Finite Group Databasebeta

← #29713000

Subgroups — #29713000

Order 15616.

1344 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840423C211yesmaximal, characteristic
2C2×#16840080C211yesmaximal, characteristic, complemented
3#21450130C211yesmaximal, characteristic, complemented
4#21447056C212yesmaximal, complemented
5#21447013C212yesmaximal, complemented
6#21446247C211yesmaximal, characteristic, complemented
7#21446111C211yesmaximal, characteristic, complemented
8#21446085C212yesmaximal, complemented
9#21445758C212yesmaximal, complemented
10#21445754C211yesmaximal, characteristic, complemented
11C22×#103×C61C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13#16840423—24no—
14#16840423—24no—
15C2×#14519766—21no—
16C2×#14519766C2211yescharacteristic
17C2×#14519753—21no—
18C2×#14519753C2211yescharacteristic
19C2×#14519751—21no—
20C2×#14519751C2211yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×C4×C22⋊D61—21no—
23C2×C4×C22⋊D61C2211yescharacteristic
24#16840080—24no—
25C2×#14519708C2212yes—
26C2×#14519701C2212yes—
27C2×#14519701C2212yes—
28C2×#14519699C2212yes—
29C2×#14519697C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519689C2212yes—
32#16839700—21no—
33#16839697—21no—
34#16839658C2212yes—
35#16839641—21no—
36#16839635—21no—
37#16839622C2211yescharacteristic
38#16839619C2212yes—
39#16839525C2212yes—
40#16839516C2212yes—
41C23×Q8×C61C2211yescharacteristic
42C23×C8×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C2×#103×C61—24no—
45C2×#103×C61C2212yes—
46C2×#103×C61C2212yes—
47C2×#11758226—22no—
48C2×#11758226—22no—
49C2×#11758226—24no—
50C2×#11758226—42no—
51C22×D244—21no—
52C22×D244C2311yescharacteristic
53C22×C22⋊D61—21no—
54C22×C22⋊D61D411yescharacteristic
55C22×C4×D61—21no—
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58#14519766—28no—
59#14519766—22no—
60#14519766—22no—
61#14519753—28no—
62#14519753—22no—
63#14519753—22no—
64#14519751—28no—
65#14519751—22no—
66#14519751D414yes—
67Q8×Dic61—22no—
68Q8×Dic61—22no—
69C4×C22⋊D61—28no—
70C4×C22⋊D61—22no—
71C4×C22⋊D61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—22no—
77C2×#11758196D411yescharacteristic
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61D411yescharacteristic
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—22no—
85C2×#11758198D411yescharacteristic
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88#14519708—24no—
89#14519701—24no—
90#14519701—42no—
91#14519699—42no—
92#14519697—24no—
93C22⋊#92418—22no—
94#14519689—42no—
95#14519684—21no—
96#14519684—22no—
97#14519684—22no—
98C22×Q8×C61—21no—
99C22×Q8×C61—22no—
100C22×Q8×C61—24no—
101C22×Q8×C61—42no—
102C22×Q8×C61D414yes—
103C22×Q8×C61C2311yescharacteristic
104C22×Q8×C61C2311yescharacteristic
105C23×C4×C61—21no—
106C23×C4×C61—21no—
107C23×C4×C61C2311yescharacteristic
108C22×C8×C61—22no—
109C22×C8×C61—42no—
110C22×C8×C61C2312yes—
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2312yes—
115#103×C61—24no—
116#103×C61—24no—
117C24×C61C2×D411yescharacteristic
118C23×D61—21no—
119C23×D61C2×D411yescharacteristic
120C2×Q8×C61—24no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—22no—
124C2×Q8×C61—28no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—44no—
128C2×Q8×C61—44no—
129C2×Q8×C61—48no—
130C2×Q8×C61C2×D412yes—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61#4114yes—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—24no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—44no—
140C22×C4×C61—42no—
141C22×C4×C61—82no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—24no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—22no—
149C2×C22⋊D61—22no—
150C2×C22⋊D61—42no—
151C2×C22⋊D61—42no—
152C22×Dic61—21no—
153C22×Dic61—22no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157#11758226—48no—
158#11758226—44no—
159#11758226—44no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—24no—
163C2×D244—21no—
164C2×D244—44no—
165C2×D244C2×D412yes—
