Finite Group Databasebeta

← #29833832

Subgroups — #29833832

Order 15616.

1140 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519800C211yesmaximal, characteristic
2C2×#16840204C212yesmaximal, complemented
3#467×D61C211yesmaximal, characteristic, complemented
4#21447762C212yesmaximal, complemented
5#21447758C212yesmaximal, complemented
6#21447753C211yesmaximal, characteristic, complemented
7#21447750C211yesmaximal, characteristic, complemented
8#21447747C211yesmaximal, characteristic, complemented
9#21447530C212yesmaximal, complemented
10#21447428C211yesmaximal, characteristic, complemented
11C61×#2188C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13C2×#14519800—28no—
14C2×#14519800—21no—
15C2×#14519800—21no—
16C2×#14519800C2212yes—
17C2×#14519800C2212yes—
18C2×Q16×D61C2211yescharacteristic, complemented
19C22×#11758201—21no—
20C22×D488C2211yescharacteristic
21C22×C8×D61C2211yescharacteristic
22C22⋊Q8×D61—21no—
23#16840204—28no—
24C2×#14519723C2212yescomplemented
25C2×#14519722C2212yescomplemented
26C2×#14519720C2212yescomplemented
27C2×#14519716C2212yescomplemented
28#16840004C2211yescharacteristic, complemented
29#16840003C2211yescharacteristic
30#16840002—21no—
31#16840001—21no—
32#16840000C2211yescharacteristic, complemented
33#107×D61C2211yescharacteristic, complemented
34#103×D61—21no—
35#16839566C2212yescomplemented
36#16839564C2212yescomplemented
37#16839562C2212yescomplemented
38C22⋊#11758185C2211yescharacteristic
39#16839541C2212yescomplemented
40C22×Q16×C61C2211yescharacteristic
41C2×Q32×C61C2212yescomplemented
42C61×#467C2211yescharacteristic
43C61×#358C2212yescomplemented
44C61×C22⋊C16C2211yescharacteristic
45C2×#11758226—28no—
46C2×#11758226—22no—
47C2×#11758226—22no—
48C2×#11758226—44no—
49C2×#11758226—41no—
50C2×#11758226—41no—
51C2×Q8×D61—21no—
52C22×D244—21no—
53C22×D244—41no—
54C22×C4×D61—41no—
55C22×C4×D61D411yescharacteristic
56#14519800—28no—
57#14519800—28no—
58#14519800—24no—
59#14519800—24no—
60#14519800—24no—
61#14519800D418yes—
62C2×Dic244C2311yescharacteristic
63C2×#11758202—21no—
64Q16×D61—24no—
65C2×#11758201—28no—
66C2×#11758201—22no—
67C2×#11758201—22no—
68C2×D488—21no—
69C2×D488—24no—
70C2×D488C2312yes—
71C22×#92418D411yescharacteristic
72C2×C8×D61—24no—
73C2×C8×D61D411yescharacteristic
74C2×C8×D61D411yescharacteristic
75C2×C8×D61C2311yescharacteristic
76C2×C8×D61C2311yescharacteristic
77#14519757—21no—
78#14519756—21no—
79#14519755—41no—
80#14519754—41no—
81#14519753—21no—
82C4⋊C4×D61—21no—
83C4⋊C4×D61—41no—
84C22⋊C4×D61—41no—
85#14519723—24no—
86#14519722—24no—
87#14519720—24no—
88#14519716—24no—
89C2×#11758185C2312yes—
90#14519711C2311yescharacteristic
91#14519710—21no—
92#14519702—21no—
93#14519701—21no—
94#14519700—21no—
95C22×Q8×C61—21no—
96C2×Q16×C61—24no—
97C2×Q16×C61—21no—
98C2×Q16×C61C2311yescharacteristic
99C2×Q16×C61C2312yes—
100C22×C8×C61C2311yescharacteristic
101C22⋊Q8×C61—21no—
102Q32×C61—24no—
103C2×C16×C61C2312yes—
104#107×C61C2311yescharacteristic
105#103×C61—21no—
106C23×D61—41no—
107C23×D61D811yescharacteristic
108C2×Q8×C61—24no—
109C2×Q8×C61—22no—
110C2×Q8×C61—21no—
111C2×Q8×C61—42no—
112C2×Q8×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61C2×D411yescharacteristic
115C22×Dic61D811yescharacteristic
116#11758226—28no—
117#11758226—28no—
118#11758226—48no—
119#11758226—44no—
120#11758226—48no—
121#11758226—44no—
122Q8×D61—42no—
123Q8×D61—42no—
124C2×D244—24no—
125C2×D244—22no—
126C2×D244—42no—
127C2×D244—48no—
128C2×D244—42no—
129C2×D244—42no—
130C2×D244—41no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—48no—
134C2×C4×D61—42no—
135C2×C4×D61—42no—
136C2×C4×D61—41no—
