Finite Group Databasebeta

← #22570581

Subgroups — #22570581

Order 8576.

484 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 129 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#16863949C211yesmaximal, characteristic, complemented
2C2×#14524039C211yesmaximal, characteristic
3#16863759C214yesmaximal, complemented
4#16863741C214yesmaximal, complemented
5#16863402C211yesmaximal, characteristic, complemented
6#16863401C211yesmaximal, characteristic, complemented
7#16863400C211yesmaximal, characteristic, complemented
8M16×Dic67C211yesmaximal, characteristic, complemented
9#418×C67C211yesmaximal, characteristic, complemented
10C22×D268C2211yescharacteristic
11C22×C22⋊D67—22no—
12C2×#11761612C2211yescharacteristic
13#14524042C2212yescomplemented
14#14524039—22no—
15#14524039C2212yes—
16#14524039C2212yes—
17#14524034C2212yescomplemented
18#14524020—22no—
19C4×Dic134—22no—
20#14524002C2211yescharacteristic
21C2×C4×Dic67C2211yescharacteristic
22#14523965C2214yescomplemented
23#14523964C2214yescomplemented
24#14523963C2214yescomplemented
25#14523953C2212yescomplemented
26C8×Dic67C2212yescomplemented
27C22×D4×C67C2211yescharacteristic
28C2×M16×C67C2211yescharacteristic
29#117×C67C2211yescharacteristic
30#102×C67C2214yescomplemented
31C23×D67—21no—
32C2×Dic134—22no—
33C2×D268—22no—
34C2×D268—22no—
35C2×D268—21no—
36C2×D268C2312yes—
37C2×C22⋊D67—24no—
38C2×C22⋊D67—24no—
39C2×C22⋊D67—48no—
40C22×Dic67D412yes—
41#11761612—22no—
42C2×C67⋊C8C2312yes—
43C4⋊Dic67—22no—
44C4⋊Dic67C2312yes—
45Dic67⋊C4—22no—
46Dic67⋊C4—22no—
47C22⋊Dic67—21no—
48C4×Dic67—21no—
49C4×Dic67D412yes—
50C4×Dic67C2311yescharacteristic
51C4×Dic67C2311yescharacteristic
52C24×C67—21no—
53C2×D4×C67—21no—
54C2×D4×C67—22no—
55C2×D4×C67—22no—
56C2×D4×C67C2312yes—
57C22×C4×C67C2311yescharacteristic
58M16×C67—22no—
59C2×C8×C67C2312yes—
60C4⋊C4×C67C2312yes—
61C22⋊C4×C67—21no—
62C42×C67—21no—
63C23×C67—22no—
64C23×C67—44no—
65C23×C67—42no—
66C23×C67C2×D411yescharacteristic
67C22×D67—22no—
68C22×D67—44no—
69C22×D67—42no—
70D4×C67—22no—
71D4×C67—22no—
72D4×C67—42no—
73C2×C4×C67—22no—
74C2×C4×C67C2×D411yescharacteristic
75C2×C4×C67#4112yes—
76C2×C4×C67#4112yes—
77C2×C4×C67C2411yescharacteristic
78C22⋊D67—48no—
79C22⋊D67—48no—
80C2×Dic67—24no—
81C2×Dic67—22no—
82C2×Dic67C2×D412yes—
83C2×Dic67C2×D412yes—
84D268—22no—
85D268—22no—
86D268—42no—
87Dic134—44no—
88C8×C67—22no—
89C67⋊C8—22no—
90C22×C67—22no—
91C22×C67—42no—
92C22×C67—44no—
93C22×C67—84no—
94C22×C67#12412yes—
95C22×C67C22×D411yescharacteristic
96C2×D67—42no—
97C2×D67—44no—
98C2×D67—84no—
99C4×C67—42no—
100C4×C67#12412yes—
101C4×C67C2×#4111yescharacteristic
102C4×C67C2×#4111yescharacteristic
103Dic67—22no—
104Dic67—42no—
105Dic67AGL(1,ℤ/8)14yes—
106C2×C67—22no—
107C2×C67—82no—
108C2×C67C2×#12411yescharacteristic