166C2×C4×D61—24no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—44no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—44no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—42no—
179C22⋊Dic61—22no—
180C22⋊Dic61—42no—
181#11758198—24no—
182#11758198—24no—
183#11758198—24no—
184#11758198—24no—
185#11758198—22no—
186#11758198—82no—
187C4⋊Dic61—24no—
188C4⋊Dic61—21no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61C2×D412yes—
192#11758196—24no—
193#11758196—22no—
194#11758196—82no—
195C4×Dic61—24no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61#12012yes—
204C23×C61#13512yes—
205C23×C61#13712yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—21no—
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—22no—
213C22×D61—42no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—42no—
217Q8×C61—22no—
218Q8×C61—24no—
219Q8×C61—22no—
220Q8×C61—22no—
221Q8×C61—48no—
222Q8×C61—42no—
223Q8×C61#12314yes—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—22no—
231C2×C4×C61—48no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—84no—
237C2×C4×C61#12012yes—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4112yes—
241C2×C4×C61H(5,𝔽2)12yes—
242C22⋊D61—44no—
243C22⋊D61—48no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257D244—24no—
258D244—48no—
259D244—41no—
260D244—41no—
261C4×D61—48no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—42no—
265C8×C61—42no—
266#92418—42no—
267#24036—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—24no—
271C22×C61—24no—
272C22×C61#45214yes—
273C22×C61C2×#12011yescharacteristic
274C22×C61#54612yes—
275C22×C61C2×#13511yescharacteristic
276C22×C61C2×#13711yescharacteristic
277C22×C61#57712yes—
278C2×D61—42no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—42no—
282C2×D61—41no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—82no—
287C2×D61—82no—
288C4×C61—24no—
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—82no—
292C4×C61C2×#12311yescharacteristic
293C4×C61#53411yescharacteristic
294C4×C61#54612yes—
295Dic61—41no—
296Dic61—41no—
297Dic61—42no—
298Dic61—41no—
299Dic61—82no—
300C2×#548—611no—
301C2×#481—611no—
302C22×#103D6111yescharacteristic, complemented
303#2487—611no—
304#2042—612no—
305#2035—612no—
306#1893—611no—
307#1869—611no—
308#1865—612no—
309#1788—612no—
310#1787—611no—
311C2×C61—24no—
312C2×C61#240511yescharacteristic
313C2×C61C2×#45211yescharacteristic
314C2×C61#301011yescharacteristic
315C2×C61#330112yes—
316C2×C61#331612yes—
317D61—81no—
318D61—82no—
319C22×#41—611no—
320C23×Q8C2×D6111yescharacteristic
321C23×C8C2×D6111yescharacteristic
322#548—1224no—
323#548—1224no—
324C2×#124—611no—
325C2×#124—1221no—
326C2×C22⋊Q8—611no—
327C2×C22⋊Q8—1221no—
328C2×#121—611no—
329C2×#121—1221no—
330C2×C4×Q8—611no—
331C2×C4×D4—611no—
332C2×C4×D4—1221no—
333C22×C4⋊C4C2×D6111yescharacteristic
334#481—1224no—
335C2×#106—612no—
336C2×#103—24no—
337C2×#103—612no—
338C2×#103—612no—
339C2×#103C2×D6112yes—
340C2×#103C2×D6112yes—
341C2×#102—612no—
342C2×#102—612no—
343C2×C22⋊C8—611no—
344C2×C22⋊C8—612no—
345#400—1221no—
346#399—1221no—
347#389—612no—
348#386—1221no—
349#385—1221no—
350#380—611no—
351#380—612no—
352#340—612no—
353#336—612no—
354C61#2403611yescharacteristic, complemented, Sylow
355C2×#41—1222no—
356C2×#41—1222no—
357C2×#41—1224no—
358C2×#41—2442no—
359C22×Q8—21no—
360C22×Q8—22no—
361C22×Q8—24no—
362C22×Q8—42no—
363C22×Q8C22⋊D6114yes—
364C22×Q8C22×D6111yescharacteristic
365C22×Q8C22×D6111yescharacteristic
366C22×D4—611no—
367C22×D4—611no—
368C22×D4—1221no—
369C22×D4—1221no—
370C23×C4—21no—
371C23×C4—21no—
372C23×C4—611no—
373C23×C4—1221no—
374C23×C4C22×D6111yescharacteristic
375C22×C8—22no—
376C22×C8—42no—
377C22×C8—612no—
378C22×C8C22×D6112yes—
379#124—1222no—
380#124—1228no—
381#124—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1228no—
384C22⋊Q8—1222no—
385#121—614no—
386#121—1228no—
387#121—1222no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×D4—1228no—
391C4×D4—1222no—
392C4×D4—1222no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4C22×D6112yes—
