Finite Group Databasebeta

← #29656280

Subgroups — #29656280

Order 15616.

1068 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16840478C211yesmaximal, characteristic, complemented
2#21455906C212yesmaximal, complemented
3#21444962C214yesmaximal, complemented
4#21444900C214yesmaximal, complemented
5#21444545C211yesmaximal, characteristic, complemented
6#21444263C212yesmaximal, complemented
7C61×#3152C211yesmaximal, characteristic, complemented
8#16840478—24no—
9#16840478—22no—
10#16840478C2214yes—
11C2×#14519766—22no—
12C2×#14519752—22no—
13C2×C4×D244C2212yes—
14C2×#14519735—22no—
15#16840078—22no—
16#16840049C2214yescomplemented
17#16839954C2214yescomplemented
18#16839895—22no—
19#16839872—24no—
20#16839819C2212yes—
21#16839511C2214yescomplemented
22#16839491C2218yescomplemented
23C2×#137×C61—22no—
24C61×#532C2211yescharacteristic
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#497—22no—
27C61×#494C2212yescomplemented
28C61×#492C2212yescomplemented
29C4×M16×C61C2211yescharacteristic
30C22×D244—22no—
31C22×D244D412yes—
32C22×C4×D61—22no—
33C2×#11758199—24no—
34#14519766—28no—
35#14519766—28no—
36#14519752—28no—
37#14519752—28no—
38C4×D244—22no—
39C4×D244—22no—
40C4×D244D414yes—
41C4×D244C2314yes—
42#14519735—28no—
43#14519735—28no—
44C2×C4⋊Dic61D412yes—
45C2×#11758198—24no—
46C2×#11758198—22no—
47C2×C4×Dic61—22no—
48#14519708—24no—
49C4⋊#92418C2314yes—
50#14519698—24no—
51#14519697—24no—
52C22⋊#92418—22no—
53#14519688—22no—
54C2×#41×C61—21no—
55C22×Q8×C61D412yes—
56#137×C61—48no—
57C2×Q16×C61—24no—
58C2×SD16×C61—24no—
59C2×M16×C61—21no—
60C4⋊Q8×C61—22no—
61C4⋊Q8×C61D412yes—
62C4⋊Q8×C61C2311yescharacteristic
63C4⋊Q8×C61C2312yes—
64#127×C61C2311yescharacteristic
65#124×C61—21no—
66#121×C61—22no—
67C4×D4×C61—22no—
68C2×C4⋊C4×C61—22no—
69C2×C42×C61C2311yescharacteristic
70#97×C61—21no—
71C4×C8×C61C2312yes—
72C23×D61—22no—
73#41×C61—44no—
74C2×Q8×C61—22no—
75C2×Q8×C61—21no—
76C2×Q8×C61—44no—
77C2×Q8×C61C2×D414yes—
78C2×D4×C61—22no—
79C2×D4×C61—21no—
80C2×D4×C61—41no—
81C22×C4×C61—22no—
82C22×C4×C61—41no—
83C22×C4×C61C2×D412yes—
84C22×C4×C61C2×D411yescharacteristic
85C22×Dic61—22no—
86C2×D244—22no—
87C2×D244—28no—
88C2×D244—44no—
89C2×D244—84no—
90C2×D244C2×D414yes—
91C2×C4×D61—24no—
92C2×C4×D61—24no—
93C2×C4×D61—48no—
94Q16×C61—48no—
95SD16×C61—48no—
96M16×C61—24no—
97C2×C8×C61—22no—
98C4⋊C4×C61—24no—
99C4⋊C4×C61—24no—
100C4⋊C4×C61—22no—
101C22⋊C4×C61—42no—
102C42×C61C2×D412yes—
103C42×C61C2×D411yescharacteristic
104C42×C61C2411yescharacteristic
105#11758199—48no—
106#11758198—216no—
107#11758198—24no—
108#11758198—24no—
109C4⋊Dic61—22no—
110C4⋊Dic61C2×D414yes—
111C4×Dic61—28no—
112C2×#92418—24no—
113C23×C61—41no—
114C23×C61#12011yescharacteristic
115C22×D61—24no—
116C22×D61—44no—
117C22×D61—44no—
118C22×D61—44no—
119Q8×C61—44no—
120Q8×C61—42no—
121Q8×C61—82no—
122D4×C61—42no—
123D4×C61—44no—
124D4×C61—82no—
125C2×C4×C61—22no—
126C2×C4×C61—24no—
127C2×C4×C61—24no—
128C2×C4×C61—44no—
129C2×C4×C61—41no—
130C2×C4×C61—82no—
131C2×C4×C61#12012yes—
132C2×C4×C61#12011yescharacteristic
133C2×C4×C61C22×D411yescharacteristic
134C2×C4×C61C22×D412yes—
135C2×Dic61—24no—
136C2×Dic61—44no—
137D244—24no—
138D244—24no—
139D244—44no—
140D244—88no—
141C4×D61—48no—
142C4×D61—48no—
143C8×C61—24no—
144#92418—44no—
145#18547—611nomaximal, Sylow
146C22×C61—22no—
147C22×C61—82no—
148C22×C61—81no—
149C22×C61#45312yes—
150C22×C61C2×#12011yescharacteristic
151C2×D61—44no—
152C2×D61—44no—
153C2×D61—44no—
154C2×D61—88no—
155C4×C61—22no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—44no—
159C4×C61—81no—
160C4×C61#57514yes—
161Dic61—44no—
162C2×#555—611no—
163#3152D6111yescharacteristic, complemented
164#3109—612no—
165#1634—614no—
166#1624—614no—
167#1558—611no—
