Finite Group Databasebeta

← #29835312

Subgroups — #29835312

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456332C211yesmaximal, characteristic, complemented
2#21456244C211yesmaximal, characteristic
3#21448053C211yesmaximal, characteristic, complemented
4#21448051C211yesmaximal, characteristic, complemented
5#21448050C211yesmaximal, characteristic, complemented
6#21448045C212yesmaximal, complemented
7#21448041C212yesmaximal, complemented
8#21447525C212yesmaximal, complemented
9#21447523C212yesmaximal, complemented
10C4×C16×D61C211yesmaximal, characteristic, complemented
11C61×#2242C211yesmaximal, characteristic, complemented
12C2×#14519800C2211yescharacteristic
13C2×#14519798—21no—
14C2×#14519796C2211yescharacteristic
15#16840318—21no—
16#16840311—21no—
17C2×C16×D61C2212yes—
18#16840166C2212yes—
19#16840164C2211yescharacteristic
20#16840161—21no—
21#16840159—21no—
22#16840146C2211yescharacteristic
23#16840144—21no—
24#16840140C2211yescharacteristic
25#16840137C2211yescharacteristic
26#16840133—21no—
27#16839799C2212yes—
28#16839797—21no—
29C4×C8×D61C2211yescharacteristic
30#16839567C2212yes—
31#16839566C2212yes—
32#16839565C2212yes—
33#16839564C2212yes—
34#16839563C2212yes—
35#16839562C2212yes—
36C4×#11758185C2211yescharacteristic
37C16×Dic61C2212yes—
38C61×#499C2211yescharacteristic
39C61×#495C2211yescharacteristic
40C61×#358C2212yes—
41C61×#357C2212yes—
42C4×C16×C61C2211yescharacteristic
43C2×#11758226—21no—
44C2×#11758224—21no—
45#14519800—24no—
46#14519798—44no—
47#14519796—24no—
48C2×Dic244C2311yescharacteristic
49C2×#11758202—21no—
50C2×#11758190—21no—
51C2×#11758200—21no—
52C2×#11758201—21no—
53C2×D488C2311yescharacteristic
54C2×#11758199—21no—
55C2×C8×D61D411yescharacteristic
56C2×C8×D61C2311yescharacteristic
57#14519776—22no—
58#14519775—21no—
59#14519773—21no—
60#14519769—21no—
61#14519765—21no—
62#14519755—42no—
63#14519750—42no—
64#14519742—42no—
65#14519737—42no—
66#14519735—22no—
67C42×D61D411yescharacteristic
68C2×#11758185C2312yes—
69C16×D61—24no—
70#14519711C2312yes—
71#14519710—22no—
72#14519709—21no—
73#14519708—22no—
74C4×#92418D411yescharacteristic
75C8×Dic61D411yescharacteristic
76C8×Dic61C2311yescharacteristic
77C2×Q16×C61C2311yescharacteristic
78C2×SD16×C61—21no—
79C2×D8×C61C2311yescharacteristic
80#125×C61—21no—
81#124×C61—21no—
82C2×C16×C61C2312yes—
83#107×C61C2312yes—
84#106×C61—21no—
85C4×C8×C61C2311yescharacteristic
86C2×Q8×C61—21no—
87C2×D4×C61—21no—
88C2×C22⋊D61—41no—
89C2×C22⋊D61—42no—
90C22×Dic61—41no—
91#11758226—42no—
92#11758226—42no—
93#11758224—42no—
94#11758224—42no—
95C2×D244—21no—
96C2×D244—41no—
97C2×C4×D61—41no—
98C2×C4×D61D812yes—
99C2×C4×D61C2×D411yescharacteristic
100C2×Dic122—21no—
101C2×Dic122—42no—
102Q16×C61—22no—
103SD16×C61—42no—
104D8×C61—22no—
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61C2411yescharacteristic
107C4⋊C4×C61—22no—
