Finite Group Databasebeta

← #30571065

Subgroups — #30571065

Order 15616.

1814 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#21459684C211yesmaximal, characteristic, complemented
2#21458827C214yesmaximal, complemented
3#21458811C214yesmaximal, complemented
4#21458511C214yesmaximal, complemented
5#21458475C214yesmaximal, complemented
6#21456784C214yesmaximal, complemented
7#21456760C211yesmaximal, characteristic, complemented
8#21456737C211yesmaximal, characteristic
9#21455866C214yesmaximal, complemented
10#21455858C211yesmaximal, characteristic, complemented
11#21455850C211yesmaximal, characteristic, complemented
12#21454900C211yesmaximal, characteristic, complemented
13C2×C61×#559C211yesmaximal, characteristic, complemented
14#16840497C2212yescomplemented
15#16840490C2212yescomplemented
16#16840485C2212yescomplemented
17#16840484—22no—
18#16840472C2212yescomplemented
19#16840453—22no—
20#16840359—22no—
21#16840336—21no—
22#16840319C2211yescharacteristic
23#16840312C2211yescharacteristic
24#16840296C2214yes—
25#16840243—21no—
26#16840144C2218yescomplemented
27#16840138C2212yes—
28#16840134C2212yes—
29#16840116C2218yescomplemented
30#16840108C2212yes—
31#16840108C2212yes—
32#16840097C2212yes—
33#16840080C2212yes—
34#16840062C2218yescomplemented
35#16840057C2212yescomplemented
36#16840047C2212yescomplemented
37#16839981C2218yescomplemented
38#16839977C2218yescomplemented
39#16839974C2218yescomplemented
40#16839972C2212yescomplemented
41#16839962C2212yescomplemented
42#16839959C2214yes—
43#16839954C2214yes—
44#16839918C2212yes—
45#16839913C2214yes—
46#16839901C2212yes—
47#16839896C2214yes—
48Q8×#92418C2214yescomplemented
49Q8×#92418C2214yescomplemented
50C4×#11758202C2214yescomplemented
51C4×#11758201C2214yescomplemented
52C4×#11758200C2218yescomplemented
53C4×#11758199C2218yescomplemented
54#16839815C2211yescharacteristic
55#16839815C2211yescharacteristic
56#16839815C2214yes—
57C61×#559—22no—
58C61×#559—22no—
59C61×#559C2214yes—
60C61×#559C2214yes—
61C2×#127×C61C2211yescharacteristic
62C2×#124×C61—21no—
63C2×#124×C61C2212yes—
64C2×#123×C61—22no—
65C2×C4×Q8×C61C2211yescharacteristic
66C2×C4×Q8×C61C2211yescharacteristic
67C2×C4×D4×C61—21no—
68C2×C4×D4×C61C2212yes—
69C2×#11758222—21no—
70C2×#11758202C2314yes—
71C2×#11758200C2318yes—
72C2×#11758201C2314yes—
73C2×#11758199C2318yes—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519775—22no—
78#14519774C2311yescharacteristic
79#14519772—24no—
80#14519771—24no—
81#14519769—21no—
82#14519769C2314yes—
83#14519767—24no—
84#14519766—22no—
85#14519765—21no—
86#14519765C2314yes—
87#14519762—24no—
88#14519760—24no—
89#14519756—21no—
90#14519756C2312yes—
91#14519755—24no—
92#14519753—22no—
93#14519753—22no—
94#14519751—21no—
95#14519751C2312yes—
96C4×Dic122—21no—
97C4×Dic122C2312yes—
98C4×Dic122C2314yes—
99#14519742—22no—
100C4×D244—21no—
101C4×D244C2312yes—
102C4×D244C2314yes—
103#14519739—22no—