168#1490—612no—
169C2×C61—22no—
170C2×C61—161no—
171C2×C61#300512yes—
172C2×C61C2×#45311yescharacteristic
173D61—84no—
174C2×#137—22no—
175#555—614no—
176#555—1224no—
177#555—1222no—
178#532C2×D6111yescharacteristic
179C2×C4⋊Q8C2×D6111yescharacteristic
180C2×#124—1222no—
181C2×#121—1222no—
182C2×C4×D4—612no—
183C2×#117—1222no—
184#497—22no—
185#494C2×D6112yescomplemented
186#492C2×D6112yescomplemented
187#481—1222no—
188#476—614no—
189#459—614no—
190#447—1222no—
191#442—1224no—
192#432—612no—
193C4×M16C2×D6111yescharacteristic
194#333—614no—
195D4⋊C8—618no—
196C61#1854711yescharacteristic, complemented, Sylow
197C2×#41—21no—
198C22×Q8C22⋊D6112yes—
199C22×D4—612no—
200C22×D4—1222no—
201C23×C4—1222no—
202#137—48no—
203C2×Q16—24no—
204C2×SD16—24no—
205C2×D8—1224no—
206C2×M16—21no—
207C4⋊Q8—22no—
208C4⋊Q8C22⋊D6112yes—
209C4⋊Q8C22×D6111yescharacteristic
210C4⋊Q8C22×D6112yes—
211#127C22×D6111yescharacteristic
212#124—21no—
213#124—1228no—
214#124—1228no—
215#121—22no—
216#121—1228no—
217#121—1228no—
218C4×D4—22no—
219C4×D4—614no—
220C4×D4—614no—
221C4×D4—1222no—
222C4×D4—1222no—
223#117—1228no—
224#117—1228no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—612no—
227C2×C22⋊C4—1224no—
228C2×C22⋊C4—1222no—
229C2×C42—1222no—
230C2×C42C22×D6111yescharacteristic
231#106—1224no—
232C4⋊C8—614no—
233#102—1224no—
234#102—1224no—
235C22⋊C8—1222no—
236#97—21no—
237#97—1222no—
238C4×C8C22×D6112yes—
239C24—1222no—
240#41—44no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—44no—
244C2×Q8C2×C22⋊D6114yes—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—614no—
249C2×D4—1228no—
250C2×D4—1222no—
251C2×D4—4884no—
252C2×D4—2444no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—1222no—
256C22×C4—1224no—
257C22×C4—1224no—
258C22×C4—2448no—
259C22×C4D4×D6112yes—
260C22×C4C2×C22⋊D6111yescharacteristic
261Q16—48no—
262SD16—48no—
263D8—2448no—
264M16—24no—
265C2×C8—22no—
266C2×C8—1224no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—614no—
271C4⋊C4—1222no—
272C22⋊C4—42no—
273C22⋊C4—12216no—
274C22⋊C4—1224no—
275C22⋊C4—1224no—
276C42—1228no—
277C42D4×D6112yes—
278C42C2×C22⋊D6111yescharacteristic
279C42C23×D6111yescharacteristic
280C23—41no—
281C23—1224no—
282C23—2444no—
283C23—2444no—
284C23—2444no—
285C23#1451974511yescharacteristic
286Q8—44no—
287Q8—42no—
288Q8—82no—
289D4—42no—
290D4—44no—
291D4—82no—
292D4—1224no—
293D4—1224no—
294D4—4888no—
295D4—2444no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—1224no—
303C2×C4—2448no—
304C2×C4—2444no—
305C2×C4—2448no—
306C2×C4#1451974511yescharacteristic
307C2×C4#1451974512yes—
308C2×C4C2×D4×D6112yes—
309C2×C4C22×C22⋊D6111yescharacteristic
310C8—24no—
311C8—2444no—
312C22—22no—
313C22—81no—
314C22—82no—
315C22—4888no—
316C22—2444no—
317C22—2444no—
318C22—2444no—
319C22#1683992912yes—
320C22C2×#1451974511yescharacteristic
321C4—21no—
322C4—21no—
323C4—22no—
324C4—44no—
325C4—81no—
326C4—2444no—
327C4#1684056114yes—
328C2—22no—
329C2—161no—
330C2—4884no—
331C2#2145506012yes—
332C2C2×#1683992911yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3152C2
socleC22×C61C2×#120
hypercenter#3152D61
hypocenterC61#18547
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#16840478C2semidirect
2. #21455906C2semidirect
3. #21444962C2semidirect
4. #21444900C2semidirect
5. #21444545C2semidirect
6. #21444263C2semidirect
7. C61×#3152C2semidirect
16. #16840049C22semidirect
17. #16839954C22semidirect
21. #16839511C22semidirect
22. #16839491C22semidirect
27. C61×#494C22semidirect
28. C61×#492C22semidirect
163. #3152D61semidirect
185. #494C2×D61semidirect
186. #492C2×D61semidirect
196. C61#18547semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168404782
#214559062
#214449622
#214449002
#214445452
#214442632
C61×#31522
#1854761