Finite Group Databasebeta

← #22566037

Subgroups — #22566037

Order 8576.

488 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14524042C211yesmaximal, characteristic, complemented
2C2×#14524001C211yesmaximal, characteristic
3#16863335C211yesmaximal, characteristic, complemented
4#16863333C211yesmaximal, characteristic
5#16863331C211yesmaximal, characteristic
6#16863317C212yesmaximal, complemented
7#16863309C212yesmaximal
8#16863293C212yesmaximal, complemented
9#16863260C211yesmaximal, characteristic, complemented
10#16863255C212yesmaximal
11#400×C67C211yesmaximal, characteristic, complemented
12C22×Dic134C2212yes—
13C22×C4×D67C2211yescharacteristic
14#14524042—24no—
15#14524001—24no—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×C4⋊Dic67C2211yescharacteristic
18C2×C4⋊Dic67C2212yes—
19C2×Dic67⋊C4C2212yes—
20C2×Dic67⋊C4C2212yes—
21C2×Dic67⋊C4C2212yes—
22C2×(C2×D67)⋊C4C2211yescharacteristic
23C2×(C2×D67)⋊C4C2211yescharacteristic
24C2×(C2×D67)⋊C4C2212yes—
25C2×(C2×D67)⋊C4C2212yes—
26C2×C4×Dic67C2211yescharacteristic
27C2×C4×Dic67C2211yescharacteristic
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
38C2×Dic134—24no—
39C2×Dic134—42no—
40C2×Dic134—42no—
41C22×Dic67C2312yes—
42C22×Dic67C2312yes—
43C22×Dic67C2311yescharacteristic
44C22×Dic67C2311yescharacteristic
45C22×Dic67C2311yescharacteristic
46C2×C4×D67—22no—
47C2×C4×D67—22no—
48C2×C4×D67Q814yescomplemented
49C4⋊Dic67—22no—
50C4⋊Dic67—22no—
51C4⋊Dic67—42no—
52Dic67⋊C4—24no—
53Dic67⋊C4—24no—
54Dic67⋊C4—42no—
55(C2×D67)⋊C4—22no—
56(C2×D67)⋊C4—22no—
57(C2×D67)⋊C4—42no—
58(C2×D67)⋊C4—42no—
59C4×Dic67—22no—
60C4×Dic67D414yes—
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2312yes—
63C22×C4×C67C2312yes—
64C22×C4×C67C2311yescharacteristic
65C22×C4×C67C2311yescharacteristic
66C4⋊C4×C67—22no—
67C4⋊C4×C67—22no—
68C23×C67C2411yescharacteristic
69C22×D67—21no—
70C22×D67—21no—
71C22×D67—22no—
72C22×D67—22no—
73C22×D67C2×Q812yes—
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—22no—
77C2×C4×C67—21no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—22no—
84C2×C4×C67—22no—
85C2×C4×C67C2×D412yes—
86C2×C4×C67C2×Q812yes—
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×Dic67—22no—
96C2×Dic67C2×D412yes—
97C2×Dic67C2×D412yes—
98C2×Dic67C2×Q812yes—
99C2×Dic67#4112yes—
100C4×D67—28no—
101Dic134—48no—
102C22×C67C22×D411yescharacteristic
103C22×C67C22×Q811yescharacteristic
104C22×C67C2×#4111yescharacteristic
105C22×C67H(5,𝔽2)12yes—
106C22×C672−1+412yes—
107C2×D67—22no—
108C2×D67—24no—
109C2×D67—41no—
110C2×D67—41no—
111C2×D67—42no—
112C4×C67—22no—
113C4×C67—22no—
114C4×C67—42no—
115C4×C67—41no—
116C4×C67—42no—
117Dic67—22no—
118Dic67—42no—
119Dic67—41no—
120Dic67—42no—
121Dic67C22⋊Q814yes—
122C2×C67C2×C22⋊Q811yescharacteristic
123C2×C67#53511yescharacteristic
