Finite Group Databasebeta

← #29680845

Subgroups — #29680845

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459020C211yesmaximal, characteristic
2C2×#16839802C211yesmaximal, characteristic, complemented
3C2×#16839744C211yesmaximal, characteristic
4#21448412C211yesmaximal, characteristic
5#21446032C214yesmaximal, complemented
6#21446027C214yesmaximal, complemented
7#21445530C211yesmaximal, characteristic, complemented
8#21445528C211yesmaximal, characteristic, complemented
9C2×C61×#428C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840243—21no—
13#16840243—21no—
14#16840242—21no—
15#16840242—21no—
16#16840239C2212yes—
17#16840236C2212yes—
18#16840235—22no—
19C4×#11758222—22no—
20#16840230—22no—
21C2×#14519736C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#117×D61—22no—
24#16839802—22no—
25#16839802C2214yes—
26C2×#14519710—21no—
27C2×#14519708—21no—
28#16839744C418yescomplemented
29C2×#14519695C2212yes—
30C2×#14519693C2212yes—
31#16839667C2212yes—
32#16839663C2212yes—
33#16839641—22no—
34#16839619—22no—
35#16839523C2214yes—
36#16839521C2214yes—
37C22×M16×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#428—22no—
41C61×#428C2214yes—
42C2×#107×C61—21no—
43C2×#106×C61—21no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—21no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222C2×C414yes—
51C2×#11758222Q814yes—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61D412yes—
55C22×C4×D61D412yes—
56C2×#11758194—21no—
57C2×#11758194—22no—
58C2×#11758194C2×C414yescomplemented
59C2×#11758194C2312yes—
60C22×#92418—21no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63#14519742—22no—
64#14519742—22no—
65C4×D244—22no—
66C4×D244—22no—
67#14519739—22no—
68#14519739—22no—
69C4×C22⋊D61—24no—
70C4×C22⋊D61—24no—
71#14519737—24no—
72#14519737—24no—
73#14519736—21no—
74#14519736—21no—
75#14519736—22no—
76#14519736—22no—
77#14519736D412yes—
78#14519736C2312yes—
79#14519735—22no—
80#14519735—22no—
81#14519734—24no—
82#14519733—24no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61D411yescharacteristic
85C2×C4⋊Dic61D411yescharacteristic
86C4⋊C4×D61—22no—
87C4⋊C4×D61—22no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×#11758198—24no—
91C22⋊C4×D61—24no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2312yes—
94C42×D61—24no—
95#14519710—24no—
96#14519708—24no—
97#14519695C2×C418yescomplemented
98#14519693C2×C418yescomplemented
99#14519684—24no—
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C2×M16×C61—21no—
103C2×M16×C61—22no—
104C2×M16×C61C2×C414yescomplemented
105C2×M16×C61C2312yes—
106C22×C8×C61—21no—
107#117×C61—21no—
108#117×C61—21no—
109#117×C61D412yes—
110#117×C61C2312yes—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61C2312yes—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121#107×C61—24no—
122#106×C61—24no—
123C24×C61C2×D411yescharacteristic
124C23×D61C2×D412yes—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—41no—
129C22×C4×C61—41no—
130C22×C4×C61—82no—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×Q812yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—22no—
142C2×C22⋊D61—22no—
143C2×C22⋊D61—24no—
144C2×C22⋊D61C4⋊C418yes—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61C2×D412yes—
149#11758222—24no—
150#11758222—24no—
151#11758222—24no—
152#11758222—24no—
153#11758222—24no—
154#11758222—24no—
155#11758222—24no—
156#11758222—24no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C22×C412yes—
162C2×D244C2×Q812yes—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—24no—
166C2×C4×D61—24no—
167C2×C4×D61—24no—
168C2×C4×D61C4⋊C418yes—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C22×C412yes—
174C2×Dic122C2×Q812yes—
175M16×C61—24no—
176M16×C61—22no—
177M16×C61C22×C414yes—
178C2×C8×C61—24no—
179C2×C8×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61C2×D412yes—
187C4⋊C4×C61C2×D412yes—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C42×C61—22no—
191C42×C61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—42no—
195#11758198—24no—
196#11758198—24no—
197#11758198—84no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—42no—
201C4⋊Dic61C2×D412yes—
202C4⋊Dic61C2×D412yes—
203#11758196—24no—
204#11758196—24no—
205C4×Dic61—24no—