104C4×C22⋊D61—24no—
105#14519736—21no—
106#14519735—22no—
107#14519734—24no—
108C4⋊C4×D61—22no—
109#14519710C2314yes—
110#14519708C2314yes—
111C4⋊#92418C2312yes—
112C4⋊#92418C2312yes—
113C4⋊#92418C2318yes—
114#14519701C2314yes—
115#14519701C2314yes—
116#14519700C2314yes—
117#14519698C2318yes—
118#14519698C2318yes—
119#14519697C2314yes—
120C22⋊#92418C2312yes—
121C22⋊#92418C2312yes—
122C4×#92418C2318yes—
123C4×#92418C2312yes—
124C4×#92418C2312yes—
125C22×Q8×C61D411yescharacteristic
126C22×Q8×C61C2311yescharacteristic
127C22×D4×C61—21no—
128C22×D4×C61—21no—
129C22×D4×C61C2312yes—
130C23×C4×C61—22no—
131#127×C61—21no—
132#127×C61—21no—
133#127×C61C2312yes—
134#127×C61C2312yes—
135#124×C61—24no—
136#124×C61—24no—
137#124×C61—24no—
138#124×C61C2318yes—
139#123×C61—28no—
140#123×C61—28no—
141C4×Q8×C61—21no—
142C4×Q8×C61—21no—
143C4×Q8×C61D412yes—
144C4×Q8×C61D412yes—
145C4×Q8×C61C2312yes—
146C4×Q8×C61C2312yes—
147C4×Q8×C61C2312yes—
148C4×Q8×C61C2312yes—
149C4×D4×C61—24no—
150C4×D4×C61—24no—
151C4×D4×C61—24no—
152C4×D4×C61C2318yes—
153C2×C4⋊C4×C61—21no—
154C2×C4⋊C4×C61D411yescharacteristic
155C2×C4⋊C4×C61D412yes—
156C2×C4⋊C4×C61C2311yescharacteristic
157C2×C22⋊C4×C61—22no—
158C2×C22⋊C4×C61—22no—
159C2×C22⋊C4×C61—22no—
160C2×C42×C61—21no—
161C2×C42×C61C2311yescharacteristic
162C2×C42×C61C2312yes—
163C24×C61—22no—
164C2×Q8×C61—24no—
165C2×Q8×C61—21no—
166C2×Q8×C61—42no—
167C2×Q8×C61C2×D412yes—
168C2×Q8×C61C2×D412yes—
169C2×Q8×C61C2412yes—
170C2×D4×C61—22no—
171C2×D4×C61—22no—
172C2×D4×C61—22no—
173C2×D4×C61—22no—
174C2×D4×C61—22no—
175C2×D4×C61—44no—
176C2×D4×C61—44no—
177C2×D4×C61—82no—
178C2×D4×C61C2414yes—
179C22×C4×C61—21no—
180C22×C4×C61—21no—
181C22×C4×C61—28no—
182C22×C4×C61—84no—
183C22×C4×C61C2×D412yes—
184C22×C4×C61C2×D412yes—
185C22×C4×C61C2×D411yescharacteristic
186C22×C4×C61C2×D411yescharacteristic
187C22×C4×C61C2411yescharacteristic
188C2×C22⋊D61—22no—
189#11758222—44no—
190C2×D244—21no—
191C2×D244—21no—
192C2×D244C2412yes—
193C2×C4×D61—22no—
194C2×C4×D61—24no—
195C2×Dic122—21no—
196C2×Dic122—21no—
197C2×Dic122C2412yes—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61C2×D412yes—
202C4⋊C4×C61C2×D412yes—
203C4⋊C4×C61C2×D414yes—
204C4⋊C4×C61C2×D414yes—
205C4⋊C4×C61C2412yes—
206C22⋊C4×C61—28no—
207C22⋊C4×C61—28no—
208C22⋊C4×C61—28no—
209C42×C61—22no—
210C42×C61—22no—
211C42×C61C2×D414yes—
212C42×C61C2×D411yescharacteristic
213C42×C61C2×D411yescharacteristic
214C42×C61C2414yes—
215C42×C61C2411yescharacteristic
216C42×C61C2411yescharacteristic
217C22⋊Dic61—22no—
218#11758202—44no—
219#11758201—44no—
220#11758200—48no—
221#11758199—48no—
222#11758198—22no—
223#11758198—24no—
224#11758198—24no—
225#11758198—22no—
226#11758198—24no—
227C4⋊Dic61—21no—
228C4⋊Dic61—21no—
229C4⋊Dic61—21no—
230C4⋊Dic61—21no—
231C4⋊Dic61C2412yes—
232C4⋊Dic61C2412yes—
233#11758196—24no—
234#11758196—22no—
235#11758196—24no—
236#11758196—24no—
237#11758196—22no—
