Finite Group Databasebeta

← #22566012

Subgroups — #22566012

Order 8576.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14524001C211yesmaximal, characteristic
3#16863336C211yesmaximal, characteristic, complemented
4#16863335C211yesmaximal, characteristic, complemented
5#16863327C211yesmaximal, characteristic, complemented
6#16863325C212yesmaximal
7#16863322C212yesmaximal
8#16863300C212yesmaximal, complemented
9#16863291C212yesmaximal, complemented
10#16863255C211yesmaximal, characteristic
11#398×C67C211yesmaximal, characteristic, complemented
12C22×Dic134C2211yescharacteristic
13C22×C4×D67C2211yescharacteristic
14#14524021—24no—
15#14524001—24no—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×Dic67⋊C4C2212yes—
18C2×Dic67⋊C4C2211yescharacteristic
19C2×Dic67⋊C4C2211yescharacteristic
20C2×Dic67⋊C4C2212yes—
21C2×(C2×D67)⋊C4C2211yescharacteristic
22C2×(C2×D67)⋊C4C2212yes—
23C2×(C2×D67)⋊C4C2211yescharacteristic
24C2×(C2×D67)⋊C4C2212yes—
25C2×C4×Dic67C2212yes—
26C67⋊#95C2211yescharacteristic
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2211yescharacteristic
34C2×C4⋊C4×C67C2212yes—
35#95×C67C2211yescharacteristic
36#95×C67C2212yes—
37C23×D67C2311yescharacteristic
38C2×Dic134—22no—
39C2×Dic134—24no—
40C22×Dic67C2312yes—
41C22×Dic67C2312yes—
42C22×Dic67C2311yescharacteristic
43C22×Dic67C2311yescharacteristic
44C22×Dic67C2311yescharacteristic
45C2×C4×D67—24no—
46C2×C4×D67D414yes—
47C4⋊Dic67—22no—
48Dic67⋊C4—22no—
49Dic67⋊C4—22no—
50Dic67⋊C4—24no—
51Dic67⋊C4—42no—
52(C2×D67)⋊C4—24no—
53(C2×D67)⋊C4—22no—
54(C2×D67)⋊C4—22no—
55(C2×D67)⋊C4—42no—
56C4×Dic67—24no—
57C22×C4×C67C2311yescharacteristic
58C22×C4×C67C2312yes—
59C22×C4×C67C2312yes—
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2311yescharacteristic
62C4⋊C4×C67—22no—
63C4⋊C4×C67—22no—
64C4⋊C4×C67—42no—
65C23×C67C2411yescharacteristic
66C22×D67—22no—
67C22×D67—21no—
68C22×D67—21no—
69C22×D67—22no—
70C22×D67C2×D412yes—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—21no—
74C2×C4×C67—21no—
75C2×C4×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67—21no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—21no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67C2×D412yes—
84C2×Dic67—22no—
85C2×Dic67—21no—
86C2×Dic67—21no—
87C2×Dic67—22no—
88C2×Dic67—22no—
89C2×Dic67—22no—
90C2×Dic67—21no—
91C2×Dic67—21no—
92C2×Dic67—24no—
93C2×Dic67—22no—
94C2×Dic67C2×D412yes—
95C2×Dic67#4114yes—
96C4×D67—28no—
97Dic134—44no—
98C22×C67C22×D411yescharacteristic
99C22×C67C2×#4112yes—
100C22×C67H(5,𝔽2)11yescharacteristic
101C22×C672−1+411yescharacteristic
102C22×C672−1+412yes—
103C2×D67—22no—
104C2×D67—22no—
105C2×D67—22no—
106C2×D67—42no—
107C2×D67—42no—
108C4×C67—22no—
109C4×C67—41no—
110C4×C67—42no—
111C4×C67—41no—
112C4×C67—42no—
113Dic67—41no—
