Finite Group Databasebeta

← #29834910

Subgroups — #29834910

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455430C211yesmaximal, characteristic
2#430×D61C211yesmaximal, characteristic, complemented
3#21448037C212yesmaximal, complemented
4#21448036C211yesmaximal, characteristic, complemented
5#21448035C212yesmaximal, complemented
6#21448034C211yesmaximal, characteristic, complemented
7#21448033C211yesmaximal, characteristic, complemented
8#21447590C212yesmaximal, complemented
9#21447567C212yesmaximal, complemented
10#349×D61C211yesmaximal, characteristic, complemented
11C61×#2239C211yesmaximal, characteristic, complemented
12C2×#14519800C2211yescharacteristic
13C2×M16×D61—21no—
14C2×M16×D61C2211yescharacteristic
15#16840275—21no—
16M32×D61C2212yes—
17#16840017C2211yescharacteristic
18#16840016C2211yescharacteristic
19#16840015C2211yescharacteristic
20#16840011—21no—
21#16840010—21no—
22#16839811C2211yescharacteristic
23#16839810—21no—
24#16839809—21no—
25#16839809C2212yes—
26#108×D61—21no—
27#108×D61C2212yes—
28#16839797C2211yescharacteristic
29#16839757—21no—
30#16839586C2212yescomplemented
31#16839585C2212yescomplemented
32#16839584C2212yescomplemented
33#16839578C2212yescomplemented
34#16839576C2212yescomplemented
35#16839574C2212yescomplemented
36#16839544C2211yescharacteristic
37#16839543C2212yes—
38C61×#470C2211yescharacteristic
39C61×#430C2211yescharacteristic
40C61×#365C2212yescomplemented
41C61×#362C2212yescomplemented
42C61×#349C2211yescharacteristic
43C2×#11758226—21no—
44C22×C4×D61D411yescharacteristic
45#14519800—22no—
46#14519800—22no—
47C2×#11758201—21no—
48C2×D488C2311yescharacteristic
49C2×#11758194—21no—
50C2×#11758194D411yescharacteristic
51C2×#11758189—41no—
52C2×#11758189D411yescharacteristic
53C2×#11758189C2311yescharacteristic
54M16×D61—22no—
55M16×D61—22no—
56M16×D61—22no—
57M16×D61—22no—
58M16×D61—41no—
59M16×D61—41no—
60C2×C8×D61—41no—
61C2×C8×D61D411yescharacteristic
62C2×C8×D61C2311yescharacteristic
63#14519762—41no—
64#14519753—21no—
65#14519752—21no—
66#14519751—21no—
67#14519747—41no—
68#14519739—41no—
69#14519737—41no—
70#14519735—21no—
71#14519717C2312yes—
72#14519716—22no—
73C16×D61—22no—
74#14519713—21no—
75#14519713C2312yes—
76#14519712—22no—
77#14519712—22no—
78#14519710—21no—
79#14519709C2311yescharacteristic
80#14519701—21no—
81#14519699—21no—
82#14519697—21no—
83C2×Q16×C61C2311yescharacteristic
84C2×M16×C61—21no—
85C2×M16×C61C2311yescharacteristic
86C22⋊Q8×C61—21no—
87M32×C61C2312yes—
88#108×C61—21no—
89#108×C61C2312yes—
90#106×C61C2311yescharacteristic
91#103×C61—21no—
92C23×D61SD1611yescharacteristic
93C2×Q8×C61—21no—
94C22×C4×C61C2×D411yescharacteristic
95C2×C22⋊D61—41no—
96C22×Dic61SD1611yescharacteristic
97#11758226—41no—
98#11758226—41no—
99#11758226—41no—
100#11758226—41no—
101C2×D244—21no—
102C2×D244—41no—
103C2×D244—41no—
104C2×C4×D61—21no—
105C2×C4×D61—21no—
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61—41no—
109C2×C4×D61SD1611yescharacteristic
110C2×C4×D61SD1611yescharacteristic
111C2×C4×D61C2×D411yescharacteristic
112C2×C4×D61C2×D411yescharacteristic
113Q16×C61—22no—
114M16×C61—22no—
115M16×C61—22no—
116M16×C61—41no—
117C2×C8×C61—41no—
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2411yescharacteristic
120C4⋊C4×C61—21no—
121C4⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123#11758201—42no—
124#11758198—41no—
125#11758198—41no—
126#11758198—41no—
127#11758198—41no—
128C4⋊Dic61—21no—
129#11758196—41no—
130C4×Dic61—41no—
