Finite Group Databasebeta

← #30564780

Subgroups — #30564780

Order 15616.

2194 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459671C211yesmaximal, characteristic, complemented
2#21457769C214yesmaximal, complemented
3#21457758C214yesmaximal, complemented
4#21457575C214yesmaximal, complemented
5#21457549C214yesmaximal, complemented
6#21456760C211yesmaximal, characteristic, complemented
7#21456737C211yesmaximal, characteristic, complemented
8#21456584C214yesmaximal, complemented
9#21455858C211yesmaximal, characteristic, complemented
10#21455850C211yesmaximal, characteristic, complemented
11#21455037C214yesmaximal, complemented
12#21454843C211yesmaximal, characteristic, complemented
13C2×C61×#550C211yesmaximal, characteristic, complemented
14C2×#14519797C2218yescomplemented
15#16840495—22no—
16#16840482C2212yescomplemented
17#16840477C2212yescomplemented
18#16840462—22no—
19#16840433C2212yescomplemented
20#16840417C2212yescomplemented
21#16840350—22no—
22#16840329—21no—
23#16840319C2211yescharacteristic
24#16840312C2211yescharacteristic
25#16840273C2214yes—
26#16840243—21no—
27#16840138C2212yescomplemented
28#16840134C2212yescomplemented
29#16840108C2212yescomplemented
30#16840108C2212yescomplemented
31#16840097C2212yes—
32#16840083C2218yescomplemented
33#16840080C2212yes—
34#16840057C2212yescomplemented
35#16840047C2212yescomplemented
36#16839996C2218yescomplemented
37#16839989C2218yescomplemented
38#16839972C2212yescomplemented
39#16839962C2212yescomplemented
40#16839959C2214yescomplemented
41#16839954C2214yescomplemented
42#16839924C2218yescomplemented
43#16839918C2212yes—
44#16839913C2214yes—
45#16839907C2218yescomplemented
46#16839905C2218yescomplemented
47#16839901C2212yes—
48#16839896C2214yes—
49C4×#11758202C2214yescomplemented
50C4×#11758201C2214yescomplemented
51D4×#92418C2214yescomplemented
52D4×#92418C2214yescomplemented
53#16839815C2211yescharacteristic
54#16839815C2211yescharacteristic
55#16839758C2218yescomplemented
56C2×C22⋊#92418C2214yes—
57C22×#41×C61C2212yes—
58C61×#550—22no—
59C61×#550—22no—
60C61×#550C2214yes—
61C61×#550C2214yes—
62C2×#127×C61—21no—
63C2×#127×C61C2211yescharacteristic
64C2×#121×C61—22no—
65C2×#121×C61C2212yes—
66C2×C4×Q8×C61C2211yescharacteristic
67C2×C4×D4×C61—21no—
68C2×C4×D4×C61C2211yescharacteristic
69C2×#11758222—21no—
70C2×#11758222C2314yes—
71#14519797—416no—
72C2×#11758202C2314yes—
73C2×#11758200C2318yes—
74C2×#11758201C2314yes—
75C2×#11758199C2318yes—
76C22×#92418C2318yes—
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519774C2311yescharacteristic
81#14519773—22no—
82#14519770—24no—
83#14519769—21no—
84#14519765—21no—