108C4⋊C4×C61—42no—
109C22⋊C4×C61—42no—
110C42×C61C2×D411yescharacteristic
111#11758202—42no—
112#11758201—42no—
113#11758200—42no—
114#11758199—42no—
115#11758198—42no—
116#11758198—42no—
117C4⋊Dic61—22no—
118#11758196—42no—
119#11758196—42no—
120#11758196—42no—
121C4×Dic61—42no—
122C4×Dic61D812yes—
123C4×Dic61C2×D411yescharacteristic
124C2×#92418C2×D411yescharacteristic
125C2×#92418C2×D411yescharacteristic
126Dic244—22no—
127D488—22no—
128#11758190—42no—
129C8×D61SD1614yes—
130C8×D61#4114yes—
131C16×C61—22no—
132#11758185—22no—
133C23×C61—41no—
134C22×D61—41no—
135C22×D61C2×D811yescharacteristic
136Q8×C61—41no—
137Q8×C61—41no—
138D4×C61—41no—
139D4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—41no—
142C2×C4×C61C2×D812yes—
143C2×C4×C61C22×D411yescharacteristic
144C22⋊D61—82no—
145C22⋊D61—84no—
146C2×Dic61—42no—
147C2×Dic61—41no—
148C2×Dic61—82no—
149C2×Dic61C2×D811yescharacteristic
150C2×Dic61C2×D812yes—
151D244—41no—
152D244—41no—
153D244—82no—
154C4×D61—24no—
155C4×D61—82no—
156C4×D61#12412yes—
157C4×D61C2×SD1612yes—
158Dic122—41no—
159Dic122—41no—
160Dic122—84no—
161C8×C61C2×SD1612yes—
162C8×C61C2×#4112yes—
163#92418#12412yes—
164#92418C2×SD1612yes—
165#33697—611nomaximal, Sylow
166C22×C61—81no—
167C22×C61—82no—
168C22×C61C22×D811yescharacteristic
169C2×D61—22no—
170C2×D61—81no—
171C2×D61—82no—
172C2×D61#48612yes—
173C4×C61—22no—
174C4×C61—81no—
175C4×C61—82no—
176C4×C61C2×#12411yescharacteristic
177C4×C61C22×SD1611yescharacteristic
178Dic61—81no—
179Dic61—82no—
180Dic61#48612yes—
181Dic61#50814yes—
182#3160—611no—
183#3147—611no—
184#2242—611no—
185#2242—611no—
186#2242—611no—
187#2242D6111yescharacteristic, complemented
188#2241—612no—
189#2240—612no—
190#2138—612no—
191#2138—612no—
192C2×C4×C16—611no—
193C2×C61—161no—
194C2×C61C2×#48611yescharacteristic
195C2×C61C2×#50812yes—
196D61—22no—
197D61—161no—
198C2×#135—611no—
199C2×#135—611no—
200C2×#135—1221no—
201#533—1221no—
202#532—1221no—
203C22×C16—612no—
204#500—612no—
205#499—611no—
206#499—1221no—
207#499C2×D6111yescharacteristic
208#498—1221no—
209#495—611no—
210#495—1221no—
211#495C2×D6111yescharacteristic
212#494—611no—
213#493—611no—
214#492—1221no—
215#427—612no—
216#427—1221no—
217C2×C4×C8—611no—
218#358—612no—
219#358—612no—
220#358—612no—
221#358C2×D6112yes—
222#357—612no—
223#357—612no—
224#357—612no—
225#357C2×D6112yes—
226C4×C16—612no—
227C4×C16—611no—
228C4×C16C2×D6111yescharacteristic
229C61#3369711yescharacteristic, complemented, Sylow
230C2×#41—1221no—
231C2×#41—1221no—
232#135—1224no—
233#135—1224no—
234#135—2444no—
235C2×Q16—611no—
236C2×Q16—1221no—
237C2×Q16C22×D6111yescharacteristic
238C2×SD16—21no—
239C2×SD16—1221no—
240C2×SD16—1221no—
241C2×SD16—1221no—
242C2×D8—611no—
243C2×D8—1221no—
244C2×D8C22×D6111yescharacteristic
245C22×C8—611no—
246C22×C8—611no—
247C4⋊Q8—1222no—
248C4⋊Q8—1221no—
249#127—1221no—
250#125—21no—
251#125—1221no—