137C2×C4×D61—41no—
138C2×C4×D61D811yescharacteristic
139C2×C4×D61D811yescharacteristic
140C2×C4×D61C2×D411yescharacteristic
141C2×C4×D61C2×D411yescharacteristic
142C2×Dic122—21no—
143C2×Dic122—41no—
144Q16×C61—24no—
145Q16×C61—22no—
146Q16×C61—22no—
147Q16×C61C2×D414yes—
148C2×C8×C61—22no—
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61C2411yescharacteristic
151C4⋊C4×C61—21no—
152C4⋊C4×C61—41no—
153C22⋊C4×C61—41no—
154C22⋊Dic61—41no—
155#11758202—42no—
156#11758201—28no—
157#11758201—28no—
158#11758198—41no—
159#11758198—41no—
160C4⋊Dic61—21no—
161C4⋊Dic61—41no—
162#11758196—41no—
163#11758196—41no—
164#11758196—41no—
165C2×#92418—22no—
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168Dic244—22no—
169D488—24no—
170D488—22no—
171D488C2×D414yes—
172C8×D61—22no—
173C8×D61—22no—
174C8×D61—22no—
175C8×D61C2×D414yes—
176C16×C61—22no—
177#11758185—22no—
178C23×C61C2×D811yescharacteristic
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—44no—
183C22×D61—42no—
184C22×D61—82no—
185C22×D61—82no—
186C22×D61C2×D811yescharacteristic
187C22×D61C2×D811yescharacteristic
188Q8×C61—24no—
189Q8×C61—44no—
190Q8×C61—42no—
191Q8×C61—41no—
192Q8×C61—41no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—44no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61C2×D811yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×Dic61—22no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61C2×D811yescharacteristic
205C2×Dic61C2×D811yescharacteristic
206D244—24no—
207D244—44no—
208D244—48no—
209D244—48no—
210D244—42no—
211C4×D61—22no—
212C4×D61—21no—
213C4×D61—21no—
214C4×D61—22no—
215C4×D61—48no—
216C4×D61—48no—
217C4×D61—82no—
218C4×D61—82no—
219C4×D61#12012yes—
220C4×D61AGL(1,ℤ/8)12yes—
221Dic122—41no—
222Dic122—41no—
223Dic122—82no—
224C8×C61—21no—
225C8×C61C22×D412yes—
226#92418—21no—
227#92418#12012yes—
228#33521—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61#51012yescomplemented
232C22×C61C22×D811yescharacteristic
233C2×D61—21no—
234C2×D61—21no—
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—21no—
238C2×D61—44no—
239C2×D61—44no—
240C2×D61—88no—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61#44712yes—
244C4×C61—21no—
245C4×C61—44no—
246C4×C61—81no—
247C4×C61—81no—
248C4×C61C2×#12011yescharacteristic
249C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
250Dic61—21no—
251Dic61—81no—
252Dic61—81no—
253Dic61#44712yes—
254C22×#135—611no—
255C2×#510—612no—
256C2×#467—611no—
257#2189—612no—
258#2189—612no—
259#2188—611no—
260#2188D6111yescharacteristic, complemented
261#2187—611no—
262#2187—611no—
263#2139—612no—
264#2112—611no—
265C2×C61—22no—
266C2×C61C2×#44711yescharacteristic
267C2×C61C2×#51011yescharacteristic
268C2×C61#341411yescharacteristic
269D61—22no—
270D61—41no—
271D61—84no—
272C22×#41—1221no—
273C2×#135—612no—
274C2×#135—612no—
275C2×#135—1221no—
276C2×#135—1221no—
277C2×#135—1228no—
278C22×Q16—611no—
279C22×Q16C2×D6111yescharacteristic
280C22×SD16—1221no—
281C22×D8—611no—
282C23×C8—611no—
283C2×C22⋊Q8—1221no—
284#510—1228no—
285C2×Q32—612no—
286C2×Q32C2×D6112yescomplemented
287C2×SD32—612no—
288C2×SD32—612no—
289C2×M32—612no—
290#467—611no—
291#467—611no—
292#467—611no—
293#467—1221no—
294#467—1221no—
295#467C2×D6111yescharacteristic
296C2×#107—611no—
297C2×#103—1221no—
298#358—612no—
299#358C2×D6112yescomplemented
300#357—612no—
301#357—612no—
302C22⋊C16—611no—
303C22⋊C16—612no—
304C22⋊C16C2×D6111yescharacteristic
305C61#3352111yescharacteristic, complemented, Sylow
306C2×#41—1222no—
307C2×#41—1222no—
308C2×#41—1228no—
309C2×#41—2441no—
310C2×#41—2441no—
311C2×#41—2444no—
312C22×Q8—21no—
313C22×Q8—1221no—