85#14519764—22no—
86#14519763—24no—
87#14519760—24no—
88#14519759—24no—
89#14519756—21no—
90#14519756C2314yes—
91#14519756C2312yes—
92#14519751—21no—
93#14519751C2314yes—
94#14519751C2312yes—
95#14519750—24no—
96#14519748—22no—
97#14519748—22no—
98#14519746—24no—
99C4×Dic122—21no—
100C4×Dic122C2312yes—
101#14519742—22no—
102C4×D244—21no—
103C4×D244C2312yes—
104#14519739—22no—
105C4×C22⋊D61—24no—
106C4×C22⋊D61—24no—
107#14519736—21no—
108#14519736C2314yes—
109#14519735—22no—
110C4⋊C4×D61—22no—
111#14519710C2314yes—
112#14519708C2314yes—
113C4⋊#92418C2312yes—
114C4⋊#92418C2312yes—
115#14519701C2314yes—
116#14519701C2314yes—
117#14519700C2314yes—
118#14519698C2318yes—
119#14519698C2318yes—
120#14519697C2314yes—
121C22⋊#92418C2312yes—
122C22⋊#92418C2312yes—
123C22⋊#92418C2318yes—
124C4×#92418C2312yes—
125C4×#92418C2312yes—
126C2×#41×C61—24no—
127C2×#41×C61—48no—
128C2×#41×C61C2318yes—
129C22×Q8×C61C2311yescharacteristic
130C22×D4×C61—22no—
131C22×D4×C61—22no—
132C22×D4×C61—21no—
133C22×D4×C61—21no—
134C22×D4×C61D411yescharacteristic
135C22×D4×C61C2312yes—
136C23×C4×C61—22no—
137C23×C4×C61C2312yes—
138#127×C61—24no—
139#127×C61—24no—
140#127×C61—21no—
141#127×C61—21no—
142#127×C61C2312yes—
143#127×C61C2312yes—
144#121×C61—24no—
145#121×C61—28no—
146#121×C61—28no—
147#121×C61C2318yes—
148C4×Q8×C61—21no—
149C4×Q8×C61—21no—
150C4×Q8×C61C2312yes—
151C4×Q8×C61C2312yes—
152C4×D4×C61—24no—
153C4×D4×C61—24no—
154C4×D4×C61D412yes—
155C4×D4×C61D412yes—
156C4×D4×C61C2312yes—
157C4×D4×C61C2312yes—
158C2×C4⋊C4×C61—21no—
159C2×C4⋊C4×C61D411yescharacteristic
160C2×C4⋊C4×C61C2311yescharacteristic
161C2×C22⋊C4×C61—22no—
162C2×C22⋊C4×C61D412yes—
163C2×C42×C61—21no—
164C2×C42×C61C2311yescharacteristic
165C24×C61—22no—
166C24×C61C2×D412yes—
167#41×C61—216no—
168#41×C61—48no—
169#41×C61—416no—
170C2×Q8×C61—24no—
171C2×Q8×C61—21no—
172C2×Q8×C61C2412yes—
173C2×D4×C61—28no—
174C2×D4×C61—28no—
175C2×D4×C61—24no—
176C2×D4×C61—22no—
177C2×D4×C61—22no—
178C2×D4×C61—22no—
179C2×D4×C61—22no—
180C2×D4×C61—22no—
181C2×D4×C61—44no—
182C2×D4×C61—48no—
183C2×D4×C61—44no—
184C2×D4×C61—82no—
185C2×D4×C61—84no—
186C2×D4×C61C2×D412yes—
187C2×D4×C61C2×D412yes—
188C2×D4×C61C2414yes—
189C22×C4×C61—21no—
190C22×C4×C61—21no—
191C22×C4×C61—28no—
192C22×C4×C61—44no—
193C22×C4×C61—48no—
194C22×C4×C61C2×D411yescharacteristic
195C22×C4×C61C2×D412yes—
196C22×C4×C61C2×D414yes—
197C22×C4×C61C2×D411yescharacteristic
198C22×C4×C61C2414yes—
199C22×C4×C61C2411yescharacteristic
200C2×C22⋊D61—22no—
201C2×C22⋊D61—24no—
202C2×C22⋊D61—24no—
203#11758222—48no—
204#11758222—44no—
205C2×D244—21no—
206C2×D244—21no—
207C2×D244C2412yes—