131#11758194—22no—
132#11758194—22no—
133#11758194—41no—
134C2×#92418—41no—
135C2×#92418C2×D411yescharacteristic
136C2×#92418C2×D411yescharacteristic
137D488—22no—
138#11758189—22no—
139#11758189—22no—
140#11758189—42no—
141#11758189—42no—
142C8×D61—21no—
143C8×D61—21no—
144C8×D61—42no—
145C8×D61—42no—
146C8×D61#4112yes—
147C8×D61#4112yes—
148C16×C61—22no—
149#11758185—22no—
150C23×C61C2×SD1611yescharacteristic
151C22×D61—21no—
152C22×D61—21no—
153C22×D61—41no—
154C22×D61—81no—
155C22×D61C2×SD1611yescharacteristic
156C22×D61C2×SD1611yescharacteristic
157Q8×C61—41no—
158Q8×C61—41no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61C2×SD1611yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C22⋊D61—82no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—41no—
169C2×Dic61C2×SD1611yescharacteristic
170C2×Dic61C2×SD1611yescharacteristic
171D244—41no—
172D244—41no—
173D244—82no—
174D244—81no—
175D244—81no—
176C4×D61—21no—
177C4×D61—21no—
178C4×D61—21no—
179C4×D61—21no—
180C4×D61—21no—
181C4×D61—21no—
182C4×D61—81no—
183C4×D61—81no—
184C4×D61#12311yescharacteristic
185C4×D61#12311yescharacteristic
186C4×D61AGL(1,ℤ/8)11yescharacteristic
187C4×D61AGL(1,ℤ/8)11yescharacteristic
188C8×C61—21no—
189C8×C61—42no—
190C8×C61C2×#4112yes—
191#92418—21no—
192#92418—42no—
193#92418#12312yes—
194#33656—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—41no—
197C22×C61C22×SD1611yescharacteristic
198C2×D61—21no—
199C2×D61—41no—
200C2×D61—42no—
201C2×D61—41no—
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—81no—
205C2×D61—81no—
206C2×D61#48111yescharacteristic
207C2×D61#48111yescharacteristic
208C4×C61—21no—
209C4×C61—81no—
210C4×C61—81no—
211C4×C61C2×#12311yescharacteristic
212C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
213Dic61—21no—
214Dic61—81no—
215Dic61#48111yescharacteristic
216Dic61#48111yescharacteristic
217#3054—611no—
218C2×#430—611no—
219#2239—611no—
220#2239—612no—
221#2239D6111yescharacteristic, complemented
222#2238—611no—
223#2238—612no—
224#2238—611no—
225#2155—612no—
226#2148—612no—
227C2×#349—611no—
228C2×C61—21no—
229C2×C61—41no—
230C2×C61C2×#48111yescharacteristic
231D61—21no—
232D61—21no—
233D61—41no—
234D61—161no—
235C2×#135—611no—
236C22×M16—611no—
237C22×M16—1221no—
238#525—1221no—
239C2×M32—612no—
240#470—611no—
241#470C2×D6111yescharacteristic
242#469—611no—
243#469—611no—
244#468—1221no—
245#468—1221no—
246#430—611no—
247#430—612no—
248#430—1221no—
249#430—1221no—
250#430C2×D6111yescharacteristic
251C2×#108—612no—
252C2×#108—1221no—
253#427—611no—
254#416—1221no—
255#365—612no—
256#365—612no—
257#365—612no—
258#365C2×D6112yescomplemented
259#362—612no—
260#362C2×D6112yescomplemented
261#361—612no—
262#361—612no—
263#349—611no—
264#349—612no—
265#349C2×D6111yescharacteristic
266C61#3365611yescharacteristic, complemented, Sylow
267C2×#41—1221no—
268C23×C4—611no—
269#135—1222no—
270#135—1222no—
271C2×Q16C22×D6111yescharacteristic
272C2×SD16—1221no—
273C2×D8—611no—
274C2×M16—21no—
275C2×M16—611no—
276C2×M16—611no—
277C2×M16—611no—
278C2×M16—1222no—
279C2×M16—1222no—
280C2×M16—1222no—
281C2×M16—1222no—
282C2×M16—1221no—
283C2×M16—2441no—
284C2×M16—2441no—
285C2×M16—2441no—
286C2×M16C22×D6111yescharacteristic
287C22×C8—611no—
288C22×C8—611no—
289C22×C8—2441no—
290#123—2441no—
291C22⋊Q8—21no—
292C22⋊Q8—1221no—
293#121—1221no—
294#121—1221no—
295#120—2441no—
296C4×D4—2441no—
297C4×D4—2441no—
298#117—1221no—
299M32—612no—
300M32—1222no—
301M32C22×D6112yes—
302C2×C16—1222no—
303#108—21no—
