Finite Group Databasebeta

← #22566021

Subgroups — #22566021

Order 8576.

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

normal subgroups only

This group has 508 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×#14524019C211yesmaximal, characteristic
2C2×C4⋊C4×D67C211yesmaximal, characteristic
3#16863336C211yesmaximal, characteristic, complemented
4#16863335C211yesmaximal, characteristic, complemented
5#16863330C212yesmaximal, complemented
6#16863327C212yesmaximal, complemented
7#16863315C211yesmaximal, characteristic, complemented
8#16863267C212yesmaximal
9#16863258C212yesmaximal
10#16863255C211yesmaximal, characteristic
11#395×C67C211yesmaximal, characteristic, complemented
12C22×C4×D67C2211yescharacteristic
13C22×C4×D67C2211yescharacteristic
14C22×C4×D67C2211yescharacteristic
15#14524019—24no—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×C4⋊Dic67C2211yescharacteristic
18C2×Dic67⋊C4C2211yescharacteristic
19C2×Dic67⋊C4C2211yescharacteristic
20C2×Dic67⋊C4C2212yes—
21C2×Dic67⋊C4C2212yes—
22C2×Dic67⋊C4C2212yes—
23C4⋊C4×D67—24no—
24C2×(C2×D67)⋊C4C2212yes—
25C2×(C2×D67)⋊C4C2212yes—
26C67⋊#95C2211yescharacteristic
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C22×Q8×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2211yescharacteristic
34C2×C4⋊C4×C67C2212yes—
35#95×C67C2211yescharacteristic
36#95×C67C2212yes—
37C23×D67C2311yescharacteristic
38C22×Dic67C2312yes—
39C22×Dic67C2312yes—
40C22×Dic67C2311yescharacteristic
41C22×Dic67C2311yescharacteristic
42C22×Dic67C2311yescharacteristic
43C2×C4×D67—22no—
44C2×C4×D67—22no—
45C2×C4×D67—22no—
46C2×C4×D67—42no—
47C2×C4×D67—42no—
48C2×C4×D67D414yes—
49C2×C4×D67Q814yescomplemented
50C4⋊Dic67—22no—
51C4⋊Dic67—22no—
52Dic67⋊C4—22no—
53Dic67⋊C4—22no—
54Dic67⋊C4—24no—
55Dic67⋊C4—42no—
56Dic67⋊C4—42no—
57(C2×D67)⋊C4—24no—
58(C2×D67)⋊C4—24no—
59C2×Q8×C67—22no—
60C2×Q8×C67—42no—
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2312yes—
63C22×C4×C67C2312yes—
64C22×C4×C67C2311yescharacteristic
65C22×C4×C67C2311yescharacteristic
66C4⋊C4×C67—22no—
67C4⋊C4×C67—42no—
68C23×C67C2411yescharacteristic
69C22×D67—22no—
70C22×D67—22no—
71C22×D67C2×D412yes—
72C22×D67C2×Q812yes—
73C22×D67#4112yes—
74Q8×C67—44no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—21no—
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×C67—22no—
86C2×C4×C67C2×D412yes—
87C2×C4×C67C2×Q812yes—
88C2×Dic67—21no—
89C2×Dic67—21no—
90C2×Dic67—22no—
91C2×Dic67—22no—
92C2×Dic67—21no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67—22no—
96C2×Dic67—24no—
97C2×Dic67—22no—
98C2×Dic67C2×D412yes—
99C2×Dic67C2×Q812yes—
100C4×D67—28no—
101C4×D67—44no—
102C4×D67—44no—
103C22×C67C22×D411yescharacteristic
104C22×C67C22×Q811yescharacteristic
105C22×C67C2×#4111yescharacteristic
106C22×C67H(5,𝔽2)12yes—
107C22×C672−1+412yes—
108C2×D67—22no—
109C2×D67—22no—
110C2×D67—24no—
111C2×D67—24no—
112C2×D67C22⋊Q814yes—
113C4×C67—22no—
114C4×C67—22no—
115C4×C67—41no—
116C4×C67—42no—
117C4×C67—42no—