208C2×C4×D61—22no—
209C2×C4×D61—24no—
210C2×Dic122—21no—
211C2×Dic122—21no—
212C2×Dic122C2412yes—
213C4⋊C4×C61—22no—
214C4⋊C4×C61—22no—
215C4⋊C4×C61—21no—
216C4⋊C4×C61C2×D412yes—
217C4⋊C4×C61C2×D412yes—
218C4⋊C4×C61C2412yes—
219C22⋊C4×C61—28no—
220C22⋊C4×C61C2×D414yes—
221C22⋊C4×C61C2×D414yes—
222C42×C61—22no—
223C42×C61—22no—
224C42×C61C2×D411yescharacteristic
225C42×C61C2×D411yescharacteristic
226C42×C61C2411yescharacteristic
227C42×C61C2411yescharacteristic
228C22⋊Dic61—22no—
229C22⋊Dic61—24no—
230C22⋊Dic61—24no—
231#11758202—28no—
232#11758201—28no—
233#11758200—48no—
234#11758199—48no—
235#11758198—24no—
236#11758198—22no—
237#11758198—22no—
238C4⋊Dic61—21no—
239C4⋊Dic61—21no—
240C4⋊Dic61—21no—
241C4⋊Dic61—21no—
242C4⋊Dic61C2412yes—
243C4⋊Dic61C2412yes—
244#11758196—24no—
245#11758196—22no—
246#11758196—22no—
247C4×Dic61—24no—
248C4×Dic61—22no—
249C2×#92418—48no—
250C2×#92418C2414yes—
251C2×#92418C2414yes—
252C23×C61—24no—
253C23×C61—24no—
254C23×C61—42no—
255C23×C61—42no—
256C23×C61—44no—
257C23×C61—44no—
258C23×C61—44no—
259C23×C61C22×D414yes—
260C23×C61C22×D411yescharacteristic
261C22×D61—22no—
262Q8×C61—24no—
263Q8×C61—22no—
264Q8×C61H(5,𝔽2)14yes—
265D4×C61—24no—
266D4×C61—24no—
267D4×C61—28no—
268D4×C61—416no—
269D4×C61—44no—
270D4×C61—44no—
271D4×C61—44no—
272D4×C61—82no—
273D4×C61—82no—
274D4×C61—88no—
275C2×C4×C61—24no—
276C2×C4×C61—22no—
277C2×C4×C61—28no—
278C2×C4×C61—22no—
279C2×C4×C61—22no—
280C2×C4×C61—42no—
281C2×C4×C61—44no—
282C2×C4×C61—41no—
283C2×C4×C61—416no—
284C2×C4×C61—42no—
285C2×C4×C61#13514yescomplemented
286C2×C4×C61C22×D414yes—
287C2×C4×C61C22×D411yescharacteristic
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C22×D411yescharacteristic
290C2×C4×C61C22×D411yescharacteristic
291C2×C4×C61C22×D411yescharacteristic
292C2×C4×C61C2511yescharacteristic
293C22⋊D61—44no—
294C22⋊D61—84no—
295C22⋊D61—84no—
296C2×Dic61—22no—
297C2×Dic61—22no—
298C2×Dic61—22no—
299D244—24no—
300D244—42no—
301D244—81no—
302C4×D61—44no—
303C4×D61—84no—
304Dic122—24no—
305Dic122—42no—
306Dic122—81no—
307#92418—24no—
308#92418—24no—
309#61566—611nomaximal, Sylow
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—24no—
313C22×C61—24no—
314C22×C61—24no—
315C22×C61—44no—
316C22×C61—44no—
317C22×C61—44no—
318C22×C61—84no—
319C22×C61—88no—
320C22×C61C2×#13512yes—
321C22×C61C23×D411yescharacteristic
322C2×D61—42no—
323C2×D61—42no—
324C2×D61—42no—
325C4×C61—21no—
326C4×C61—21no—
327C4×C61—24no—
328C4×C61—24no—
329C4×C61—21no—
330C4×C61—42no—
331C4×C61C2×#13512yes—
332C4×C61C2×#13512yes—
333C4×C61C2×H(5,𝔽2)12yes—
334Dic61—42no—
335Dic61—42no—
336Dic61—42no—
337#3506—611no—
