Finite Group Databasebeta

← #29836600

Subgroups — #29836600

Order 15616.

712 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 139 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×#16840199C212yesmaximal, complemented
2#21457503C212yesmaximal, complemented
3#21447929C212yesmaximal, complemented
4#21447871C212yesmaximal, complemented
5#21447758C212yesmaximal, complemented
6#21447729C212yesmaximal, complemented
7C4×D976C211yesmaximal, characteristic, complemented
8#21447652C211yesmaximal, characteristic, complemented
9C61×#2249C211yesmaximal, characteristic, complemented
10C2×#14519800C2212yes—
11C2×D8×D61C2212yescomplemented
12#16840405—22no—
13#16840199—28no—
14C2×#14519720C2212yescomplemented
15C2×D976C2211yescharacteristic
16C2×#14519718C2212yes—
17C2×#14519716C2212yescomplemented
18#16840000C2212yescomplemented
19#16839995C2212yes—
20#16839974—22no—
21#16839911—22no—
22#16839906—22no—
23C4×D488C2212yescomplemented
24C8×D244C2211yescharacteristic
25#16839588C2211yescharacteristic, complemented
26C4⋊#11758185C2211yescharacteristic, complemented
27#16839566C2212yescomplemented
28#16839564C2212yescomplemented
29#16839563C2212yescomplemented
30#16839541C2212yescomplemented
31C2×Q32×C61C2211yescharacteristic, complemented
32C2×SD32×C61C2212yescomplemented
33C2×D16×C61C2211yescharacteristic
34C61×#495C2212yescomplemented
35C4×C16×C61C2211yescharacteristic, complemented
36C2×#11758226—22no—
37C2×D4×D61—22no—
38#14519800—24no—
39#14519800—42no—
40C2×#11758201—22no—
41C2×D488C2312yes—
42C2×#11758199—22no—
43D8×D61—24no—
44D8×D61—42no—
45C2×C8×D61C2312yes—
46#14519759—42no—
47#14519753—22no—
48#14519750—42no—
49#14519748—22no—
50#14519747—42no—
51C4×D244—22no—
52C4×D244D411yescharacteristic
53C22⋊C4×D61—42no—
54#14519720—24no—
55D976D414yescomplemented
56#14519718—24no—
57#14519716—24no—
58C2×#11758185C2312yes—
59#14519711C2312yes—
60C4⋊#92418D411yescharacteristic
61#14519701—22no—
62#14519699—22no—
63#14519698—22no—
64#14519689D412yes—
65C2×Q16×C61C2312yes—
66C2×SD16×C61—22no—
67C2×D8×C61C2312yes—
68#124×C61—22no—
69Q32×C61—41no—
70SD32×C61—24no—
71D16×C61—41no—
72C2×C16×C61C2311yescharacteristic
73C2×C16×C61C2311yescharacteristic
74C4×C8×C61C2311yescharacteristic
75C23×D61—42no—
76C2×Q8×C61—22no—
77C2×D4×C61—22no—
78C2×C22⋊D61—42no—
79C2×C22⋊D61—42no—
80#11758226—44no—
81#11758226—82no—
82D4×D61—44no—
83D4×D61—82no—
84C2×D244—22no—
85C2×D244—42no—
86C2×D244D811yescharacteristic
87C2×C4×D61—42no—
88C2×C4×D61—42no—
89C2×C4×D61C2×D412yes—
90Q16×C61—22no—
91Q16×C61—22no—
92SD16×C61—82no—
93D8×C61—22no—
94D8×C61—22no—
95C2×C8×C61C2×D411yescharacteristic
96C2×C8×C61C2411yescharacteristic
97C22⋊C4×C61—42no—
98C22⋊C4×C61—42no—
99C42×C61C2×D411yescharacteristic
100C22⋊Dic61—42no—
101#11758201—44no—
102#11758199—44no—
103#11758198—42no—
104#11758198—42no—
105#11758198D812yes—
106C4⋊Dic61—22no—
