Finite Group Databasebeta

← #22571363

Subgroups — #22571363

Order 8576.

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

normal subgroups only

This group has 326 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#16863951C211yesmaximal, characteristic
2#16863951C211yesmaximal, characteristic
3#16863799C211yesmaximal, characteristic, complemented
4#16863794C211yesmaximal, characteristic, complemented
5#16863746C211yesmaximal, characteristic, complemented
6#16863735C214yesmaximal
7#16863701C214yesmaximal, complemented
8#16863447C211yesmaximal, characteristic
9#487×C67C211yesmaximal, characteristic, complemented
10#14524043C2211yescharacteristic
11#14524043C2211yescharacteristic
12#14524034—21no—
13#14524034C2212yes—
14#14524023—21no—
15#14524023—21no—
16#14524019—21no—
17#14524019C2212yes—
18Q8×Dic67—21no—
19Q8×Dic67C2212yes—
20#14524008—21no—
21#14524008—21no—
22#14524001—21no—
23#14524001C2212yes—
24C42×D67C2211yescharacteristic
25#14523976C2214yes—
26#14523975C2214yes—
27Dic67⋊C8C2212yes—
28#14523967C2214yes—
29(C2×D67)⋊C8C2212yes—
30C4×C67⋊C8C2211yescharacteristic
31C4⋊Q8×C67C2211yescharacteristic
32C4⋊Q8×C67C2211yescharacteristic
33#103×C67C2214yes—
34C4×C8×C67C2211yescharacteristic
35C2×Dic134—21no—
36C2×Dic134—21no—
37C2×C4×D67D411yescharacteristic
38C2×C4×D67D411yescharacteristic
39C2×C4×D67C2311yescharacteristic
40C2×C67⋊C8C2312yes—
41C4⋊Dic67—21no—
42C4⋊Dic67—21no—
43C4⋊Dic67—22no—
44C4⋊Dic67—22no—
45C4⋊Dic67C2312yes—
46C4⋊Dic67C2312yes—
47Dic67⋊C4—22no—
48Dic67⋊C4—22no—
49(C2×D67)⋊C4—22no—
50(C2×D67)⋊C4—22no—
51C4×Dic67—22no—
52C4×Dic67—22no—
53C4×Dic67D411yescharacteristic
54C4×Dic67D411yescharacteristic
55C4×Dic67C2311yescharacteristic
56C2×Q8×C67—21no—
57C2×Q8×C67C2312yes—
58C2×C8×C67C2312yes—
59C4⋊C4×C67—21no—
60C4⋊C4×C67—21no—
61C4⋊C4×C67—21no—
62C4⋊C4×C67C2312yes—
63C42×C67C2311yescharacteristic
64C22×D67C2×D411yescharacteristic
65Q8×C67—24no—
66Q8×C67—42no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67C2×D411yescharacteristic
70C2×C4×C67C2×D411yescharacteristic
71C2×C4×C67C2411yescharacteristic
72C2×Dic67—22no—
73C2×Dic67—22no—
74C2×Dic67C2×D411yescharacteristic
75C2×Dic67C2×D411yescharacteristic
76C2×Dic67C2×D411yescharacteristic
77C4×D67—22no—
78C4×D67—22no—
79C4×D67—22no—
80Dic134—42no—
81Dic134—42no—
82C8×C67—22no—
83C67⋊C8—22no—
84C22×C67C22×D411yescharacteristic
85C2×D67—21no—
86C2×D67—21no—
87C2×D67—21no—
88C4×C67—42no—
89C4×C67—42no—
90C4×C67#13512yes—
91C4×C67#13712yes—
92C4×C672−1+412yes—
93Dic67—21no—
94Dic67—21no—
95Dic67—21no—
96Dic67—42no—
97Dic67—42no—
98C2×C67#53611yescharacteristic
99C2×C67C2×#13511yescharacteristic
100C2×C67C2×#13711yescharacteristic
101D67—41no—
102#3231—671nomaximal, Sylow
103C67#323111yescharacteristic, complemented, Sylow
104#533—671no—
105#533—671no—
106#500—671no—
107#498—671no—
108#487—671no—
109#487D6711yescharacteristic, complemented
110#485—674no—
111#479—674no—
112#431—671no—
113C4⋊Q8—671no—
114C4⋊Q8—671no—
115C4⋊Q8C2×D6711yescharacteristic
116C4⋊Q8C2×D6711yescharacteristic
117#125—672no—
118#125—1341no—
119C22⋊Q8—672no—
120C22⋊Q8—1341no—
121C22⋊Q8—1341no—
122C22⋊Q8—1341no—
123C4×Q8—672no—
124C4×Q8—1341no—
125C4×Q8—1341no—
126C4×Q8—1341no—
127#117—672no—
128#117—1341no—
129C2×C42—671no—
130#107—674no—
131#106—674no—
132C4⋊C8—672no—
133#103—674no—
134#103C2×D6714yes—
135C22⋊C8—672no—
136C4×C8—671no—
137C4×C8C2×D6711yescharacteristic
138C2×Q8—21no—
139C2×Q8—1341no—
140C2×Q8—1341no—
141C2×Q8C22×D6712yes—
142C22×C4—671no—
143C22×C4—671no—
144C22×C4—671no—
145C2×C8—672no—
146C2×C8C22×D6712yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—672no—
151C4⋊C4—672no—
152C4⋊C4—1342no—
153C4⋊C4—1342no—
154C4⋊C4—1342no—
155C4⋊C4—1342no—
156C4⋊C4—1341no—
157C4⋊C4—1341no—
158C4⋊C4C22×D6712yes—
159C22⋊C4—1342no—
160C22⋊C4—1342no—
161C42—671no—
162C42—671no—
163C42—671no—
164C42—1342no—
165C42—1342no—
166C42C22×D6711yescharacteristic
167C23—671no—
168Q8—24no—
169Q8—42no—
170Q8—2682no—
171Q8—2682no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—671no—
175C2×C4—671no—
176C2×C4—671no—
177C2×C4—1342no—
178C2×C4—1342no—
179C2×C4—1342no—
180C2×C4—1342no—
181C2×C4—1342no—
182C2×C4D4×D6711yescharacteristic
183C2×C4D4×D6711yescharacteristic
184C2×C4C23×D6711yescharacteristic
185C8—22no—
186C8—1342no—
187C22—1341no—
188C22—1341no—
189C22—1341no—
190C22C2×D4×D6711yescharacteristic
191C4—42no—
192C4—42no—
193C4—2682no—
194C4—2682no—
195C4—1341no—
196C4—1341no—
197C4—1341no—
198C4#1452406412yes—
199C4#1452407512yes—
200C4#1452409012yes—
201C2—2681no—
202C2#1686396611yescharacteristic
203C2C2×#1452406411yescharacteristic
204C2C2×#1452407511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#487×C67C2
socleC22×C67C22×D4
hypercenter#487D67
hypocenterC67#3231
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
3. #16863799C2semidirect
4. #16863794C2semidirect
5. #16863746C2semidirect
7. #16863701C2semidirect
9. #487×C67C2semidirect
103. C67#3231semidirect
109. #487D67semidirect