118Dic67—22no—
119Dic67—22no—
120Dic67—41no—
121Dic67—42no—
122Dic67—42no—
123C2×C67C2×C22⋊Q811yescharacteristic
124C2×C67#53411yescharacteristic
125C2×C67#53611yescharacteristic
126C2×C67#53912yes—
127C2×C67#54412yes—
128D67—24no—
129#2627—671nomaximal, Sylow
130C67#262711yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—671no—
132C22×C4⋊C4—671no—
133#400—671no—
134#400—671no—
135#398—672no—
136#398—672no—
137#395—671no—
138#395D6711yescharacteristic, complemented
139#385—672no—
140#381—672no—
141#380—671no—
142C22×Q8C2×D6711yescharacteristic
143C23×C4—671no—
144C23×C4—671no—
145C23×C4—671no—
146C22⋊Q8—1344no—
147C2×C4⋊C4—672no—
148C2×C4⋊C4—672no—
149C2×C4⋊C4—671no—
150C2×C4⋊C4—671no—
151C2×C4⋊C4—671no—
152C2×C4⋊C4—672no—
153C2×C4⋊C4—671no—
154C2×C4⋊C4—1344no—
155C2×C4⋊C4C2×D6711yescharacteristic
156C2×C4⋊C4C2×D6712yes—
157C2×C22⋊C4—672no—
158C2×C22⋊C4—672no—
159#95—671no—
160#95—672no—
161#95—672no—
162#95—672no—
163#95—672no—
164#95—672no—
165#95C2×D6711yescharacteristic
166#95C2×D6712yes—
167C24—671no—
168C2×Q8—22no—
169C2×Q8—42no—
170C22×C4—2682no—
171C22×C4—2682no—
172C22×C4—674no—
173C22×C4—671no—
174C22×C4—671no—
175C22×C4—674no—
176C22×C4—672no—
177C22×C4—672no—
178C22×C4—671no—
179C22×C4—1342no—
180C22×C4—1342no—
181C22×C4—1342no—
182C22×C4C22×D6712yes—
183C22×C4C22×D6711yescharacteristic
184C22×C4C22×D6712yes—
185C22×C4C22×D6711yescharacteristic
186C22×C4C22×D6711yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—2682no—
190C4⋊C4—2682no—
191C4⋊C4—1342no—
192C4⋊C4—1342no—
193C4⋊C4—1344no—
194C4⋊C4—1342no—
195C4⋊C4—1342no—
196C22⋊C4—1344no—
197C22⋊C4—1344no—
198C23—672no—
199C23—672no—
200C23—672no—
201C23—1342no—
202C23—1342no—
203C23C23×D6711yescharacteristic
204Q8—44no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—2684no—
217C2×C4—2684no—
218C2×C4—672no—
219C2×C4—672no—
220C2×C4—1348no—
221C2×C4—1342no—
222C2×C4—1341no—
223C2×C4—1342no—
224C2×C4—1342no—
225C2×C4—1342no—
226C2×C4—1344no—
227C2×C4—1342no—
228C2×C4—1341no—
229C2×C4—1341no—
230C2×C4—1342no—
231C2×C4D4×D6712yes—
232C2×C4Q8×D6712yes—
233C22—674no—
234C22—1344no—
235C22—1342no—
236C22—1342no—
237C22—1344no—
238C22C2×D4×D6711yescharacteristic
239C22C2×Q8×D6711yescharacteristic
240C22#41×D6711yescharacteristic
241C22#1452408812yes—
242C22#1452409112yes—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—42no—
247C4—42no—
248C4—2681no—
249C4—2682no—
250C4—2682no—
251C4—1342no—
252C4—1342no—
253C2—1344no—
254C2C22⋊Q8×D6711yescharacteristic
255C2#1686395611yescharacteristic
256C2#1686397111yescharacteristic
257C2#1686399212yes—
258C2#1686402312yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#395×C67C2
hypercenter#395D67
hypocenterC67#2627
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. #16863336C2semidirect
4. #16863335C2semidirect
5. #16863330C2semidirect
6. #16863327C2semidirect
7. #16863315C2semidirect
11. #395×C67C2semidirect
49. C2×C4×D67Q8semidirect
130. C67#2627semidirect
138. #395D67semidirect