Finite Group Databasebeta

← #22563848

Subgroups — #22563848

Order 8576.

800 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×Q8×D67C211yesmaximal, characteristic
2C2×#14524042C212yesmaximal, complemented
3C2×#14524021C211yesmaximal, characteristic
4#16863317C211yesmaximal, characteristic, complemented
5#16863316C211yesmaximal, characteristic, complemented
6#16863315C211yesmaximal, characteristic, complemented
7#16863300C212yesmaximal, complemented
8#16863270C212yesmaximal
9#16863267C212yesmaximal
10#16863243C211yesmaximal, characteristic
11#395×C67C211yesmaximal, characteristic, complemented
12C22×Dic134C2211yescharacteristic
13C22×Dic134C2211yescharacteristic
14C22×Dic134C2212yes—
15C2×Q8×D67—28no—
16C2×Q8×D67—24no—
17C22×C4×D67C2212yes—
18C22×C4×D67C2211yescharacteristic
19#14524042—28no—
20#14524021—24no—
21C2×C4⋊Dic67C2212yes—
22C2×Dic67⋊C4C2212yes—
23C2×Dic67⋊C4C2212yes—
24C2×Dic67⋊C4C2211yescharacteristic
25C2×Dic67⋊C4C2211yescharacteristic
26C2×(C2×D67)⋊C4C2211yescharacteristic
27C2×(C2×D67)⋊C4C2211yescharacteristic
28C2×(C2×D67)⋊C4C2212yes—
29C2×C4×Dic67C2212yes—
30C67⋊#95C2211yescharacteristic
31C67⋊#95C2212yes—
32C67⋊#95C2212yes—
33C67⋊#95C2212yes—
34C22×Q8×C67C2211yescharacteristic
35C2×C4⋊C4×C67C2212yes—
36C2×C4⋊C4×C67C2211yescharacteristic
37#95×C67C2211yescharacteristic
38#95×C67C2212yes—
39C23×D67C2311yescharacteristic
40C2×Dic134—22no—
41C2×Dic134—24no—
42C2×Dic134—24no—
43C2×Dic134—24no—
44C2×Dic134—22no—
45C2×Dic134—24no—
46C2×Dic134D418yes—
47Q8×D67—232no—
48C22×Dic67C2311yescharacteristic
49C22×Dic67C2311yescharacteristic
50C22×Dic67C2312yes—
51C22×Dic67C2312yes—
52C22×Dic67C2311yescharacteristic
53C2×C4×D67—24no—
54C2×C4×D67—24no—
55C2×C4×D67—24no—
56C2×C4×D67—24no—
57C2×C4×D67D414yes—
58C4⋊Dic67—24no—
59Dic67⋊C4—22no—
60Dic67⋊C4—24no—
61Dic67⋊C4—22no—
62Dic67⋊C4—24no—
63(C2×D67)⋊C4—22no—
64(C2×D67)⋊C4—22no—
65(C2×D67)⋊C4Q818yescomplemented
66C4×Dic67—24no—
67C2×Q8×C67—24no—
68C2×Q8×C67—22no—
69C22×C4×C67C2311yescharacteristic
70C22×C4×C67C2311yescharacteristic
71C22×C4×C67C2312yes—
72C22×C4×C67C2311yescharacteristic
73C22×C4×C67C2312yes—
74C4⋊C4×C67—22no—
75C4⋊C4×C67—24no—
76C23×C67C2411yescharacteristic
77C22×D67—22no—
78C22×D67—21no—
79C22×D67—21no—
80C22×D67C2×D412yes—
81C22×D67C2×Q814yes—
82Q8×C67—28no—
83C2×C4×C67—22no—
84C2×C4×C67—22no—
85C2×C4×C67—22no—
86C2×C4×C67—22no—
87C2×C4×C67—22no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—22no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67C2×D414yes—
94C2×C4×C67C2×D412yes—
95C2×C4×C67C2×Q814yescomplemented
96C2×Dic67—22no—
97C2×Dic67—22no—
98C2×Dic67—22no—
99C2×Dic67—22no—
100C2×Dic67—21no—
101C2×Dic67—21no—
102C2×Dic67—22no—
103C2×Dic67—21no—
104C2×Dic67—21no—
105C2×Dic67C2×D414yes—
106C2×Dic67C2×D414yes—
107C2×Dic67C2×D412yes—
108C2×Dic67C2×Q814yescomplemented
109C4×D67—216no—
110C4×D67—28no—
111Dic134—28no—
112Dic134—216no—
113Dic134—44no—
114C22×C67C22×D411yescharacteristic
115C22×C67C22×D412yes—
116C22×C67C22×Q812yes—
117C22×C672−1+411yescharacteristic
118C22×C672−1+411yescharacteristic
119C2×D67—22no—
120C2×D67—22no—
121C2×D67—24no—
122C2×D67—24no—
123C2×D67C4⋊Q814yescomplemented
124C4×C67—22no—
125C4×C67—24no—
126C4×C67—24no—
127C4×C67—41no—
128C4×C67—41no—
129Dic67—24no—
130Dic67—24no—
131Dic67—41no—