206C4×Dic61—24no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211#11758194—24no—
212#11758194—22no—
213#11758194C22×C414yes—
214C2×#92418—24no—
215C2×#92418—22no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61C2×C4⋊C412yes—
219C23×C61#12011yescharacteristic
220C23×C61#12011yescharacteristic
221C23×C61#12312yes—
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61C2×C4⋊C414yes—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—44no—
231C2×C4×C61—42no—
232C2×C4×C61—81no—
233C2×C4×C61—81no—
234C2×C4×C61—82no—
235C2×C4×C61—82no—
236C2×C4×C61C2×C4⋊C412yes—
237C2×C4×C61C4×D412yes—
238C2×C4×C61C4×D412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C4×D412yes—
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61C22⋊Q812yes—
244C2×C4×C61C22⋊Q812yes—
245C2×C4×C61C22⋊Q812yes—
246C2×C4×C61C22⋊Q812yes—
247C2×C4×C61#12312yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×Q811yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C22⋊D61—28no—
255C22⋊D61—28no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—81no—
264C2×Dic61—81no—
265C2×Dic61C2×C4⋊C414yes—
266D244—22no—
267D244—22no—
268D244—22no—
269D244—22no—
270C4×D61—28no—
271C4×D61—28no—
272C4×D61—84no—
273C4×D61—84no—
274Dic122—22no—
275Dic122—22no—
276Dic122—22no—
277Dic122—22no—
278C8×C61—24no—
279#92418—24no—
280#20858—611nomaximal, Sylow
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61#38512yes—
290C22×C61#38512yes—
291C22×C61C22×C4⋊C411yescharacteristic
292C22×C61C2×#12011yescharacteristic
293C22×C61C2×#12311yescharacteristic
294C2×D61—24no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—44no—
298C2×D61—82no—
299C4×C61—42no—
300C4×C61—42no—
301C4×C61—82no—
302C4×C61#38512yes—
303C4×C61#38512yes—
304C4×C61C2×C4×D411yescharacteristic
305C4×C61C2×C4×D411yescharacteristic
306C4×C61C2×C22⋊Q811yescharacteristic
307C4×C61C2×C22⋊Q811yescharacteristic
308Dic61—24no—
309Dic61—42no—
310Dic61—42no—
311Dic61—82no—
312#3428—611no—
313C2×#428—611no—
314C2×#428D6111yescharacteristic, complemented
315C2×#413—611no—
316#2311—611no—
317#1857—614no—
318#1856—614no—
319#1733—611no—
320#1733—611no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—22no—
324C2×C61—22no—
325C2×C61C2×#38511yescharacteristic
326C2×C61#302111yescharacteristic
327C2×C61#302211yescharacteristic
328D61—44no—
329C22×#41—611no—
330C22×M16—611no—
331C22×M16C2×D6111yescharacteristic
332#520—1221no—
333#520—1221no—
334#520—1221no—
335#520—1221no—
336#519—612no—
337#518—612no—
338#518—1222no—
339C4×#41—1222no—
340C4×#41—1222no—
341C2×#117—611no—
342C2×#117—611no—
343C2×#117—1222no—
344C2×#117C2×D6111yescharacteristic
345C22×C4⋊C4C2×D6111yescharacteristic
346#428—22no—
347#428—614no—
348#428—1222no—
349#428C2×D6114yes—
350C2×#107—21no—
351C2×#107—1221no—
352C2×#106—21no—
353C2×#106—1221no—
354#413—618no—
355C2×#101—612no—
356C2×#100—612no—
357#391—612no—
358#390—612no—
359#386—1222no—
360#380—1222no—
361#339—614no—
362#339—614no—
363C61#2085811yescharacteristic, complemented, Sylow
364C2×#41—614no—
365C2×#41—614no—
366C2×#41—1222no—
367C2×#41—1221no—
368C2×#41—1221no—
369C2×#41—1222no—
370C2×#41—1221no—
371C2×#41—1221no—
372C22×Q8—611no—
373C22×D4—612no—
374C22×D4—611no—
375C23×C4—21no—
376C23×C4—612no—
377C23×C4C22×D6111yescharacteristic
378C2×M16—22no—
379C2×M16—21no—
380C2×M16—614no—
381C2×M16—612no—
382C2×M16—1221no—
383C2×M16—1222no—
384C2×M16C4×D6114yescomplemented
385C2×M16C22×D6112yes—
386C22×C8—21no—
387C22×C8—1221no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×Q8—1222no—
391C4×Q8—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394C4×D4—1224no—
395C4×D4—1224no—
396C4×D4—1224no—
397C4×D4—1224no—
398C4×D4—1222no—
399C4×D4—1222no—
400#117—21no—
401#117—21no—
402#117—612no—
403#117—612no—
404#117—1224no—
405#117—1224no—
406#117—1222no—
407#117—1222no—
408#117—1221no—
409#117—1221no—
410#117—1222no—
411#117—1222no—
412#117C22⋊D6112yes—
413#117C22×D6112yes—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—1222no—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4C22⋊D6111yescharacteristic
425C2×C4⋊C4C22⋊D6111yescharacteristic
426C2×C4⋊C4C22×D6112yes—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1224no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1224no—
431C2×C22⋊C4—1221no—
432C2×C42—21no—
433C2×C42—612no—
434C2×C42—1224no—
