C2×C67⋊C8C2312yes—
44C4⋊Dic67D412yes—
45C4⋊Dic67C2312yes—
46Dic67⋊C4—24no—
47C22⋊Dic67—21no—
48C22⋊Dic67—24no—
49C22⋊Dic67D412yes—
50(C2×D67)⋊C4—24no—
51C4×Dic67—21no—
52C4×Dic67C2312yes—
53C24×C67—21no—
54C2×Q8×C67—21no—
55C2×Q8×C67—22no—
56C2×Q8×C67—22no—
57C2×Q8×C67C2312yes—
58C2×D4×C67—21no—
59C2×D4×C67—22no—
60C2×D4×C67D414yes—
61C2×D4×C67C2312yes—
62C22×C4×C67—21no—
63C22×C4×C67C2311yescharacteristic
64SD16×C67—28no—
65C2×C8×C67—22no—
66C2×C8×C67C2312yes—
67C23×C67—22no—
68C23×C67—24no—
69C23×C67—82no—
70C23×C67C2×D411yescharacteristic
71C22×D67—21no—
72Q8×C67—22no—
73Q8×C67—22no—
74Q8×C67—42no—
75D4×C67—24no—
76D4×C67—22no—
77D4×C67C2×D414yes—
78C2×C4×C67—22no—
79C2×C4×C67—42no—
80C2×C4×C67C2×D411yescharacteristic
81C2×C4×C67C2×D412yes—
82C2×C4×C67#4112yes—
83C2×C4×C67C2411yescharacteristic
84C22⋊D67—42no—
85C2×Dic67—22no—
86C2×Dic67—21no—
87C2×Dic67—84no—
88C2×Dic67C2×D414yes—
89D268—21no—
90D268—21no—
91C4×D67—42no—
92Dic134—21no—
93Dic134—21no—
94C8×C67—22no—
95C67⋊C8—22no—
96C22×C67—22no—
97C22×C67—24no—
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—41no—
110Dic67—42no—
111C2×C67—22no—
112C2×C67—44no—
113C2×C67C2×#12111yescharacteristic
114C2×C67#57712yes—
115D67—81no—
116#3042—671nomaximal, Sylow
117C67#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—671no—
119C22×SD16D6711yescharacteristic, complemented
120#536—671no—
121#518—671no—
122#463—674no—
123#461—674no—
124#424—671no—
125#419—671no—
126#416—671no—
127C2×#41—671no—
128C22×Q8C2×D6711yescharacteristic
129C22×D4C2×D6711yescharacteristic
130AGL(1,ℤ/8)—1344no—
131C2×SD16—24no—
132C2×SD16—672no—
133C2×SD16C2×D6714yes—
134C2×D8—672no—
135C2×M16—671no—
136C22×C8C2×D6711yescharacteristic
137C4⋊Q8—672no—
138#124—672no—
139#123—1342no—
140C22⋊Q8—1342no—
141C4×D4—674no—
142C4×D4—1342no—
143#117—671no—
144#117—671no—
145C2×C22⋊C4—1342no—
146C4⋊C8—674no—
147#103—674no—
148#103—672no—
149#102—672no—
150C24—21no—
151#41—1342no—
152#41—1342no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—671no—
157C2×Q8C22×D6712yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—671no—
161C2×D4—1341no—
162C2×D4C22⋊D6714yes—
163C2×D4C22×D6712yes—
164C22×C4—21no—
165C22×C4—1341no—
166C22×C4—1344no—
167C22×C4C22×D6711yescharacteristic
168SD16—28no—
169SD16—1344no—
170D8—1344no—
171M16—1342no—
172C2×C8—22no—
173C2×C8—672no—
174C2×C8C22×D6712yes—
175C4⋊C4—672no—
176C4⋊C4—672no—
177C4⋊C4—1344no—
178C22⋊C4—672no—
179C22⋊C4—1344no—
180C22⋊C4—1344no—
181C22⋊C4—1341no—
182C42—672no—
183C42—1341no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—1341no—
188C23D4×D6711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—1341no—
193Q8—1341no—
194D4—24no—
195D4—22no—
196D4—2682no—
197D4—1341no—
198D4—1341no—
199D4C2×C22⋊D6714yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—2682no—
203C2×C4—5364no—
204C2×C4—674no—
205C2×C4—1342no—
206C2×C4—1341no—
207C2×C4D4×D6711yescharacteristic
208C2×C4C2×C22⋊D6712yes—
209C2×C4#1176162712yes—
210C2×C4C23×D6711yescharacteristic
211C8—22no—
212C8—1342no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—2681no—
218C22—2682no—
219C22#1452401512yes—
220C22C2×D4×D6711yescharacteristic
221C4—42no—
222C4—2681no—
223C4—2682no—
224C4—1344no—
225C4#1452401512yes—
226C4C22×C22⋊D6711yescharacteristic
227C4C2×#1176162711yescharacteristic
228C2—22no—
229C2—44no—
230C2—5361no—
231C2C2×#1452401511yescharacteristic
232C2#1686420212yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
FittingC22×SD16×C67C2
socleC22×C67C22×D4
hypercenterC22×SD16D67
hypocenterC67#3042
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #16863970C2semidirect
3. #16863871C2semidirect
4. #16863613C2semidirect
5. #16863599C2semidirect
6. #16863422C2semidirect
7. #16863405C2semidirect
8. #16863393C2semidirect
9. C22×SD16×C67C2semidirect
15. #14524041C22semidirect
16. #14524032C22semidirect
24. Dic67⋊C8C22semidirect
25. #14523968C22semidirect
26. #14523967C22semidirect
27. #14523965C22semidirect
117. C67#3042semidirect
119. C22×SD16D67semidirect