Finite Group Databasebeta

← #22565786

Subgroups — #22565786

Order 8576.

492 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 145 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×Dic134C211yesmaximal, characteristic, complemented
2#16863326C211yesmaximal, characteristic, complemented
3#16863321C211yesmaximal, characteristic
4#16863316C211yesmaximal, characteristic, complemented
5#16863308C211yesmaximal, characteristic
6#16863284C212yesmaximal
7#16863280C212yesmaximal, complemented
8#16863277C212yesmaximal, complemented
9#16863269C212yesmaximal
10#16863264C211yesmaximal, characteristic, complemented
11C2×#117×C67C211yesmaximal, characteristic
12C22×Dic134C2211yescharacteristic
13C4×Dic134—24no—
14C2×C4⋊Dic67C2211yescharacteristic
15C2×Dic67⋊C4C2212yes—
16C2×Dic67⋊C4C2212yes—
17C2×Dic67⋊C4C2212yes—
18C2×C22⋊Dic67C2212yes—
19C2×C22⋊Dic67C2212yes—
20C2×C4×Dic67C2212yes—
21C67⋊#95C2212yes—
22C67⋊#95C2212yes—
23C67⋊#95C2211yescharacteristic
24C67⋊#95C2211yescharacteristic
25C67⋊#95C2212yes—
26C67⋊#95C2212yes—
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C23×C4×C67C2211yescharacteristic
30#117×C67—24no—
31C2×C4⋊C4×C67C2211yescharacteristic
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C22⋊C4×C67C2211yescharacteristic
34C2×C22⋊C4×C67C2211yescharacteristic
35C2×C42×C67C2211yescharacteristic
36C2×C42×C67C2211yescharacteristic
37C2×Dic134—22no—
38C2×Dic134—24no—
39C22×Dic67C2312yes—
40C22×Dic67C2312yes—
41C22×Dic67C2312yes—
42C22×Dic67C2312yes—
43C4⋊Dic67—22no—
44Dic67⋊C4—24no—
45Dic67⋊C4—24no—
46Dic67⋊C4—24no—
47C22⋊Dic67—24no—
48C22⋊Dic67—24no—
49C4×Dic67—24no—
50C24×C67C2311yescharacteristic
51C22×C4×C67—22no—
52C22×C4×C67—22no—
53C22×C4×C67—22no—
54C22×C4×C67C2311yescharacteristic
55C22×C4×C67C2311yescharacteristic
56C22×C4×C67C2311yescharacteristic
57C22×C4×C67C2311yescharacteristic
58C22×C4×C67C2311yescharacteristic
59C22×C4×C67C2311yescharacteristic
60C4⋊C4×C67—22no—
61C4⋊C4×C67—22no—
62C22⋊C4×C67—22no—
63C22⋊C4×C67—22no—
64C42×C67—22no—
65C42×C67D414yes—
66C23×C67—21no—
67C23×C67—21no—
68C23×C67—22no—
69C23×C67—22no—
70C23×C67#4112yes—
71C23×C67C2411yescharacteristic
72C2×C4×C67—28no—
73C2×C4×C67—21no—
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67C2×D412yes—
84C2×C4×C67C2×D412yes—
85C2×C4×C67C2×D412yes—
86C2×C4×C67#4112yes—
87C2×Dic67—22no—
88C2×Dic67—22no—
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—22no—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67—22no—
96C2×Dic67—22no—
97C2×Dic67—22no—
98C2×Dic67#4114yes—
99Dic134—28no—
100C22×C67—22no—
101C22×C67—24no—
102C22×C67—41no—
103C22×C67—41no—
104C22×C67—42no—
105C22×C67C22×D411yescharacteristic
106C22×C67C2×#4112yes—
107C22×C67C2×#4111yescharacteristic
108C22×C67C2×#4111yescharacteristic
109C22×C672−1+412yes—
110C4×C67—22no—
111C4×C67—22no—
112C4×C67—22no—
113C4×C67—22no—
114C4×C67—41no—
115C4×C67—41no—
116Dic67—24no—
117Dic67—42no—
118Dic67—42no—
119Dic67—42no—