114Dic67—42no—
115Dic67—41no—
116Dic67—42no—
117Dic67#12314yes—
118C2×C67C2×#12311yescharacteristic
119C2×C67#53611yescharacteristic
120C2×C67#53711yescharacteristic
121C2×C67#54112yes—
122C2×C67#54212yes—
123D67—42no—
124#2625—671nomaximal, Sylow
125C67#262511yescharacteristic, complemented, Sylow
126C2×C22⋊Q8—671no—
127C2×#117—671no—
128#400—671no—
129#400—671no—
130#398—671no—
131#398D6711yescharacteristic, complemented
132#397—672no—
133#397—672no—
134#391—672no—
135#389—672no—
136#380—671no—
137C22×Q8—671no—
138C23×C4—671no—
139C22⋊Q8—1344no—
140#117—1344no—
141C2×C4⋊C4—672no—
142C2×C4⋊C4—671no—
143C2×C4⋊C4—671no—
144C2×C4⋊C4—671no—
145C2×C4⋊C4—672no—
146C2×C4⋊C4C2×D6711yescharacteristic
147C2×C4⋊C4C2×D6711yescharacteristic
148C2×C4⋊C4C2×D6712yes—
149C2×C22⋊C4—672no—
150C2×C22⋊C4—672no—
151C2×C22⋊C4—671no—
152C2×C22⋊C4—671no—
153C2×C42—672no—
154#95—671no—
155#95—672no—
156#95—672no—
157#95—672no—
158#95—672no—
159#95—672no—
160#95C2×D6711yescharacteristic
161#95C2×D6712yes—
162C24—671no—
163C2×Q8—1342no—
164C2×Q8—1344no—
165C22×C4—671no—
166C22×C4—674no—
167C22×C4—672no—
168C22×C4—672no—
169C22×C4—671no—
170C22×C4—671no—
171C22×C4—1344no—
172C22×C4C22×D6712yes—
173C22×C4C22×D6711yescharacteristic
174C22×C4C22×D6711yescharacteristic
175C22×C4C22×D6712yes—
176C22×C4C22×D6711yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—2682no—
181C4⋊C4—1342no—
182C4⋊C4—1344no—
183C4⋊C4—1342no—
184C4⋊C4—1342no—
185C22⋊C4—2682no—
186C22⋊C4—1344no—
187C22⋊C4—1342no—
188C22⋊C4—1342no—
189C42—1344no—
190C23—672no—
191C23—1342no—
192C23—1341no—
193C23—1341no—
194C23—1342no—
195C23C23×D6711yescharacteristic
196Q8—2684no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—672no—
210C2×C4—674no—
211C2×C4—1348no—
212C2×C4—1342no—
213C2×C4—1341no—
214C2×C4—1341no—
215C2×C4—1342no—
216C2×C4—1342no—
217C2×C4—1342no—
218C2×C4—1341no—
219C2×C4—1341no—
220C2×C4—1342no—
221C2×C4—1344no—
222C2×C4D4×D6712yes—
223C22—2682no—
224C22—2682no—
225C22—1342no—
226C22—1342no—
227C22—1342no—
228C22C2×D4×D6711yescharacteristic
229C22#41×D6712yes—
230C22#1452408811yescharacteristic
231C22#1452409011yescharacteristic
232C22#1452409112yes—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—41no—
238C4—2682no—
239C4—2681no—
240C4—2682no—
241C4—2681no—
242C4—674no—
243C2—2682no—
244C2#123×D6711yescharacteristic
245C2#1686397111yescharacteristic
246C2#1686397511yescharacteristic
247C2#1686400512yes—
248C2#1686401612yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#398×C67C2
hypercenter#398D67
hypocenterC67#2625
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #16863336C2semidirect
4. #16863335C2semidirect
5. #16863327C2semidirect
8. #16863300C2semidirect
9. #16863291C2semidirect
11. #398×C67C2semidirect
125. C67#2625semidirect
131. #398D67semidirect