252#124—21no—
253#124—1221no—
254C22⋊Q8—2442no—
255#121—2442no—
256C4×Q8—2442no—
257C4×D4—2442no—
258#117—1222no—
259C2×C42—611no—
260C2×C16—612no—
261C2×C16—1224no—
262C2×C16C22×D6112yes—
263#107—612no—
264#107—1222no—
265#107C22×D6112yes—
266#106—21no—
267#106—1222no—
268#106—1221no—
269C4×C8—611no—
270C4×C8—611no—
271C4×C8—611no—
272C4×C8C22×D6111yescharacteristic
273#41—2442no—
274#41—2442no—
275#41—2442no—
276#41—2442no—
277C2×Q8—21no—
278C2×Q8—1221no—
279C2×Q8—2442no—
280C2×D4—21no—
281C2×D4—1221no—
282C2×D4—2441no—
283C2×D4—2442no—
284C2×D4—2441no—
285C22×C4—611no—
286C22×C4—612no—
287C22×C4—2441no—
288C22×C4—2441no—
289Q16—22no—
290Q16—1222no—
291Q16—2442no—
292SD16—42no—
293SD16—2442no—
294SD16—2442no—
295SD16—2442no—
296D8—22no—
297D8—1222no—
298D8—2442no—
299C2×C8—611no—
300C2×C8—611no—
301C2×C8—614no—
302C2×C8—614no—
303C2×C8D4×D6111yescharacteristic
304C2×C8C23×D6111yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4—1222no—
308C4⋊C4—2442no—
309C4⋊C4—2442no—
310C4⋊C4—2442no—
311C22⋊C4—42no—
312C22⋊C4—2442no—
313C22⋊C4—2442no—
314C42—611no—
315C42—612no—
316C42—2442no—
317C42D4×D6111yescharacteristic
318C16—22no—
319C16—1222no—
320C23—41no—
321C23—611no—
322C23—2441no—
323Q8—41no—
324Q8—41no—
325Q8—4884no—
326Q8—2441no—
327Q8—2441no—
328D4—41no—
329D4—41no—
330D4—4882no—
331D4—4884no—
332D4—4882no—
333D4—2441no—
334D4—2441no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—611no—
338C2×C4—612no—
339C2×C4—612no—
340C2×C4—612no—
341C2×C4—1224no—
342C2×C4—4882no—
343C2×C4—4882no—
344C2×C4—2442no—
345C2×C4—2441no—
346C2×C4D8×D6112yes—
347C2×C4C2×D4×D6111yescharacteristic
348C8—612no—
349C8—612no—
350C8SD16×D6112yes—
351C8#41×D6112yes—
352C22—82no—
353C22—81no—
354C22—612no—
355C22—1222no—
356C22—4882no—
357C22—4881no—
358C22C2×D8×D6111yescharacteristic
359C4—22no—
360C4—82no—
361C4—81no—
362C4—614no—
363C4—612no—
364C4—4882no—
365C4—4881no—
366C4#124×D6111yescharacteristic
367C4C2×SD16×D6111yescharacteristic
368C2—161no—
369C2—1222no—
370C2—9761no—
371C2#486×D6111yescharacteristic
372C2#508×D6112yes—
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2242C2
socleC22×C61C22×D8
prime radical#21456244C2
hypercenter#2242D61
hypocenterC61#33697
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2242

Complements

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

NH ≅ G/Ntype
1. #21456332C2semidirect
3. #21448053C2semidirect
4. #21448051C2semidirect
5. #21448050C2semidirect
6. #21448045C2semidirect
7. #21448041C2semidirect
8. #21447525C2semidirect
9. #21447523C2semidirect
10. C4×C16×D61C2semidirect
11. C61×#2242C2semidirect
187. #2242D61semidirect
229. C61#33697semidirect

Maximal subgroups

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

subgroupindex
#214563322
#214562442
#214480532
#214480512
#214480502
#214480452
#214480412
#214475252
#214475232
C4×C16×D612
C61×#22422
#3369761