107C4⋊Dic61D811yescharacteristic
108#11758196—42no—
109C2×#92418C2×D412yes—
110D488—22no—
111D488C2×D414yes—
112C8×D61—22no—
113C8×D61—22no—
114C16×C61—21no—
115C16×C61C2×D412yes—
116#11758185—22no—
117C23×C61—42no—
118C22×D61—42no—
119C22×D61—84no—
120C22×D61—84no—
121C22×D61—162no—
122C22×D61C2×D812yes—
123Q8×C61—42no—
124Q8×C61—42no—
125D4×C61—42no—
126D4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61C2×D811yescharacteristic
129C2×C4×C61C2×D811yescharacteristic
130C2×C4×C61C22×D411yescharacteristic
131C22⋊D61—84no—
132C22⋊D61—162no—
133C2×Dic61—42no—
134C2×Dic61C2×D812yes—
135D244—24no—
136D244—41no—
137D244—42no—
138D244—84no—
139C4×D61—22no—
140C4×D61—22no—
141C4×D61—84no—
142C4×D61—162no—
143C8×C61—21no—
144C8×C61C22×D411yescharacteristic
145C8×C61H(5,𝔽2)11yescharacteristic
146#92418—22no—
147#33839—611nomaximal, Sylow
148C22×C61—82no—
149C22×C61—82no—
150C22×C61—82no—
151C22×C61C22×D811yescharacteristic
152C2×D61—22no—
153C2×D61—22no—
154C2×D61—22no—
155C2×D61—44no—
156C2×D61—82no—
157C2×D61—82no—
158C2×D61—168no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—82no—
162C4×C61D4211yescharacteristic
163C4×C61#57611yescharacteristic
164Dic61—22no—
165Dic61—82no—
166C2×#509—612no—
167#3280—612no—
168#2249D6111yescharacteristic, complemented
169#2220—612no—
170#2210—612no—
171#2189—612no—
172#2184—612no—
173C4×D16—611no—
174#2169—611no—
175C2×C61—162no—
176C2×C61D4×D811yescharacteristic
177C2×C61#341411yescharacteristic
178C2×C61#341511yescharacteristic
179D61—42no—
180D61—84no—
181C2×#135—612no—
182C22×D8—612no—
183#546—1222no—
184#509—1228no—
185C2×Q32C2×D6111yescharacteristic, complemented
186C2×SD32—612no—
187C2×SD32C2×D6112yescomplemented
188C2×D16—612no—
189C2×D16—611no—
190C2×D16C2×D6111yescharacteristic
191C2×M32—612no—
192#495C2×D6112yescomplemented
193#467—612no—
194#466—612no—
195#463—1222no—
196#450—1222no—
197#449—1222no—
198C4×D8—612no—
199C8×D4—611no—
200#366—611no—
201C4⋊C16—611no—
202#358—612no—
203#357—612no—
204#357—612no—
205C22⋊C16—612no—
206C4×C16C2×D6111yescharacteristic, complemented
207C61#3383911yescharacteristic, complemented, Sylow
208C2×#41—1222no—
209C22×D4—1222no—
210#135—1224no—
211#135—2442no—
212C2×Q16C22×D6112yes—
213C2×SD16—22no—
214C2×SD16—1222no—
215C2×D8—612no—
216C2×D8—1222no—
217C2×D8—1224no—
218C2×D8—2442no—
219C2×D8C22×D6112yes—
220C22×C8—612no—
221#124—22no—
222#123—2442no—
223C22⋊Q8—1222no—
224#121—1222no—
225#121—2442no—
226#120—2442no—
227C4×D4—611no—
228C4×D4—1222no—
229C2×C22⋊C4—2442no—
230Q32—41no—
231SD32—24no—
232SD32—1224no—
233D16—41no—
234D16—614no—
235D16—1224no—
236M32—1224no—
237C2×C16—612no—
238C2×C16C22×D6111yescharacteristic
239C2×C16C22×D6111yescharacteristic
240#107—612no—
241C4⋊C8—611no—
242#103—1222no—
243#102—1222no—
244#102—1222no—
245C22⋊C8—612no—
246C4×C8C22×D6111yescharacteristic
247C24—2442no—
248#41—4882no—
