Finite Group Databasebeta

← #22564520

Subgroups — #22564520

Order 8576.

584 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×C4×C22⋊D67C211yesmaximal, characteristic
2#16863334C212yesmaximal, complemented
3#16863331C212yesmaximal, complemented
4#16863292C212yesmaximal, complemented
5#16863275C212yesmaximal
6#16863269C212yesmaximal, complemented
7#16863256C211yesmaximal, characteristic, complemented
8#16863253C212yesmaximal, complemented
9C22×C4⋊C4×C67C211yesmaximal, characteristic, complemented
10C22×C22⋊D67C2211yescharacteristic
11C22×C4×D67C2211yescharacteristic
12C4×C22⋊D67—24no—
13C2×C4⋊Dic67C2212yes—
14C2×Dic67⋊C4C2211yescharacteristic
15C2×Dic67⋊C4C2214yes—
16C2×C22⋊Dic67C2212yes—
17C2×C22⋊Dic67C2211yescharacteristic
18C2×(C2×D67)⋊C4C2211yescharacteristic
19C2×(C2×D67)⋊C4C2214yes—
20C2×C4×Dic67C2211yescharacteristic
21C67⋊#95C2214yes—
22C67⋊#95C2212yes—
23C67⋊#95C2214yes—
24C23×C4×C67C2211yescharacteristic
25C23×C4×C67C2212yes—
26C2×C4⋊C4×C67—24no—
27C2×C4⋊C4×C67C2212yes—
28C2×C4⋊C4×C67C2212yes—
29C23×D67C2311yescharacteristic
30C2×C22⋊D67—22no—
31C2×C22⋊D67—22no—
32C2×C22⋊D67Q814yes—
33C22×Dic67C2311yescharacteristic
34C22×Dic67C2314yes—
35C22×Dic67C2311yescharacteristic
36C22×Dic67C2311yescharacteristic
37C2×C4×D67—22no—
38C2×C4×D67—24no—
39C4⋊Dic67—24no—
40Dic67⋊C4—22no—
41Dic67⋊C4—44no—
42C22⋊Dic67—22no—
43C22⋊Dic67—24no—
44(C2×D67)⋊C4—22no—
45(C2×D67)⋊C4—44no—
46C4×Dic67—22no—
47C24×C67C2311yescharacteristic
48C22×C4×C67—24no—
49C22×C4×C67—24no—
50C22×C4×C67—44no—
51C22×C4×C67D414yes—
52C22×C4×C67C2314yes—
53C22×C4×C67C2311yescharacteristic
54C22×C4×C67C2311yescharacteristic
55C4⋊C4×C67—24no—
56C4⋊C4×C67—24no—
57C23×C67—22no—
58C23×C67—22no—
59C23×C67C2×D412yes—
60C23×C67C2×Q812yes—
61C23×C67#4112yes—
62C23×C67C2411yescharacteristic
63C22×D67—21no—
64C22×D67—22no—
65C22×D67—21no—
66C22×D67—22no—
67C22×D67C2×Q812yes—
68C2×C4×C67—22no—
69C2×C4×C67—28no—
70C2×C4×C67—24no—
71C2×C4×C67—24no—
72C2×C4×C67—24no—
73C2×C4×C67—48no—
74C2×C4×C67C2×D412yes—
75C2×C4×C67C2×D412yes—
76C2×C4×C67#4114yes—
77C22⋊D67—28no—
78C2×Dic67—21no—
79C2×Dic67—21no—
80C2×Dic67—21no—
81C2×Dic67—21no—
82C2×Dic67—24no—
83C2×Dic67—24no—
84C2×Dic67—22no—
85C2×Dic67—24no—
86C2×Dic67—22no—
87C2×Dic67C2×Q812yes—
88C4×D67—44no—
89C22×C67—22no—
90C22×C67—22no—
91C22×C67—24no—
92C22×C67—24no—
93C22×C67C22⋊Q814yes—
94C22×C67C22×D411yescharacteristic
95C22×C67C22×Q811yescharacteristic
96C22×C67C2×#4112yes—
97C22×C67C2×#4111yescharacteristic
98C22×C67H(5,𝔽2)12yes—
99C2×D67—22no—
100C2×D67—24no—
101C2×D67—41no—
102C2×D67—41no—
103C2×D67—42no—
104C4×C67—22no—
105C4×C67—44no—
106C4×C67#12314yes—
107Dic67—22no—
108Dic67—41no—
109Dic67—41no—
110Dic67—44no—
111C2×C67—24no—
112C2×C67C2×C22⋊Q811yescharacteristic
113C2×C67C2×#12311yescharacteristic
114C2×C67#53511yescharacteristic