238C4×Dic61—24no—
239C4×Dic61—22no—
240C2×#92418C2414yes—
241C2×#92418C2414yes—
242C23×C61—24no—
243C23×C61—44no—
244C23×C61—44no—
245C23×C61—44no—
246C23×C61C22×D411yescharacteristic
247C22×D61—22no—
248Q8×C61—24no—
249Q8×C61—24no—
250Q8×C61—22no—
251Q8×C61—22no—
252Q8×C61—42no—
253D4×C61—28no—
254D4×C61—44no—
255D4×C61—44no—
256D4×C61—44no—
257D4×C61—82no—
258D4×C61—82no—
259C2×C4×C61—24no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—22no—
263C2×C4×C61—42no—
264C2×C4×C61—41no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—88no—
268C2×C4×C61#13514yes—
269C2×C4×C61C22×D414yes—
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D414yes—
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C2511yescharacteristic
277C22⋊D61—44no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281D244—24no—
282D244—42no—
283D244—81no—
284C4×D61—44no—
285C4×D61—84no—
286Dic122—24no—
287Dic122—42no—
288Dic122—81no—
289#92418—24no—
290#92418—24no—
291#61885—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—44no—
295C22×C61—44no—
296C22×C61—44no—
297C22×C61—88no—
298C22×C61C2×#13512yes—
299C22×C61C23×D411yescharacteristic
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C4×C61—21no—
304C4×C61—24no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—24no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61C2×#13512yes—
311C4×C61C2×#13512yes—
312C4×C61#58512yes—
313Dic61—42no—
314Dic61—42no—
315Dic61—42no—
316#3507—611no—
317C2×#559D6111yescharacteristic, complemented
318#3396—614no—
319#3394—614no—
320#3365—614no—
321#3361—614no—
322#3211—614no—
323#3208—611no—
324#3205—611no—
325#3105—614no—
326#3104—611no—
327#3103—611no—
328#2982—611no—
329C2×C61—22no—
330C2×C61—84no—
331C2×C61#348711yescharacteristic
332C2×C61C22×#13511yescharacteristic
333C2×C61#358511yescharacteristic
334D61—82no—
335#559—22no—
336#559—22no—
337#559—612no—
338#559C2×D6114yes—
339#559C2×D6114yes—
340#557—612no—
341#556—612no—
342#556—1222no—
343#554—612no—
344#552—1222no—
345#539—1222no—
346#536—1221no—
347#533—611no—
348#532—611no—
349C2×#127C2×D6111yescharacteristic
350#529—614no—
351C2×#124—21no—
352C2×#124C2×D6112yes—
353C2×#123—22no—
354#520—1221no—
355C2×C4×Q8C2×D6111yescharacteristic
356C2×C4×Q8C2×D6111yescharacteristic
357C2×C4×D4—21no—
358C2×C4×D4C2×D6112yes—
359#495—618no—
360#493—612no—
361#492—612no—
362#488—618no—
363#486—612no—
364#486—612no—
365#484—612no—
366#481—612no—
367#478—618no—
368#477—612no—
369#475—612no—
370#464—618no—
371#463—618no—
372#463—618no—
373#462—612no—
374#460—614no—
375#460—612no—
376#459—614no—
377#451—612no—
378#450—614no—
379#448—612no—
380#447—614no—
381C8×Q8—614no—
382C8×Q8—614no—
383C4×Q16—614no—
384C4×SD16—614no—
385C4×SD16—618no—
386C4×D8—618no—
387#431—611no—
388#431—611no—
389#431—614no—
390C61#6188511yescharacteristic, complemented, Sylow
391C2×#41—1221no—
392C22×Q8C22⋊D6111yescharacteristic