249#41—2444no—
250C2×Q8—22no—
251C2×D4—22no—
252C2×D4—611no—
253C2×D4—1222no—
254C2×D4—4882no—
255C2×D4—2442no—
256C2×D4—2442no—
257C2×D4—2444no—
258C2×D4—2442no—
259C22×C4—612no—
260C22×C4—2442no—
261C22×C4—2442no—
262Q16—22no—
263Q16—22no—
264SD16—82no—
265SD16—2444no—
266D8—22no—
267D8—22no—
268D8—614no—
269D8—1222no—
270D8—2444no—
271C2×C8—612no—
272C2×C8—1222no—
273C2×C8—1222no—
274C2×C8D4×D6111yescharacteristic
275C2×C8C23×D6111yescharacteristic
276C4⋊C4—611no—
277C4⋊C4—1222no—
278C4⋊C4—2442no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—612no—
282C22⋊C4—2442no—
283C22⋊C4—2442no—
284C22⋊C4—2442no—
285C42D4×D6111yescharacteristic
286C16—21no—
287C16—1222no—
288C16D4×D6112yes—
289C23—42no—
290C23—612no—
291C23—9762no—
292C23—4884no—
293C23—4884no—
294C23—2442no—
295Q8—42no—
296Q8—42no—
297D4—42no—
298D4—42no—
299D4—1224no—
300D4—9762no—
301D4—4884no—
302D4—4884no—
303D4—2441no—
304D4—2442no—
305C2×C4—42no—
306C2×C4—612no—
307C2×C4—1222no—
308C2×C4—1222no—
309C2×C4—9762no—
310C2×C4—4884no—
311C2×C4—2442no—
312C2×C4D8×D6111yescharacteristic
313C2×C4D8×D6111yescharacteristic
314C2×C4C2×D4×D6111yescharacteristic
315C8—21no—
316C8—1222no—
317C8C2×D4×D6111yescharacteristic
318C8#1451982211yescharacteristic
319C22—82no—
320C22—82no—
321C22—82no—
322C22—1222no—
323C22—1222no—
324C22—1222no—
325C22—9768no—
326C22—4882no—
327C22—4882no—
328C22—2444no—
329C22C2×D8×D6111yescharacteristic
330C4—21no—
331C4—21no—
332C4—82no—
333C4—1222no—
334C4—4882no—
335C4#1684039711yescharacteristic
336C4#1684056311yescharacteristic
337C2—162no—
338C2—4884no—
339C2—2442no—
340C2#2145749411yescharacteristic
341C2#2145895211yescharacteristic
342C2#2145895711yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2249C2
socleC22×C61C22×D8
hypercenter#2249D61
hypocenterC61#33839
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2249

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. #21457503C2semidirect
3. #21447929C2semidirect
4. #21447871C2semidirect
5. #21447758C2semidirect
6. #21447729C2semidirect
7. C4×D976C2semidirect
8. #21447652C2semidirect
9. C61×#2249C2semidirect
11. C2×D8×D61C22semidirect
14. C2×#14519720C22semidirect
17. C2×#14519716C22semidirect
18. #16840000C22semidirect
23. C4×D488C22semidirect
25. #16839588C22semidirect
26. C4⋊#11758185C22semidirect
27. #16839566C22semidirect
28. #16839564C22semidirect
29. #16839563C22semidirect
30. #16839541C22semidirect
31. C2×Q32×C61C22semidirect
32. C2×SD32×C61C22semidirect
34. C61×#495C22semidirect
35. C4×C16×C61C22semidirect
55. D976D4semidirect
168. #2249D61semidirect
185. C2×Q32C2×D61semidirect
187. C2×SD32C2×D61semidirect
192. #495C2×D61semidirect
206. C4×C16C2×D61semidirect
207. C61#33839semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
#214575032
#214479292
#214478712
#214477582
#214477292
C4×D9762
#214476522
C61×#22492
#3383961