Finite Group Databasebeta

← #30549690

Subgroups — #30549690

Order 15616.

1834 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459635C211yesmaximal, characteristic
2#21459631C211yesmaximal, characteristic
3#21456365C211yesmaximal, characteristic, complemented
4#21456362C212yesmaximal, complemented
5#21456360C212yesmaximal, complemented
6#21456349C214yesmaximal, complemented
7#21456346C218yesmaximal, complemented
8#21456330C212yesmaximal, complemented
9#21456329C211yesmaximal, characteristic, complemented
10C2×#16840158C211yesmaximal, characteristic, complemented
11#498×D61C212yesmaximal, complemented
12#21454413C214yesmaximal
13C4×#14519785C211yesmaximal, characteristic
14C61×#3165C211yesmaximal, characteristic, complemented
15C2×#14519785C2212yes—
16Q8×Dic122—21no—
17#16840492—21no—
18#16840490—21no—
19#16840487—21no—
20#16840473—21no—
21#16840472—21no—
22#16840470—22no—
23#16840467—22no—
24Q8×D244—21no—
25#16840427—21no—
26Q8×C22⋊D61—22no—
27#16840425—22no—
28#16840393C2212yes—
29#16840392C2214yes—
30#16840390C2212yes—
31#16840389C2214yes—
32#16840319C2211yescharacteristic
33#16840317C2212yes—
34#16840317C2212yes—
35C2×#14519777C2211yescharacteristic
36C4⋊Q8×D61C2212yes—
37C4⋊Q8×D61C2212yes—
38#16840243C2212yes—
39#16840242C2212yes—
40C4×#11758222C2211yescharacteristic
41#16840171C2214yes—
42#16840170C2218yes—
43#16840169C2218yes—
44#16840167C2212yes—
45#16840165C2211yescharacteristic
46#16840165C2211yescharacteristic
47#16840163C2218yes—
48#16840161C2214yes—
49#16840160C2212yes—
50#16840159C2218yes—
51#16840158C2211yescharacteristic
52#16840158C2212yes—
53#16840158C2214yes—
54#16840158C2212yes—
55#16840158C2211yescharacteristic
56#16839804C2214yes—
57#16839803C2218yes—
58#16839801C2218yes—
59#16839799C2218yes—
60#16839798C2214yes—
61C2×#14519708C2214yes—
62#106×D61C2218yes—
63#16839720C2212yes—
64#16839719C2214yes—
65C4×#11758189C2212yes—
66C2×C4×#92418C2211yescharacteristic
67C4×C8×D61C2212yes—
68C61×#533C2211yescharacteristic
69C2×C4⋊Q8×C61C2211yescharacteristic
70C61×#501C2214yes—
71C61×#500C2212yes—
72C61×#498C2212yes—
73C61×#428C2214yes—
74C4×M16×C61C2211yescharacteristic
75C2×#11758222C2311yescharacteristic
76C22×Dic122—21no—
77C2×Q8×D61—22no—
78C2×Q8×D61—22no—
79#14519785Q8116yescomplemented
80C2×#11758189C2314yes—
81C22×#92418C2312yes—
82C2×C8×D61C2314yes—
83#14519777—21no—
84#14519777—21no—
85#14519777—22no—
86#14519777C2311yescharacteristic
87#14519777C2312yes—
88#14519777C2312yes—
89#14519777C2311yescharacteristic
90#14519776—22no—
91#14519776—24no—
92#14519776—22no—
93#14519776—22no—
94#14519776C2314yes—
95#14519776C2314yes—
96#14519775—24no—
97#14519775—22no—
98#14519775—22no—
99#14519769—22no—
100#14519769C2312yes—
101#14519768—22no—
102#14519768C2314yes—
103#14519768C2314yes—
104#14519767—24no—
105#14519767—24no—
106#14519757—22no—
107#14519757—22no—
108#14519757—24no—
109#14519756—21no—
110#14519756—21no—
111#14519756—24no—
112#14519755—24no—
113#14519755—24no—
114#14519754—24no—
115#14519753—22no—
116#14519753—22no—
117#14519753—24no—
118Q8×Dic61—22no—
119Q8×Dic61—22no—