393C22×Q8C22×D6111yescharacteristic
394C22×D4—21no—
395C22×D4—21no—
396C22×D4C22×D6112yes—
397C23×C4—22no—
398C2×Q16—614no—
399C2×SD16—614no—
400C2×SD16—618no—
401C2×D8—618no—
402C4⋊Q8—611no—
403C4⋊Q8—611no—
404C4⋊Q8—611no—
405C4⋊Q8—1222no—
406#127—21no—
407#127—21no—
408#127—611no—
409#127C22×D6112yes—
410#127C22×D6112yes—
411#126—1224no—
412#126—1224no—
413#125—614no—
414#125—1224no—
415#125—1221no—
416#124—24no—
417#124—24no—
418#124—24no—
419#124—614no—
420#124—1222no—
421#124—1221no—
422#124C22×D6118yes—
423#123—28no—
424#123—28no—
425#123—1224no—
426#123—1224no—
427C22⋊Q8—612no—
428C22⋊Q8—1222no—
429C22⋊Q8—1222no—
430C22⋊Q8—1221no—
431C22⋊Q8—1224no—
432#121—612no—
433#121—1221no—
434C4×Q8—21no—
435C4×Q8—21no—
436C4×Q8—614no—
437C4×Q8—612no—
438C4×Q8—1222no—
439C4×Q8—1221no—
440C4×Q8C22⋊D6112yes—
441C4×Q8C22⋊D6112yes—
442C4×Q8C22×D6112yes—
443C4×Q8C22×D6112yes—
444C4×Q8C22×D6112yes—
445C4×Q8C22×D6112yes—
446C4×D4—24no—
447C4×D4—24no—
448C4×D4—24no—
449C4×D4—614no—
450C4×D4—612no—
451C4×D4—1224no—
452C4×D4—1222no—
453C4×D4—1221no—
454C4×D4C22×D6118yes—
455#117—1221no—
456#117—1222no—
457#117—1224no—
458C2×C4⋊C4—21no—
459C2×C4⋊C4—1222no—
460C2×C4⋊C4C22⋊D6111yescharacteristic
461C2×C4⋊C4C22⋊D6112yes—
462C2×C4⋊C4C22×D6111yescharacteristic
463C2×C22⋊C4—22no—
464C2×C22⋊C4—22no—
465C2×C22⋊C4—22no—
466C2×C42—21no—
467C2×C42C22×D6112yes—
468C2×C42C22×D6111yescharacteristic
469#107—614no—
470#106—614no—
471C4⋊C8—612no—
472C4⋊C8—612no—
473C4⋊C8—618no—
474#103—614no—
475#103—614no—
476#103—614no—
477#102—618no—
478#102—618no—
479#102—614no—
480C22⋊C8—612no—
481C22⋊C8—612no—
482C4×C8—618no—
483C4×C8—612no—
484C4×C8—612no—
485C24—22no—
486#41—2444no—
487C2×Q8—21no—
488C2×Q8—24no—
489C2×Q8—42no—
490C2×Q8—612no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8C2×C22⋊D6112yes—
494C2×Q8C2×C22⋊D6112yes—
495C2×Q8C23×D6112yes—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—22no—
500C2×D4—22no—
501C2×D4—44no—
502C2×D4—44no—
503C2×D4—82no—
504C2×D4—612no—
505C2×D4—1222no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4C23×D6114yes—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—28no—
512C22×C4—84no—
513C22×C4—1224no—
514C22×C4—1222no—
515C22×C4C2×C22⋊D6112yes—
516C22×C4C2×C22⋊D6112yes—
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C2×C22⋊D6111yescharacteristic
519C22×C4C23×D6111yescharacteristic
520Q16—2444no—
521SD16—2448no—
522SD16—2444no—
523D8—2448no—
524C2×C8—614no—
525C2×C8—614no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—1221no—
532C4⋊C4—1224no—
533C4⋊C4—1222no—
534C4⋊C4—1224no—
535C4⋊C4—1222no—
536C4⋊C4—1224no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4C2×C22⋊D6112yes—
541C4⋊C4C2×C22⋊D6112yes—
542C4⋊C4C2×C22⋊D6114yes—
543C4⋊C4C2×C22⋊D6114yes—
544C4⋊C4C23×D6112yes—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—28no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—1222no—
553C22⋊C4—1224no—
554C42—22no—