314C22×D4—1221no—
315C22×D4—2441no—
316C23×C4—611no—
317C23×C4—2441no—
318#135—618no—
319#135—1224no—
320#135—1224no—
321#135—1228no—
322#135—1228no—
323#135—1224no—
324C2×Q16—24no—
325C2×Q16—21no—
326C2×Q16—611no—
327C2×Q16—1224no—
328C2×Q16—1221no—
329C2×Q16C22×D6111yescharacteristic
330C2×Q16C22×D6112yes—
331C2×SD16—1222no—
332C2×SD16—1222no—
333C2×SD16—1228no—
334C2×D8—612no—
335C2×D8—1224no—
336C2×D8—1221no—
337C22×C8—611no—
338C22×C8—611no—
339C22×C8—611no—
340C22×C8—611no—
341C22×C8—611no—
342C22×C8—1224no—
343C22×C8C22×D6111yescharacteristic
344C22⋊Q8—21no—
345C22⋊Q8—1221no—
346C22⋊Q8—1221no—
347C22⋊Q8—1221no—
348C22⋊Q8—2441no—
349C22⋊Q8—2441no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—2441no—
352C2×C22⋊C4—2441no—
353Q32—24no—
354Q32—1224no—
355SD32—1224no—
356SD32—1224no—
357M32—1224no—
358C2×C16—612no—
359C2×C16C22×D6112yes—
360#107—611no—
361#107—1221no—
362#107C22×D6111yescharacteristic
363#103—21no—
364#103—1221no—
365#103—1221no—
366#103—1221no—
367C24—611no—
368C24—2441no—
369#41—1228no—
370#41—1228no—
371#41—2444no—
372#41—2448no—
373#41—2448no—
374#41—2444no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—42no—
379C2×Q8—41no—
380C2×Q8—1221no—
381C2×Q8—2441no—
382C2×Q8—2442no—
383C2×Q8—2442no—
384C2×D4—1224no—
385C2×D4—1222no—
386C2×D4—2442no—
387C2×D4—2442no—
388C2×D4—2442no—
389C2×D4—2448no—
390C2×D4—2441no—
391C22×C4—41no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—2442no—
400C22×C4—2442no—
401C22×C4—2448no—
402C22×C4—2441no—
403C22×C4—2441no—
404C22×C4D4×D6111yescharacteristic
405Q16—24no—
406Q16—22no—
407Q16—22no—
408Q16—1222no—
409Q16—2442no—
410Q16D4×D6114yes—
411SD16—1228no—
412SD16—1228no—
413D8—614no—
414D8—1224no—
415D8—1222no—
416C2×C8—22no—
417C2×C8—611no—
418C2×C8—614no—
419C2×C8—611no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8D4×D6111yescharacteristic
425C2×C8C23×D6111yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—1221no—
429C4⋊C4—2441no—
430C4⋊C4—2441no—
431C4⋊C4—2441no—
432C4⋊C4—2441no—
433C22⋊C4—41no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C16—22no—
438C16—1222no—
439C23—611no—
440C23—611no—
441C23—1222no—
442C23—1222no—
443C23—4882no—
444C23—4882no—
445C23—2441no—
446C23—2444no—
447C23—2442no—
448C23D8×D6111yescharacteristic
449Q8—24no—
450Q8—44no—
451Q8—42no—
452Q8—41no—
453Q8—41no—
454Q8—4882no—
455Q8—2441no—
456Q8—2441no—
457D4—1224no—
458D4—2444no—
459D4—2448no—
460D4—2448no—
461D4—2442no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—611no—
469C2×C4—612no—
470C2×C4—612no—
471C2×C4—611no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—1221no—
476C2×C4—1221no—
477C2×C4—4882no—
478C2×C4—4882no—
479C2×C4—2448no—
480C2×C4—2448no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4D8×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C8—21no—
486C8—612no—
487C8—1221no—
488C8C2×D4×D6112yes—
489C22—21no—
490C22—21no—
491C22—612no—
492C22—1221no—
493C22—1221no—
494C22—1221no—
495C22—1221no—
496C22—1221no—
497C22—4888no—
498C22—4882no—
499C22—4882no—
500C22—2444no—
501C22—2444no—
502C22#1684020412yescomplemented
503C22C2×D8×D6111yescharacteristic
504C4—21no—
505C4—44no—
506C4—81no—
507C4—81no—
508C4—612no—
509C4—1221no—
510C4—4881no—
511C4—4881no—
512C4#120×D6111yescharacteristic
513C4AGL(1,ℤ/8)×D6111yescharacteristic
514C2—22no—
515C2—1222no—
516C2—4884no—
517C2—2441no—
518C2#447×D6111yescharacteristic
519C2C2×#1684020411yescharacteristic
520C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2188C2
socleC22×C61C22×D8
prime radicalC22×#14519800C2
hypercenter#2188D61
hypocenterC61#33521
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2188

