Finite Group Databasebeta

← #29713111

Subgroups — #29713111

Order 15616.

1410 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840078C211yesmaximal, characteristic, complemented
2C2×#16839763C211yesmaximal, characteristic, complemented
3#21450127C211yesmaximal, characteristic, complemented
4#21447050C212yesmaximal, complemented
5#21447005C212yesmaximal, complemented
6#21446258C211yesmaximal, characteristic, complemented
7#21446117C211yesmaximal, characteristic, complemented
8#21446087C212yesmaximal, complemented
9#21445759C211yesmaximal, characteristic, complemented
10#21445759C212yesmaximal, complemented
11C2×C61×#550C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13C2×#14519758—21no—
14C22×C4⋊Dic61C2211yescharacteristic
15#16840078—24no—
16C2×#14519708C2212yes—
17#16839763—24no—
18C2×#14519701C2212yes—
19C2×#14519701C2212yes—
20C2×#14519698C2212yes—
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×C22⋊#92418C2211yescharacteristic
25C2×C22⋊#92418C2212yes—
26#16839692—21no—
27#16839687—21no—
28#16839660C2212yes—
29#16839623—21no—
30#16839622C2212yes—
31#16839622C2211yescharacteristic
32#16839524C2212yes—
33#16839517C2212yes—
34C22×#41×C61C2211yescharacteristic
35C22×#41×C61C2211yescharacteristic
36C61×#550—24no—
37C61×#550—24no—
38C2×#127×C61—21no—
39C2×#127×C61C2211yescharacteristic
40C2×#121×C61—21no—
41C2×#121×C61—21no—
42C2×#121×C61C2211yescharacteristic
43C2×#121×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C2×C4×D4×C61—21no—
46C2×C4×D4×C61C2211yescharacteristic
47C23×Dic61—21no—
48C2×#11758194—22no—
49C2×#11758194—42no—
50C22×#92418C2312yes—
51C22×#92418C2312yes—
52#14519758—44no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2312yes—
58C2×#11758196—22no—
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61—21no—
61#14519708—24no—
62#14519701—24no—
63#14519701—24no—
64#14519698—24no—
65#14519698—24no—
66#14519698—42no—
67#14519698—42no—
68C22⋊#92418—22no—
69C22⋊#92418—42no—
70#14519684—21no—
71#14519684—22no—
72#14519684—22no—
73C2×#41×C61—22no—
74C2×#41×C61—22no—
75C2×#41×C61—22no—
76C2×#41×C61—24no—
77C2×#41×C61—24no—
78C2×#41×C61—42no—
79C2×#41×C61—42no—
80C2×#41×C61D414yes—
81C22×Q8×C61C2311yescharacteristic
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C22×D4×C61—21no—
88C22×D4×C61D411yescharacteristic
89C22×D4×C61C2311yescharacteristic
90C22×D4×C61C2311yescharacteristic
91C23×C4×C61—21no—
92C23×C4×C61—21no—
93C23×C4×C61C2311yescharacteristic
94C23×C4×C61C2311yescharacteristic
95#127×C61—28no—
96#127×C61—22no—
97#127×C61—22no—
98#121×C61—28no—
99#121×C61—28no—
100#121×C61—22no—
101#121×C61—22no—
102#121×C61—22no—
103#121×C61D414yes—
104C4×Q8×C61—22no—
105C4×Q8×C61—22no—
106C4×D4×C61—28no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C22⋊C4×C61D411yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C24×C61—21no—
119C24×C61—21no—
120C24×C61C2×D411yescharacteristic
121C24×C61C2×D411yescharacteristic
122#41×C61—28no—
123#41×C61—44no—
124#41×C61—48no—
125#41×C61—48no—
126#41×C61—44no—
127#41×C61—44no—
128C2×Q8×C61—21no—
129C2×Q8×C61—22no—
130C2×Q8×C61C2×D412yes—
131C2×Q8×C61#4114yes—
132C2×D4×C61—24no—
133C2×D4×C61—24no—
134C2×D4×C61—22no—
135C2×D4×C61—24no—
136C2×D4×C61—24no—
137C2×D4×C61—24no—
138C2×D4×C61—24no—
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—41no—
147C2×D4×C61—44no—
148C2×D4×C61—44no—
149C2×D4×C61—42no—
150C2×D4×C61—42no—
151C2×D4×C61—82no—
152C2×D4×C61—82no—
153C2×D4×C61—82no—
154C2×D4×C61C2×D412yes—
155C2×D4×C61C2×D412yes—
156C22×C4×C61—21no—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—24no—
160C22×C4×C61—21no—
161C22×C4×C61—24no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—22no—
165C22×C4×C61—22no—
166C22×C4×C61—44no—
167C22×C4×C61—44no—
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D412yes—
