Finite Group Databasebeta

← #21455218

Subgroups — #21455218

Order 7808.

800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14519822C211yesmaximal, characteristic, complemented
2#16840324C212yesmaximal, complemented
3#16840321C211yesmaximal, characteristic, complemented
4#16839953C214yesmaximal, complemented
5#16839950C214yesmaximal, complemented
6C2×#14519691C212yesmaximal, complemented
7C2×#99×C61C211yesmaximal, characteristic, complemented
8#14519822—22no—
9#14519822—24no—
10#14519822C2214yes—
11C2×#11758222—21no—
12C2×#11758224—22no—
13C22×C22⋊D61—21no—
14C22×C22⋊D61C2212yes—
15C22×C22⋊D61C2211yescharacteristic, complemented
16C2×D4×D61—22no—
17#14519761—22no—
18#14519761C2214yescomplemented
19#14519760—21no—
20#14519758C2212yescomplemented
21#14519751—24no—
22#14519749—22no—
23#14519747—22no—
24#14519747C2214yescomplemented
25#14519746—21no—
26#14519745C2212yescomplemented
27C2×C22⋊Dic61C2211yescharacteristic, complemented
28#14519691C2218yescomplemented
29C22×D4×C61C2211yescharacteristic
30C2×C22⋊C4×C61C2212yescomplemented
31#99×C61C2214yescomplemented
32C24×C61D411yescharacteristic, complemented
33C24×C61C2311yescharacteristic
34C23×D61—21no—
35C23×D61D412yescomplemented
36C23×D61C2311yescharacteristic, complemented
37C2×D4×C61—22no—
38C2×D4×C61—22no—
39C2×D4×C61D412yescomplemented
40C2×D4×C61C2312yes—
41C22×C4×C61—22no—
42C22×C4×C61D411yescharacteristic, complemented
43C2×C22⋊D61—22no—
44C2×C22⋊D61—24no—
45C2×C22⋊D61—22no—
46C2×C22⋊D61—22no—
47C2×C22⋊D61—24no—
48C2×C22⋊D61—22no—
49C2×C22⋊D61—22no—
50C2×C22⋊D61—21no—
51C2×C22⋊D61—22no—
52C2×C22⋊D61—21no—
53C2×C22⋊D61—22no—
54C2×C22⋊D61—22no—
55C2×C22⋊D61D414yescomplemented
56C2×C22⋊D61C2314yes—
57C22×Dic61—21no—
58C22×Dic61—21no—
59C22×Dic61D412yescomplemented
60C22×Dic61C2311yescharacteristic, complemented
61#11758224—28no—
62#11758224—28no—
63D4×D61—28no—
64D4×D61—28no—
65#11758222—24no—
66#11758222—24no—
67C2×D244—21no—
68C2×D244—24no—
69C2×C4×D61—22no—
70C2×Dic122—21no—
71C22⋊C4×C61—22no—
72C22⋊C4×C61C2314yescomplemented
73C22⋊Dic61—21no—
74C22⋊Dic61—22no—
75C22⋊Dic61C2312yescomplemented
76#11758198—24no—
77#11758198—22no—
78#11758198—24no—
79C4⋊Dic61—24no—
80#11758196—22no—
81C23×C61—21no—
82C23×C61—21no—
83C23×C61—81no—
84C23×C61—81no—
85C23×C61C2×D412yes—
86C23×C61C2×D411yescharacteristic
87C23×C61C2×D411yescharacteristic
88C23×C61C2×D412yes—
89C23×C61C2×D412yes—
90C23×C61C2411yescharacteristic
91C22×D61—22no—
92C22×D61—24no—
93C22×D61—22no—
94C22×D61—22no—
95C22×D61—21no—
96C22×D61—81no—
97C22×D61—82no—
98C22×D61—82no—
99C22×D61C2×D414yes—
100D4×C61—24no—
101D4×C61—24no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—24no—
105C2×C4×C61—82no—
106C2×C4×C61C2×D412yes—
107C22⋊D61—28no—
108C22⋊D61—24no—
109C22⋊D61—24no—
110C22⋊D61—24no—
111C22⋊D61—24no—
112C22⋊D61—41no—
113C22⋊D61—42no—
114C22⋊D61—41no—
115C22⋊D61—82no—
116C2×Dic61—24no—
117C2×Dic61—22no—
118C2×Dic61—22no—
119C2×Dic61—21no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—81no—
123C2×Dic61C2×D414yes—
124D244—24no—
125D244—84no—
126C4×D61—28no—
127Dic122—24no—
128C22×C61—22no—
129C22×C61—41no—
130C22×C61—42no—
131C22×C61—42no—
132C22×C61—41no—
133C22×C61—41no—
134C22×C61—42no—
135C22×C61—42no—
136C22×C61—81no—
137C22×C61#12011yescharacteristic
138C22×C61#12012yes—
139C22×C61#12011yescharacteristic
140C22×C61C22×D412yes—
141C22×C61C22×D411yescharacteristic
142C2×D61—24no—
143C2×D61—44no—
144C2×D61—44no—
145C2×D61—44no—
146C2×D61—42no—
147C2×D61—41no—
148C2×D61—82no—
149C2×D61—84no—
150C4×C61—22no—
151C4×C61—44no—
152Dic61—24no—
153Dic61—42no—
154Dic61—41no—
155#3026—611nomaximal, Sylow
156C2×C61—21no—
157C2×C61—22no—
158C2×C61—21no—
159C2×C61—22no—
160C2×C61—21no—
161C2×C61—42no—
162C2×C61C2×#12011yescharacteristic
163D61—44no—
164D61—81no—
165C2×H(5,𝔽2)—611no—
166#534—611no—
167#534—612no—
168#458—614no—
169B(4,𝔽2)—614no—
170C2×#99—612no—