Complements

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

NH ≅ G/Ntype
2. C2×#16840204C2semidirect
3. #467×D61C2semidirect
4. #21447762C2semidirect
5. #21447758C2semidirect
6. #21447753C2semidirect
7. #21447750C2semidirect
8. #21447747C2semidirect
9. #21447530C2semidirect
10. #21447428C2semidirect
11. C61×#2188C2semidirect
18. C2×Q16×D61C22semidirect
24. C2×#14519723C22semidirect
25. C2×#14519722C22semidirect
26. C2×#14519720C22semidirect
27. C2×#14519716C22semidirect
28. #16840004C22semidirect
32. #16840000C22semidirect
33. #107×D61C22semidirect
35. #16839566C22semidirect
36. #16839564C22semidirect
37. #16839562C22semidirect
39. #16839541C22semidirect
41. C2×Q32×C61C22semidirect
43. C61×#358C22semidirect
231. C22×C61#510semidirect
260. #2188D61semidirect
286. C2×Q32C2×D61semidirect
299. #358C2×D61semidirect
305. C61#33521semidirect
502. C22#16840204semidirect

Maximal subgroups

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

subgroupindex
C22×#145198002
C2×#168402042
#467×D612
#214477622
#214477582
#214477532
#214477502
#214477472
#214475302
#214474282
C61×#21882
#3352161