435C2×C42—1221no—
436#107—24no—
437#107—1224no—
438#106—24no—
439#106—1224no—
440#101—618no—
441#100—618no—
442#95—1224no—
443C24—612no—
444C24D4×D6111yescharacteristic
445#41—1224no—
446#41—1224no—
447#41—1224no—
448#41—1224no—
449#41—1224no—
450#41—1224no—
451#41—1224no—
452#41—1224no—
453C2×Q8—612no—
454C2×Q8—612no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×D4—618no—
460C2×D4—612no—
461C2×D4—612no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1224no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—82no—
475C22×C4—612no—
476C22×C4—618no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1224no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4C2×C4×D6112yes—
486C22×C4#1175822212yes—
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4D4×D6111yescharacteristic
490C22×C4Q8×D6112yes—
491C22×C4#1175822612yes—
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C23×D6111yescharacteristic
495M16—22no—
496M16—24no—
497M16—614no—
498M16—1222no—
499M16—1224no—
500M16C2×C4×D6114yes—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—1222no—
504C2×C8—1224no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—612no—
512C4⋊C4—612no—
513C4⋊C4—1224no—
514C4⋊C4—1224no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—2442no—
518C4⋊C4C2×C22⋊D6112yes—
519C4⋊C4C2×C22⋊D6112yes—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—1224no—
523C22⋊C4—1224no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—4884no—
527C22⋊C4—2442no—
528C42—22no—
529C42—22no—
530C42—1224no—
531C42—1224no—
532C42—1222no—
533C42—1222no—
534C42—1222no—
535C42—1222no—
536C23—41no—
537C23—41no—
538C23—614no—
539C23—1222no—
540C23—1222no—
541C23—2442no—
542C23—2442no—
543C23C4⋊C4×D6112yes—
544C23#1451974511yescharacteristic
545C23#1451974511yescharacteristic
546C23#1451976212yes—
547C23C2×D4×D6111yescharacteristic
548Q8—1222no—
549Q8—1222no—
550Q8—1222no—
551Q8—1222no—
552D4—1228no—
553D4—1228no—
554D4—1222no—
555D4—1222no—
556D4—1222no—
557D4—1222no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—614no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1228no—
570C2×C4—1228no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—4884no—
574C2×C4—4884no—
575C2×C4—4881no—
576C2×C4—4881no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4C4⋊C4×D6112yes—
581C2×C4C4×C22⋊D6112yes—
582C2×C4C4×C22⋊D6112yes—
583C2×C4#1451973912yes—
584C2×C4#1451973912yes—
585C2×C4#1451974511yescharacteristic
586C2×C4#1451974511yescharacteristic
587C2×C4#1451975312yes—
588C2×C4#1451975312yes—
589C2×C4#1451975512yes—
590C2×C4#1451975512yes—
591C2×C4#1451976212yes—
592C2×C4C22×C4×D6111yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4C22×C22⋊D6111yescharacteristic
595C2×C4C2×Q8×D6111yescharacteristic
596C2×C4C2×#1175822211yescharacteristic
597C2×C4C2×#1175822611yescharacteristic
598C8—24no—
599C8—1224no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—42no—
607C22—42no—
608C22—1224no—
609C22—4882no—
610C22—2444no—
611C22—2442no—
612C22—2442no—
613C22#1683963312yes—
614C22#1683963312yes—
615C22C2×C4⋊C4×D6111yescharacteristic
616C22C2×#1451974511yescharacteristic
617C22C2×#1451976211yescharacteristic
618C4—42no—
619C4—42no—
620C4—82no—
621C4—1224no—
622C4—4882no—
623C4—2442no—
624C4—2442no—
625C4#1683963312yes—
626C4#1683963312yes—
627C4C2×C4×C22⋊D6111yescharacteristic
628C4C2×#1451973911yescharacteristic
629C4C2×#1451975311yescharacteristic
630C4C2×#1451975511yescharacteristic
631C2—21no—
632C2—21no—
633C2—22no—
634C2—22no—
635C2—2444no—
636C2C2×#1683963311yescharacteristic
637C2#2145518811yescharacteristic
638C2#2145519511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#428D61
hypocenterC61#20858
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#16839802C2semidirect
5. #21446032C2semidirect
6. #21446027C2semidirect
7. #21445530C2semidirect
8. #21445528C2semidirect
9. C2×C61×#428C2semidirect
28. #16839744C4semidirect
58. C2×#11758194C2×C4semidirect
97. #14519695C2×C4semidirect
98. #14519693C2×C4semidirect
104. C2×M16×C61C2×C4semidirect
314. C2×#428D61semidirect
363. C61#20858semidirect
384. C2×M16C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214590202
C2×#168398022
C2×#168397442
#214484122
#214460322
#214460272
#214455302
#214455282
C2×C61×#4282
#2085861