171C2×#99D6111yescharacteristic, complemented
172C61#302611yescharacteristic, complemented, Sylow
173H(5,𝔽2)—614no—
174H(5,𝔽2)—1224no—
175H(5,𝔽2)—1222no—
176C2×#41—1222no—
177C2×#41—1221no—
178C22×D4—612no—
179C22×D4—611no—
180C22×D4—1222no—
181C22×D4—1221no—
182C22×D4C2×D6111yescharacteristic
183#123—612no—
184#123—614no—
185#123—1222no—
186#123—1221no—
187#121—1224no—
188#121—1222no—
189#120—612no—
190#120—614no—
191#120—1222no—
192#120—1221no—
193C2×C22⋊C4—611no—
194C2×C22⋊C4C2×D6112yescomplemented
195#99—618no—
196#99C2×D6114yescomplemented
197C24—612no—
198C24—611no—
199C24—1221no—
200C24D24411yescharacteristic, complemented
201C24C22×D6111yescharacteristic
202#41—1224no—
203#41—1228no—
204#41—1228no—
205#41—1224no—
206C2×Q8—1221no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—614no—
210C2×D4—614no—
211C2×D4—1221no—
212C2×D4—1224no—
213C2×D4—1228no—
214C2×D4—1228no—
215C2×D4—1222no—
216C2×D4—1224no—
217C2×D4—1222no—
218C2×D4—1222no—
219C2×D4—1224no—
220C2×D4—1222no—
221C2×D4—1222no—
222C2×D4—1222no—
223C2×D4—1221no—
224C2×D4—1221no—
225C2×D4—1222no—
226C2×D4—1222no—
227C2×D4D24412yescomplemented
228C2×D4C22×D6112yes—
229C22×C4—22no—
230C22×C4—612no—
231C22×C4—611no—
232C22×C4—1222no—
233C22×C4—1221no—
234C22×C4—1221no—
235C22×C4D24411yescharacteristic, complemented
236C4⋊C4—1224no—
237C4⋊C4—1222no—
238C22⋊C4—22no—
239C22⋊C4—612no—
240C22⋊C4—1224no—
241C22⋊C4—1224no—
242C22⋊C4—1221no—
243C22⋊C4—1222no—
244C22⋊C4—1222no—
245C22⋊C4C22×D6114yescomplemented
246C23—21no—
247C23—21no—
248C23—81no—
249C23—81no—
250C23—614no—
251C23—1224no—
252C23—1222no—
253C23—1222no—
254C23—1222no—
255C23—1221no—
256C23—4882no—
257C23—4882no—
258C23—4881no—
259C23C2×D24412yes—
260C23C2×D24411yescharacteristic
261C23C2×D24411yescharacteristic
262C23D4×D6112yes—
263C23D4×D6112yes—
264C23C23×D6111yescharacteristic
265Q8—1224no—
266D4—24no—
267D4—24no—
268D4—1224no—
269D4—1228no—
270D4—1224no—
271D4—1224no—
272D4—1224no—
273D4—1224no—
274D4—4884no—
275D4—4882no—
276D4—2441no—
277D4—2441no—
278D4—2442no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—82no—
283C2×C4—614no—
284C2×C4—1228no—
285C2×C4—1224no—
286C2×C4—1222no—
287C2×C4—1222no—
288C2×C4—1222no—
289C2×C4—1222no—
290C2×C4—1221no—
291C2×C4—4881no—
292C2×C4C2×D24412yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—1224no—
303C22—4882no—
304C22—4884no—
305C22—2444no—
306C22—2442no—
307C22—2444no—
308C22—2444no—
309C22—2441no—
310C22#1451974711yescharacteristic
311C22#1451974711yescharacteristic
312C22#1451974712yes—
313C22C2×D4×D6112yes—
314C22C22×D24411yescharacteristic
315C4—22no—
316C4—44no—
317C4—1224no—
318C4—2442no—
319C4—2441no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—4881no—
327C2—2444no—
328C2C2×#1451974711yescharacteristic
subgroupG/H
centerC2C2×#14519747
commutatorC23×C61C24
FrattiniC23C23×D61
FittingC2×#99×C61C2
socleC2×C61C2×#120
hypercenterC2×#99D61
hypocenterC61#3026
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#14519822C2semidirect
2. #16840324C2semidirect
3. #16840321C2semidirect
4. #16839953C2semidirect
5. #16839950C2semidirect
6. C2×#14519691C2semidirect
7. C2×#99×C61C2semidirect
15. C22×C22⋊D61C22semidirect
18. #14519761C22semidirect
20. #14519758C22semidirect
24. #14519747C22semidirect
26. #14519745C22semidirect
27. C2×C22⋊Dic61C22semidirect
28. #14519691C22semidirect
30. C2×C22⋊C4×C61C22semidirect
31. #99×C61C22semidirect
32. C24×C61D4semidirect
35. C23×D61D4semidirect
36. C23×D61C23semidirect
39. C2×D4×C61D4semidirect
42. C22×C4×C61D4semidirect
55. C2×C22⋊D61D4semidirect
59. C22×Dic61D4semidirect
60. C22×Dic61C23semidirect
72. C22⋊C4×C61C23semidirect
75. C22⋊Dic61C23semidirect
171. C2×#99D61semidirect
172. C61#3026semidirect
194. C2×C22⋊C4C2×D61semidirect
196. #99C2×D61semidirect
200. C24D244semidirect
227. C2×D4D244semidirect
235. C22×C4D244semidirect
245. C22⋊C4C22×D61semidirect