Finite Group Databasebeta

← #21449205

Subgroups — #21449205

Order 7808.

786 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14519745C211yesmaximal, characteristic
2C2×#14519736C211yesmaximal, characteristic, complemented
3C22×#11758198C211yesmaximal, characteristic, complemented
4#16839684C212yesmaximal, complemented
5#16839678C212yesmaximal, complemented
6#16839662C212yesmaximal, complemented
7#16839652C212yesmaximal, complemented
8#16839642C211yesmaximal, characteristic, complemented
9#16839634C211yesmaximal, characteristic, complemented
10#16839617C211yesmaximal, characteristic, complemented
11C2×#123×C61C211yesmaximal, characteristic, complemented
12C24×D61C2211yescharacteristic
13C22×C22⋊D61C2212yes—
14C23×Dic61C2211yescharacteristic
15#14519745—24no—
16#14519736—24no—
17C2×C4⋊Dic61C2212yescomplemented
18C2×#11758196C2212yescomplemented
19C2×C22⋊Dic61C2211yescharacteristic
20C2×C22⋊Dic61C2211yescharacteristic
21C2×C22⋊Dic61C2212yescomplemented
22C2×#11758198—24no—
23C2×#11758198C2212yes—
24C2×#11758198C2212yes—
25C2×#11758198C2212yes—
26C2×#11758198C2212yes—
27C2×#11758198C2212yes—
28C2×C4×Dic61C2212yescomplemented
29#14519684C2212yescomplemented
30#14519684C2212yescomplemented
31C22×D4×C61C2211yescharacteristic
32C23×C4×C61C2211yescharacteristic
33#123×C61—24no—
34C2×C4⋊C4×C61C2211yescharacteristic, complemented
35C2×C4⋊C4×C61C2211yescharacteristic, complemented
36C2×C22⋊C4×C61C2211yescharacteristic
37C2×C22⋊C4×C61C2211yescharacteristic, complemented
38C2×C22⋊C4×C61C2211yescharacteristic, complemented
39C24×C61C2311yescharacteristic
40C24×C61C2311yescharacteristic
41C23×D61—22no—
42C23×D61—24no—
43C23×D61—22no—
44C23×D61—22no—
45C23×D61—42no—
46C23×D61C2312yes—
47C2×D4×C61—22no—
48C2×D4×C61—22no—
49C2×D4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61D414yescomplemented
53C22×C4×C61C2311yescharacteristic
54C22×C4×C61C2311yescharacteristic
55C22×C4×C61C2311yescharacteristic
56C22×C4×C61C2311yescharacteristic
57C22×C4×C61C2311yescharacteristic
58C2×C22⋊D61—24no—
59C2×C22⋊D61—24no—
60C2×C22⋊D61—42no—
61C22×Dic61—22no—
62C22×Dic61—42no—
63C22×Dic61C2312yes—
64C22×Dic61C2312yes—
65C22×Dic61C2312yes—
66C4⋊C4×C61—22no—
67C4⋊C4×C61—22no—
68C22⋊C4×C61—22no—
69C22⋊C4×C61—22no—
70C22⋊C4×C61—22no—
71C22⋊Dic61—22no—
72C22⋊Dic61—42no—
73C22⋊Dic61D414yes—
74#11758198—24no—
75#11758198—24no—
76#11758198—24no—
77#11758198—24no—
78#11758198—42no—
79C4⋊Dic61—24no—
80#11758196—42no—
81C4×Dic61—24no—
82C23×C61—22no—
83C23×C61—22no—
84C23×C61—21no—
85C23×C61—21no—
86C23×C61—22no—
87C23×C61—21no—
88C23×C61—22no—
89C23×C61C2×D412yes—
90C23×C61C2×D412yes—
91C23×C61#4112yes—
92C23×C61C2411yescharacteristic
93C22×D61—28no—
94C22×D61—22no—
95C22×D61—22no—
96C22×D61—22no—
97C22×D61—22no—
98C22×D61—22no—
99C22×D61—22no—
100C22×D61—48no—
101C22×D61—48no—
102C22×D61—44no—
103C22×D61—44no—
104C22×D61#4114yes—
105D4×C61—44no—
106C2×C4×C61—28no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—22no—
116C2×C4×C61—21no—
117C2×C4×C61—22no—
118C2×C4×C61C2×D412yes—
119C2×C4×C61C2×D412yes—
120C22⋊D61—48no—
121C2×Dic61—22no—
122C2×Dic61—22no—
123C2×Dic61—22no—
124C2×Dic61—22no—
125C2×Dic61—22no—
126C2×Dic61—22no—
127C2×Dic61—22no—
128C2×Dic61—22no—
129C2×Dic61—44no—
130C2×Dic61C2×D414yes—
131C22×C61—22no—
132C22×C61—22no—
133C22×C61—24no—
134C22×C61—24no—
135C22×C61—41no—
136C22×C61—41no—
137C22×C61—41no—
138C22×C61—42no—
139C22×C61—42no—
140C22×C61#12114yes—
141C22×C61C22×D411yescharacteristic
142C22×C61C22×D411yescharacteristic
143C22×C61C2×#4112yes—
144C22×C61C2×#4111yescharacteristic
145C22×C61H(5,𝔽2)12yes—
146C2×D61—24no—
147C2×D61—24no—
148C2×D61—24no—
149C2×D61—416no—
150C2×D61—42no—
151C2×D61—42no—
152C2×D61—42no—
153C2×D61—42no—
154C4×C61—22no—
155C4×C61—22no—
156C4×C61—41no—
157C4×C61—41no—
158C4×C61—41no—
159Dic61—24no—
160Dic61—42no—
161Dic61—42no—
162#2411—611nomaximal, Sylow
163C2×C61—24no—
164C2×C61—81no—
165C2×C61C2×#12111yescharacteristic
166C2×C61#52511yescharacteristic
167C2×C61#53511yescharacteristic
168C2×C61#54612yes—
169C2×C61#55112yes—
170D61—44no—
171C2×#123D6111yescharacteristic, complemented
