Finite Group Databasebeta

← #22570935

Subgroups — #22570935

Order 8576.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#16863925C212yesmaximal, complemented
2#16863798C211yesmaximal, characteristic, complemented
3#16863750C212yesmaximal, complemented
4#16863651C211yesmaximal, characteristic, complemented
5#16863646C211yesmaximal, characteristic
6#16863639C212yesmaximal, complemented
7#16863508C211yesmaximal, characteristic, complemented
8#16863503C211yesmaximal, characteristic, complemented
9#16863497C212yesmaximal, complemented
10C4×#11761611C211yesmaximal, characteristic, complemented
11#495×C67C211yesmaximal, characteristic, complemented
12C2×#11761611C2211yescharacteristic
13C2×#11761611C2211yescharacteristic
14#14524036—22no—
15#14524035C2212yes—
16#14524025—22no—
17#14524019C2212yes—
18#14524014C2212yes—
19Q8×Dic67C2212yes—
20D4×Dic67C2212yes—
21(C22⋊D67)⋊C4—22no—
22#14523999—22no—
23C42×D67C2211yescharacteristic
24#14523977C2212yes—
25#14523975C2211yescharacteristic
26#14523975C2211yescharacteristic
27#14523967C2212yes—
28#14523967C2212yes—
29#14523964C2212yes—
30#14523964C2212yes—
31#14523953C2211yescharacteristic
32#14523953C2211yescharacteristic
33C4×C67⋊C8C2211yescharacteristic
34C2×Q16×C67C2211yescharacteristic
35C2×SD16×C67C2212yes—
36C2×D8×C67C2211yescharacteristic
37#124×C67C2212yes—
38C4×C8×C67C2211yescharacteristic
39C2×C22⋊D67—22no—
40C22×Dic67—22no—
41C2×C4×D67D412yes—
42C2×C4×D67C2311yescharacteristic
43#11761611—22no—
44#11761611—22no—
45C2×C67⋊C8C2311yescharacteristic
46C2×C67⋊C8C2311yescharacteristic
47C4⋊Dic67—22no—
48C4⋊Dic67C2312yes—
49C4⋊Dic67C2312yes—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52Dic67⋊C4—22no—
53C22⋊Dic67—22no—
54C22⋊Dic67—22no—
55(C2×D67)⋊C4—22no—
56C4×Dic67—22no—
57C4×Dic67D412yes—
58C4×Dic67C2311yescharacteristic
59C2×Q8×C67C2312yes—
60C2×D4×C67C2312yes—
61Q16×C67—22no—
62SD16×C67—24no—
63D8×C67—22no—
64C2×C8×C67C2311yescharacteristic
65C2×C8×C67C2311yescharacteristic
66C22⋊C4×C67—22no—
67C22⋊C4×C67—22no—
68C42×C67C2311yescharacteristic
69C23×C67—22no—
70C22×D67C2×D411yescharacteristic
71Q8×C67—22no—
72Q8×C67—22no—
73D4×C67—22no—
74D4×C67—22no—
75C2×C4×C67—22no—
76C2×C4×C67C2×D412yes—
77C2×C4×C67C2411yescharacteristic
78C22⋊D67—44no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—44no—
82C2×Dic67C2×D412yes—
83C2×Dic67C2×D411yescharacteristic
84C4×D67—21no—
85C4×D67—21no—
86C4×D67—42no—
87C8×C67#4112yes—
88C8×C67#4112yes—
89C67⋊C8—21no—
90C67⋊C8—21no—
91C22×C67—42no—
92C22×C67—42no—
93C22×C67—42no—
94C22×C67C22×D411yescharacteristic
95C2×D67—21no—
96C2×D67—22no—
97C4×C67—22no—
98C4×C67—42no—
99C4×C67C2×#4111yescharacteristic
100C4×C67C2×#4111yescharacteristic
101Dic67—21no—
102Dic67—22no—
103Dic67—42no—
104Dic67—42no—
105C2×C67—82no—
106C2×C67#53211yescharacteristic
107C2×C67#57512yes—
108D67—41no—
109#3183—671nomaximal, Sylow
110C67#318311yescharacteristic, complemented, Sylow
111#529—672no—
112#499—671no—
113#495D6711yescharacteristic, complemented
114#488—672no—
115#470—671no—
116#469—671no—
117#468—672no—
118#442—671no—
119#441—671no—
120#440—672no—
121C4×M16—671no—
122C2×Q16C2×D6711yescharacteristic
123C2×SD16C2×D6712yes—
124C2×D8C2×D6711yescharacteristic
125C2×M16—671no—
126C2×M16—671no—
127#126—1342no—
128#125—672no—
129#124C2×D6712yes—
130#123—1342no—
131C22⋊Q8—672no—
132#121—672no—
133C4×Q8—672no—
134C4×D4—672no—
135C4×D4—1342no—
136#117—1342no—
137C2×C42—671no—
138#107—672no—
139#106—671no—
140#106—671no—
141#103—672no—
142#103—672no—
143#102—672no—
144#102—672no—
145#97—671no—
146#97—671no—
147C4×C8—671no—
148C4×C8C2×D6711yescharacteristic
149C2×Q8C22×D6712yes—
150C2×D4—1342no—
151C2×D4C22×D6712yes—
152C22×C4—672no—
153C22×C4—671no—
154C22×C4—1342no—
155Q16—22no—
156SD16—24no—
157D8—22no—
158M16—1342no—
159M16—1342no—
160C2×C8—671no—
161C2×C8—671no—
162C2×C8C22×D6711yescharacteristic
163C2×C8C22×D6711yescharacteristic
164C4⋊C4—672no—
165C4⋊C4—672no—
166C4⋊C4—1342no—
167C4⋊C4—1342no—
168C4⋊C4—1342no—
169C4⋊C4—1342no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—1342no—
173C22⋊C4—1342no—
174C22⋊C4—1342no—
175C42—672no—
176C42—671no—
177C42—1342no—
178C42C22×D6711yescharacteristic
179C23—22no—
180C23—671no—
181Q8—22no—
182Q8—22no—
183D4—22no—
184D4—22no—
185D4—2684no—
186C2×C4—22no—
187C2×C4—2682no—
188C2×C4—2684no—
189C2×C4—672no—
190C2×C4—671no—
191C2×C4—1342no—
192C2×C4—1342no—
193C2×C4—1341no—
194C2×C4—1341no—
195C2×C4D4×D6712yes—
196C2×C4C23×D6711yescharacteristic
197C8—1341no—
198C8—1341no—
199C8#1176162712yes—
200C8#1176162712yes—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—1342no—
205C22—1341no—
206C22C2×D4×D6711yescharacteristic
207C4—22no—
208C4—42no—
209C4—2682no—
210C4—2682no—
211C4—1342no—
212C4—1341no—
213C4C2×#1176162711yescharacteristic
214C4C2×#1176162711yescharacteristic
215C2—82no—
216C2—2681no—
217C2#1686394411yescharacteristic
218C2#1686419312yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#495×C67C2
socleC22×C67C22×D4
hypercenter#495D67
hypocenterC67#3183
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #16863925C2semidirect
2. #16863798C2semidirect
3. #16863750C2semidirect
4. #16863651C2semidirect
6. #16863639C2semidirect
7. #16863508C2semidirect
8. #16863503C2semidirect
9. #16863497C2semidirect
10. C4×#11761611C2semidirect
11. #495×C67C2semidirect
110. C67#3183semidirect
113. #495D67semidirect