Finite Group Databasebeta

← #30564741

Subgroups — #30564741

Order 15616.

2090 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459620C211yesmaximal, characteristic
2C2×#16840427C211yesmaximal, characteristic
3#21457698C214yesmaximal, complemented
4#21457694C214yesmaximal, complemented
5#21457569C214yesmaximal, complemented
6#21457566C214yesmaximal, complemented
7#21456767C211yesmaximal, characteristic, complemented
8#21456592C214yesmaximal, complemented
9#21455859C211yesmaximal, characteristic, complemented
10#21455854C211yesmaximal, characteristic, complemented
11#451×D61C214yesmaximal, complemented
12#21454842C211yesmaximal, characteristic
13C61×#3209C211yesmaximal, characteristic, complemented
14C22×Q8×D61C2212yes—
15C2×Q16×D61C2218yescomplemented
16#16840473C2212yes—
17#16840472—22no—
18#16840472C2212yes—
19#16840468—22no—
20Q8×D244C2212yes—
21#16840427—22no—
22#16840427—22no—
23#16840427C2214yes—
24#16840427C2214yes—
25#16840427C2212yes—
26#16840390—21no—
27#16840387—22no—
28#16840319C2211yescharacteristic
29C2×#14519776—21no—
30C2×#14519755—22no—
31C2×#14519753C2212yes—
32C22⋊Q8×D61C2214yes—
33#16840236—21no—
34C2×C4×Dic122C2211yescharacteristic
35C2×C4×D244C2211yescharacteristic
36C2×#14519739—21no—
37#16840141C2212yescomplemented
38#16840112C2212yescomplemented
39#16840098C2212yes—
40#16840096C2218yescomplemented
41#16840095C2212yes—
42#16840058C2212yescomplemented
43#16840054C2212yescomplemented
44#16840002C2218yescomplemented
45#16840000C2218yescomplemented
46#16839973C2212yescomplemented
47#16839969C2214yescomplemented
48#16839969C2212yescomplemented
49#16839920C2212yes—
50#16839919C2218yescomplemented
51#16839918C2214yes—
52#16839917C2212yes—
53#16839916C2218yescomplemented
54#16839916C2218yescomplemented
55C4×Dic244C2214yescomplemented
56C4×#11758202C2214yescomplemented
57C8×D244C2214yes—
58#16839827C2214yes—
59#16839815C2211yescharacteristic
60#103×D61C2218yescomplemented
61C22⋊C8×D61C2214yes—
62C61×#533C2211yescharacteristic
63C2×C4⋊Q8×C61C2211yescharacteristic
64C61×#494C2212yescomplemented
65C61×#487C2212yescomplemented
66C61×#462C2214yescomplemented
67C61×#451C2214yes—
68C61×#431C2211yescharacteristic
69C22×Dic122—22no—
70C22×Dic122C2311yescharacteristic
71C2×Q8×D61—24no—
72C2×Q8×D61—48no—
73C2×Q8×D61C2318yes—
74C2×Q8×D61C2314yes—
75C22×D244D411yescharacteristic
76C22×C4×D61—22no—
77C22×C4×D61C2312yes—
78C2×Dic244C2314yes—
79C2×#11758202C2314yes—
80C2×#11758202C2318yes—
81Q16×D61—416no—
82C2×C8×D61C2318yes—
83#14519777—22no—
84#14519777C2311yescharacteristic
85#14519777C2311yescharacteristic
86#14519776—22no—
87#14519776—24no—
88#14519776—24no—
89#14519769—21no—
90#14519769—22no—
91#14519768—22no—
92#14519757—24no—
93#14519757C2314yes—
94#14519756—21no—
95#14519756C2312yes—
96#14519755—24no—
97#14519755—24no—
98#14519755—24no—
99#14519755—28no—
100#14519755—28no—
101#14519754—24no—
102#14519753—24no—
