Finite Group Databasebeta

← #22570604

Subgroups — #22570604

Order 8576.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#16863947C211yesmaximal, characteristic
2C4⋊Q8×D67C211yesmaximal, characteristic, complemented
3#16863757C214yesmaximal, complemented
4#16863746C211yesmaximal, characteristic, complemented
5#16863744C211yesmaximal, characteristic, complemented
6#16863743C211yesmaximal, characteristic, complemented
7#16863396C214yesmaximal, complemented
8C4×#11761611C211yesmaximal, characteristic, complemented
9#487×C67C211yesmaximal, characteristic, complemented
10C2×#11761629C2212yes—
11C2×Q8×D67—21no—
12C2×#11761611C2212yes—
13#14524043C2211yescharacteristic
14#14524042—21no—
15#14524042C2212yes—
16#14524041—21no—
17#14524040C2211yescharacteristic
18#14524032C2212yes—
19#14524018—21no—
20#14524018—21no—
21#14524005—21no—
22#14524005—21no—
23C4⋊C4×D67—21no—
24C4⋊C4×D67C2212yes—
25C42×D67C2211yescharacteristic
26#14523967C2214yes—
27#14523965C2214yes—
28#14523963C2214yes—
29#14523953C2212yes—
30C4×C67⋊C8C2211yescharacteristic
31C4⋊Q8×C67C2211yescharacteristic
32C4⋊Q8×C67C2211yescharacteristic
33#103×C67C2214yes—
34C4×C8×C67C2211yescharacteristic
35#11761629—28no—
36C2×Dic134—21no—
37C2×Dic134—22no—
38Q8×D67—44no—
39C2×D268—22no—
40C2×D268—21no—
41C2×D268—21no—
42C2×D268C2312yes—
43C2×C4×D67—22no—
44C2×C4×D67—22no—
45C2×C4×D67D411yescharacteristic
46C2×C4×D67D411yescharacteristic
47C2×C4×D67C2311yescharacteristic
48#11761611—24no—
49C2×C67⋊C8C2312yes—
50C4⋊Dic67—21no—
51C4⋊Dic67C2312yes—
52Dic67⋊C4—22no—
53Dic67⋊C4—22no—
54(C2×D67)⋊C4—22no—
55(C2×D67)⋊C4—22no—
56C4×Dic67D411yescharacteristic
57C4×Dic67D411yescharacteristic
58C4×Dic67C2311yescharacteristic
59C2×Q8×C67—21no—
60C2×Q8×C67C2312yes—
61C2×C8×C67C2312yes—
62C4⋊C4×C67—21no—
63C4⋊C4×C67—21no—
64C4⋊C4×C67—21no—
65C4⋊C4×C67C2312yes—
66C42×C67C2311yescharacteristic
67C22×D67—22no—
68C22×D67C2×D411yescharacteristic
69Q8×C67—24no—
70Q8×C67—42no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2×D411yescharacteristic
75C2×C4×C67C2411yescharacteristic
76C2×Dic67—22no—
77C2×Dic67C2×D411yescharacteristic
78C2×Dic67C2×D411yescharacteristic
79C2×Dic67C2×D411yescharacteristic
80D268—24no—
81D268—44no—
82D268—42no—
83D268—42no—
84C4×D67—22no—
85C4×D67—22no—
86C4×D67—44no—
87C4×D67—44no—
88C4×D67#4114yes—
89Dic134—44no—
90Dic134—42no—
91C8×C67—22no—
92C67⋊C8—22no—
93C22×C67C22×D411yescharacteristic
94C2×D67—21no—
95C2×D67—21no—
96C2×D67—42no—
97C2×D67—42no—
98C2×D67—42no—
99C2×D67#12412yes—
100C4×C67—42no—
101C4×C67—42no—
102C4×C67AGL(1,ℤ/8)12yes—
103C4×C67#13712yes—
104C4×C67C2×#4112yes—
105Dic67—21no—
106Dic67—21no—
107Dic67—42no—
108Dic67#12412yes—
