Finite Group Databasebeta

← #22571326

Subgroups — #22571326

Order 8576.

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

normal subgroups only

This group has 404 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#16863944C211yesmaximal, characteristic
2C4⋊Q8×D67C211yesmaximal, characteristic, complemented
3#16863801C211yesmaximal, characteristic, complemented
4#16863799C211yesmaximal, characteristic, complemented
5#16863742C211yesmaximal, characteristic, complemented
6#16863708C214yesmaximal, complemented
7#16863674C214yesmaximal, complemented
8#16863447C211yesmaximal, characteristic, complemented
9#486×C67C211yesmaximal, characteristic, complemented
10C2×#11761627—21no—
11C2×Q8×D67—21no—
12#14524043C2211yescharacteristic
13#14524043C2211yescharacteristic
14#14524042—21no—
15#14524042C2212yes—
16#14524041—21no—
17#14524030—21no—
18#14524014—21no—
19#14524014C2212yes—
20D4×Dic67—21no—
21D4×Dic67C2212yes—
22C4⋊C4×D67—21no—
23C4⋊C4×D67C2212yes—
24C42×D67C2211yescharacteristic
25#14523977C2214yes—
26#14523976C2214yes—
27Dic67⋊C8C2212yes—
28#14523964C2214yes—
29(C2×D67)⋊C8C2212yes—
30C4×C67⋊C8C2211yescharacteristic
31C4⋊Q8×C67C2211yescharacteristic
32#127×C67C2211yescharacteristic
33#102×C67C2214yes—
34C4×C8×C67C2211yescharacteristic
35#11761627—44no—
36C2×Dic134—21no—
37C2×Dic134—21no—
38C2×Dic134—22no—
39Q8×D67—44no—
40C2×C22⋊D67—22no—
41C22×Dic67—22no—
42C2×C4×D67—22no—
43C2×C4×D67D411yescharacteristic
44C2×C4×D67D411yescharacteristic
45C2×C4×D67C2311yescharacteristic
46C2×C67⋊C8C2312yes—
47C4⋊Dic67—21no—
48C4⋊Dic67—21no—
49C4⋊Dic67C2312yes—
50C4⋊Dic67C2312yes—
51Dic67⋊C4—22no—
52Dic67⋊C4—22no—
53C22⋊Dic67—22no—
54C22⋊Dic67—22no—
55C4×Dic67D411yescharacteristic
56C4×Dic67D411yescharacteristic
57C4×Dic67C2311yescharacteristic
58C2×Q8×C67—21no—
59C2×D4×C67—21no—
60C2×D4×C67—21no—
61C2×D4×C67C2312yes—
62C2×C8×C67C2312yes—
63C4⋊C4×C67—21no—
64C4⋊C4×C67C2312yes—
65C42×C67C2311yescharacteristic
66C23×C67—22no—
67C22×D67C2×D411yescharacteristic
68Q8×C67—42no—
69D4×C67—24no—
70D4×C67—42no—
71D4×C67—42no—
72C2×C4×C67—22no—
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2×D411yescharacteristic
75C2×C4×C67C2411yescharacteristic
76C22⋊D67—44no—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—44no—
80C2×Dic67C2×D411yescharacteristic
81C2×Dic67C2×D411yescharacteristic
82C2×Dic67C2×D411yescharacteristic
83C4×D67—22no—
84C4×D67—22no—
85C4×D67—44no—
86C4×D67D814yes—
87Dic134—42no—
88Dic134—42no—
89Dic134—44no—
90C8×C67—22no—
91C67⋊C8—22no—
92C22×C67—42no—
93C22×C67—42no—
94C22×C67—42no—
95C22×C67C22×D411yescharacteristic
96C2×D67—21no—
97C2×D67—21no—
98C2×D67C2×D812yes—
99C4×C67—42no—
100C4×C67C2×D812yes—
101C4×C67#13712yes—
102C4×C672−1+412yes—