120C4×Dic122—22no—
121C4×Dic122—21no—
122C4×Dic122—21no—
123C4×Dic122D411yescharacteristic
124C4×Dic122D411yescharacteristic
125#14519742—22no—
126#14519742—22no—
127#14519742—22no—
128C4×D244—22no—
129C4×D244D411yescharacteristic
130C4×D244D411yescharacteristic
131#14519739—22no—
132C4×C22⋊D61—24no—
133C4×C22⋊D61D412yes—
134C4×C22⋊D61D412yes—
135#14519737—24no—
136#14519736C2312yes—
137#14519736C2311yescharacteristic
138#14519735C2314yes—
139#14519735C2314yes—
140#14519734C2312yes—
141C2×C4⋊Dic61—21no—
142C2×C4⋊Dic61C2312yes—
143C4⋊C4×D61—22no—
144C4⋊C4×D61—22no—
145C4⋊C4×D61C2314yes—
146C4⋊C4×D61C2314yes—
147C42×D61C2312yes—
148#14519711C2318yes—
149#14519710C2314yes—
150#14519710C2314yes—
151#14519709C2318yes—
152#14519708C2314yes—
153#14519708C2318yes—
154#14519708C2318yes—
155#14519708C2314yes—
156#14519687C2314yes—
157C4×#92418C2311yescharacteristic
158C4×#92418C2311yescharacteristic
159C4×#92418C2312yes—
160C8×Dic61C2314yes—
161C22×Q8×C61—21no—
162C2×M16×C61C2312yes—
163C4⋊Q8×C61—21no—
164C4⋊Q8×C61—21no—
165C4⋊Q8×C61C2312yes—
166C4⋊Q8×C61C2312yes—
167C4⋊Q8×C61C2311yescharacteristic
168C4⋊Q8×C61C2311yescharacteristic
169#125×C61C2312yes—
170C22⋊Q8×C61—21no—
171C22⋊Q8×C61—21no—
172C4×Q8×C61—21no—
173C4×Q8×C61—21no—
174#117×C61C2312yes—
175C2×C4⋊C4×C61—21no—
176C2×C4⋊C4×C61C2312yes—
177C2×C42×C61C2311yescharacteristic
178#107×C61C2318yes—
179#106×C61C2318yes—
180#97×C61C2312yes—
181C4×C8×C61C2312yes—
182C2×Q8×C61—21no—
183C2×Q8×C61—22no—
184C2×Q8×C61—21no—
185C2×Q8×C61—22no—
186C2×Q8×C61—82no—
187C22×C4×C61—22no—
188C22×C4×C61C2×D411yescharacteristic
189C22×C4×C61C2×D411yescharacteristic
190C22×C4×C61C2411yescharacteristic
191C2×C22⋊D61C2×D412yes—
192C22×Dic61—22no—
193Q8×D61—84no—
194Q8×D61—84no—
195#11758222C2×Q818yes—
196C2×D244C2×D411yescharacteristic
197C2×C4×D61—24no—
198C2×C4×D61—24no—
199C2×C4×D61C2×D412yes—
200C2×C4×D61C2×D412yes—
201C2×C4×D61C2412yes—
202C2×Dic122—21no—
203C2×Dic122—22no—
204C2×Dic122—21no—
205C2×Dic122—22no—
206C2×Dic122—24no—
207C2×Dic122—24no—
208C2×Dic122—82no—
209C2×Dic122C2×D411yescharacteristic
210M16×C61C2×Q818yescomplemented
211C2×C8×C61C2414yes—
212C4⋊C4×C61—22no—
213C4⋊C4×C61—22no—
214C4⋊C4×C61—22no—
215C4⋊C4×C61—22no—
216C4⋊C4×C61—22no—
217C4⋊C4×C61—21no—
218C4⋊C4×C61—21no—
219C4⋊C4×C61C2412yes—
220C4⋊C4×C61C2414yes—
221C4⋊C4×C61C2412yes—
222C22⋊C4×C61—22no—
223C42×C61—22no—
224C42×C61C2×D411yescharacteristic
225C42×C61C2×D411yescharacteristic
226C42×C61C2411yescharacteristic
227C42×C61C2411yescharacteristic
228C22⋊Dic61—22no—
229C22⋊Dic61C2×D412yes—
230#11758198—24no—
231#11758198—24no—
232#11758198C2×D412yes—
233#11758198C2×D412yes—
234C4⋊Dic61—21no—
235C4⋊Dic61—22no—
236C4⋊Dic61—22no—
237C4⋊Dic61—22no—
238C4⋊Dic61—21no—
239C4⋊Dic61—24no—
240C4⋊Dic61—24no—
241C4⋊Dic61C2×D411yescharacteristic
242C4⋊Dic61C2×D411yescharacteristic
243C4⋊Dic61C2414yes—
244C4⋊Dic61C2412yes—
245C4⋊Dic61C2412yes—
246#11758196—22no—
247#11758196—22no—
248#11758196—24no—
249#11758196—24no—
250#11758196—24no—
251#11758196—24no—
252#11758196—24no—
253#11758196—24no—
254#11758196C2×D412yes—
