Finite Group Databasebeta

← #29835442

Subgroups — #29835442

Order 15616.

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

normal subgroups only

This group has 918 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
1C2×#16840199C214yesmaximal, complemented
2#21456248C211yesmaximal, characteristic, complemented
3#493×D61C211yesmaximal, characteristic, complemented
4#21448084C214yesmaximal, complemented
5#21448076C211yesmaximal, characteristic, complemented
6#21448069C211yesmaximal, characteristic, complemented
7#21448068C211yesmaximal, characteristic, complemented
8C4×#14519716C211yesmaximal, characteristic, complemented
9C61×#2248C211yesmaximal, characteristic, complemented
10C2×#14519800—21no—
11C2×#14519800C2212yes—
12C2×D8×D61—21no—
13C2×D8×D61C2212yes—
14#16840313—21no—
15#127×D61—21no—
16#16840199—216no—
17C2×#14519720C2214yescomplemented
18C2×D976C2214yescomplemented
19C2×#14519718C2214yescomplemented
20C2×#14519716C2212yes—
21#16840147—21no—
22#16840147C2212yescomplemented
23#16840139C2211yescharacteristic
24#16840139C2211yescharacteristic, complemented
25#16840138—21no—
26#16840137—21no—
27#16840137C2212yes—
28#16840136—21no—
29C4×C8×D61C2211yescharacteristic
30#16839537C2212yescomplemented
31C4×#11758185C2211yescharacteristic, complemented
32C2×SD32×C61C2214yescomplemented
33C61×#494C2211yescharacteristic, complemented
34C61×#493C2211yescharacteristic
35C4×C16×C61C2211yescharacteristic, complemented
36C2×#11758226—22no—
37C2×D4×D61—22no—
38C2×D4×D61—42no—
39#14519800—28no—
40#14519800—44no—
41C2×#11758201—22no—
42C2×#11758201—22no—
43C2×D488—21no—
44C2×D488—21no—
45C2×D488C2312yes—
46C2×D488C2312yes—
47C2×#11758199—22no—
48C2×#11758199—22no—
49D8×D61—28no—
50D8×D61—44no—
51C2×C8×D61D411yescharacteristic
52C2×C8×D61C2311yescharacteristic
53#14519774—21no—
54#14519774—21no—
55#14519773—22no—
56#14519773—42no—
57#14519766—22no—
58#14519752—42no—
59#14519739—42no—
60C42×D61D411yescharacteristic
61#14519720—28no—
62D976—28no—
63#14519718—28no—
64#14519716D418yescomplemented
65C2×#11758185C2312yes—
66C4×#92418D411yescharacteristic
67C8×Dic61D411yescharacteristic
68C8×Dic61C2311yescharacteristic
69C2×Q16×C61—21no—
70C2×Q16×C61C2312yes—
71C2×D8×C61—21no—
72C2×D8×C61C2312yes—
73C4⋊Q8×C61—21no—
74#127×C61—21no—
75SD32×C61—28no—
76C2×C16×C61C2312yes—
77C4×C8×C61C2311yescharacteristic
78C23×D61—42no—
79C2×Q8×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61—42no—
82C2×C22⋊D61—42no—
83C2×C22⋊D61—44no—
84#11758226—44no—
85#11758226—44no—
86D4×D61—44no—
87D4×D61—44no—
88D4×D61—88no—
89C2×D244—22no—
90C2×D244—22no—
91C2×D244—42no—
92C2×D244—42no—
93C2×D244—42no—
94C2×C4×D61—42no—
95C2×C4×D61D812yes—
96C2×C4×D61C2×D411yescharacteristic
97Q16×C61—24no—
98Q16×C61—42no—
99D8×C61—24no—
100D8×C61—42no—
101C2×C8×C61C2×D411yescharacteristic
102C2×C8×C61C2411yescharacteristic
103C4⋊C4×C61—42no—
104C42×C61C2×D411yescharacteristic