103#14519753—24no—
104#14519753C2314yes—
105#14519753C2318yes—
106C4×Dic122—21no—
107C4×Dic122—21no—
108C4×Dic122—21no—
109C4×Dic122C2312yes—
110C4×Dic122C2312yes—
111C4×Dic122C2312yes—
112C4×D244—22no—
113C4×D244—22no—
114C4×D244D412yes—
115C4×D244D412yes—
116C4×D244C2312yes—
117C4×D244C2312yes—
118#14519739—22no—
119#14519739—22no—
120#14519739—24no—
121#14519739—24no—
122#14519736—21no—
123C2×C4⋊Dic61D411yescharacteristic
124C2×C4⋊Dic61C2311yescharacteristic
125C2×#11758196—21no—
126C2×#11758196—22no—
127C4⋊C4×D61—24no—
128C4⋊C4×D61C2314yes—
129C2×#11758198—22no—
130C2×#11758198D412yes—
131C22⋊C4×D61—24no—
132C2×C4×Dic61—21no—
133#14519711C2314yes—
134#14519710C2314yes—
135C4⋊#92418C2312yes—
136#14519702C2314yes—
137#14519702C2314yes—
138#14519701C2314yes—
139#14519701C2318yes—
140#14519700C2314yes—
141C22⋊#92418C2312yes—
142#14519689C2318yes—
143C4×#92418C2312yes—
144C22×Q8×C61C2312yes—
145C2×Q16×C61C2318yes—
146C4⋊Q8×C61—21no—
147C4⋊Q8×C61—21no—
148C4⋊Q8×C61C2312yes—
149C4⋊Q8×C61C2312yes—
150C4⋊Q8×C61C2311yescharacteristic
151C4⋊Q8×C61C2311yescharacteristic
152#125×C61—21no—
153C22⋊Q8×C61—21no—
154C22⋊Q8×C61C2312yes—
155C4×Q8×C61—21no—
156C4×Q8×C61C2312yes—
157#117×C61—21no—
158C2×C4⋊C4×C61—21no—
159C2×C4⋊C4×C61—21no—
160C2×C42×C61C2311yescharacteristic
161C4⋊C8×C61C2312yes—
162#103×C61C2318yes—
163C22⋊C8×C61C2312yes—
164C4×C8×C61C2312yes—
165C23×D61C2×D412yes—
166C2×Q8×C61—22no—
167C2×Q8×C61—21no—
168C2×Q8×C61—44no—
169C2×Q8×C61C2414yes—
170C2×Q8×C61C2412yes—
171C22×C4×C61—22no—
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2411yescharacteristic
175C22×Dic61—21no—
176C22×Dic61—21no—
177C22×Dic61C2×D412yes—
178Q8×D61—216no—
179Q8×D61—48no—
180Q8×D61—416no—
181Q8×D61—48no—
182C2×D244Q1618yescomplemented
183C2×D244C2×D412yes—
184C2×D244C2×D412yes—
185C2×C4×D61—28no—
186C2×C4×D61—24no—
187C2×C4×D61—24no—
188C2×C4×D61—44no—
189C2×C4×D61—48no—
190C2×C4×D61C2×D414yes—
191C2×C4×D61C2414yes—
192C2×Dic122—21no—
193C2×Dic122—24no—
194C2×Dic122—28no—
195C2×Dic122—21no—
196C2×Dic122—24no—
197C2×Dic122—48no—
198C2×Dic122C2412yes—
199C2×Dic122C2412yes—
200Q16×C61—48no—
201C2×C8×C61C2414yes—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61—22no—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—21no—
207C4⋊C4×C61—21no—
208C4⋊C4×C61—21no—
209C4⋊C4×C61—22no—
210C4⋊C4×C61—22no—
211C4⋊C4×C61C2412yes—
212C22⋊C4×C61—22no—
213C42×C61—22no—
214C42×C61C2×D411yescharacteristic
215C42×C61C2×D411yescharacteristic
216C42×C61C2411yescharacteristic
217C42×C61C2411yescharacteristic
218C22⋊Dic61—22no—
219#11758202—28no—
220#11758202—48no—
221#11758198—24no—
222#11758198—24no—
223#11758198—28no—
224#11758198C2×D414yes—
225#11758198C2×D414yes—
226C4⋊Dic61—21no—
227C4⋊Dic61—21no—
228C4⋊Dic61—21no—
229C4⋊Dic61—21no—
230C4⋊Dic61—24no—
231C4⋊Dic61C2×D412yes—
232C4⋊Dic61C2×D412yes—
233C4⋊Dic61C2412yes—
234C4⋊Dic61C2412yes—
235#11758196—22no—
236#11758196—22no—
