Finite Group Databasebeta

← #22564689

Subgroups — #22564689

Order 8576.

512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14524021C211yesmaximal, characteristic
2C2×C4⋊C4×D67C211yesmaximal, characteristic
3#16863336C211yesmaximal, characteristic, complemented
4#16863335C211yesmaximal, characteristic, complemented
5#16863334C211yesmaximal, characteristic, complemented
6#16863300C212yesmaximal, complemented
7#16863260C212yesmaximal, complemented
8#16863258C212yesmaximal
9#16863258C212yesmaximal
10#16863243C211yesmaximal, characteristic
11#400×C67C211yesmaximal, characteristic, complemented
12C22×Dic134C2211yescharacteristic
13C22×C4×D67C2212yes—
14C22×C4×D67C2211yescharacteristic
15#14524021—24no—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×Dic67⋊C4C2211yescharacteristic
18C2×Dic67⋊C4C2212yes—
19C2×Dic67⋊C4C2211yescharacteristic
20C4⋊C4×D67—24no—
21C2×(C2×D67)⋊C4C2212yes—
22C2×(C2×D67)⋊C4C2211yescharacteristic
23C2×(C2×D67)⋊C4C2211yescharacteristic
24C2×C4×Dic67C2212yes—
25C67⋊#95C2211yescharacteristic
26C67⋊#95C2212yes—
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2211yescharacteristic
34#95×C67C2211yescharacteristic
35#95×C67C2212yes—
36#95×C67C2212yes—
37C23×D67C2311yescharacteristic
38C2×Dic134—22no—
39C2×Dic134—24no—
40C22×Dic67C2312yes—
41C22×Dic67C2311yescharacteristic
42C22×Dic67C2311yescharacteristic
43C22×Dic67C2312yes—
44C22×Dic67C2311yescharacteristic
45C2×C4×D67—24no—
46C2×C4×D67—24no—
47C2×C4×D67—28no—
48C2×C4×D67D414yes—
49C4⋊Dic67—22no—
50Dic67⋊C4—22no—
51Dic67⋊C4—24no—
52Dic67⋊C4—22no—
53(C2×D67)⋊C4—22no—
54(C2×D67)⋊C4—22no—
55(C2×D67)⋊C4—24no—
56C4×Dic67—24no—
57C22×C4×C67C2311yescharacteristic
58C22×C4×C67C2311yescharacteristic
59C22×C4×C67C2312yes—
60C22×C4×C67C2312yes—
61C22×C4×C67C2311yescharacteristic
62C4⋊C4×C67—22no—
63C4⋊C4×C67—22no—
64C23×C67C2411yescharacteristic
65C22×D67—22no—
66C22×D67—21no—
67C22×D67—21no—
68C22×D67C2×D412yes—
69C22×D67#4114yes—
70C2×C4×C67—22no—
71C2×C4×C67—21no—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—22no—
76C2×C4×C67—24no—
77C2×C4×C67—22no—
78C2×C4×C67—21no—
79C2×C4×C67—22no—
80C2×C4×C67—21no—
81C2×C4×C67C2×D412yes—
82C2×Dic67—21no—
83C2×Dic67—21no—
84C2×Dic67—22no—
85C2×Dic67—22no—
86C2×Dic67—22no—
87C2×Dic67—24no—
88C2×Dic67—22no—
89C2×Dic67—21no—
90C2×Dic67—21no—
91C2×Dic67—22no—
92C2×Dic67C2×D412yes—
93C2×Dic67#4114yes—
94C4×D67—28no—
95C4×D67—48no—
96Dic134—44no—
97C22×C67C22×D411yescharacteristic
98C22×C67C2×#4112yes—
99C22×C67C2×#4112yes—
100C22×C672−1+411yescharacteristic
101C22×C672−1+411yescharacteristic
102C2×D67—22no—
103C2×D67—22no—
104C2×D67—24no—
105C2×D67—24no—
106C2×D67#12414yes—
107C4×C67—22no—
108C4×C67—41no—
109C4×C67—42no—
110C4×C67—41no—
111C4×C67—42no—
112Dic67—41no—
113Dic67—42no—
114Dic67—41no—
