Finite Group Databasebeta

← #29712989

Subgroups — #29712989

Order 15616.

1364 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840423C211yesmaximal, characteristic
2C2×#16840075C211yesmaximal, characteristic, complemented
3#21450121C211yesmaximal, characteristic, complemented
4#21447056C212yesmaximal, complemented
5#21447020C212yesmaximal, complemented
6#21446247C211yesmaximal, characteristic, complemented
7#21446111C211yesmaximal, characteristic, complemented
8#21446071C212yesmaximal, 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×#14519760—21no—
18C2×#14519753—21no—
19C2×#14519753C2211yescharacteristic
20C2×#14519751—21no—
21C2×#14519751C2211yescharacteristic
22C2×Q8×Dic61C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×C4×C22⋊D61C2211yescharacteristic
25#16840075—24no—
26C2×#14519710C2212yes—
27C2×#14519701C2212yes—
28C2×#14519701C2212yes—
29C2×#14519699C2212yes—
30C2×#14519697C2212yes—
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519689C2212yes—
33#16839693—21no—
34#16839688—21no—
35#16839658C2212yes—
36#16839625—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#14519760—44no—
62#14519753—28no—
63#14519753—22no—
64#14519753—22no—
65#14519751—28no—
66#14519751—22no—
67#14519751D414yes—
68Q8×Dic61—22no—
69Q8×Dic61—22no—
70C4×C22⋊D61—28no—
71C4×C22⋊D61—22no—
72C4×C22⋊D61—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—22no—
78C2×#11758196D411yescharacteristic
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61D411yescharacteristic
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—22no—
86C2×#11758198D411yescharacteristic
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89#14519710—24no—
90#14519701—24no—
91#14519701—42no—
92#14519699—42no—
93#14519697—24no—
94C22⋊#92418—22no—
95#14519689—42no—
96#14519684—21no—
97#14519684—22no—
98#14519684—22no—
99C22×Q8×C61—21no—
100C22×Q8×C61—22no—
101C22×Q8×C61—24no—
102C22×Q8×C61—42no—
103C22×Q8×C61D414yes—
104C22×Q8×C61C2311yescharacteristic
105C22×Q8×C61C2311yescharacteristic
106C23×C4×C61—21no—
107C23×C4×C61—21no—
108C23×C4×C61C2311yescharacteristic
109C22×C8×C61—22no—
110C22×C8×C61—42no—
111C22×C8×C61C2312yes—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61—22no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61C2312yes—
116#103×C61—24no—
117#103×C61—24no—
118C24×C61C2×D411yescharacteristic
119C23×D61—21no—
120C23×D61C2×D411yescharacteristic
121C2×Q8×C61—24no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—28no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—44no—
129C2×Q8×C61—44no—
130C2×Q8×C61—48no—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61C2×D412yes—
133C2×Q8×C61#4114yes—
134C22×C4×C61—21no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—24no—
138C22×C4×C61—21no—
139C22×C4×C61—22no—
140C22×C4×C61—44no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—24no—
149C2×C22⋊D61—22no—
150C2×C22⋊D61—22no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—42no—
153C2×C22⋊D61—42no—
154C22×Dic61—21no—
155C22×Dic61—22no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159#11758226—48no—
160#11758226—44no—
161#11758226—44no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—24no—
165C2×D244—21no—
166C2×D244—44no—
167C2×D244C2×D412yes—
168C2×C4×D61—24no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—44no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—44no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—42no—
181C22⋊Dic61—22no—
182C22⋊Dic61—42no—
183#11758198—24no—
184#11758198—24no—
185#11758198—24no—
186#11758198—24no—
187#11758198—22no—
188#11758198—44no—
189C4⋊Dic61—24no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61C2×D412yes—
194#11758196—24no—
195#11758196—22no—
196#11758196—44no—