403C2×C22⋊C4—611no—
404C2×C22⋊C4—611no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1222no—
410C2×C22⋊C4—1221no—
411C2×C42—611no—
412C2×C42—1221no—
413#106—1224no—
414#103—24no—
415#103—24no—
416#103—1224no—
417#103—2442no—
418#102—1224no—
419#102—2442no—
420C22⋊C8—1222no—
421C22⋊C8—2442no—
422#95—1222no—
423#95—1222no—
424#95—1221no—
425C24—611no—
426C24—1221no—
427C24D4×D6111yescharacteristic
428#41—2448no—
429#41—2444no—
430#41—2444no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—24no—
435C2×Q8—28no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—48no—
441C2×Q8#1175822214yes—
442C2×Q8C2×C22⋊D6112yes—
443C2×Q8C2×C22⋊D6112yes—
444C2×D4—612no—
445C2×D4—1224no—
446C2×D4—1222no—
447C2×D4—1224no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—1221no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—2444no—
454C2×D4—2442no—
455C2×D4—2442no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—1224no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—2444no—
476C22×C4D4×D6112yes—
477C22×C4D4×D6111yescharacteristic
478C22×C4C2×C22⋊D6112yes—
479C22×C4C23×D6111yescharacteristic
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—44no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—612no—
491C4⋊C4—1224no—
492C4⋊C4—1224no—
493C4⋊C4—1221no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—4882no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C22⋊C4—1224no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C22⋊C4—1224no—
504C22⋊C4—4882no—
505C22⋊C4—2442no—
506C42—1224no—
507C42—1221no—
508C42—1221no—
509C23—22no—
510C23—22no—
511C23—1222no—
512C23—1221no—
513C23—1221no—
514C23—1221no—
515C23—1222no—
516C23—1222no—
517C23—2441no—
518C23—2441no—
519C23—2442no—
520C23—2442no—
521C23#1451974512yes—
522C23#1451980012yes—
523C23#1451980912yes—
524C23C2×D4×D6111yescharacteristic
525Q8—22no—
526Q8—24no—
527Q8—22no—
528Q8—22no—
529Q8—48no—
530Q8—42no—
531Q8#1451976014yes—
532D4—1224no—
533D4—2448no—
534D4—2444no—
535D4—2448no—
536D4—2441no—
537D4—2441no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4—1221no—
552C2×C4—1222no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—2448no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2442no—
568C2×C4#1451974512yes—
569C2×C4C2×D4×D6111yescharacteristic
570C2×C4C22×C22⋊D6111yescharacteristic
571C2×C4C2×#1175822212yes—
572C2×C4#1451982212yes—
573C8—42no—
574C8—2442no—
575C22—24no—
576C22—24no—
577C22—22no—
578C22—22no—
579C22—4882no—
580C22—4882no—
581C22—2442no—
582C22—2442no—
583C22—2441no—
584C22—2442no—
585C22—2442no—
586C22—2441no—
587C22—2441no—
588C22—2442no—
589C22#1683992314yes—
590C22C2×#1451974511yescharacteristic
591C22#1684040212yes—
592C22C2×#1451980011yescharacteristic
593C22C2×#1451980911yescharacteristic
594C22#1684056812yes—
595C4—22no—
596C4—24no—
597C4—42no—
598C4—82no—
599C4—4882no—
600C4—2441no—
601C4—2442no—
602C4—2441no—
603C4—2441no—
604C4C2×#1451976011yescharacteristic
605C4#1684032111yescharacteristic
606C4#1684040212yes—
607C2—24no—
608C2—4882no—
609C2—4881no—
610C2#2144914611yescharacteristic
611C2C2×#1683992311yescharacteristic
612C2#2145510111yescharacteristic
613C2#2145771712yes—
614C2#2145792612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#103×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840423C2
hypercenterC22×#103D61
hypocenterC61#24036
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
2. C2×#16840080C2semidirect
3. #21450130C2semidirect
4. #21447056C2semidirect
5. #21447013C2semidirect
6. #21446247C2semidirect
7. #21446111C2semidirect
8. #21446085C2semidirect
9. #21445758C2semidirect
10. #21445754C2semidirect
11. C22×#103×C61C2semidirect
302. C22×#103D61semidirect
354. C61#24036semidirect

Maximal subgroups

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

subgroupindex
C2×#168404232
C2×#168400802
#214501302
#214470562
#214470132
#214462472
#214461112
#214460852
#214457582
#214457542
C22×#103×C612
#2403661