Finite Group Databasebeta

← #22564682

Subgroups — #22564682

Order 8576.

544 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 161 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#16863328C211yesmaximal, characteristic, complemented
4#16863327C211yesmaximal, characteristic, complemented
5#16863326C211yesmaximal, characteristic, complemented
6#16863300C212yesmaximal, complemented
7#16863270C212yesmaximal
8#16863267C212yesmaximal
9#16863265C212yesmaximal, complemented
10#16863243C211yesmaximal, characteristic
11#398×C67C211yesmaximal, characteristic, complemented
12C22×Dic134C2211yescharacteristic
13C22×C4×D67C2211yescharacteristic
14C22×C4×D67C2212yes—
15#14524021—24no—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×C4⋊Dic67C2212yes—
18C2×Dic67⋊C4C2211yescharacteristic
19C2×Dic67⋊C4C2212yes—
20C2×Dic67⋊C4C2211yescharacteristic
21C2×Dic67⋊C4C2212yes—
22C2×Dic67⋊C4C2212yes—
23C4⋊C4×D67—24no—
24C2×(C2×D67)⋊C4C2211yescharacteristic
25C2×(C2×D67)⋊C4C2211yescharacteristic
26C2×(C2×D67)⋊C4C2212yes—
27C2×C4×Dic67C2212yes—
28C67⋊#95C2211yescharacteristic
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×Dic67C2311yescharacteristic
41C22×Dic67C2312yes—
42C22×Dic67C2311yescharacteristic
43C22×Dic67C2312yes—
44C22×Dic67C2311yescharacteristic
45C2×C4×D67—24no—
46C2×C4×D67—24no—
47C2×C4×D67—42no—
48C2×C4×D67—42no—
49C2×C4×D67D414yes—
50C4⋊Dic67—22no—
51C4⋊Dic67—42no—
52Dic67⋊C4—22no—
53Dic67⋊C4—24no—
54Dic67⋊C4—22no—
55Dic67⋊C4—42no—
56Dic67⋊C4—42no—
57(C2×D67)⋊C4—22no—
58(C2×D67)⋊C4—22no—
59(C2×D67)⋊C4Q818yescomplemented
60C4×Dic67—24no—
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2312yes—
63C22×C4×C67C2311yescharacteristic
64C22×C4×C67C2311yescharacteristic
65C22×C4×C67C2312yes—
66C4⋊C4×C67—22no—
67C4⋊C4×C67—22no—
68C4⋊C4×C67—42no—
69C23×C67C2411yescharacteristic
70C22×D67—22no—
71C22×D67—21no—
72C22×D67—21no—
73C22×D67C2×D412yes—
74C22×D67C2×Q814yes—
75C2×C4×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67—22no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—21no—
83C2×C4×C67—22no—
84C2×C4×C67—21no—
85C2×C4×C67—22no—
86C2×C4×C67C2×D412yes—
87C2×C4×C67C2×Q814yes—
88C2×Dic67—22no—
89C2×Dic67—22no—
90C2×Dic67—21no—
91C2×Dic67—21no—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—21no—
95C2×Dic67—22no—
96C2×Dic67—21no—
97C2×Dic67—22no—
98C2×Dic67C2×D412yes—
99C2×Dic67C2×Q814yes—
100C2×Dic67#4114yes—
101C4×D67—28no—
102C4×D67—48no—
103Dic134—44no—
104C22×C67C22×D411yescharacteristic
105C22×C67C22×Q812yes—
106C22×C67C2×#4112yes—
107C22×C67H(5,𝔽2)11yescharacteristic
108C22×C67H(5,𝔽2)11yescharacteristic
109C2×D67—22no—
110C2×D67—22no—
111C2×D67—24no—
112C2×D67—24no—
113C2×D67C4⋊Q814yes—
114C4×C67—22no—
115C4×C67—24no—
116C4×C67—41no—
117C4×C67—42no—
118C4×C67—41no—
119Dic67—24no—
120Dic67—41no—
121Dic67—42no—
122Dic67—41no—
123Dic67#12314yes—
124C2×C67C2×#12311yescharacteristic