338C2×#550D6111yescharacteristic, complemented
339#3305—614no—
340#3304—614no—
341#3287—614no—
342#3284—614no—
343#3208—611no—
344#3205—611no—
345#3188—614no—
346#3104—611no—
347#3103—611no—
348#3002—614no—
349#2972—611no—
350C2×C61—22no—
351C2×C61—44no—
352C2×C61—84no—
353C2×C61#348411yescharacteristic
354C2×C61C22×#13511yescharacteristic
355C2×C61#358511yescharacteristic
356D61—82no—
357C22×#41C2×D6112yes—
358C2×#135—618no—
359#559—1222no—
360#556—612no—
361#555—612no—
362#553—1222no—
363#550—22no—
364#550—22no—
365#550—612no—
366#550C2×D6114yes—
367#550C2×D6114yes—
368#547—612no—
369#538—1222no—
370#535—1221no—
371#533—611no—
372#532—611no—
373C2×#127—21no—
374C2×#127C2×D6111yescharacteristic
375#525—614no—
376C2×#121—22no—
377C2×#121C2×D6112yes—
378#520—1221no—
379C2×C4×Q8C2×D6111yescharacteristic
380C2×C4×D4—21no—
381C2×C4×D4C2×D6111yescharacteristic
382#493—612no—
383#492—612no—
384#486—612no—
385#486—612no—
386#484—612no—
387#482—618no—
388#481—612no—
389#477—612no—
390#475—612no—
391#466—618no—
392#465—618no—
393#462—612no—
394#460—614no—
395#460—612no—
396#459—614no—
397#452—618no—
398#451—612no—
399#450—614no—
400#449—618no—
401#449—618no—
402#448—612no—
403#447—614no—
404C4×Q16—614no—
405C4×SD16—614no—
406C8×D4—614no—
407C8×D4—614no—
408#431—611no—
409#431—611no—
410#416—618no—
411C2×C22⋊C8—614no—
412C61#6156611yescharacteristic, complemented, Sylow
413C2×#41—24no—
414C2×#41—48no—
415C2×#41—614no—
416C2×#41—1221no—
417C2×#41C22×D6118yes—
418C22×Q8C22×D6111yescharacteristic
419C22×D4—22no—
420C22×D4—22no—
421C22×D4—21no—
422C22×D4—21no—
423C22×D4C22⋊D6111yescharacteristic
424C22×D4C22×D6112yes—
425C23×C4—22no—
426C23×C4C22×D6112yes—
427#135—24416no—
428C2×Q16—614no—
429C2×SD16—614no—
430C2×SD16—618no—
431C2×D8—618no—
432C22×C8—618no—
433C4⋊Q8—611no—
434C4⋊Q8—611no—
435C4⋊Q8—611no—
436#127—24no—
437#127—24no—
438#127—21no—
439#127—21no—
440#127—611no—
441#127—1222no—
442#127C22×D6112yes—
443#127C22×D6112yes—
444#126—1224no—
445#125—1221no—
446#124—1222no—
447#124—1224no—
448#124—1221no—
449#123—1224no—
450#123—1224no—
451C22⋊Q8—612no—
452C22⋊Q8—614no—
453C22⋊Q8—1221no—
454#121—24no—
455#121—28no—
456#121—28no—
457#121—612no—
458#121—614no—
459#121—1222no—
460#121—1222no—
461#121—1221no—
462#121—1224no—
463#121C22×D6118yes—
464#120—1224no—
465C4×Q8—21no—
466C4×Q8—21no—
467C4×Q8—612no—
468C4×Q8—1222no—
469C4×Q8—1221no—
470C4×Q8C22×D6112yes—
471C4×Q8C22×D6112yes—
472C4×D4—24no—
473C4×D4—24no—
474C4×D4—612no—
475C4×D4—1224no—
476C4×D4—1224no—
477C4×D4—1222no—
478C4×D4—1221no—
479C4×D4C22⋊D6112yes—
480C4×D4C22⋊D6112yes—
481C4×D4C22×D6112yes—
482C4×D4C22×D6112yes—