172C2×#120—611no—
173C2×#117—611no—
174C22×C22⋊C4—611no—
175#396—612no—
176#394—612no—
177#390—612no—
178#388—612no—
179#386—611no—
180#385—611no—
181#379—611no—
182C61#241111yescharacteristic, complemented, Sylow
183C25—611no—
184C22×D4—612no—
185C22×D4C2×D6111yescharacteristic
186C23×C4—611no—
187C23×C4C2×D6111yescharacteristic
188#123—24no—
189#120—1224no—
190#117—1224no—
191C2×C4⋊C4—612no—
192C2×C4⋊C4—612no—
193C2×C4⋊C4C2×D6111yescharacteristic, complemented
194C2×C4⋊C4C2×D6111yescharacteristic, complemented
195C2×C22⋊C4—611no—
196C2×C22⋊C4—611no—
197C2×C22⋊C4—612no—
198C2×C22⋊C4—612no—
199C2×C22⋊C4—612no—
200C2×C22⋊C4—612no—
201C2×C22⋊C4—612no—
202C2×C22⋊C4—612no—
203C2×C22⋊C4—1224no—
204C2×C22⋊C4C2×D6111yescharacteristic
205C2×C22⋊C4C2×D6111yescharacteristic, complemented
206C2×C22⋊C4C2×D6111yescharacteristic, complemented
207C2×C42—612no—
208#95—612no—
209#95—612no—
210C24—612no—
211C24—1224no—
212C24—1222no—
213C24—1222no—
214C24—1222no—
215C24—2442no—
216C24C22×D6111yescharacteristic
217C24C22×D6111yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—1224no—
222C2×D4—1224no—
223C2×D4—2442no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—612no—
227C22×C4—612no—
228C22×C4—612no—
229C22×C4—1222no—
230C22×C4—2442no—
231C22×C4C22⋊D6114yescomplemented
232C22×C4C22×D6111yescharacteristic
233C22×C4C22×D6111yescharacteristic
234C22×C4C22×D6111yescharacteristic
235C22×C4C22×D6111yescharacteristic
236C22×C4C22×D6111yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—1224no—
240C4⋊C4—2442no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—614no—
245C22⋊C4—1224no—
246C22⋊C4—1224no—
247C22⋊C4—1222no—
248C22⋊C4—1224no—
249C22⋊C4—1224no—
250C22⋊C4—2442no—
251C22⋊C4—2442no—
252C42—1224no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—21no—
259C23—22no—
260C23—614no—
261C23—1228no—
262C23—1222no—
263C23—1222no—
264C23—1222no—
265C23—1222no—
266C23—1222no—
267C23—1222no—
268C23—2448no—
269C23—2448no—
270C23—2444no—
271C23—2444no—
272C23D4×D6112yes—
273C23#1175822412yes—
274C23C2×C22⋊D6112yes—
275C23C23×D6111yescharacteristic
276D4—44no—
277D4—2448no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—28no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—614no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1222no—
294C2×C4—1222no—
295C2×C4—1222no—
296C2×C4—1222no—
297C2×C4—1222no—
298C2×C4—1222no—
299C2×C4—2444no—
300C2×C4C2×C22⋊D6112yes—
301C2×C4C2×C22⋊D6112yes—
302C22—22no—
303C22—22no—
304C22—24no—
305C22—24no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—42no—
310C22—42no—
311C22—1224no—
312C22—1224no—
313C22—1224no—
314C22—24416no—
315C22—2442no—
316C22—2442no—
317C22—2442no—
318C22—2442no—
319C22#1451974914yes—
320C22C2×D4×D6111yescharacteristic
321C22C22×C22⋊D6111yescharacteristic
322C22#41×D6112yes—
323C22C2×#1175822411yescharacteristic
324C22#1451982312yes—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—1224no—
331C4—2442no—
332C4—2442no—
333C2—24no—
334C2—81no—
335C2—2444no—
336C2C2×#1451974911yescharacteristic
337C2#1684026911yescharacteristic
338C2#1684032611yescharacteristic
339C2#1684040412yes—
340C2#1684043512yes—
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC2×#123×C61C2
hypercenterC2×#123D61
hypocenterC61#2411
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#14519736C2semidirect
3. C22×#11758198C2semidirect
4. #16839684C2semidirect
5. #16839678C2semidirect
6. #16839662C2semidirect
7. #16839652C2semidirect
8. #16839642C2semidirect
9. #16839634C2semidirect
10. #16839617C2semidirect
11. C2×#123×C61C2semidirect
17. C2×C4⋊Dic61C22semidirect
18. C2×#11758196C22semidirect
21. C2×C22⋊Dic61C22semidirect
28. C2×C4×Dic61C22semidirect
29. #14519684C22semidirect
30. #14519684C22semidirect
34. C2×C4⋊C4×C61C22semidirect
35. C2×C4⋊C4×C61C22semidirect
37. C2×C22⋊C4×C61C22semidirect
38. C2×C22⋊C4×C61C22semidirect
52. C22×C4×C61D4semidirect
171. C2×#123D61semidirect
182. C61#2411semidirect
193. C2×C4⋊C4C2×D61semidirect
194. C2×C4⋊C4C2×D61semidirect
205. C2×C22⋊C4C2×D61semidirect
206. C2×C22⋊C4C2×D61semidirect
231. C22×C4C22⋊D61semidirect