237#11758196—22no—
238#11758196—22no—
239#11758196—24no—
240#11758196—24no—
241#11758196—28no—
242C4×Dic61—22no—
243C4×Dic61—22no—
244C4×Dic61—22no—
245C2×#92418C2414yes—
246Dic244—28no—
247C8×D61—48no—
248C23×C61C22×D411yescharacteristic
249C22×D61—42no—
250C22×D61—42no—
251C22×D61C2×Q1618yescomplemented
252C22×D61C22×D414yes—
253Q8×C61—24no—
254Q8×C61—28no—
255Q8×C61—44no—
256Q8×C61—81no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—24no—
261C2×C4×C61—41no—
262C2×C4×C61—44no—
263C2×C4×C61C2×Q1614yescomplemented
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C2511yescharacteristic
270C2×Dic61—24no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61C22×D414yes—
278D244C2×Q1618yescomplemented
279D244C2×Q1618yescomplemented
280C4×D61—28no—
281C4×D61—44no—
282C4×D61—416no—
283C4×D61—84no—
284C4×D61—84no—
285Dic122—24no—
286Dic122—22no—
287Dic122—24no—
288Dic122—416no—
289Dic122—84no—
290Dic122—81no—
291Dic122H(5,𝔽2)14yes—
292C8×C61—24no—
293#92418—24no—
294#61563—611nomaximal, Sylow
295C22×C61—21no—
296C22×C61—21no—
297C22×C61C22×Q1612yes—
298C22×C61C23×D411yescharacteristic
299C2×D61—24no—
300C2×D61—24no—
301C2×D61—48no—
302C2×D61C22×Q1618yes—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—42no—
307C4×C61—44no—
308C4×C61—42no—
309C4×C61C22×Q1612yes—
310C4×C61C22×Q1612yes—
311C4×C61C2×H(5,𝔽2)12yes—
312Dic61—24no—
313Dic61—24no—
314Dic61—42no—
315Dic61—42no—
316Dic61—42no—
317#3501—611no—
318C2×D4×Q8—611no—
319#3299—614no—
320#3299—614no—
321D4×Q16—614no—
322D4×Q16—614no—
323#3209—611no—
324#3209D6111yescharacteristic, complemented
325#3189—614no—
326#3104—611no—
327#3104—611no—
328C2×#451—614no—
329#2972—611no—
330C2×C61—22no—
331C2×C61#348411yescharacteristic
332C2×C61C23×Q1611yescharacteristic
333C2×C61#358611yescharacteristic
334D61—28no—
335C23×Q8—612no—
336C22×Q16—618no—
337#554—612no—
338#554—612no—
339#554—1222no—
340#554—1222no—
341D4×Q8—614no—
342D4×Q8—614no—
343D4×Q8—612no—
344D4×Q8—612no—
345D4×Q8—1222no—
346D4×Q8—1222no—
347#544—1221no—
348#543—1222no—
349#533—611no—
350#533C2×D6111yescharacteristic
351C2×C4⋊Q8—1221no—
352C2×C4⋊Q8C2×D6111yescharacteristic
353C2×C22⋊Q8—612no—
354C2×C22⋊Q8—614no—
355C2×C22⋊Q8—1222no—
356#518—1221no—
357C2×C4×Q8—611no—
358C2×C4×D4—611no—
359C2×C4×D4—1221no—
360#494—612no—
361#494C2×D6112yescomplemented
362#487—612no—
363#487C2×D6112yescomplemented
364#484—612no—
365#484—618no—
366#484—612no—
367#477—612no—
368#477—612no—
369#467—618no—
370#467—618no—
371#462—612no—
372#462—614no—
373#462—612no—
374#462C2×D6114yescomplemented
375#451—614no—
376#451—618no—
377#451—618no—
378#451—612no—
379#451—618no—
380#451—612no—
381#451C2×D6114yes—
382C4×Q16—614no—
383C4×Q16—614no—
384C8×D4—614no—
385C8×D4—614no—
386#431—611no—
387#431C2×D6111yescharacteristic
388C2×#103—618no—
389C2×C22⋊C8—614no—
390C61#6156311yescharacteristic, complemented, Sylow
391C22×Q8—618no—
392C22×Q8—614no—