555C42—22no—
556C42—1224no—
557C42—1222no—
558C42C2×C22⋊D6114yes—
559C42C2×C22⋊D6111yescharacteristic
560C42C2×C22⋊D6111yescharacteristic
561C42C23×D6111yescharacteristic
562C42C23×D6114yes—
563C42C23×D6111yescharacteristic
564C23—24no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—1222no—
569C23C22×C22⋊D6111yescharacteristic
570Q8—24no—
571Q8—24no—
572Q8—22no—
573Q8—22no—
574Q8—42no—
575Q8—1224no—
576Q8—4881no—
577Q8—2442no—
578D4—28no—
579D4—44no—
580D4—44no—
581D4—44no—
582D4—82no—
583D4—82no—
584D4—1224no—
585D4—4881no—
586D4—2444no—
587D4—2442no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—88no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—4884no—
601C2×C4—2444no—
602C2×C4#1451979714yes—
603C2×C4C22×C22⋊D6114yes—
604C2×C4C22×C22⋊D6111yescharacteristic
605C2×C4C22×C22⋊D6114yes—
606C2×C4C22×C22⋊D6111yescharacteristic
607C2×C4C22×C22⋊D6111yescharacteristic
608C2×C4C22×C22⋊D6111yescharacteristic
609C2×C4C22×C22⋊D6111yescharacteristic
610C2×C4C24×D6111yescharacteristic
611C8—1224no—
612C8—1224no—
613C22—21no—
614C22—21no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—88no—
619C22—2442no—
620C22—2442no—
621C22—2442no—
622C22C2×#1451979712yes—
623C22C23×C22⋊D6111yescharacteristic
624C4—21no—
625C4—24no—
626C4—24no—
627C4—21no—
628C4—21no—
629C4—42no—
630C4—42no—
631C4—2442no—
632C4—2442no—
633C4—2442no—
634C4C2×#1451979712yes—
635C4C2×#1451979712yes—
636C4#1684059812yes—
637C2—22no—
638C2—84no—
639C2—4882no—
640C2#2145947711yescharacteristic
641C2C22×#1451979711yescharacteristic
642C2#2146059911yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#559C2
socleC22×C61C23×D4
prime radical#21456737C2
hypercenterC2×#559D61
hypocenterC61#61885
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
1. #21459684C2semidirect
2. #21458827C2semidirect
3. #21458811C2semidirect
4. #21458511C2semidirect
5. #21458475C2semidirect
6. #21456784C2semidirect
7. #21456760C2semidirect
9. #21455866C2semidirect
10. #21455858C2semidirect
11. #21455850C2semidirect
12. #21454900C2semidirect
13. C2×C61×#559C2semidirect
14. #16840497C22semidirect
15. #16840490C22semidirect
16. #16840485C22semidirect
18. #16840472C22semidirect
26. #16840144C22semidirect
29. #16840116C22semidirect
34. #16840062C22semidirect
35. #16840057C22semidirect
36. #16840047C22semidirect
37. #16839981C22semidirect
38. #16839977C22semidirect
39. #16839974C22semidirect
40. #16839972C22semidirect
41. #16839962C22semidirect
48. Q8×#92418C22semidirect
49. Q8×#92418C22semidirect
50. C4×#11758202C22semidirect
51. C4×#11758201C22semidirect
52. C4×#11758200C22semidirect
53. C4×#11758199C22semidirect
317. C2×#559D61semidirect
390. C61#61885semidirect

Maximal subgroups

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

subgroupindex
#214596842
#214588272
#214588112
#214585112
#214584752
#214567842
#214567602
#214567372
#214558662
#214558582
#214558502
#214549002
C2×C61×#5592
#6188561