124C2×C67#53611yescharacteristic
125C2×C67#54212yes—
126C2×C67#54412yes—
127D67—42no—
128#2628—671nomaximal, Sylow
129C67#262811yescharacteristic, complemented, Sylow
130C2×C4⋊Q8—671no—
131C2×#117—671no—
132#400—671no—
133#400D6711yescharacteristic, complemented
134#399—671no—
135#399—671no—
136#395—672no—
137#393—672no—
138#389—672no—
139#382—671no—
140#380—672no—
141C22×Q8—672no—
142C23×C4—671no—
143C4⋊Q8—1344no—
144#117—1344no—
145C2×C4⋊C4—672no—
146C2×C4⋊C4—672no—
147C2×C4⋊C4—671no—
148C2×C4⋊C4—672no—
149C2×C4⋊C4—671no—
150C2×C4⋊C4—672no—
151C2×C4⋊C4C2×D6711yescharacteristic
152C2×C4⋊C4C2×D6711yescharacteristic
153C2×C22⋊C4—672no—
154C2×C22⋊C4—672no—
155C2×C22⋊C4—671no—
156C2×C22⋊C4—671no—
157C2×C42—671no—
158C2×C42—671no—
159#95—672no—
160#95—672no—
161#95—671no—
162#95—672no—
163#95C2×D6712yes—
164#95C2×D6711yescharacteristic
165#95C2×D6712yes—
166C24—671no—
167C2×Q8—2682no—
168C2×Q8—2682no—
169C2×Q8—1344no—
170C22×C4—671no—
171C22×C4—671no—
172C22×C4—674no—
173C22×C4—672no—
174C22×C4—672no—
175C22×C4—671no—
176C22×C4—1342no—
177C22×C4—1342no—
178C22×C4C22×D6711yescharacteristic
179C22×C4C22×D6712yes—
180C22×C4C22×D6712yes—
181C22×C4C22×D6711yescharacteristic
182C22×C4C22×D6711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—2682no—
186C4⋊C4—2682no—
187C4⋊C4—1342no—
188C4⋊C4—1344no—
189C4⋊C4—1344no—
190C4⋊C4—1342no—
191C22⋊C4—2682no—
192C22⋊C4—2682no—
193C22⋊C4—1342no—
194C22⋊C4—1342no—
195C42—674no—
196C42—1342no—
197C23—672no—
198C23—1341no—
199C23—1341no—
200C23—1342no—
201C23—1342no—
202C23C23×D6711yescharacteristic
203Q8—2688no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—672no—
216C2×C4—672no—
217C2×C4—672no—
218C2×C4—672no—
219C2×C4—1348no—
220C2×C4—1341no—
221C2×C4—1342no—
222C2×C4—1342no—
223C2×C4—1342no—
224C2×C4—1342no—
225C2×C4—1342no—
226C2×C4—1342no—
227C2×C4—1342no—
228C2×C4—1342no—
229C2×C4D4×D6712yes—
230C2×C4Q8×D6712yes—
231C22—2682no—
232C22—2681no—
233C22—2681no—
234C22—1344no—
235C22—1342no—
236C22C2×D4×D6711yescharacteristic
237C22C2×Q8×D6711yescharacteristic
238C22#41×D6711yescharacteristic
239C22#1452408912yes—
240C22#1452409012yes—
241C4—22no—
242C4—22no—
243C4—42no—
244C4—42no—
245C4—41no—
246C4—2682no—
247C4—2682no—
248C4—2681no—
249C4—674no—
250C4—1342no—
251C2—2682no—
252C2C22⋊Q8×D6711yescharacteristic
253C2#1686396411yescharacteristic
254C2#1686396811yescharacteristic
255C2#1686401212yes—
256C2#1686402612yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#400×C67C2
hypercenter#400D67
hypocenterC67#2628
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#14524042C2semidirect
3. #16863335C2semidirect
6. #16863317C2semidirect
8. #16863293C2semidirect
9. #16863260C2semidirect
11. #400×C67C2semidirect
48. C2×C4×D67Q8semidirect
129. C67#2628semidirect
133. #400D67semidirect