Finite Group Databasebeta

← #22563875

Subgroups — #22563875

Order 8576.

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

normal subgroups only

This group has 780 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
1C2×#14524020C212yesmaximal, complemented
2C22×C22⋊Dic67C212yesmaximal
3#16863332C211yesmaximal, characteristic, complemented
4#16863319C212yesmaximal, complemented
5#16863308C211yesmaximal, characteristic, complemented
6#16863271C212yesmaximal, complemented
7#16863268C212yesmaximal, complemented
8#16863254C212yesmaximal, complemented
9#379×C67C211yesmaximal, characteristic
10C22×Dic134C2211yescharacteristic, complemented
11C23×Dic67C2212yes—
12C23×Dic67C2212yes—
13#14524020—28no—
14C2×C4⋊Dic67C2212yescomplemented
15C2×C4⋊Dic67C2211yescharacteristic, complemented
16C2×Dic67⋊C4C2212yescomplemented
17C2×Dic67⋊C4C2212yescomplemented
18C2×Dic67⋊C4C2212yescomplemented
19C2×C22⋊Dic67—28no—
20C2×C22⋊Dic67C2212yes—
21C2×C22⋊Dic67C2212yes—
22C2×C22⋊Dic67C2212yes—
23C2×C22⋊Dic67C2212yes—
24C67⋊#95C2211yescharacteristic, complemented
25C67⋊#95C2212yescomplemented
26C67⋊#95C2211yescharacteristic, complemented
27C67⋊#95C2212yescomplemented
28C25×C67C2211yescharacteristic
29C2×C22⋊C4×C67C2212yes—
30C2×C22⋊C4×C67C2212yes—
31C2×C22⋊C4×C67C2211yescharacteristic
32C2×C22⋊C4×C67C2211yescharacteristic
33C2×Dic134—24no—
34C2×Dic134—41no—
35C22×Dic67—24no—
36C22×Dic67—24no—
37C22×Dic67—24no—
38C22×Dic67—24no—
39C22×Dic67—42no—
40C22×Dic67—42no—
41C22×Dic67C2312yes—
42C22×Dic67C2312yes—
43C22×Dic67C2312yes—
44C22×Dic67C2312yes—
45C4⋊Dic67—24no—
46C4⋊Dic67—41no—
47Dic67⋊C4—24no—
48Dic67⋊C4—24no—
49Dic67⋊C4—42no—
50C22⋊Dic67—24no—
51C22⋊Dic67—24no—
52C22⋊Dic67—24no—
53C22⋊Dic67D418yes—
54C24×C67—24no—
55C24×C67—24no—
56C24×C67—41no—
57C24×C67—41no—
58C24×C67Q814yescomplemented
59C24×C67C2311yescharacteristic
60C24×C67C2312yes—
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2312yes—
63C22×C4×C67C2311yescharacteristic
64C22⋊C4×C67—24no—
65C22⋊C4×C67—24no—
66C22⋊C4×C67—41no—
67C22⋊C4×C67—41no—
68C23×C67—22no—
69C23×C67—22no—
70C23×C67—22no—
71C23×C67—22no—
72C23×C67—216no—
73C23×C67—22no—
74C23×C67—22no—
75C23×C67—21no—
76C23×C67—21no—
77C23×C67—44no—
78C23×C67—44no—
79C23×C67—48no—
80C23×C67—44no—
81C23×C67C2×D414yes—
82C23×C67C2×Q812yes—
83C23×C67C2×Q814yes—
84C23×C67#4114yes—
85C23×C67C2411yescharacteristic
86C2×C4×C67—22no—
87C2×C4×C67—22no—
88C2×C4×C67—22no—
89C2×C4×C67—22no—
90C2×C4×C67—22no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67—22no—
96C2×Dic67—22no—
97C2×Dic67—22no—
98C2×Dic67—22no—
99C2×Dic67—22no—
100C2×Dic67—22no—
101C2×Dic67—22no—
102C2×Dic67—22no—
103C2×Dic67—48no—
104C2×Dic67—48no—
105C2×Dic67C2×D414yes—
106C2×Dic67C2×D414yes—
107Dic134—44no—
108C22×C67—24no—
109C22×C67—24no—
110C22×C67—24no—
111C22×C67—24no—
112C22×C67—24no—
113C22×C67—24no—
114C22×C67—24no—
115C22×C67—22no—
116C22×C67—416no—
117C22×C67—41no—
118C22×C67—42no—
119C22×C67—41no—
120C22×C67C22⋊Q818yescomplemented
121C22×C67C22×D412yes—
122C22×C67C22×Q811yescharacteristic
123C22×C67C2×#4112yes—
124C22×C67H(5,𝔽2)11yescharacteristic
125C22×C67H(5,𝔽2)11yescharacteristic
126C4×C67—41no—
127C4×C67—42no—
128C4×C67—41no—
129Dic67—24no—
130Dic67—24no—
131Dic67—42no—
132Dic67—42no—
133C2×C67—28no—
134C2×C67—42no—
135C2×C67C2×C22⋊Q812yes—
136C2×C67#54311yescharacteristic
