Finite Group Databasebeta

← #22570592

Subgroups — #22570592

Order 8576.

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

normal subgroups only

This group has 400 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#16863952C211yesmaximal, characteristic, complemented
2#124×D67C211yesmaximal, characteristic
3#16863757C212yesmaximal, complemented
4#16863754C212yesmaximal, complemented
5#16863753C211yesmaximal, characteristic, complemented
6#16863751C211yesmaximal, characteristic, complemented
7#16863750C211yesmaximal, characteristic, complemented
8#16863403C212yesmaximal, complemented
9#16863399C212yesmaximal
10C4×#11761611C211yesmaximal, characteristic, complemented
11#488×C67C211yesmaximal, characteristic, complemented
12C2×Q8×D67C2211yescharacteristic
13C2×D4×D67C2211yescharacteristic
14C2×#11761611C2212yes—
15#14524042C2212yescomplemented
16#14524035C2211yescharacteristic, complemented
17#14524032C2211yescharacteristic
18#14524031C2211yescharacteristic
19#14524030C2211yescharacteristic
20#14524029—22no—
21#14524021—22no—
22#14524008—22no—
23#14524001C2212yes—
24C22⋊C4×D67—22no—
25C42×D67C2211yescharacteristic
26#14523968C2212yescomplemented
27#14523967C2212yescomplemented
28#14523966C2212yescomplemented
29#14523965C2212yes—
30#14523964C2212yes—
31#14523963C2212yes—
32#14523953C2212yescomplemented
33C4×C67⋊C8C2211yescharacteristic, complemented
34#125×C67C2211yescharacteristic, complemented
35#124×C67C2211yescharacteristic
36#103×C67C2212yescomplemented
37#102×C67C2212yes—
38C4×C8×C67C2211yescharacteristic, complemented
39C23×D67—21no—
40C2×Dic134—22no—
41C2×Dic134—21no—
42C2×Dic134C2311yescharacteristic
43Q8×D67—24no—
44C2×D268C2311yescharacteristic
45C2×C22⋊D67—21no—
46D4×D67—24no—
47C2×C4×D67—21no—
48C2×C4×D67D412yes—
49C2×C4×D67C2311yescharacteristic
50#11761611—24no—
51C2×C67⋊C8C2312yes—
52C4⋊Dic67C2312yes—
53Dic67⋊C4—22no—
54Dic67⋊C4—22no—
55C22⋊Dic67—22no—
56(C2×D67)⋊C4—22no—
57(C2×D67)⋊C4—22no—
58(C2×D67)⋊C4—22no—
59C4×Dic67—22no—
60C4×Dic67D412yes—
61C4×Dic67C2311yescharacteristic
62C2×Q8×C67C2311yescharacteristic
63C2×D4×C67C2311yescharacteristic
64C2×C8×C67C2312yes—
65C4⋊C4×C67—22no—
66C4⋊C4×C67C2312yes—
67C22⋊C4×C67—22no—
68C42×C67C2311yescharacteristic
69C23×C67—21no—
70C22×D67—21no—
71C22×D67—44no—
72C22×D67—42no—
73C22×D67C2×D411yescharacteristic
74Q8×C67—22no—
75D4×C67—22no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67C2×D412yes—
79C2×C4×C67C2411yescharacteristic
80C22⋊D67—42no—
81C2×Dic67—22no—
82C2×Dic67—21no—
83C2×Dic67C2×D412yes—
84C2×Dic67C2×D411yescharacteristic
85D268—22no—
86C4×D67—42no—
87C4×D67—42no—
88C4×D67#4114yes—
89Dic134—22no—
90Dic134—44no—
91Dic134—42no—
92C8×C67—22no—
93C67⋊C8—22no—
94C22×C67—41no—
95C22×C67—42no—
96C22×C67C22×D411yescharacteristic
97C2×D67—22no—
98C2×D67—41no—
99C2×D67—42no—
100C2×D67—84no—
101C2×D67#12412yes—
102C4×C67—22no—
103C4×C67—41no—
104C4×C67—42no—
105C4×C67C2×#4112yes—
106Dic67—22no—
107Dic67—41no—
108Dic67—42no—
109Dic67#12412yes—
110C2×C67—81no—
