Finite Group Databasebeta

← #22570533

Subgroups — #22570533

Order 8576.

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

normal subgroups only

This group has 400 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#16863951C211yesmaximal, characteristic, complemented
2#16863947C211yesmaximal, characteristic
3#16863752C214yesmaximal, complemented
4#16863746C211yesmaximal, characteristic, complemented
5#16863744C211yesmaximal, characteristic, complemented
6#16863743C211yesmaximal, characteristic, complemented
7#16863391C214yesmaximal, complemented
8C4×C8×D67C211yesmaximal, characteristic, complemented
9#487×C67C211yesmaximal, characteristic, complemented
10C2×#11761629C2212yes—
11C2×C8×D67C2212yes—
12#14524043C2211yescharacteristic
13#14524040C2211yescharacteristic
14#14524034C2212yes—
15#14524032C2212yes—
16#14524023—21no—
17#14524019—21no—
18#14524018—21no—
19#14524018—21no—
20Q8×Dic67—21no—
21#14524008—21no—
22#14524005—21no—
23#14524005—21no—
24#14524001C2212yes—
25C42×D67C2211yescharacteristic
26#14523967C2214yes—
27#14523965C2214yes—
28#14523963C2214yes—
29C4×C67⋊C8C2211yescharacteristic
30C8×Dic67C2212yes—
31C4⋊Q8×C67C2211yescharacteristic
32C4⋊Q8×C67C2211yescharacteristic
33#103×C67C2214yes—
34C4×C8×C67C2211yescharacteristic
35#11761629—28no—
36C2×Dic134—21no—
37C2×D268—22no—
38C2×D268—21no—
39C2×D268—21no—
40C2×D268C2312yes—
41C2×C4×D67—22no—
42C2×C4×D67D411yescharacteristic
43C2×C4×D67D411yescharacteristic
44C2×C4×D67C2311yescharacteristic
45C2×C67⋊C8C2312yes—
46C8×D67—24no—
47C4⋊Dic67—21no—
48C4⋊Dic67—22no—
49C4⋊Dic67C2312yes—
50Dic67⋊C4—22no—
51(C2×D67)⋊C4—22no—
52(C2×D67)⋊C4—22no—
53(C2×D67)⋊C4—22no—
54C4×Dic67—22no—
55C4×Dic67D411yescharacteristic
56C4×Dic67D411yescharacteristic
57C4×Dic67C2311yescharacteristic
58C2×Q8×C67—21no—
59C2×Q8×C67C2312yes—
60C2×C8×C67C2312yes—
61C4⋊C4×C67—21no—
62C4⋊C4×C67—21no—
63C4⋊C4×C67—21no—
64C4⋊C4×C67C2312yes—
65C42×C67C2311yescharacteristic
66C22×D67—22no—
67C22×D67C2×D411yescharacteristic
68Q8×C67—24no—
69Q8×C67—42no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2411yescharacteristic
75C2×Dic67—22no—
76C2×Dic67C2×D411yescharacteristic
77C2×Dic67C2×D411yescharacteristic
78C2×Dic67C2×D411yescharacteristic
79D268—24no—
80D268—44no—
81D268—42no—
82D268—42no—
83C4×D67—22no—
84C4×D67—22no—
85C4×D67—44no—
86C4×D67#4114yes—
87Dic134—42no—
88C8×C67—22no—
89C67⋊C8—22no—
90C22×C67C22×D411yescharacteristic
91C2×D67—21no—
92C2×D67—21no—
93C2×D67—42no—
94C2×D67—42no—
95C2×D67—42no—
96C2×D67#12412yes—
97C4×C67—42no—
98C4×C67—42no—
99C4×C67#13512yes—
100C4×C67#13512yes—
101C4×C67C2×#4112yes—
102Dic67—21no—
103Dic67—21no—
104Dic67—42no—
105Dic67#12412yes—
106C2×C67C2×#12411yescharacteristic
107C2×C67C2×#13511yescharacteristic
108C2×C67C2×#13511yescharacteristic
109D67—22no—
110D67—82no—
111#3131—671nomaximal, Sylow
112C67#313111yescharacteristic, complemented, Sylow
113#533—671no—
114#532—671no—
115#488—674no—
116#487—671no—
117#487D6711yescharacteristic, complemented
118#486—671no—
119#486—671no—
120#416—674no—
121C2×C4×C8—671no—
122C2×#41—672no—
123C22×C8—672no—
124C4⋊Q8—671no—
125C4⋊Q8C2×D6711yescharacteristic
126C4⋊Q8C2×D6711yescharacteristic
127#127—671no—
128#125—672no—
129#124—672no—
130C22⋊Q8—1341no—
131C22⋊Q8—1341no—
132#121—1341no—
133#121—1341no—
134C4×Q8—1341no—
135C4×Q8—1341no—
136C4×D4—1341no—
137C4×D4—1341no—
138#117—672no—
139C2×C42—671no—
140#103—674no—
141#103C2×D6714yes—
142#102—674no—
143#102—674no—
144C4×C8—672no—
145C4×C8—671no—
146C4×C8C2×D6711yescharacteristic
147#41—1348no—
148C2×Q8—21no—
149C2×Q8—1341no—
150C2×Q8C22×D6712yes—
151C2×D4—672no—
152C2×D4—1342no—
153C2×D4—1341no—
154C2×D4—1341no—
155C22×C4—671no—
156C22×C4—671no—
157C22×C4—671no—
158C22×C4—1342no—
159C2×C8—672no—
160C2×C8—1344no—
161C2×C8C22×D6712yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—672no—
166C4⋊C4—1342no—
167C4⋊C4—1342no—
168C4⋊C4—1341no—
169C4⋊C4C22×D6712yes—
170C22⋊C4—1342no—
171C22⋊C4—1342no—
172C22⋊C4—1342no—
173C42—671no—
174C42—671no—
175C42—671no—
176C42—1342no—
177C42C22×D6711yescharacteristic
178C23—671no—
179C23—1342no—
180Q8—24no—
181Q8—42no—
182Q8—2682no—
183D4—2684no—
184D4—2682no—
185D4—2682no—
186D4—1344no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—2684no—
190C2×C4—671no—
191C2×C4—674no—
192C2×C4—671no—
193C2×C4—671no—
194C2×C4—1342no—
195C2×C4—1342no—
196C2×C4—1342no—
197C2×C4D4×D6711yescharacteristic
198C2×C4D4×D6711yescharacteristic
199C2×C4C23×D6711yescharacteristic
200C8—22no—
201C8—1342no—
202C22—2682no—
203C22—2682no—
204C22—2682no—
205C22—672no—
206C22—1341no—
207C22—1341no—
208C22C2×D4×D6711yescharacteristic
209C4—42no—
210C4—42no—
211C4—2682no—
212C4—672no—
213C4—1341no—
214C4—1341no—
215C4#1452406412yes—
216C4#1452406612yes—
217C4#41×D6712yes—
218C2—5362no—
219C2—1342no—
220C2#124×D6711yescharacteristic
221C2C2×#1452406411yescharacteristic
222C2C2×#1452406611yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#487×C67C2
socleC22×C67C22×D4
hypercenter#487D67
hypocenterC67#3131
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #16863951C2semidirect
3. #16863752C2semidirect
4. #16863746C2semidirect
5. #16863744C2semidirect
6. #16863743C2semidirect
7. #16863391C2semidirect
8. C4×C8×D67C2semidirect
9. #487×C67C2semidirect
112. C67#3131semidirect
117. #487D67semidirect