Finite Group Databasebeta

← #29656282

Subgroups — #29656282

Order 15616.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 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
1#21456276C211yesmaximal, characteristic, complemented
2#21455910C212yesmaximal, complemented
3#21444964C214yesmaximal, complemented
4#21444902C214yesmaximal, complemented
5#21444545C211yesmaximal, characteristic, complemented
6#21444262C212yesmaximal, complemented
7C2×C61×#555C211yesmaximal, characteristic, complemented
8C2×#14519808—22no—
9#16840312C2211yescharacteristic
10#16840154—22no—
11#16840141C2212yescomplemented
12#16840136C2212yescomplemented
13#16840080—22no—
14#16840052C2214yescomplemented
15#16839957C2214yescomplemented
16#16839896—22no—
17#16839872—24no—
18#16839819C2212yes—
19C4×#11758194C2211yescharacteristic
20#16839511C2214yescomplemented
21D4⋊#92418C2218yescomplemented
22C61×#555—22no—
23C61×#555—24no—
24C61×#555C2214yes—
25C2×C4⋊Q8×C61C2211yescharacteristic
26C2×#124×C61—22no—
27C2×#121×C61—22no—
28C2×C4×D4×C61C2212yes—
29C2×#117×C61—22no—
30C2×#11758222—21no—
31#14519808—48no—
32C2×#11758202—24no—
33C2×#11758201—24no—
34C2×#11758199—24no—
35C2×#11758194—21no—
36#14519777C2311yescharacteristic
37#14519774C2311yescharacteristic
38#14519765—21no—
39#14519751—22no—
40C4×D244—22no—
41#14519708—24no—
42C4⋊#92418C2314yes—
43#14519698—24no—
44#14519697—24no—
45C22⋊#92418—22no—
46#14519688—21no—
47#14519688—22no—
48C4×#92418C2312yes—
49C22×Q8×C61D412yes—
50C22×D4×C61—22no—
51C22×D4×C61D412yes—
52C23×C4×C61—22no—
53C4⋊Q8×C61—22no—
54C4⋊Q8×C61D412yes—
55C4⋊Q8×C61C2312yes—
56#124×C61—28no—
57#124×C61—28no—
58#121×C61—28no—
59#121×C61—28no—
60C4×D4×C61—22no—
61C4×D4×C61—22no—
62C4×D4×C61D414yes—
63C4×D4×C61C2314yes—
64#117×C61—28no—
65#117×C61—28no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61D412yes—
68C2×C22⋊C4×C61—24no—
69C2×C22⋊C4×C61—22no—
70C2×C42×C61—22no—
71C2×C42×C61C2311yescharacteristic
72C24×C61—22no—
73C2×Q8×C61—22no—
74C2×Q8×C61—44no—
75C2×Q8×C61C2×D414yes—
76C2×D4×C61—28no—
77C2×D4×C61—22no—
78C2×D4×C61—44no—
79C2×D4×C61—84no—
80C2×D4×C61C2×D414yes—
81C22×C4×C61—22no—
82C22×C4×C61—22no—
83C22×C4×C61—24no—
84C22×C4×C61—24no—
85C22×C4×C61—48no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D412yes—
88C2×C22⋊D61—41no—
89#11758222—44no—
90C2×D244—21no—
91C2×D244—22no—
92C2×C4×D61—41no—
93C2×Dic122—21no—
94C4⋊C4×C61—24no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61C2×D414yes—
98C22⋊C4×C61—216no—
99C22⋊C4×C61—24no—
100C22⋊C4×C61—24no—
101C42×C61—28no—
102C42×C61C2×D411yescharacteristic
103C42×C61C2×D412yes—
104C42×C61C2411yescharacteristic
105#11758202—48no—
106#11758201—48no—
107#11758199—48no—
108#11758198—42no—
109C4⋊Dic61—22no—
110#11758194—24no—
111C2×#92418—22no—
112C2×#92418—24no—
113C23×C61—24no—
114C23×C61—44no—
115C23×C61—44no—
116C23×C61—44no—
117C23×C61#12011yescharacteristic
118C22×D61—41no—
119Q8×C61—44no—
120Q8×C61—82no—
121D4×C61—24no—
122D4×C61—24no—
123D4×C61—44no—
124D4×C61—88no—
125C2×C4×C61—22no—
126C2×C4×C61—24no—
127C2×C4×C61—24no—
128C2×C4×C61—24no—
129C2×C4×C61—48no—
130C2×C4×C61—44no—
131C2×C4×C61—44no—
132C2×C4×C61—48no—
133C2×C4×C61#12012yes—
134C2×C4×C61#12011yescharacteristic
135C2×C4×C61C22×D411yescharacteristic
136C2×C4×C61C22×D412yes—
137C22⋊D61—82no—
138C2×Dic61—41no—
139D244—42no—
140D244—44no—
141C4×D61—82no—
142Dic122—42no—
143#92418—24no—
144#92418—44no—
145#18547—611nomaximal, Sylow
146C22×C61—22no—
147C22×C61—44no—
148C22×C61—44no—
149C22×C61—44no—
150C22×C61—88no—
151C22×C61#45312yes—
152C22×C61C2×#12011yescharacteristic
153C2×D61—82no—
154C2×D61—81no—
155C4×C61—22no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—44no—
159C4×C61—44no—
160C4×C61#57514yes—
161Dic61—81no—