393C22×Q8—611no—
394C22×Q8—1222no—
395C22×Q8—1224no—
396C22×Q8—2448no—
397C22×Q8C22×D6112yes—
398C22×D4—611no—
399C23×C4—612no—
400C23×C4—1222no—
401C2×Q16—614no—
402C2×Q16—618no—
403C2×Q16—614no—
404C2×Q16—24416no—
405C2×Q16C22×D6118yes—
406C22×C8—618no—
407C4⋊Q8—21no—
408C4⋊Q8—21no—
409C4⋊Q8—611no—
410C4⋊Q8—611no—
411C4⋊Q8—1222no—
412C4⋊Q8—1224no—
413C4⋊Q8—1224no—
414C4⋊Q8—1222no—
415C4⋊Q8C22×D6112yes—
416C4⋊Q8C22×D6112yes—
417C4⋊Q8C22×D6111yescharacteristic
418C4⋊Q8C22×D6111yescharacteristic
419#125—21no—
420#125—1222no—
421#125—1222no—
422#125—1221no—
423C22⋊Q8—21no—
424C22⋊Q8—618no—
425C22⋊Q8—614no—
426C22⋊Q8—612no—
427C22⋊Q8—614no—
428C22⋊Q8—1224no—
429C22⋊Q8—1228no—
430C22⋊Q8—1228no—
431C22⋊Q8—1224no—
432C22⋊Q8—1224no—
433C22⋊Q8—1224no—
434C22⋊Q8—1224no—
435C22⋊Q8—1224no—
436C22⋊Q8—1224no—
437C22⋊Q8—1221no—
438C22⋊Q8C22×D6112yes—
439C4×Q8—21no—
440C4×Q8—612no—
441C4×Q8—612no—
442C4×Q8—612no—
443C4×Q8—1221no—
444C4×Q8—1221no—
445C4×Q8—1221no—
446C4×Q8C22×D6112yes—
447C4×D4—612no—
448C4×D4—612no—
449C4×D4—612no—
450C4×D4—612no—
451C4×D4—1222no—
452C4×D4—1222no—
453C4×D4—1224no—
454C4×D4—1224no—
455C4×D4—1222no—
456C4×D4—1222no—
457#117—21no—
458#117—1221no—
459C2×C4⋊C4—21no—
460C2×C4⋊C4—21no—
461C2×C4⋊C4—611no—
462C2×C4⋊C4—611no—
463C2×C4⋊C4—614no—
464C2×C4⋊C4—1224no—
465C2×C4⋊C4—1222no—
466C2×C4⋊C4—1221no—
467C2×C22⋊C4—612no—
468C2×C22⋊C4—1224no—
469C2×C22⋊C4—1222no—
470C2×C42—1221no—
471C2×C42C22×D6111yescharacteristic
472#107—614no—
473#107—614no—
474C4⋊C8—612no—
475C4⋊C8C22×D6112yes—
476#103—618no—
477#103—614no—
478#103—614no—
479#103—614no—
480#103—614no—
481#103C22×D6118yes—
482C22⋊C8—612no—
483C22⋊C8—618no—
484C22⋊C8C22×D6112yes—
485C4×C8—612no—
486C4×C8C22×D6112yes—
487C24—612no—
488C2×Q8—22no—
489C2×Q8—21no—
490C2×Q8—44no—
491C2×Q8—612no—
492C2×Q8—612no—
493C2×Q8—1224no—
494C2×Q8—1228no—
495C2×Q8—1224no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—12216no—
499C2×Q8—2448no—
500C2×Q8—2448no—
501C2×Q8—2448no—
502C2×Q8—24416no—
503C2×Q8C23×D6114yes—
504C2×Q8C23×D6112yes—
505C2×D4—612no—
506C2×D4—612no—
507C2×D4—618no—
508C22×C4—22no—
509C22×C4—612no—
510C22×C4—614no—
511C22×C4—614no—
512C22×C4—1224no—
513C22×C4—1224no—
514C22×C4—1228no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—2444no—
518C22×C4—2448no—
519C22×C4D4×D6111yescharacteristic
520C22×C4D4×D6111yescharacteristic
521C22×C4C23×D6111yescharacteristic
522Q16—48no—
523Q16—1228no—
524Q16—1228no—
525Q16—2448no—
526C2×C8—614no—
527C2×C8—2448no—
528C2×C8C23×D6114yes—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—612no—
539C4⋊C4—612no—
540C4⋊C4—612no—
541C4⋊C4—612no—
542C4⋊C4—1224no—
543C4⋊C4—1224no—
544C4⋊C4—1228no—
545C4⋊C4—1224no—
546C4⋊C4—1221no—
547C4⋊C4—1222no—
548C4⋊C4—1222no—
549C4⋊C4—1222no—
550C4⋊C4—1222no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4C23×D6112yes—
555C22⋊C4—22no—
556C22⋊C4—614no—
557C22⋊C4—614no—
558C22⋊C4—1224no—