109C2×C67C2×AGL(1,ℤ/8)12yes—
110D67—82no—
111#3138—671nomaximal, Sylow
112C67#313811yescharacteristic, complemented, Sylow
113#533—671no—
114C2×#127—671no—
115#490—674no—
116#486—674no—
117#418—671no—
118#418—671no—
119#418—671no—
120#418D6711yescharacteristic, complemented
121C4×M16—671no—
122C22×D4—671no—
123C22×D4—1342no—
124C22×D4C2×D6711yescharacteristic
125C2×M16—671no—
126C2×M16C2×D6711yescharacteristic
127C4⋊Q8—672no—
128#127—672no—
129#127—672no—
130#127—1342no—
131#125—672no—
132C22⋊Q8—1342no—
133C4×Q8—1342no—
134#117—671no—
135#117C2×D6711yescharacteristic
136C2×C42—671no—
137#102—674no—
138#102—674no—
139#102—674no—
140#102C2×D6714yescomplemented
141#97—672no—
142C4×C8—672no—
143C24—21no—
144C24—1341no—
145C2×Q8—1342no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—2688no—
150C2×D4—672no—
151C2×D4—1344no—
152C2×D4—1344no—
153C2×D4—1341no—
154C2×D4—1342no—
155C2×D4—1342no—
156C2×D4C22×D6712yes—
157C22×C4—672no—
158C22×C4C22×D6711yescharacteristic
159M16—22no—
160M16—1342no—
161C2×C8—672no—
162C2×C8C22×D6712yes—
163C4⋊C4—672no—
164C4⋊C4—1342no—
165C4⋊C4—1342no—
166C4⋊C4—1342no—
167C4⋊C4C22×D6712yes—
168C22⋊C4—21no—
169C22⋊C4—1341no—
170C42—21no—
171C42—672no—
172C42—671no—
173C42—671no—
174C42—1341no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—2684no—
179C23—2682no—
180C23—1342no—
181C23D4×D6711yescharacteristic
182Q8—2684no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—2688no—
187D4—2688no—
188D4—2682no—
189D4—1342no—
190D4—1342no—
191C2×C4—22no—
192C2×C4—672no—
193C2×C4—672no—
194C2×C4—1342no—
195C2×C4—1344no—
196C2×C4D4×D6711yescharacteristic
197C2×C4#1176162712yes—
198C2×C4#1176162812yes—
199C2×C4C23×D6711yescharacteristic
200C8—22no—
201C8—1342no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—2682no—
207C22—2684no—
208C22—5364no—
209C22#1452402912yes—
210C22C2×D4×D6711yescharacteristic
211C4—42no—
212C4—2682no—
213C4—674no—
214C4—1342no—
215C4#1452402912yes—
216C4C2×#1176162711yescharacteristic
217C4C2×#1176162811yescharacteristic
218C2—22no—
219C2—82no—
220C2—5362no—
221C2C2×#1452402911yescharacteristic
222C2AGL(1,ℤ/8)×D6712yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#418×C67C2
socleC22×C67C22×D4
hypercenter#418D67
hypocenterC67#3138
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. #16863949C2semidirect
3. #16863759C2semidirect
4. #16863741C2semidirect
5. #16863402C2semidirect
6. #16863401C2semidirect
7. #16863400C2semidirect
8. M16×Dic67C2semidirect
9. #418×C67C2semidirect
13. #14524042C22semidirect
17. #14524034C22semidirect
22. #14523965C22semidirect
23. #14523964C22semidirect
24. #14523963C22semidirect
25. #14523953C22semidirect
26. C8×Dic67C22semidirect
30. #102×C67C22semidirect
112. C67#3138semidirect
120. #418D67semidirect
140. #102C2×D67semidirect