Finite Group Databasebeta

← #22564497

Subgroups — #22564497

Order 8576.

512 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 139 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×#14524001C211yesmaximal, characteristic
2C2×C4⋊C4×D67C211yesmaximal, characteristic
3#16863334C211yesmaximal, characteristic, complemented
4#16863328C211yesmaximal, characteristic, complemented
5#16863326C211yesmaximal, characteristic, complemented
6#16863283C212yesmaximal, complemented
7#16863279C212yesmaximal, complemented
8#16863270C212yesmaximal
9#16863258C211yesmaximal, characteristic
10#16863258C212yesmaximal
11#400×C67C211yesmaximal, characteristic, complemented
12C22×C4×D67C2211yescharacteristic
13C22×C4×D67C2211yescharacteristic
14C22×C4×D67C2211yescharacteristic
15#14524001—24no—
16C2×C4⋊Dic67C2212yes—
17C2×C4⋊Dic67C2211yescharacteristic
18C2×C4⋊Dic67C2211yescharacteristic
19C2×Dic67⋊C4C2212yes—
20C2×Dic67⋊C4C2211yescharacteristic
21C2×Dic67⋊C4C2211yescharacteristic
22C4⋊C4×D67—24no—
23C2×(C2×D67)⋊C4C2212yes—
24C2×(C2×D67)⋊C4C2212yes—
25C2×C4×Dic67C2212yes—
26C67⋊#95C2212yes—
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2211yescharacteristic
31C67⋊#95C2212yes—
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2211yescharacteristic
34#95×C67C2212yes—
35#95×C67C2211yescharacteristic
36#95×C67C2212yes—
37C23×D67C2311yescharacteristic
38C22×Dic67C2312yes—
39C22×Dic67C2311yescharacteristic
40C22×Dic67C2312yes—
41C22×Dic67C2311yescharacteristic
42C22×Dic67C2311yescharacteristic
43C2×C4×D67—22no—
44C2×C4×D67—24no—
45C2×C4×D67—22no—
46C2×C4×D67—22no—
47C2×C4×D67—22no—
48C2×C4×D67—24no—
49C2×C4×D67D414yes—
50C4⋊Dic67—22no—
51C4⋊Dic67—22no—
52C4⋊Dic67—24no—
53Dic67⋊C4—22no—
54Dic67⋊C4—24no—
55Dic67⋊C4—22no—
56(C2×D67)⋊C4—24no—
57(C2×D67)⋊C4—24no—
58C4×Dic67—24no—
59C22×C4×C67C2312yes—
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2312yes—
62C22×C4×C67C2311yescharacteristic
63C22×C4×C67C2311yescharacteristic
64C4⋊C4×C67—22no—
65C4⋊C4×C67—22no—
66C23×C67C2411yescharacteristic
67C22×D67—21no—
68C22×D67—21no—
69C22×D67—22no—
70C22×D67C2×D412yes—
71C22×D67#4114yes—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—21no—
75C2×C4×C67—22no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—24no—
81C2×C4×C67—22no—
82C2×C4×C67C2×D412yes—
83C2×C4×C67#4112yes—
84C2×Dic67—21no—
85C2×Dic67—22no—
86C2×Dic67—21no—
87C2×Dic67—22no—
88C2×Dic67—21no—
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—22no—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67C2×D412yes—
96C2×Dic67#4112yes—
97C4×D67—28no—
98C4×D67—28no—
99C4×D67—44no—
100C22×C67C22×D411yescharacteristic
101C22×C67C2×#4112yes—
102C22×C67C2×#4111yescharacteristic
103C22×C67C2×#4111yescharacteristic
104C22×C672−1+412yes—
105C2×D67—22no—
106C2×D67—22no—
107C2×D67—24no—
108C2×D67—24no—
109C2×D67#12314yes—
110C4×C67—22no—
111C4×C67—22no—
112C4×C67—41no—
113C4×C67—42no—