109C2×C67C2×#12411yescharacteristic
110C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C67C2×#13711yescharacteristic
112D67—22no—
113D67—82no—
114#3141—671nomaximal, Sylow
115C67#314111yescharacteristic, complemented, Sylow
116#532—671no—
117C2×C4⋊Q8—671no—
118#489—674no—
119#487—671no—
120#487D6711yescharacteristic, complemented
121#486—671no—
122#486—671no—
123#417—674no—
124C4×M16—671no—
125C2×#41—672no—
126C22×Q8—1341no—
127C2×M16—672no—
128C4⋊Q8—672no—
129C4⋊Q8—671no—
130C4⋊Q8—1341no—
131C4⋊Q8—1341no—
132C4⋊Q8C2×D6711yescharacteristic
133C4⋊Q8C2×D6711yescharacteristic
134#127—671no—
135#124—672no—
136#121—1341no—
137#121—1341no—
138C4×D4—1341no—
139C4×D4—1341no—
140C2×C4⋊C4—672no—
141C2×C4⋊C4—1341no—
142C2×C42—671no—
143#103—674no—
144#103C2×D6714yes—
145#102—674no—
146#102—674no—
147#97—672no—
148C4×C8—671no—
149C4×C8C2×D6711yescharacteristic
150#41—1348no—
151C2×Q8—21no—
152C2×Q8—2684no—
153C2×Q8—1342no—
154C2×Q8—1341no—
155C2×Q8C22×D6712yes—
156C2×D4—672no—
157C2×D4—1342no—
158C2×D4—1341no—
159C2×D4—1341no—
160C22×C4—671no—
161C22×C4—671no—
162C22×C4—671no—
163C22×C4—1342no—
164C22×C4—1342no—
165M16—1344no—
166C2×C8—672no—
167C2×C8C22×D6712yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—672no—
172C4⋊C4—1342no—
173C4⋊C4—1342no—
174C4⋊C4—1341no—
175C4⋊C4C22×D6712yes—
176C22⋊C4—1342no—
177C22⋊C4—1342no—
178C42—671no—
179C42—671no—
180C42—671no—
181C42C22×D6711yescharacteristic
182C23—671no—
183C23—1342no—
184Q8—24no—
185Q8—42no—
186Q8—2684no—
187Q8—2682no—
188D4—2684no—
189D4—2682no—
190D4—2682no—
191D4—1344no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—2684no—
195C2×C4—2684no—
196C2×C4—671no—
197C2×C4—674no—
198C2×C4—671no—
199C2×C4—671no—
200C2×C4—1342no—
201C2×C4—1342no—
202C2×C4—1342no—
203C2×C4D4×D6711yescharacteristic
204C2×C4D4×D6711yescharacteristic
205C2×C4C23×D6711yescharacteristic
206C8—22no—
207C8—1342no—
208C22—2682no—
209C22—2682no—
210C22—2682no—
211C22—672no—
212C22—1341no—
213C22—1341no—
214C22C2×D4×D6711yescharacteristic
215C4—42no—
216C4—42no—
217C4—2682no—
218C4—672no—
219C4—1341no—
220C4—1341no—
221C4#1452407012yes—
222C4#1452407512yes—
223C4#41×D6712yes—
224C2—5362no—
225C2—1342no—
226C2#124×D6711yescharacteristic
227C2C2×#1452407011yescharacteristic
228C2C2×#1452407511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#487×C67C2
socleC22×C67C22×D4
hypercenter#487D67
hypocenterC67#3141
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
2. C4⋊Q8×D67C2semidirect
3. #16863757C2semidirect
4. #16863746C2semidirect
5. #16863744C2semidirect
6. #16863743C2semidirect
7. #16863396C2semidirect
8. C4×#11761611C2semidirect
9. #487×C67C2semidirect
115. C67#3141semidirect
120. #487D67semidirect