483#117—614no—
484#117—1221no—
485#117—1222no—
486C2×C4⋊C4—21no—
487C2×C4⋊C4—1222no—
488C2×C4⋊C4C22⋊D6111yescharacteristic
489C2×C4⋊C4C22×D6111yescharacteristic
490C2×C22⋊C4—22no—
491C2×C22⋊C4C22⋊D6112yes—
492C2×C42—21no—
493C2×C42C22×D6111yescharacteristic
494#107—614no—
495#106—614no—
496C4⋊C8—612no—
497C4⋊C8—612no—
498#103—614no—
499#103—614no—
500#103—614no—
501#102—618no—
502#102—618no—
503#102—614no—
504C22⋊C8—612no—
505C22⋊C8—612no—
506C22⋊C8—618no—
507C4×C8—612no—
508C4×C8—612no—
509C24—22no—
510C24C2×C22⋊D6112yes—
511#41—216no—
512#41—48no—
513#41—416no—
514#41—2448no—
515#41—2444no—
516C2×Q8—24no—
517C2×Q8—21no—
518C2×Q8—612no—
519C2×Q8—1221no—
520C2×Q8—1221no—
521C2×Q8C23×D6112yes—
522C2×D4—28no—
523C2×D4—28no—
524C2×D4—22no—
525C2×D4—22no—
526C2×D4—22no—
527C2×D4—22no—
528C2×D4—22no—
529C2×D4—24no—
530C2×D4—44no—
531C2×D4—48no—
532C2×D4—44no—
533C2×D4—82no—
534C2×D4—84no—
535C2×D4—612no—
536C2×D4—1224no—
537C2×D4—1224no—
538C2×D4—1222no—
539C2×D4—1221no—
540C2×D4—1221no—
541C2×D4C2×C22⋊D6112yes—
542C2×D4C2×C22⋊D6112yes—
543C2×D4C23×D6114yes—
544C22×C4—21no—
545C22×C4—21no—
546C22×C4—28no—
547C22×C4—44no—
548C22×C4—48no—
549C22×C4—1224no—
550C22×C4—1222no—
551C22×C4C2×C22⋊D6111yescharacteristic
552C22×C4C2×C22⋊D6112yes—
553C22×C4C2×C22⋊D6114yes—
554C22×C4C2×C22⋊D6111yescharacteristic
555C22×C4C23×D6111yescharacteristic
556C22×C4C23×D6114yes—
557Q16—1228no—
558SD16—1228no—
559SD16—2448no—
560D8—2448no—
561C2×C8—614no—
562C2×C8—614no—
563C2×C8—2448no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—21no—
567C4⋊C4—612no—
568C4⋊C4—612no—
569C4⋊C4—1221no—
570C4⋊C4—1224no—
571C4⋊C4—1222no—
572C4⋊C4—1222no—
573C4⋊C4—1221no—
574C4⋊C4—1221no—
575C4⋊C4—1221no—
576C4⋊C4C2×C22⋊D6112yes—
577C4⋊C4C2×C22⋊D6112yes—
578C4⋊C4C23×D6112yes—
579C22⋊C4—28no—
580C22⋊C4—1224no—
581C22⋊C4—1222no—
582C22⋊C4—1224no—
583C22⋊C4—1224no—
584C22⋊C4—1222no—
585C22⋊C4—1222no—
586C22⋊C4C2×C22⋊D6114yes—
587C22⋊C4C2×C22⋊D6114yes—
588C42—22no—
589C42—22no—
590C42—1224no—
591C42—1222no—
592C42C2×C22⋊D6111yescharacteristic
593C42C2×C22⋊D6111yescharacteristic
594C42C23×D6111yescharacteristic
595C42C23×D6111yescharacteristic
596C23—24no—
597C23—24no—
598C23—42no—
599C23—42no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—1222no—
604C23C22×C22⋊D6114yes—
605C23C22×C22⋊D6111yescharacteristic
606Q8—24no—
607Q8—22no—
608Q8—1224no—
609Q8—4881no—
610Q8—2442no—
611Q8#1451982214yes—
612D4—24no—
613D4—24no—
614D4—28no—
615D4—416no—
616D4—44no—
617D4—44no—
618D4—44no—
619D4—82no—
620D4—82no—
621D4—88no—
622D4—1224no—
623D4—4881no—
624D4—4884no—
625D4—4884no—
626D4—2444no—