115C2×C67#55312yes—
116C2×C67#55412yes—
117D67—42no—
118#2487—671nomaximal, Sylow
119C67#248711yescharacteristic, complemented, Sylow
120C2×C4×D4—671no—
121C22×C4⋊C4D6711yescharacteristic, complemented
122#400—672no—
123#399—672no—
124#389—672no—
125#386—672no—
126#385—672no—
127#380—671no—
128#380—672no—
129C22×D4—671no—
130C23×C4—671no—
131C23×C4C2×D6711yescharacteristic
132C23×C4C2×D6712yes—
133C4×D4—1344no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—674no—
136C2×C4⋊C4—671no—
137C2×C4⋊C4—672no—
138C2×C4⋊C4C2×D6712yes—
139C2×C4⋊C4C2×D6712yes—
140C2×C22⋊C4—674no—
141C2×C22⋊C4—671no—
142C2×C22⋊C4—672no—
143C2×C22⋊C4—671no—
144C2×C42—671no—
145#95—674no—
146#95—672no—
147#95—674no—
148C24—671no—
149C24C22×D6711yescharacteristic
150C2×D4—674no—
151C2×D4—1342no—
152C2×D4—1342no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—44no—
156C22×C4—671no—
157C22×C4—671no—
158C22×C4—674no—
159C22×C4—671no—
160C22×C4—1342no—
161C22×C4—1344no—
162C22×C4C22⋊D6714yes—
163C22×C4C22×D6714yes—
164C22×C4C22×D6711yescharacteristic
165C22×C4C22×D6711yescharacteristic
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—2684no—
169C4⋊C4—1342no—
170C4⋊C4—1344no—
171C22⋊C4—2684no—
172C22⋊C4—1342no—
173C22⋊C4—1344no—
174C22⋊C4—1342no—
175C42—1342no—
176C23—22no—
177C23—22no—
178C23—672no—
179C23—1341no—
180C23—1342no—
181C23—1341no—
182C23—1342no—
183C23C2×C22⋊D6712yes—
184C23Q8×D6712yes—
185C23#1176162912yes—
186C23C23×D6711yescharacteristic
187D4—1348no—
188C2×C4—22no—
189C2×C4—28no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—48no—
194C2×C4—2684no—
195C2×C4—672no—
196C2×C4—1341no—
197C2×C4—1341no—
198C2×C4—1344no—
199C2×C4—1342no—
200C2×C4—1344no—
201C2×C4—1344no—
202C2×C4—1342no—
203C2×C4—1341no—
204C2×C4—1341no—
205C2×C4C2×C22⋊D6712yes—
206C2×C4C2×C22⋊D6712yes—
207C2×C4#1176162814yes—
208C22—24no—
209C22—24no—
210C22—22no—
211C22—22no—
212C22—2682no—
213C22—2681no—
214C22—2681no—
215C22—1344no—
216C22—1342no—
217C22#1452401914yes—
218C22C22×C22⋊D6711yescharacteristic
219C22C2×Q8×D6711yescharacteristic
220C22C2×#1176162812yes—
221C22C2×#1176162911yescharacteristic
222C22#1452408812yes—
223C4—22no—
224C4—44no—
225C4—2684no—
226C4—2681no—
227C4—2681no—
228C4—1342no—
229C4#1452402614yes—
230C2—24no—
231C2—2682no—
232C2C2×#1452401911yescharacteristic
233C2C2×#1452402611yescharacteristic
234C2#1686396311yescharacteristic
235C2#1686409412yes—
236C2#1686410112yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC22×C4⋊C4×C67C2
hypercenterC22×C4⋊C4D67
hypocenterC67#2487
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #16863334C2semidirect
3. #16863331C2semidirect
4. #16863292C2semidirect
6. #16863269C2semidirect
7. #16863256C2semidirect
8. #16863253C2semidirect
9. C22×C4⋊C4×C67C2semidirect
119. C67#2487semidirect
121. C22×C4⋊C4D67semidirect