137C2×C67D4211yescharacteristic
138C2×C67#54612yes—
139C2×C67#55111yescharacteristic
140#2437—671nomaximal, Sylow
141C67#243711yescharacteristic, complemented, Sylow
142C2×C22⋊Q8—672no—
143C22×C22⋊C4—672no—
144#399—671no—
145#396—672no—
146#393—671no—
147#385—672no—
148#385—672no—
149#380—672no—
150#379D6711yescharacteristic
151C25C2×D6711yescharacteristic
152C22×Q8—671no—
153C23×C4—672no—
154C23×C4—672no—
155C22⋊Q8—1348no—
156C2×C4⋊C4—672no—
157C2×C4⋊C4—671no—
158C2×C4⋊C4—672no—
159C2×C4⋊C4—672no—
160C2×C4⋊C4—672no—
161C2×C22⋊C4—672no—
162C2×C22⋊C4—672no—
163C2×C22⋊C4—672no—
164C2×C22⋊C4—672no—
165C2×C22⋊C4—1348no—
166C2×C22⋊C4C2×D6712yes—
167C2×C22⋊C4C2×D6712yes—
168C2×C22⋊C4C2×D6711yescharacteristic
169C2×C22⋊C4C2×D6711yescharacteristic
170#95—671no—
171#95—672no—
172#95—672no—
173#95—671no—
174C24—24no—
175C24—24no—
176C24—41no—
177C24—41no—
178C24Dic13414yescomplemented
179C24C22×D6711yescharacteristic
180C24C22×D6712yes—
181C2×Q8—2681no—
182C2×Q8—1344no—
183C22×C4—2682no—
184C22×C4—2682no—
185C22×C4—672no—
186C22×C4—672no—
187C22×C4—672no—
188C22×C4—672no—
189C22×C4—1344no—
190C22×C4—1344no—
191C22×C4—1344no—
192C22×C4—1344no—
193C22×C4C22×D6711yescharacteristic
194C22×C4C22×D6712yes—
195C22×C4C22×D6711yescharacteristic
196C4⋊C4—2682no—
197C4⋊C4—2681no—
198C4⋊C4—1344no—
199C4⋊C4—1344no—
200C4⋊C4—1344no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—678no—
206C22⋊C4—1344no—
207C22⋊C4—1344no—
208C22⋊C4—1344no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—216no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—44no—
219C23—44no—
220C23—48no—
221C23—44no—
222C23D4×D6714yes—
223C23C2×Dic13412yes—
224C23C2×Dic13414yes—
225C23#1176162714yes—
226C23C23×D6711yescharacteristic
227Q8—2684no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—2688no—
236C2×C4—2688no—
237C2×C4—674no—
238C2×C4—674no—
239C2×C4—1342no—
240C2×C4—1342no—
241C2×C4—1342no—
242C2×C4—1342no—
243C2×C4—1342no—
244C2×C4—1342no—
245C2×C4—1342no—
246C2×C4—1342no—
247C2×C4—1342no—
248C2×C4—1342no—
249C22—24no—
250C22—24no—
251C22—24no—
252C22—24no—
253C22—24no—
254C22—24no—
255C22—22no—
256C22—24no—
257C22—416no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22#1452402018yescomplemented
262C22C2×D4×D6712yes—
263C22C22×Dic13411yescharacteristic
264C22C2×#1176162712yes—
265C22#1452408811yescharacteristic
266C22#1452408811yescharacteristic
267C4—41no—
268C4—42no—
269C4—41no—
270C4—2682no—
271C4—2682no—
272C4—1344no—
273C4—1344no—
274C2—28no—
275C2—42no—
276C2C2×#1452402012yes—
277C2#1686402211yescharacteristic
278C2#1686402811yescharacteristic
279C2#1686403312yes—
280C2#1686407711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#379×C67C2
hypercenter#379D67
hypocenterC67#2437
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#14524020C2semidirect
3. #16863332C2semidirect
4. #16863319C2semidirect
5. #16863308C2semidirect
6. #16863271C2semidirect
7. #16863268C2semidirect
8. #16863254C2semidirect
10. C22×Dic134C22semidirect
14. C2×C4⋊Dic67C22semidirect
15. C2×C4⋊Dic67C22semidirect
16. C2×Dic67⋊C4C22semidirect
17. C2×Dic67⋊C4C22semidirect
18. C2×Dic67⋊C4C22semidirect
24. C67⋊#95C22semidirect
25. C67⋊#95C22semidirect
26. C67⋊#95C22semidirect
27. C67⋊#95C22semidirect
58. C24×C67Q8semidirect
120. C22×C67C22⋊Q8semidirect
141. C67#2437semidirect
178. C24Dic134semidirect
261. C22#14524020semidirect