172C22×C4×C61C2×D412yes—
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2411yescharacteristic
175C22×Dic61—22no—
176C22×Dic61—22no—
177C22×Dic61—42no—
178C22×Dic61—42no—
179C22×Dic61—42no—
180C4⋊C4×C61—24no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C2×D412yes—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—24no—
187C22⋊C4×C61—22no—
188C42×C61—24no—
189C42×C61—21no—
190C42×C61—21no—
191C22⋊Dic61—42no—
192C22⋊Dic61—44no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—42no—
198#11758196—44no—
199#11758194—44no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—22no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—22no—
210C23×C61—21no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—22no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—42no—
224C23×C61—42no—
225C23×C61#12012yes—
226C23×C61C22×D411yescharacteristic
227Q8×C61—22no—
228Q8×C61—24no—
229Q8×C61#12314yescomplemented
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—24no—
233D4×C61—44no—
234D4×C61—48no—
235D4×C61—48no—
236D4×C61—48no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—42no—
240D4×C61—84no—
241D4×C61—84no—
242D4×C61—84no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—24no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—24no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—48no—
257C2×C4×C61—48no—
258C2×C4×C61—42no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—42no—
262C2×C4×C61#12012yes—
263C2×C4×C61AGL(1,ℤ/8)12yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4112yes—
267C2×C4×C61H(5,𝔽2)12yes—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—84no—
275#92418—42no—
276#92418—42no—
277#24045—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—41no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—82no—
295C22×C61—82no—
296C22×C61—82no—
297C22×C61—82no—
298C22×C61C2×#12011yescharacteristic
299C22×C61#54612yes—
300C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57712yes—
303C4×C61—22no—
304C4×C61—24no—
305C4×C61—22no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—42no—
309C4×C61C2×#12311yescharacteristic
310C4×C61#53411yescharacteristic
311C4×C61#54612yes—
312Dic61—82no—
313Dic61—82no—
314C2×#550D6111yescharacteristic
315C2×#481—611no—
316C2×#417—611no—
317#2486—611no—
318#2041—612no—
319#2034—612no—
320#1895—611no—
321#1870—611no—
322#1865—612no—
323#1788—612no—
324#1788—611no—
325C2×C61—42no—
326C2×C61—81no—
327C2×C61—82no—
328C2×C61—82no—
329C2×C61#240511yescharacteristic
330C2×C61#300611yescharacteristic
331C2×C61#300911yescharacteristic
332C2×C61#331712yes—
333C2×C61#333212yes—
334C22×#41C2×D6111yescharacteristic
335C22×#41C2×D6111yescharacteristic
336C22×M16—611no—
337#550—24no—
338#550—24no—
339C2×#127—21no—
340C2×#127C2×D6111yescharacteristic
341C2×#123—1221no—
342C2×#121—21no—
343C2×#121—21no—
344C2×#121C2×D6111yescharacteristic
345C2×#121C2×D6111yescharacteristic
346C2×C4×Q8C2×D6111yescharacteristic
347C2×C4×D4—21no—
348C2×C4×D4C2×D6111yescharacteristic
349C22×C4⋊C4—611no—
350#481—1224no—
351C2×#106—612no—
352#417—1224no—
353C2×#103—612no—
354C2×#103—612no—
355C2×#102—612no—
356C2×#102—612no—
357C2×#102—612no—
358C2×#102—612no—
359C2×C22⋊C8—611no—
360C2×C22⋊C8—612no—
361#398—1221no—
362#397—1221no—
363#389—612no—
364#381—1221no—
365#380—611no—
366#380—612no—
367#340—612no—
368#336—612no—
369C61#2404511yescharacteristic, complemented, Sylow
370C2×#41—22no—
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—24no—
374C2×#41—24no—
375C2×#41—42no—
376C2×#41—42no—
377C2×#41C22⋊D6114yes—
378C22×Q8C22×D6111yescharacteristic
379C22×D4—21no—
380C22×D4—21no—
381C22×D4—21no—
382C22×D4—21no—
383C22×D4—21no—
384C22×D4—21no—
385C22×D4C22⋊D6111yescharacteristic
386C22×D4C22×D6111yescharacteristic