559C22⋊C4—1224no—
560C22⋊C4—1228no—
561C22⋊C4—1222no—
562C42—22no—
563C42—1222no—
564C42—1222no—
565C42—1222no—
566C42D4×D6111yescharacteristic
567C42D4×D6111yescharacteristic
568C42C23×D6111yescharacteristic
569C42C23×D6111yescharacteristic
570C23—614no—
571C23—618no—
572C23—2442no—
573C23—2442no—
574C23C2×D4×D6111yescharacteristic
575Q8—24no—
576Q8—28no—
577Q8—44no—
578Q8—81no—
579Q8—614no—
580Q8—1224no—
581Q8—1222no—
582Q8—1224no—
583Q8—4881no—
584Q8—4884no—
585Q8—24416no—
586D4—618no—
587D4—618no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—614no—
595C2×C4—1224no—
596C2×C4—1222no—
597C2×C4—1228no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—4884no—
602C2×C4—4884no—
603C2×C4—2444no—
604C2×C4—24416no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4Q16×D6114yescomplemented
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C2×D4×D6111yescharacteristic
613C2×C4C24×D6111yescharacteristic
614C8—24no—
615C8—1224no—
616C22—21no—
617C22—21no—
618C22—618no—
619C22—1224no—
620C22—1224no—
621C22—2448no—
622C22C2×Q16×D6112yes—
623C22C22×D4×D6111yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—1224no—
631C4—1224no—
632C4—2442no—
633C4—2442no—
634C4—2442no—
635C4C2×Q16×D6112yes—
636C4C2×Q16×D6112yes—
637C4H(5,𝔽2)×D6112yes—
638C2—22no—
639C2—1228no—
640C2#2145945311yescharacteristic
641C2C22×Q16×D6111yescharacteristic
642C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3209C2
socleC22×C61C23×D4
prime radicalC22×D244D4
hypercenter#3209D61
hypocenterC61#61563
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
3. #21457698C2semidirect
4. #21457694C2semidirect
5. #21457569C2semidirect
6. #21457566C2semidirect
7. #21456767C2semidirect
8. #21456592C2semidirect
9. #21455859C2semidirect
10. #21455854C2semidirect
11. #451×D61C2semidirect
13. C61×#3209C2semidirect
15. C2×Q16×D61C22semidirect
37. #16840141C22semidirect
38. #16840112C22semidirect
40. #16840096C22semidirect
42. #16840058C22semidirect
43. #16840054C22semidirect
44. #16840002C22semidirect
45. #16840000C22semidirect
46. #16839973C22semidirect
47. #16839969C22semidirect
48. #16839969C22semidirect
50. #16839919C22semidirect
53. #16839916C22semidirect
54. #16839916C22semidirect
55. C4×Dic244C22semidirect
56. C4×#11758202C22semidirect
60. #103×D61C22semidirect
64. C61×#494C22semidirect
65. C61×#487C22semidirect
66. C61×#462C22semidirect
182. C2×D244Q16semidirect
251. C22×D61C2×Q16semidirect
263. C2×C4×C61C2×Q16semidirect
278. D244C2×Q16semidirect
279. D244C2×Q16semidirect
324. #3209D61semidirect
361. #494C2×D61semidirect
363. #487C2×D61semidirect
374. #462C2×D61semidirect
390. C61#61563semidirect
607. C2×C4Q16×D61semidirect

Maximal subgroups

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

subgroupindex
#214596202
C2×#168404272
#214576982
#214576942
#214575692
#214575662
#214567672
#214565922
#214558592
#214558542
#451×D612
#214548422
C61×#32092
#6156361