105#11758201—44no—
106#11758201—44no—
107#11758199—44no—
108#11758199—44no—
109#11758198—44no—
110C4×Dic61D812yes—
111C4×Dic61C2×D411yescharacteristic
112C2×#92418C2×D411yescharacteristic
113C2×#92418C2×D411yescharacteristic
114D488—24no—
115D488—24no—
116D488—42no—
117D488—42no—
118C8×D61D814yes—
119C8×D61C2×D414yes—
120C16×C61C2×D414yescomplemented
121#11758185C2×D414yescomplemented
122C23×C61—42no—
123C22×D61—42no—
124C22×D61—42no—
125C22×D61—88no—
126C22×D61—84no—
127C22×D61C2×D811yescharacteristic
128Q8×C61—42no—
129Q8×C61—42no—
130D4×C61—42no—
131D4×C61—42no—
132D4×C61—84no—
133C2×C4×C61—42no—
134C2×C4×C61C2×D812yes—
135C2×C4×C61C22×D411yescharacteristic
136C22⋊D61—88no—
137C22⋊D61—84no—
138C2×Dic61C2×D812yes—
139C2×Dic61C2×D811yescharacteristic
140D244—42no—
141D244—42no—
142D244—42no—
143D244—42no—
144D244—84no—
145D244—84no—
146D244—84no—
147C4×D61—24no—
148C4×D61—84no—
149C4×D61#12712yes—
150C4×D61C2×D812yes—
151C8×C61C2×D812yes—
152C8×C61C22×D412yes—
153#92418#12712yes—
154#92418C2×D812yes—
155#33717—611nomaximal, Sylow
156C22×C61—82no—
157C22×C61—84no—
158C22×C61C22×D811yescharacteristic
159C2×D61—22no—
160C2×D61—82no—
161C2×D61—82no—
162C2×D61—84no—
163C2×D61—84no—
164C2×D61—168no—
165C2×D61#49312yes—
166C4×C61—82no—
167C4×C61#50914yescomplemented
168C4×C61C2×#12711yescharacteristic
169C4×C61C22×D811yescharacteristic
170Dic61—22no—
171Dic61#49312yes—
172C2×#509—614no—
173#3147—611no—
174C2×#493—611no—
175#2250—614no—
176#2248—611no—
177#2248D6111yescharacteristic, complemented
178#2246—611no—
179#2246—611no—
180C4×M32—611no—
181C2×C61—162no—
182C2×C61C2×#49311yescharacteristic
183C2×C61C2×#50912yes—
184D61—22no—
185D61—162no—
186D61—162no—
187C2×#135—612no—
188C2×#135—1221no—
189C22×D8—612no—
190C22×D8—1221no—
191#532—1221no—
192C2×#127—1221no—
193#509—12216no—
194C2×SD32—614no—
195C2×SD32C2×D6114yescomplemented
196C2×D16—614no—
197C2×D16—614no—
198C2×M32—612no—
199#495—612no—
200#495—1221no—
201#494C2×D6111yescharacteristic, complemented
202#493—612no—
203#493—611no—
204#493—611no—
205#493—1221no—
206#493—1221no—
207#493C2×D6111yescharacteristic
208#492—1221no—
209C2×C4×C8—611no—
210#346—612no—
211C4×C16—611no—
212C4×C16C2×D6111yescharacteristic, complemented
213C61#3371711yescharacteristic, complemented, Sylow
214C2×#41—1222no—
215C22×D4—1222no—
216C22×D4—2442no—
217#135—1228no—
218#135—2444no—
219C2×Q16—21no—
220C2×Q16C22×D6112yes—
221C2×SD16—1222no—
222C2×SD16—1222no—
223C2×D8—21no—
224C2×D8—612no—
225C2×D8—612no—
226C2×D8—1228no—
227C2×D8—1222no—
228C2×D8—1222no—
229C2×D8—1221no—
230C2×D8—1221no—
231C2×D8—2444no—
232C2×D8C22×D6112yes—
233C22×C8—611no—
234C22×C8—611no—
235C4⋊Q8—21no—
236#127—21no—
237#127—1222no—
238#127—1221no—
239#127—1221no—
240#127—2442no—
241#124—1222no—
242#121—2442no—
243C4×D4—2442no—
244C2×C42—611no—
245SD32—28no—
246SD32—1228no—
247D16—1228no—
248D16—1228no—
249M32—618no—