111C2×C67C2×#12411yescharacteristic
112C2×C67#57512yes—
113D67—22no—
114D67—81no—
115#3140—671nomaximal, Sylow
116C67#314011yescharacteristic, complemented, Sylow
117#533—671no—
118C2×#124—671no—
119#489—672no—
120#489—672no—
121#488—671no—
122#488—671no—
123#488—671no—
124#488D6711yescharacteristic, complemented
125#419—672no—
126#418—672no—
127C4×M16—671no—
128C22×Q8—671no—
129C22×D4—671no—
130C2×M16—672no—
131C4⋊Q8—672no—
132#125—671no—
133#125C2×D6711yescharacteristic, complemented
134#124—671no—
135#124—671no—
136#124—671no—
137#124—1342no—
138#124C2×D6711yescharacteristic
139C22⋊Q8—1342no—
140C4×Q8—1342no—
141#117—672no—
142C2×C22⋊C4—1342no—
143C2×C42—671no—
144#103—672no—
145#103—672no—
146#103—672no—
147#103C2×D6712yescomplemented
148#102—672no—
149#102—672no—
150#102—672no—
151#102C2×D6712yes—
152#97—672no—
153C4×C8—671no—
154C4×C8C2×D6711yescharacteristic, complemented
155C24—1341no—
156C2×Q8—671no—
157C2×Q8—1342no—
158C2×Q8—1341no—
159C2×Q8—1344no—
160C2×Q8C22×D6711yescharacteristic
161C2×D4—671no—
162C2×D4—1344no—
163C2×D4—1341no—
164C2×D4C22×D6711yescharacteristic
165C22×C4—672no—
166C22×C4—671no—
167C22×C4—1341no—
168M16—1344no—
169C2×C8—672no—
170C2×C8C22×D6712yes—
171C4⋊C4—22no—
172C4⋊C4—672no—
173C4⋊C4—1342no—
174C4⋊C4—1342no—
175C4⋊C4C22×D6712yes—
176C22⋊C4—22no—
177C22⋊C4—1342no—
178C22⋊C4—1342no—
179C22⋊C4—1342no—
180C22⋊C4—1342no—
181C42—672no—
182C42—671no—
183C42—1342no—
184C42C22×D6711yescharacteristic
185C23—21no—
186C23—2682no—
187C23—2684no—
188C23—671no—
189C23—1341no—
190Q8—22no—
191Q8—2682no—
192Q8—2684no—
193Q8—1342no—
194D4—22no—
195D4—2682no—
196D4—1342no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—2682no—
200C2×C4—2682no—
201C2×C4—671no—
202C2×C4—674no—
203C2×C4—672no—
204C2×C4—1342no—
205C2×C4—1341no—
206C2×C4D4×D6712yes—
207C2×C4C23×D6711yescharacteristic
208C8—22no—
209C8—1342no—
210C22—41no—
211C22—42no—
212C22—2681no—
213C22—2682no—
214C22—5364no—
215C22—672no—
216C22—1342no—
217C22C2×D4×D6711yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—2681no—
222C4—2682no—
223C4—672no—
224C4—1342no—
225C4#41×D6712yes—
226C2—81no—
227C2—5361no—
228C2—1342no—
229C2#124×D6711yescharacteristic
230C2#1686419312yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#488×C67C2
socleC22×C67C22×D4
hypercenter#488D67
hypocenterC67#3140
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #16863952C2semidirect
3. #16863757C2semidirect
4. #16863754C2semidirect
5. #16863753C2semidirect
6. #16863751C2semidirect
7. #16863750C2semidirect
8. #16863403C2semidirect
10. C4×#11761611C2semidirect
11. #488×C67C2semidirect
15. #14524042C22semidirect
16. #14524035C22semidirect
26. #14523968C22semidirect
27. #14523967C22semidirect
28. #14523966C22semidirect
32. #14523953C22semidirect
33. C4×C67⋊C8C22semidirect
34. #125×C67C22semidirect
36. #103×C67C22semidirect
38. C4×C8×C67C22semidirect
116. C67#3140semidirect
124. #488D67semidirect
133. #125C2×D67semidirect
147. #103C2×D67semidirect
154. C4×C8C2×D67semidirect