103Dic67—21no—
104Dic67—21no—
105Dic67—42no—
106Dic67—42no—
107Dic67C2×D812yes—
108C2×C67—82no—
109C2×C67#53611yescharacteristic
110C2×C67C22×D811yescharacteristic
111C2×C67C2×#13711yescharacteristic
112D67—22no—
113#3226—671nomaximal, Sylow
114C67#322611yescharacteristic, complemented, Sylow
115#532—671no—
116C2×C4⋊Q8—671no—
117#500—671no—
118#500—671no—
119#486—671no—
120#486D6711yescharacteristic, complemented
121#480—674no—
122#474—674no—
123#431—671no—
124C2×#41—1341no—
125C22×Q8—1341no—
126C4⋊Q8—672no—
127C4⋊Q8—671no—
128C4⋊Q8—671no—
129C4⋊Q8—1341no—
130C4⋊Q8—1341no—
131C4⋊Q8C2×D6711yescharacteristic
132#127C2×D6711yescharacteristic
133#124—1341no—
134#121—672no—
135#121—1341no—
136C4×D4—672no—
137C4×D4—1341no—
138C2×C4⋊C4—672no—
139C2×C4⋊C4—1341no—
140C2×C42—671no—
141#107—674no—
142#107—674no—
143C4⋊C8—672no—
144#102—674no—
145#102C2×D6714yes—
146C22⋊C8—672no—
147C4×C8—671no—
148C4×C8C2×D6711yescharacteristic
149#41—2684no—
150C2×Q8—21no—
151C2×Q8—2684no—
152C2×Q8—1342no—
153C2×Q8—1341no—
154C2×Q8—1341no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—1342no—
158C2×D4C22×D6712yes—
159C22×C4—671no—
160C22×C4—671no—
161C22×C4—671no—
162C22×C4—1342no—
163C22×C4—1342no—
164C2×C8—672no—
165C2×C8C22×D6712yes—
166C4⋊C4—21no—
167C4⋊C4—672no—
168C4⋊C4—672no—
169C4⋊C4—1342no—
170C4⋊C4—1342no—
171C4⋊C4—1341no—
172C4⋊C4—1341no—
173C4⋊C4C22×D6712yes—
174C22⋊C4—1342no—
175C22⋊C4—1342no—
176C42—671no—
177C42—671no—
178C42—671no—
179C42C22×D6711yescharacteristic
180C23—22no—
181C23—671no—
182Q8—42no—
183Q8—2684no—
184Q8—2682no—
185Q8—2682no—
186D4—24no—
187D4—42no—
188D4—42no—
189D4—2684no—
190C2×C4—22no—
191C2×C4—2684no—
192C2×C4—2684no—
193C2×C4—671no—
194C2×C4—674no—
195C2×C4—671no—
196C2×C4—671no—
197C2×C4—1342no—
198C2×C4—1342no—
199C2×C4—1342no—
200C2×C4—1342no—
201C2×C4D4×D6711yescharacteristic
202C2×C4D4×D6711yescharacteristic
203C2×C4C23×D6711yescharacteristic
204C8—22no—
205C8—1342no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—672no—
210C22—1341no—
211C22—1341no—
212C22C2×D4×D6711yescharacteristic
213C4—42no—
214C4—2682no—
215C4—2682no—
216C4—672no—
217C4—1341no—
218C4—1341no—
219C4D8×D6712yes—
220C4#1452407212yes—
221C4#1452409012yes—
222C2—82no—
223C2—1342no—
224C2#1686396611yescharacteristic
225C2C2×D8×D6711yescharacteristic
226C2C2×#1452407211yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#486×C67C2
socleC22×C67C22×D4
hypercenter#486D67
hypocenterC67#3226
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D67C2semidirect
3. #16863801C2semidirect
4. #16863799C2semidirect
5. #16863742C2semidirect
6. #16863708C2semidirect
7. #16863674C2semidirect
8. #16863447C2semidirect
9. #486×C67C2semidirect
114. C67#3226semidirect
120. #486D67semidirect