132Dic67—41no—
133Dic67#12014yes—
134C2×C67C2×#12011yescharacteristic
135C2×C67C2×C4⋊Q811yescharacteristic
136C2×C67#53611yescharacteristic
137C2×C67D4×Q812yes—
138C2×C67D4×Q812yes—
139D67—24no—
140#2434—671nomaximal, Sylow
141C67#243411yescharacteristic, complemented, Sylow
142C23×Q8—671no—
143C2×C4⋊Q8—672no—
144C2×C22⋊Q8—671no—
145#395—671no—
146#395—671no—
147#395—671no—
148#395D6711yescharacteristic, complemented
149#391—672no—
150#385—672no—
151#385—672no—
152C4×C22⋊C4—671no—
153C22×Q8—672no—
154C22×Q8—671no—
155C22×Q8—671no—
156C22×Q8—1344no—
157C22×Q8—1348no—
158C22×Q8C2×D6711yescharacteristic
159C23×C4—671no—
160C23×C4—672no—
161C4⋊Q8—1348no—
162C22⋊Q8—1344no—
163C2×C4⋊C4—671no—
164C2×C4⋊C4—671no—
165C2×C4⋊C4—672no—
166C2×C4⋊C4—672no—
167C2×C4⋊C4—672no—
168C2×C4⋊C4C2×D6711yescharacteristic
169C2×C4⋊C4C2×D6712yes—
170C2×C22⋊C4—672no—
171C2×C22⋊C4—671no—
172C2×C22⋊C4—671no—
173C2×C42—672no—
174#95—671no—
175#95—672no—
176#95—672no—
177#95—672no—
178#95C2×D6711yescharacteristic
179#95C2×D6712yes—
180C24—671no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—678no—
184C2×Q8—1342no—
185C2×Q8—1344no—
186C2×Q8—1344no—
187C2×Q8—1344no—
188C2×Q8—1344no—
189C2×Q8—13432no—
190C2×Q8—1342no—
191C22×C4—671no—
192C22×C4—672no—
193C22×C4—672no—
194C22×C4—674no—
195C22×C4—671no—
196C22×C4—671no—
197C22×C4—1344no—
198C22×C4—1344no—
199C22×C4—1344no—
200C22×C4—1344no—
201C22×C4C22×D6711yescharacteristic
202C22×C4C22×D6711yescharacteristic
203C22×C4C22×D6711yescharacteristic
204C22×C4C22×D6712yes—
205C22×C4C22×D6712yes—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—1342no—
209C4⋊C4—1344no—
210C4⋊C4—1344no—
211C4⋊C4—1342no—
212C4⋊C4—1344no—
213C22⋊C4—678no—
214C22⋊C4—1342no—
215C22⋊C4—1342no—
216C42—1344no—
217C23—672no—
218C23—674no—
219C23—1342no—
220C23—1341no—
221C23—1341no—
222C23C23×D6711yescharacteristic
223Q8—28no—
224Q8—2684no—
225Q8—13416no—
226Q8—1348no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—674no—
238C2×C4—672no—
239C2×C4—674no—
240C2×C4—674no—
241C2×C4—1348no—
242C2×C4—13416no—
243C2×C4—1341no—
244C2×C4—1341no—
245C2×C4—1342no—
246C2×C4—1341no—
247C2×C4—1341no—
248C2×C4—1342no—
249C2×C4—1342no—
250C2×C4—1342no—
251C2×C4—1342no—
252C2×C4D4×D6712yes—
253C2×C4D4×D6714yes—
254C2×C4Q8×D6714yescomplemented
255C22—674no—
256C22—1344no—
257C22—1344no—
258C22—1342no—
259C22—1342no—
260C22C2×D4×D6712yes—
261C22C2×D4×D6711yescharacteristic
262C22C2×Q8×D6712yes—
263C22#1452409011yescharacteristic
264C22#1452409111yescharacteristic
265C4—24no—
266C4—22no—
267C4—24no—
268C4—41no—
269C4—41no—
270C4—2681no—
271C4—2681no—
272C4—674no—
273C4—1344no—
274C4—1344no—
275C2—1344no—
276C2#120×D6711yescharacteristic
277C2C4⋊Q8×D6711yescharacteristic
278C2#1686396911yescharacteristic
279C2#1686405912yes—
280C2#1686406112yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#395×C67C2
hypercenter#395D67
hypocenterC67#2434
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#14524042C2semidirect
4. #16863317C2semidirect
5. #16863316C2semidirect
6. #16863315C2semidirect
7. #16863300C2semidirect
11. #395×C67C2semidirect
65. (C2×D67)⋊C4Q8semidirect
95. C2×C4×C67C2×Q8semidirect
108. C2×Dic67C2×Q8semidirect
123. C2×D67C4⋊Q8semidirect
141. C67#2434semidirect
148. #395D67semidirect
254. C2×C4Q8×D67semidirect