304#108—612no—
305#108—1222no—
306#108—1222no—
307#108—1221no—
308#108C22×D6112yes—
309#107—1221no—
310#106—611no—
311#106C22×D6111yescharacteristic
312#103—21no—
313#103—1221no—
314#102—1221no—
315#102—1221no—
316C24—611no—
317#41—2441no—
318#41—2441no—
319#41—2441no—
320#41—2441no—
321C2×Q8—21no—
322C2×D4—1221no—
323C2×D4—2441no—
324C2×D4—2441no—
325C2×D4—2441no—
326C22×C4—611no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—611no—
331C22×C4—1221no—
332C22×C4—1221no—
333C22×C4—1221no—
334C22×C4—1221no—
335C22×C4—2441no—
336C22×C4D4×D6111yescharacteristic
337Q16—22no—
338SD16—2442no—
339D8—1222no—
340M16—22no—
341M16—22no—
342M16—41no—
343M16—1222no—
344M16—1222no—
345M16—1222no—
346M16—1222no—
347M16—2442no—
348M16—2442no—
349M16—2441no—
350C2×C8—41no—
351C2×C8—611no—
352C2×C8—612no—
353C2×C8—612no—
354C2×C8—611no—
355C2×C8—1221no—
356C2×C8—1221no—
357C2×C8—2442no—
358C2×C8—2442no—
359C2×C8—2441no—
360C2×C8D4×D6111yescharacteristic
361C2×C8C23×D6111yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—41no—
364C4⋊C4—1221no—
365C4⋊C4—2441no—
366C22⋊C4—41no—
367C22⋊C4—2441no—
368C22⋊C4—2441no—
369C22⋊C4—2441no—
370C22⋊C4—2441no—
371C42—2441no—
372C16—22no—
373C16—1222no—
374C23—611no—
375C23—611no—
376C23—1221no—
377C23—1221no—
378C23—4881no—
379C23—2441no—
380C23SD16×D6111yescharacteristic
381Q8—41no—
382Q8—41no—
383D4—4881no—
384D4—4881no—
385D4—4882no—
386D4—4882no—
387D4—2441no—
388D4—2441no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—611no—
394C2×C4—611no—
395C2×C4—611no—
396C2×C4—611no—
397C2×C4—611no—
398C2×C4—611no—
399C2×C4—1221no—
400C2×C4—1221no—
401C2×C4—1221no—
402C2×C4—1221no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—1221no—
407C2×C4—4881no—
408C2×C4—4881no—
409C2×C4—2441no—
410C2×C4SD16×D6111yescharacteristic
411C2×C4C2×D4×D6111yescharacteristic
412C8—21no—
413C8—42no—
414C8—612no—
415C8—1221no—
416C8—2442no—
417C8#41×D6112yes—
418C22—21no—
419C22—41no—
420C22—611no—
421C22—611no—
422C22—1221no—
423C22—4881no—
424C22—4881no—
425C22—4881no—
426C22—4881no—
427C22—2441no—
428C22—2441no—
429C22—2442no—
430C22C2×SD16×D6111yescharacteristic
431C4—21no—
432C4—81no—
433C4—81no—
434C4—611no—
435C4—611no—
436C4—1221no—
437C4—4881no—
438C4#123×D6111yescharacteristic
439C4AGL(1,ℤ/8)×D6111yescharacteristic
440C2—21no—
441C2—41no—
442C2—1221no—
443C2—1221no—
444C2—9761no—
445C2—2441no—
446C2#481×D6111yescharacteristic
subgroupG/H
centerC2#481×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2239C2
socleC2×C61C2×#481
prime radical#21455430C2
hypercenter#2239D61
hypocenterC61#33656
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2239

Complements

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

NH ≅ G/Ntype
2. #430×D61C2semidirect
3. #21448037C2semidirect
4. #21448036C2semidirect
5. #21448035C2semidirect
6. #21448034C2semidirect
7. #21448033C2semidirect
8. #21447590C2semidirect
9. #21447567C2semidirect
10. #349×D61C2semidirect
11. C61×#2239C2semidirect
30. #16839586C22semidirect
31. #16839585C22semidirect
32. #16839584C22semidirect
33. #16839578C22semidirect
34. #16839576C22semidirect
35. #16839574C22semidirect
40. C61×#365C22semidirect
41. C61×#362C22semidirect
221. #2239D61semidirect
258. #365C2×D61semidirect
260. #362C2×D61semidirect
266. C61#33656semidirect

Maximal subgroups

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

subgroupindex
#214554302
#430×D612
#214480372
#214480362
#214480352
#214480342
#214480332
#214475902
#214475672
#349×D612
C61×#22392
#3365661