125C2×C67C2×C4⋊Q811yescharacteristic
126C2×C67#53511yescharacteristic
127C2×C67#55412yes—
128C2×C67#55412yes—
129D67—24no—
130#2502—671nomaximal, Sylow
131C67#250211yescharacteristic, complemented, Sylow
132C2×C22⋊Q8—671no—
133C22×C4⋊C4—671no—
134#398—671no—
135#398—671no—
136#398—671no—
137#398D6711yescharacteristic, complemented
138#391—672no—
139#385—672no—
140#385—672no—
141#384—672no—
142C4×C22⋊C4—671no—
143C22×Q8—671no—
144C23×C4—672no—
145C23×C4—671no—
146C22⋊Q8—1344no—
147C2×C4⋊C4—672no—
148C2×C4⋊C4—672no—
149C2×C4⋊C4—672no—
150C2×C4⋊C4—672no—
151C2×C4⋊C4—671no—
152C2×C4⋊C4—671no—
153C2×C4⋊C4—671no—
154C2×C4⋊C4—1344no—
155C2×C4⋊C4C2×D6711yescharacteristic
156C2×C4⋊C4C2×D6711yescharacteristic
157C2×C4⋊C4C2×D6712yes—
158C2×C22⋊C4—672no—
159C2×C22⋊C4—671no—
160C2×C22⋊C4—671no—
161C2×C42—672no—
162#95—671no—
163#95—672no—
164#95—672no—
165#95—672no—
166#95C2×D6711yescharacteristic
167#95C2×D6712yes—
168C24—671no—
169C2×Q8—1342no—
170C2×Q8—1344no—
171C22×C4—2682no—
172C22×C4—2682no—
173C22×C4—671no—
174C22×C4—672no—
175C22×C4—674no—
176C22×C4—671no—
177C22×C4—672no—
178C22×C4—671no—
179C22×C4—1344no—
180C22×C4—1344no—
181C22×C4C22×D6711yescharacteristic
182C22×C4C22×D6712yes—
183C22×C4C22×D6711yescharacteristic
184C22×C4C22×D6711yescharacteristic
185C22×C4C22×D6712yes—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—2682no—
190C4⋊C4—2682no—
191C4⋊C4—2682no—
192C4⋊C4—1344no—
193C4⋊C4—1342no—
194C4⋊C4—1342no—
195C4⋊C4—1342no—
196C22⋊C4—678no—
197C22⋊C4—1342no—
198C22⋊C4—1342no—
199C42—1344no—
200C23—672no—
201C23—674no—
202C23—1342no—
203C23—1341no—
204C23—1341no—
205C23C23×D6711yescharacteristic
206Q8—2684no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—2688no—
219C2×C4—672no—
220C2×C4—674no—
221C2×C4—674no—
222C2×C4—1348no—
223C2×C4—1341no—
224C2×C4—1341no—
225C2×C4—1342no—
226C2×C4—1342no—
227C2×C4—1341no—
228C2×C4—1342no—
229C2×C4—1341no—
230C2×C4—1342no—
231C2×C4—1342no—
232C2×C4—1342no—
233C2×C4D4×D6712yes—
234C2×C4Q8×D6714yes—
235C22—674no—
236C22—1344no—
237C22—1344no—
238C22—1342no—
239C22—1342no—
240C22C2×D4×D6711yescharacteristic
241C22C2×Q8×D6712yes—
242C22#41×D6712yes—
243C22#1452408811yescharacteristic
244C22#1452408911yescharacteristic
245C4—24no—
246C4—22no—
247C4—41no—
248C4—42no—
249C4—41no—
250C4—2681no—
251C4—2682no—
252C4—2681no—
253C4—674no—
254C4—1344no—
255C2—1344no—
256C2#123×D6711yescharacteristic
257C2C4⋊Q8×D6711yescharacteristic
258C2#1686396211yescharacteristic
259C2#1686410012yes—
260C2#1686410212yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#398×C67C2
hypercenter#398D67
hypocenterC67#2502
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #16863328C2semidirect
4. #16863327C2semidirect
5. #16863326C2semidirect
6. #16863300C2semidirect
9. #16863265C2semidirect
11. #398×C67C2semidirect
59. (C2×D67)⋊C4Q8semidirect
131. C67#2502semidirect
137. #398D67semidirect