115Dic67—42no—
116Dic67#12314yes—
117C2×C67C2×#12311yescharacteristic
118C2×C67C2×#12411yescharacteristic
119C2×C67#53611yescharacteristic
120C2×C67#55212yes—
121C2×C67#55212yes—
122D67—24no—
123#2503—671nomaximal, Sylow
124C67#250311yescharacteristic, complemented, Sylow
125C2×C22⋊Q8—671no—
126C22×C4⋊C4—671no—
127#400—671no—
128#400—671no—
129#400—671no—
130#400D6711yescharacteristic, complemented
131#391—672no—
132#382—672no—
133#381—672no—
134#381—672no—
135C4×C22⋊C4—671no—
136C22×Q8—671no—
137C23×C4—671no—
138C23×C4—672no—
139C22⋊Q8—1344no—
140C2×C4⋊C4—672no—
141C2×C4⋊C4—671no—
142C2×C4⋊C4—671no—
143C2×C4⋊C4—671no—
144C2×C4⋊C4—1344no—
145C2×C4⋊C4C2×D6711yescharacteristic
146C2×C4⋊C4C2×D6711yescharacteristic
147C2×C22⋊C4—671no—
148C2×C22⋊C4—671no—
149C2×C22⋊C4—672no—
150C2×C42—672no—
151#95—671no—
152#95—672no—
153#95—672no—
154#95—672no—
155#95—672no—
156#95—672no—
157#95—672no—
158#95C2×D6711yescharacteristic
159#95C2×D6712yes—
160#95C2×D6712yes—
161C24—671no—
162C2×Q8—1342no—
163C2×Q8—1344no—
164C22×C4—671no—
165C22×C4—674no—
166C22×C4—672no—
167C22×C4—671no—
168C22×C4—671no—
169C22×C4—672no—
170C22×C4—1344no—
171C22×C4—1348no—
172C22×C4—1344no—
173C22×C4C22×D6712yes—
174C22×C4C22×D6711yescharacteristic
175C22×C4C22×D6711yescharacteristic
176C22×C4C22×D6712yes—
177C22×C4C22×D6711yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—1344no—
181C4⋊C4—1342no—
182C4⋊C4—1342no—
183C4⋊C4—1342no—
184C22⋊C4—1342no—
185C22⋊C4—1342no—
186C22⋊C4—1344no—
187C42—1344no—
188C23—672no—
189C23—674no—
190C23—1342no—
191C23—1341no—
192C23—1341no—
193C23C23×D6711yescharacteristic
194Q8—2684no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—2688no—
207C2×C4—672no—
208C2×C4—674no—
209C2×C4—1348no—
210C2×C4—1341no—
211C2×C4—1341no—
212C2×C4—1342no—
213C2×C4—1342no—
214C2×C4—1342no—
215C2×C4—1344no—
216C2×C4—1342no—
217C2×C4—1341no—
218C2×C4—1342no—
219C2×C4—1341no—
220C2×C4D4×D6712yes—
221C22—674no—
222C22—1344no—
223C22—1344no—
224C22—1342no—
225C22—1342no—
226C22C2×D4×D6711yescharacteristic
227C22#41×D6712yes—
228C22#41×D6712yes—
229C22#1452409011yescharacteristic
230C22#1452409111yescharacteristic
231C4—22no—
232C4—42no—
233C4—41no—
234C4—42no—
235C4—41no—
236C4—2682no—
237C4—2681no—
238C4—2682no—
239C4—2681no—
240C4—674no—
241C2—1344no—
242C2#123×D6711yescharacteristic
243C2#124×D6711yescharacteristic
244C2#1686396911yescharacteristic
245C2#1686407812yes—
246C2#1686408212yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#400×C67C2
hypercenter#400D67
hypocenterC67#2503
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #16863336C2semidirect
4. #16863335C2semidirect
5. #16863334C2semidirect
6. #16863300C2semidirect
7. #16863260C2semidirect
11. #400×C67C2semidirect
124. C67#2503semidirect
130. #400D67semidirect