120C2×C67—42no—
121C2×C67#52511yescharacteristic
122C2×C67#52511yescharacteristic
123C2×C67#53611yescharacteristic
124C2×C67#55212yes—
125C2×C67#55612yes—
126#2599—671nomaximal, Sylow
127C67#259911yescharacteristic, complemented, Sylow
128C2×C4×Q8—671no—
129C2×#117D6711yescharacteristic
130#398—671no—
131#397—671no—
132#395—671no—
133#393—671no—
134#388—672no—
135#387—672no—
136#387—672no—
137#385—672no—
138#383—671no—
139C22×Q8—671no—
140C23×C4C2×D6711yescharacteristic
141C4×Q8—1344no—
142#117—24no—
143C2×C4⋊C4—672no—
144C2×C4⋊C4—671no—
145C2×C4⋊C4—672no—
146C2×C4⋊C4—672no—
147C2×C4⋊C4C2×D6711yescharacteristic
148C2×C4⋊C4C2×D6711yescharacteristic
149C2×C22⋊C4—672no—
150C2×C22⋊C4—672no—
151C2×C22⋊C4C2×D6711yescharacteristic
152C2×C22⋊C4C2×D6711yescharacteristic
153C2×C42—672no—
154C2×C42C2×D6711yescharacteristic
155C2×C42C2×D6711yescharacteristic
156#95—672no—
157#95—672no—
158#95—671no—
159#95—671no—
160#95—672no—
161#95—672no—
162#95—672no—
163#95—672no—
164C24C22×D6711yescharacteristic
165C2×Q8—1344no—
166C2×Q8—1342no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—672no—
171C22×C4—672no—
172C22×C4—672no—
173C22×C4—672no—
174C22×C4C22×D6711yescharacteristic
175C22×C4C22×D6711yescharacteristic
176C22×C4C22×D6711yescharacteristic
177C22×C4C22×D6711yescharacteristic
178C22×C4C22×D6711yescharacteristic
179C22×C4C22×D6711yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—1344no—
183C4⋊C4—1344no—
184C4⋊C4—1342no—
185C4⋊C4—1344no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—1344no—
189C22⋊C4—1344no—
190C42—22no—
191C42—1344no—
192C42C22⋊D6714yes—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—22no—
197C23#1176162812yes—
198C23C23×D6711yescharacteristic
199Q8—1348no—
200C2×C4—28no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—674no—
212C2×C4—1342no—
213C2×C4—1342no—
214C2×C4—1342no—
215C2×C4—1342no—
216C2×C4—1342no—
217C2×C4—1342no—
218C2×C4—1342no—
219C2×C4—1342no—
220C2×C4—1342no—
221C2×C4—1342no—
222C2×C4—1342no—
223C2×C4C2×C22⋊D6712yes—
224C2×C4C2×C22⋊D6712yes—
225C2×C4C2×C22⋊D6712yes—
226C2×C4#1176162812yes—
227C22—24no—
228C22—22no—
229C22—42no—
230C22—41no—
231C22—41no—
232C22C22×C22⋊D6711yescharacteristic
233C22#41×D6712yes—
234C22C2×#1176162811yescharacteristic
235C22C2×#1176162811yescharacteristic
236C22#1452409012yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—22no—
241C4—41no—
242C4—41no—
243C4—2682no—
244C4—2682no—
245C4—2682no—
246C4—1344no—
247C2—42no—
248C2#1686390311yescharacteristic
249C2#1686390711yescharacteristic
250C2#1686396711yescharacteristic
251C2#1686407912yes—
252C2#1686411412yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×#117×C67C2
hypercenterC2×#117D67
hypocenterC67#2599
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic134C2semidirect
2. #16863326C2semidirect
4. #16863316C2semidirect
7. #16863280C2semidirect
8. #16863277C2semidirect
10. #16863264C2semidirect
127. C67#2599semidirect