387C22×D4C22×D6111yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392C23×C4C22×D6111yescharacteristic
393C2×M16—1222no—
394C2×M16—2442no—
395C22×C8—612no—
396C22×C8—612no—
397#127—28no—
398#127—22no—
399#127—22no—
400#123—2444no—
401#121—28no—
402#121—28no—
403#121—22no—
404#121—22no—
405#121—22no—
406#121C22⋊D6114yes—
407C4×Q8—22no—
408C4×Q8—22no—
409C4×D4—28no—
410C4×D4—22no—
411C4×D4—22no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—612no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4C22⋊D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1222no—
425C2×C22⋊C4C22⋊D6111yescharacteristic
426C2×C22⋊C4C22⋊D6111yescharacteristic
427C2×C42—21no—
428C2×C42C22×D6111yescharacteristic
429#106—1224no—
430#103—1224no—
431#103—1224no—
432#102—1224no—
433#102—1224no—
434#102—2442no—
435#102—2442no—
436C22⋊C8—1222no—
437C22⋊C8—2442no—
438#95—1222no—
439#95—1221no—
440#95—1222no—
441C24—21no—
442C24—21no—
443C24C2×C22⋊D6111yescharacteristic
444C24C2×C22⋊D6111yescharacteristic
445#41—28no—
446#41—44no—
447#41—48no—
448#41—48no—
449#41—44no—
450#41—44no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8#1175822414yes—
454C2×Q8C2×C22⋊D6112yes—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—41no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—82no—
475C2×D4—82no—
476C2×D4—82no—
477C2×D4C2×C22⋊D6112yes—
478C2×D4C2×C22⋊D6112yes—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6112yes—
499C22×C4C2×C22⋊D6112yes—
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—2444no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—1222no—
510C4⋊C4—24no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—2442no—
518C4⋊C4—2444no—
519C4⋊C4C2×C22⋊D6112yes—
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—2442no—
525C22⋊C4—2444no—
526C42—24no—
527C42—21no—
528C42—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23#1451974612yes—
549C23C22×C22⋊D6111yescharacteristic
550Q8—22no—
551Q8—24no—
552Q8#1451975814yescomplemented
553D4—22no—
554D4—22no—
555D4—24no—
556D4—44no—
557D4—48no—
558D4—48no—
559D4—48no—
560D4—41no—
561D4—41no—
562D4—42no—
563D4—84no—
564D4—84no—
565D4—84no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—4884no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4#1451974612yes—
593C2×C4#1451980212yes—
594C2×C4C22×C22⋊D6111yescharacteristic
595C2×C4C22×C22⋊D6111yescharacteristic
596C2×C4C2×#1175822412yes—
597C2×C4#1451982212yes—
598C8—2442no—
599C8—2442no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22C2×#1451974611yescharacteristic
621C22#1684040612yes—
622C22C2×#1451980211yescharacteristic
623C22#1684056011yescharacteristic
624C22#1684057312yes—
625C4—22no—
626C4—24no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—4882no—
632C4—4882no—
633C4C2×#1451975811yescharacteristic
634C4#1684032111yescharacteristic
635C4#1684040612yes—
636C2—42no—
637C2—81no—
638C2—82no—
639C2—82no—
640C2#2144914211yescharacteristic
641C2#2145507111yescharacteristic
642C2#2145509611yescharacteristic
643C2#2145794512yes—
644C2#2145814612yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#550C2
socleC23×C61C22×D4
hypercenterC2×#550D61
hypocenterC61#24045
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×#16840078C2semidirect
2. C2×#16839763C2semidirect
3. #21450127C2semidirect
4. #21447050C2semidirect
5. #21447005C2semidirect
6. #21446258C2semidirect
7. #21446117C2semidirect
8. #21446087C2semidirect
9. #21445759C2semidirect
10. #21445759C2semidirect
229. Q8×C61#123semidirect
369. C61#24045semidirect
552. Q8#14519758semidirect

Maximal subgroups

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

subgroupindex
C2×#168400782
C2×#168397632
#214501272
#214470502
#214470052
#214462582
#214461172
#214460872
#214457592
C2×C61×#5502
#2404561