114C4×C67—42no—
115Dic67—22no—
116Dic67—22no—
117Dic67—41no—
118Dic67—42no—
119Dic67—42no—
120C2×C67C2×#12311yescharacteristic
121C2×C67#52511yescharacteristic
122C2×C67#53611yescharacteristic
123C2×C67#55212yes—
124C2×C67#55212yes—
125D67—24no—
126#2484—671nomaximal, Sylow
127C67#248411yescharacteristic, complemented, Sylow
128C2×#117—671no—
129C22×C4⋊C4—671no—
130#400—671no—
131#400D6711yescharacteristic, complemented
132#398—671no—
133#398—671no—
134#387—672no—
135#387—672no—
136#385—672no—
137#381—671no—
138#381—672no—
139C23×C4—671no—
140C23×C4—671no—
141C23×C4—671no—
142#117—1344no—
143C2×C4⋊C4—671no—
144C2×C4⋊C4—671no—
145C2×C4⋊C4—671no—
146C2×C4⋊C4—671no—
147C2×C4⋊C4—672no—
148C2×C4⋊C4—672no—
149C2×C4⋊C4—1344no—
150C2×C4⋊C4C2×D6711yescharacteristic
151C2×C4⋊C4C2×D6711yescharacteristic
152C2×C22⋊C4—672no—
153C2×C22⋊C4—672no—
154C2×C42—672no—
155#95—672no—
156#95—672no—
157#95—672no—
158#95—672no—
159#95—671no—
160#95—672no—
161#95C2×D6712yes—
162#95C2×D6711yescharacteristic
163#95C2×D6712yes—
164C24—671no—
165C22×C4—671no—
166C22×C4—671no—
167C22×C4—674no—
168C22×C4—672no—
169C22×C4—671no—
170C22×C4—672no—
171C22×C4—1342no—
172C22×C4—1344no—
173C22×C4—1342no—
174C22×C4—1342no—
175C22×C4—1344no—
176C22×C4—1342no—
177C22×C4C22×D6712yes—
178C22×C4C22×D6712yes—
179C22×C4C22×D6711yescharacteristic
180C22×C4C22×D6711yescharacteristic
181C22×C4C22×D6711yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—1342no—
185C4⋊C4—1342no—
186C4⋊C4—1344no—
187C4⋊C4—1342no—
188C4⋊C4—1342no—
189C4⋊C4—1344no—
190C22⋊C4—1344no—
191C22⋊C4—1344no—
192C42—1344no—
193C23—672no—
194C23—674no—
195C23—1341no—
196C23—1341no—
197C23—1342no—
198C23C23×D6711yescharacteristic
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—2684no—
210C2×C4—672no—
211C2×C4—672no—
212C2×C4—1348no—
213C2×C4—1348no—
214C2×C4—1342no—
215C2×C4—1341no—
216C2×C4—1342no—
217C2×C4—1342no—
218C2×C4—1342no—
219C2×C4—1342no—
220C2×C4—1342no—
221C2×C4—1342no—
222C2×C4—1341no—
223C2×C4—1342no—
224C2×C4—1341no—
225C2×C4D4×D6712yes—
226C2×C4#1176162912yes—
227C22—674no—
228C22—1344no—
229C22—1344no—
230C22—1342no—
231C22—1342no—
232C22C2×D4×D6711yescharacteristic
233C22#41×D6711yescharacteristic
234C22#41×D6712yes—
235C22C2×#1176162911yescharacteristic
236C22#1452409012yes—
237C4—22no—
238C4—22no—
239C4—42no—
240C4—41no—
241C4—42no—
242C4—2682no—
243C4—2681no—
244C4—2682no—
245C4—1342no—
246C4—1342no—
247C2—1344no—
248C2#123×D6711yescharacteristic
249C2#1686390911yescharacteristic
250C2#1686396611yescharacteristic
251C2#1686408212yes—
252C2#1686408412yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#400×C67C2
hypercenter#400D67
hypocenterC67#2484
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #16863334C2semidirect
4. #16863328C2semidirect
5. #16863326C2semidirect
6. #16863283C2semidirect
7. #16863279C2semidirect
11. #400×C67C2semidirect
127. C67#2484semidirect
131. #400D67semidirect