197C4×Dic61—24no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61#12012yes—
206C23×C61#13512yes—
207C23×C61AGL(1,ℤ/8)12yes—
208C23×C61C22×D411yescharacteristic
209C22×D61—21no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—22no—
215C22×D61—42no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—42no—
219Q8×C61—22no—
220Q8×C61—24no—
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—48no—
224Q8×C61—42no—
225Q8×C61#12314yescomplemented
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—24no—
232C2×C4×C61—22no—
233C2×C4×C61—48no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—84no—
239C2×C4×C61#12012yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4112yes—
243C2×C4×C61H(5,𝔽2)12yes—
244C22⋊D61—44no—
245C22⋊D61—48no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259D244—24no—
260D244—48no—
261D244—41no—
262D244—41no—
263C4×D61—48no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—42no—
267C8×C61—42no—
268#92418—42no—
269#24035—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—24no—
273C22×C61—24no—
274C22×C61#44914yes—
275C22×C61C2×#12011yescharacteristic
276C22×C61#54612yes—
277C22×C61C2×#13511yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61#57812yes—
280C2×D61—42no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—42no—
284C2×D61—41no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—82no—
289C2×D61—82no—
290C4×C61—24no—
291C4×C61—22no—
292C4×C61—42no—
293C4×C61—82no—
294C4×C61C2×#12311yescharacteristic
295C4×C61#53411yescharacteristic
296C4×C61#54612yes—
297Dic61—41no—
298Dic61—41no—
299Dic61—42no—
300Dic61—41no—
301Dic61—82no—
302C2×#548—611no—
303C2×#480—611no—
304C22×#103D6111yescharacteristic, complemented
305#2486—611no—
306#2042—612no—
307#2036—612no—
308#1893—611no—
309#1869—611no—
310#1863—612no—
311#1788—612no—
312#1787—611no—
313C2×C61—24no—
314C2×C61#240511yescharacteristic
315C2×C61C2×#44911yescharacteristic
316C2×C61#301111yescharacteristic
317C2×C61#330012yes—
318C2×C61#330812yes—
319D61—81no—
320D61—82no—
321C22×#41—611no—
322C23×Q8C2×D6111yescharacteristic
323C23×C8C2×D6111yescharacteristic
324#548—1224no—
325#548—1224no—
326C2×#124—611no—
327C2×#124—1221no—
328C2×#123—1221no—
329C2×C22⋊Q8—611no—
330C2×C22⋊Q8—1221no—
331C2×#121—611no—
332C2×#121—1221no—
333C2×C4×Q8—611no—
334C2×C4×D4—611no—
335C2×C4×D4—1221no—
336C22×C4⋊C4C2×D6111yescharacteristic
337#480—1224no—
338C2×#107—612no—
339C2×#103—24no—
340C2×#103—612no—
341C2×#103—612no—
342C2×#103C2×D6112yes—
343C2×#103C2×D6112yes—
344C2×#102—612no—
345C2×#102—612no—
346C2×C22⋊C8—611no—
347C2×C22⋊C8—612no—
348#398—1221no—
349#397—1221no—
350#389—612no—
351#381—1221no—
352#380—611no—
353#380—612no—
354#340—612no—
355#336—612no—
356C61#2403511yescharacteristic, complemented, Sylow
357C2×#41—1222no—
358C2×#41—1222no—
359C2×#41—1224no—
360C2×#41—2442no—
361C22×Q8—21no—
362C22×Q8—22no—
363C22×Q8—24no—
364C22×Q8—42no—
365C22×Q8C22⋊D6114yes—
366C22×Q8C22×D6111yescharacteristic
367C22×Q8C22×D6111yescharacteristic
368C22×D4—611no—
369C22×D4—611no—
370C22×D4—1221no—
371C22×D4—1221no—
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—611no—
375C23×C4—1221no—
376C23×C4C22×D6111yescharacteristic
377C22×C8—22no—
378C22×C8—42no—
379C22×C8—612no—
380C22×C8C22×D6112yes—
381#124—1222no—
382#124—1228no—
383#124—1222no—
384#123—2444no—
385C22⋊Q8—1222no—
386C22⋊Q8—1228no—
387C22⋊Q8—1222no—
388#121—614no—
389#121—1228no—
390#121—1222no—
391C4×Q8—1222no—
392C4×Q8—1222no—
393C4×D4—1228no—
394C4×D4—1222no—
395C4×D4—1222no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4C22×D6112yes—
406C2×C22⋊C4—611no—
407C2×C22⋊C4—611no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1222no—
413C2×C22⋊C4—1221no—
414C2×C42—611no—
415C2×C42—1221no—
416#107—1224no—
417#103—24no—
