Finite Group Databasebeta

← #22570936

Subgroups — #22570936

Order 8576.

370 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 115 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#16863928C212yesmaximal, complemented
2#16863796C211yesmaximal, characteristic, complemented
3#16863779C211yesmaximal, characteristic, complemented
4#16863749C212yesmaximal, complemented
5#16863655C211yesmaximal, characteristic, complemented
6#16863646C211yesmaximal, characteristic
7#16863641C212yesmaximal, complemented
8#16863504C211yesmaximal, characteristic, complemented
9#16863496C212yesmaximal, complemented
10#16863348C211yesmaximal, characteristic, complemented
11#442×C67C211yesmaximal, characteristic, complemented
12C2×Dic268C2211yescharacteristic
13C2×#11761616C2212yes—
14C2×#11761611C2211yescharacteristic
15C2×#11761611C2211yescharacteristic
16#14524036—22no—
17#14524034C2212yes—
18#14524030C2212yes—
19#14524025—22no—
20#14524023C2212yes—
21#14524014C2212yes—
22C4×Dic134C2212yes—
23C4×C22⋊D67—22no—
24#14524002—22no—
25C42×D67C2211yescharacteristic
26#14523976C2212yes—
27#14523975C2211yescharacteristic
28#14523968C2212yes—
29#14523966C2212yes—
30#14523964C2212yes—
31#14523954C2211yescharacteristic
32C8×Dic67C2211yescharacteristic
33C8×Dic67C2211yescharacteristic
34C2×D8×C67C2211yescharacteristic
35C4×D4×C67C2212yes—
36#106×C67C2211yescharacteristic
37#102×C67C2212yes—
38#97×C67C2211yescharacteristic
39C2×Dic134C2312yes—
40C2×C22⋊D67—22no—
41C2×C4×D67D412yes—
42C2×C4×D67C2311yescharacteristic
43Dic268—22no—
44#11761616—24no—
45#11761611—22no—
46#11761611—22no—
47C2×C67⋊C8C2311yescharacteristic
48C2×C67⋊C8C2311yescharacteristic
49C4⋊Dic67—22no—
50C4⋊Dic67—22no—
51C4⋊Dic67C2312yes—
52Dic67⋊C4—22no—
53Dic67⋊C4—22no—
54C22⋊Dic67—22no—
55C22⋊Dic67—22no—
56C22⋊Dic67—22no—
57(C2×D67)⋊C4—22no—
58C4×Dic67—22no—
59C4×Dic67D412yes—
60C4×Dic67C2311yescharacteristic
61C2×D4×C67C2312yes—
62C22×C4×C67—22no—
63D8×C67—22no—
64C2×C8×C67C2311yescharacteristic
65C2×C8×C67C2311yescharacteristic
66C4⋊C4×C67C2312yes—
67C22⋊C4×C67—22no—
68C42×C67C2311yescharacteristic
69C23×C67—22no—
70C22×D67C2×D411yescharacteristic
71D4×C67—22no—
72D4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—44no—
75C2×C4×C67C2×D412yes—
76C2×C4×C67C2411yescharacteristic
77C22⋊D67—44no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67C2×D411yescharacteristic
81C2×Dic67C2×D412yes—
82C4×D67—21no—
83C4×D67—21no—
84C4×D67—42no—
85Dic134—22no—
86Dic134—22no—
87C8×C67—21no—
88C8×C67#4112yes—
89C67⋊C8—21no—
90C67⋊C8#4112yes—
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#495—671no—
114#488—672no—
115#470—671no—
116#469—671no—
117#468—672no—
118#442D6711yescharacteristic, complemented
119#441—671no—
120#440—672no—
121C4×M16—671no—
122C2×Q16—671no—
123C2×SD16—672no—
124C2×D8C2×D6711yescharacteristic
125C2×M16—671no—
126C2×M16—671no—
127#126—1342no—
128#125—672no—
129#124—672no—
130#123—1342no—
131C22⋊Q8—672no—
132#121—672no—
133C4×Q8—672no—
134C4×D4—1342no—
135C4×D4C2×D6712yes—
136#117—1342no—
137C2×C42—671no—
138#107—672no—
139#106—671no—
140#106C2×D6711yescharacteristic
141#103—672no—
142#103—672no—
143#102—672no—
144#102C2×D6712yes—
145#97—671no—
146#97C2×D6711yescharacteristic
147C4×C8—671no—
148C4×C8—671no—
149C2×Q8—672no—
150C2×D4—1342no—
151C2×D4C22×D6712yes—
152C22×C4—22no—
153C22×C4—672no—
154C22×C4—671no—
155Q16—1342no—
156SD16—1344no—
157D8—22no—
158M16—1342no—
159M16—1342no—
160C2×C8—671no—
161C2×C8—671no—
162C2×C8C22×D6711yescharacteristic
163C2×C8C22×D6711yescharacteristic
164C4⋊C4—672no—
165C4⋊C4—1342no—
166C4⋊C4—1342no—
167C4⋊C4—1342no—
168C4⋊C4—1342no—
169C4⋊C4C22×D6712yes—
170C22⋊C4—22no—
171C22⋊C4—1342no—
172C22⋊C4—1342no—
173C22⋊C4—1342no—
174C22⋊C4—1342no—
175C42—672no—
176C42—671no—
177C42—1342no—
178C42C22×D6711yescharacteristic
179C23—22no—
180C23—671no—
181Q8—1342no—
182Q8—1342no—
183D4—22no—
184D4—22no—
185D4—2684no—
186C2×C4—22no—
187C2×C4—44no—
188C2×C4—2682no—
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—21no—
198C8—672no—
199C8—1341no—
200C8#1176162812yes—
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—
213C4#41×D6711yescharacteristic
214C4C2×#1176162811yescharacteristic
215C2—82no—
216C2—2681no—
217C2#1686394311yescharacteristic
218C2#1686419212yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#442×C67C2
socleC22×C67C22×D4
hypercenter#442D67
hypocenterC67#3183
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #16863928C2semidirect
2. #16863796C2semidirect
3. #16863779C2semidirect
4. #16863749C2semidirect
5. #16863655C2semidirect
7. #16863641C2semidirect
8. #16863504C2semidirect
9. #16863496C2semidirect
10. #16863348C2semidirect
11. #442×C67C2semidirect
110. C67#3183semidirect
118. #442D67semidirect