250C2×C16—612no—
251C2×C16C22×D6112yes—
252C4×C8—611no—
253C4×C8—611no—
254C4×C8—611no—
255C4×C8C22×D6111yescharacteristic
256C24—2442no—
257#41—2444no—
258#41—2444no—
259C2×Q8—22no—
260C2×D4—22no—
261C2×D4—42no—
262C2×D4—1222no—
263C2×D4—1222no—
264C2×D4—4888no—
265C2×D4—2442no—
266C2×D4—2444no—
267C2×D4—2444no—
268C2×D4—2444no—
269C2×D4—2442no—
270C2×D4—2442no—
271C2×D4—2442no—
272C22×C4—611no—
273C22×C4—612no—
274C22×C4—2442no—
275Q16—24no—
276Q16—42no—
277SD16—2444no—
278SD16—2444no—
279D8—24no—
280D8—42no—
281D8—1224no—
282D8—1224no—
283D8—2444no—
284D8—2444no—
285D8—2442no—
286D8—2442no—
287C2×C8—611no—
288C2×C8—611no—
289C2×C8—614no—
290C2×C8—614no—
291C2×C8D4×D6111yescharacteristic
292C2×C8C23×D6111yescharacteristic
293C4⋊C4—42no—
294C22⋊C4—2444no—
295C42—611no—
296C42—612no—
297C42D4×D6111yescharacteristic
298C16—614no—
299C16D4×D6114yescomplemented
300C23—42no—
301C23—611no—
302C23—4884no—
303C23—4888no—
304C23—2442no—
305C23—2442no—
306Q8—42no—
307Q8—42no—
308D4—42no—
309D4—42no—
310D4—84no—
311D4—4884no—
312D4—4884no—
313D4—4888no—
314D4—4884no—
315D4—4884no—
316D4—2442no—
317D4—2442no—
318D4—2442no—
319D4—2442no—
320C2×C4—42no—
321C2×C4—611no—
322C2×C4—612no—
323C2×C4—612no—
324C2×C4—612no—
325C2×C4—1224no—
326C2×C4—4884no—
327C2×C4D8×D6112yes—
328C2×C4C2×D4×D6111yescharacteristic
329C8—612no—
330C8—612no—
331C8D8×D6112yes—
332C8C2×D4×D6112yes—
333C22—82no—
334C22—84no—
335C22—612no—
336C22—1222no—
337C22—9768no—
338C22—4882no—
339C22—4882no—
340C22—4884no—
341C22—4884no—
342C22C2×D8×D6111yescharacteristic
343C4—82no—
344C4—612no—
345C4—1222no—
346C4#1684019914yescomplemented
347C4#127×D6111yescharacteristic
348C4C2×D8×D6111yescharacteristic
349C2—162no—
350C2—1222no—
351C2—9762no—
352C2—9762no—
353C2#493×D6111yescharacteristic
354C2C2×#1684019912yes—
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2248C2
socleC22×C61C22×D8
hypercenter#2248D61
hypocenterC61#33717
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. #21456248C2semidirect
3. #493×D61C2semidirect
4. #21448084C2semidirect
5. #21448076C2semidirect
6. #21448069C2semidirect
7. #21448068C2semidirect
8. C4×#14519716C2semidirect
9. C61×#2248C2semidirect
17. C2×#14519720C22semidirect
18. C2×D976C22semidirect
19. C2×#14519718C22semidirect
22. #16840147C22semidirect
24. #16840139C22semidirect
30. #16839537C22semidirect
31. C4×#11758185C22semidirect
32. C2×SD32×C61C22semidirect
33. C61×#494C22semidirect
35. C4×C16×C61C22semidirect
64. #14519716D4semidirect
120. C16×C61C2×D4semidirect
121. #11758185C2×D4semidirect
167. C4×C61#509semidirect
177. #2248D61semidirect
195. C2×SD32C2×D61semidirect
201. #494C2×D61semidirect
212. C4×C16C2×D61semidirect
213. C61#33717semidirect
299. C16D4×D61semidirect
346. C4#16840199semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
#214562482
#493×D612
#214480842
#214480762
#214480692
#214480682
C4×#145197162
C61×#22482
#3371761