627D4—2442no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—28no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—416no—
637C2×C4—42no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—1222no—
641C2×C4—4884no—
642C2×C4—2444no—
643C2×C4#1451979714yescomplemented
644C2×C4C22×C22⋊D6114yes—
645C2×C4C22×C22⋊D6111yescharacteristic
646C2×C4C22×C22⋊D6111yescharacteristic
647C2×C4C22×C22⋊D6111yescharacteristic
648C2×C4C22×C22⋊D6111yescharacteristic
649C2×C4C22×C22⋊D6111yescharacteristic
650C2×C4C24×D6111yescharacteristic
651C8—1224no—
652C8—1224no—
653C22—21no—
654C22—21no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—84no—
662C22—88no—
663C22—2442no—
664C22—2442no—
665C22—2442no—
666C22C2×#1451979712yes—
667C22C23×C22⋊D6111yescharacteristic
668C4—21no—
669C4—21no—
670C4—24no—
671C4—24no—
672C4—21no—
673C4—42no—
674C4—2442no—
675C4—2442no—
676C4—2442no—
677C4C2×#1451979712yes—
678C4C2×#1451979712yes—
679C4C2×#1451982212yes—
680C2—22no—
681C2—44no—
682C2—84no—
683C2—4882no—
684C2#2145945411yescharacteristic
685C2C22×#1451979711yescharacteristic
686C2#2146059911yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#550C2
socleC22×C61C23×D4
hypercenterC2×#550D61
hypocenterC61#61566
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #21459671C2semidirect
2. #21457769C2semidirect
3. #21457758C2semidirect
4. #21457575C2semidirect
5. #21457549C2semidirect
6. #21456760C2semidirect
7. #21456737C2semidirect
8. #21456584C2semidirect
9. #21455858C2semidirect
10. #21455850C2semidirect
11. #21455037C2semidirect
12. #21454843C2semidirect
13. C2×C61×#550C2semidirect
14. C2×#14519797C22semidirect
16. #16840482C22semidirect
17. #16840477C22semidirect
19. #16840433C22semidirect
20. #16840417C22semidirect
27. #16840138C22semidirect
28. #16840134C22semidirect
29. #16840108C22semidirect
30. #16840108C22semidirect
32. #16840083C22semidirect
34. #16840057C22semidirect
35. #16840047C22semidirect
36. #16839996C22semidirect
37. #16839989C22semidirect
38. #16839972C22semidirect
39. #16839962C22semidirect
40. #16839959C22semidirect
41. #16839954C22semidirect
42. #16839924C22semidirect
45. #16839907C22semidirect
46. #16839905C22semidirect
49. C4×#11758202C22semidirect
50. C4×#11758201C22semidirect
51. D4×#92418C22semidirect
52. D4×#92418C22semidirect
55. #16839758C22semidirect
285. C2×C4×C61#135semidirect
338. C2×#550D61semidirect
412. C61#61566semidirect
643. C2×C4#14519797semidirect

Maximal subgroups

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

subgroupindex
#214596712
#214577692
#214577582
#214575752
#214575492
#214567602
#214567372
#214565842
#214558582
#214558502
#214550372
#214548432
C2×C61×#5502
#6156661