418#103—24no—
419#103—1224no—
420#103—2442no—
421#102—1224no—
422#102—2442no—
423C22⋊C8—1222no—
424C22⋊C8—2442no—
425#95—1222no—
426#95—1222no—
427#95—1221no—
428C24—611no—
429C24—1221no—
430C24D4×D6111yescharacteristic
431#41—2448no—
432#41—2444no—
433#41—2444no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—28no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—48no—
444C2×Q8#1175822214yes—
445C2×Q8C2×C22⋊D6112yes—
446C2×Q8C2×C22⋊D6112yes—
447C2×D4—612no—
448C2×D4—1224no—
449C2×D4—1224no—
450C2×D4—1222no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—1221no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—2444no—
457C2×D4—2442no—
458C2×D4—2442no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—1224no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—2444no—
480C22×C4D4×D6112yes—
481C22×C4D4×D6111yescharacteristic
482C22×C4C2×C22⋊D6112yes—
483C22×C4C23×D6111yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—44no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1222no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—612no—
495C4⋊C4—1224no—
496C4⋊C4—1224no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—2444no—
502C22⋊C4—1224no—
503C22⋊C4—1224no—
504C22⋊C4—1224no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1224no—
508C22⋊C4—2444no—
509C22⋊C4—2442no—
510C42—1224no—
511C42—1221no—
512C42—1221no—
513C23—22no—
514C23—22no—
515C23—1222no—
516C23—1221no—
517C23—1221no—
518C23—1221no—
519C23—1222no—
520C23—1222no—
521C23—2441no—
522C23—2441no—
523C23—2442no—
524C23—2442no—
525C23#1451974512yes—
526C23#1451979812yes—
527C23#1451980412yes—
528C23C2×D4×D6111yescharacteristic
529Q8—22no—
530Q8—24no—
531Q8—22no—
532Q8—22no—
533Q8—48no—
534Q8—42no—
535Q8#1451976014yescomplemented
536D4—1224no—
537D4—2448no—
538D4—2444no—
539D4—2448no—
540D4—2441no—
541D4—2441no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—84no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—2448no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2442no—
572C2×C4#1451974512yes—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4C2×#1175822212yes—
576C2×C4#1451982212yes—
577C8—42no—
578C8—2442no—
579C22—22no—
580C22—24no—
581C22—24no—
582C22—22no—
583C22—4882no—
584C22—4882no—
585C22—2441no—
586C22—2442no—
587C22—2442no—
588C22—2442no—
589C22—2441no—
590C22—2442no—
591C22—2442no—
592C22—2441no—
593C22#1683990614yes—
594C22C2×#1451974511yescharacteristic
595C22#1684040212yes—
596C22C2×#1451979811yescharacteristic
597C22C2×#1451980411yescharacteristic
598C22#1684057512yes—
599C4—22no—
600C4—24no—
601C4—42no—
602C4—82no—
603C4—4882no—
604C4—2441no—
605C4—2441no—
606C4—2442no—
607C4—2441no—
608C4C2×#1451976011yescharacteristic
609C4#1684032111yescharacteristic
610C4#1684040212yes—
611C2—24no—
612C2—4882no—
613C2—4881no—
614C2#2144914611yescharacteristic
615C2C2×#1683990611yescharacteristic
616C2#2145511011yescharacteristic
617C2#2145770212yes—
618C2#2145780612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#103×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840423C2
hypercenterC22×#103D61
hypocenterC61#24035
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×#16840075C2semidirect
3. #21450121C2semidirect
4. #21447056C2semidirect
5. #21447020C2semidirect
6. #21446247C2semidirect
7. #21446111C2semidirect
8. #21446071C2semidirect
9. #21445758C2semidirect
10. #21445754C2semidirect
11. C22×#103×C61C2semidirect
225. Q8×C61#123semidirect
304. C22×#103D61semidirect
356. C61#24035semidirect
535. Q8#14519760semidirect

Maximal subgroups

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

subgroupindex
C2×#168404232
C2×#168400752
#214501212
#214470562
#214470202
#214462472
#214461112
#214460712
#214457582
#214457542
C22×#103×C612
#2403561