162C2×#555D6111yescharacteristic, complemented
163#3152—611no—
164#3109—612no—
165#1634—614no—
166#1624—614no—
167#1558—611no—
168#1490—612no—
169C2×C61—22no—
170C2×C61—84no—
171C2×C61#300512yes—
172C2×C61C2×#45311yescharacteristic
173D61—161no—
174C2×#137—1222no—
175#555—22no—
176#555—24no—
177#555C2×D6114yes—
178#532—611no—
179C2×C4⋊Q8C2×D6111yescharacteristic
180C2×#124—22no—
181C2×#121—22no—
182C2×C4×D4C2×D6112yes—
183C2×#117—22no—
184#497—1222no—
185#494—612no—
186#492—612no—
187#481—1222no—
188#476—614no—
189#459—614no—
190#447—1222no—
191#442—1224no—
192#432—612no—
193C4×M16—611no—
194#333—614no—
195D4⋊C8—618no—
196C61#1854711yescharacteristic, complemented, Sylow
197C2×#41—1221no—
198C22×Q8C22⋊D6112yes—
199C22×D4—22no—
200C22×D4C22⋊D6112yes—
201C23×C4—22no—
202#137—2448no—
203C2×Q16—1224no—
204C2×SD16—1224no—
205C2×D8—1224no—
206C2×M16—1221no—
207C4⋊Q8—22no—
208C4⋊Q8—611no—
209C4⋊Q8C22⋊D6112yes—
210C4⋊Q8C22×D6112yes—
211#127—611no—
212#124—28no—
213#124—28no—
214#124—1221no—
215#121—28no—
216#121—28no—
217#121—1222no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—1222no—
221C4×D4C22⋊D6114yes—
222C4×D4C22×D6114yes—
223#117—28no—
224#117—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C22⋊D6112yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C42—22no—
230C2×C42C22×D6111yescharacteristic
231#106—1224no—
232C4⋊C8—614no—
233#102—1224no—
234#102—1224no—
235C22⋊C8—1222no—
236#97—1221no—
237#97—1222no—
238C4×C8—612no—
239C24—22no—
240#41—2444no—
241C2×Q8—22no—
242C2×Q8—44no—
243C2×Q8—1221no—
244C2×Q8C2×C22⋊D6114yes—
245C2×D4—28no—
246C2×D4—22no—
247C2×D4—44no—
248C2×D4—84no—
249C2×D4—1222no—
250C2×D4—1221no—
251C2×D4—2441no—
252C2×D4C2×C22⋊D6114yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—48no—
258C22×C4—2441no—
259C22×C4C2×C22⋊D6112yes—
260C22×C4C2×C22⋊D6111yescharacteristic
261Q16—2448no—
262SD16—2448no—
263D8—2448no—
264M16—1224no—
265C2×C8—1222no—
266C2×C8—1224no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—1222no—
271C4⋊C4C2×C22⋊D6114yes—
272C22⋊C4—216no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—2442no—
276C42—28no—
277C42C2×C22⋊D6111yescharacteristic
278C42C2×C22⋊D6112yes—
279C42C23×D6111yescharacteristic
280C23—24no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—2441no—
285C23#1451974611yescharacteristic
286Q8—44no—
287Q8—82no—
288Q8—2442no—
289D4—24no—
290D4—24no—
291D4—44no—
292D4—88no—
293D4—4882no—
294D4—2442no—
295D4—2444no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—4882no—
305C2×C4—2441no—
306C2×C4#1451974611yescharacteristic
307C2×C4#1451974612yes—
308C2×C4C22×C22⋊D6111yescharacteristic
309C2×C4C22×C22⋊D6112yes—
310C8—1224no—
311C8—2444no—
312C22—22no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—88no—
317C22—4881no—
318C22—4882no—
319C22#1683993012yes—
320C22C2×#1451974611yescharacteristic
321C4—21no—
322C4—21no—
323C4—22no—
324C4—44no—
325C4—44no—
326C4—4881no—
327C4#1684056014yes—
328C2—22no—
329C2—84no—
330C2—9761no—
331C2#2145506112yes—
332C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#555C2
socleC22×C61C2×#120
hypercenterC2×#555D61
hypocenterC61#18547
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #21456276C2semidirect
2. #21455910C2semidirect
3. #21444964C2semidirect
4. #21444902C2semidirect
5. #21444545C2semidirect
6. #21444262C2semidirect
7. C2×C61×#555C2semidirect
11. #16840141C22semidirect
12. #16840136C22semidirect
14. #16840052C22semidirect
15. #16839957C22semidirect
20. #16839511C22semidirect
21. D4⋊#92418C22semidirect
162. C2×#555D61semidirect
196. C61#18547semidirect

Maximal subgroups

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

subgroupindex
#214562762
#214559102
#214449642
#214449022
#214445452
#214442622
C2×C61×#5552
#1854761