255#11758196C2×D412yes—
256C4×Dic61—22no—
257C4×Dic61—24no—
258C4×Dic61—24no—
259C4×Dic61C2×D412yes—
260C4×Dic61C2×D412yes—
261C4×Dic61C2412yes—
262C2×#92418C2×Q818yescomplemented
263C2×#92418C2412yes—
264C2×#92418C2412yes—
265#11758189C2×Q8116yescomplemented
266C8×D61C2×Q8116yescomplemented
267C23×C61C22×D411yescharacteristic
268C22×D61C22×D412yes—
269Q8×C61—42no—
270Q8×C61—44no—
271Q8×C61—42no—
272Q8×C61—82no—
273C2×C4×C61—22no—
274C2×C4×C61—22no—
275C2×C4×C61—22no—
276C2×C4×C61—24no—
277C2×C4×C61—22no—
278C2×C4×C61—82no—
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C22×Q814yes—
285C2×C4×C61C2511yescharacteristic
286C22⋊D61C4⋊Q818yes—
287C2×Dic61—22no—
288C2×Dic61—22no—
289C2×Dic61—24no—
290C2×Dic61—82no—
291C2×Dic61C22×D412yes—
292C2×Dic61C22×D412yes—
293C2×Dic61C22×D412yes—
294D244C4⋊Q814yes—
295C4×D61—24no—
296C4×D61—24no—
297C4×D61—84no—
298C4×D61—84no—
299C4×D61C22×Q818yes—
300Dic122—42no—
301Dic122—44no—
302Dic122—42no—
303Dic122—84no—
304Dic122—84no—
305Dic122—82no—
306Dic122C4⋊Q814yes—
307C8×C61C22×Q818yes—
308#92418C22×Q814yes—
309#92418C22×Q814yes—
310#60665—611nomaximal, Sylow
311C22×C61—21no—
312C22×C61—21no—
313C22×C61C2×C4⋊Q812yes—
314C22×C61C23×D411yescharacteristic
315C2×D61—22no—
316C2×D61—22no—
317C2×D61C2×C4⋊Q814yes—
318C4×C61—21no—
319C4×C61—21no—
320C4×C61—42no—
321C4×C61—44no—
322C4×C61—42no—
323C4×C61C2×C4⋊Q812yes—
324C4×C61#57712yes—
325C4×C61#57712yes—
326C4×C61C23×Q812yes—
327Dic61—22no—
328Dic61—22no—
329Dic61—42no—
330Dic61—44no—
331Dic61—42no—
332Dic61C2×C4⋊Q814yes—
333#3503—611no—
334#3503—611no—
335#3165—612no—
336#3165—612no—
337#3165—611no—
338#3165D6111yescharacteristic, complemented
339#3163—618no—
340#3163—614no—
341#3160—612no—
342#3160—611no—
343C2×#498—612no—
344C2×#498—611no—
345#2912—614no—
346C4×#131—611no—
347C2×C61—22no—
348C2×C61C22×C4⋊Q811yescharacteristic
349C2×C61C2×#57711yescharacteristic
350C2×C61C2×#57711yescharacteristic
351D61—24no—
352C2×#131—612no—
353Q82—1221no—
354Q82—1221no—
355#557—1221no—
356#557—1221no—
357#554—1222no—
358#554—1221no—
359#554—1222no—
360#554—1221no—
361D4×Q8—1221no—
362D4×Q8—1222no—
363D4×Q8—1221no—
364D4×Q8—1222no—
365#544—614no—
366#544—614no—
367#544—612no—
368#544—612no—
369#533—612no—
370#533—612no—
371#533—611no—
372#533C2×D6111yescharacteristic
373C2×C4⋊Q8—612no—
374C2×C4⋊Q8—612no—
375C2×C4⋊Q8—611no—
376C2×C4⋊Q8C2×D6111yescharacteristic
377#520—612no—
378#520—612no—
379C4×#41—611no—
380#501—618no—
381#501—618no—
382#501—614no—
383#501C2×D6114yes—
384#500—611no—
385#500—611no—
386#500—612no—
387#500C2×D6112yes—
388#499—614no—
389#499—618no—
390#498—611no—
391#498—612no—
392#498—614no—
393#498—612no—
394#498—611no—
395#498—618no—
396#498—612no—
397#498C2×D6112yes—
398#428—614no—
399#428—618no—
400#428—618no—
401#428C2×D6114yes—
402#427—614no—
403#427—618no—
404C2×#106—614no—
405C2×#106—618no—
406#405—612no—
407#405—614no—
408C4×M16—612no—
409C4×M16C2×D6111yescharacteristic
410C2×C4×C8—611no—
411C2×C4×C8—612no—
412C61#6066511yescharacteristic, complemented, Sylow
413C2×#41—611no—
414C22×Q8—21no—
415C22×Q8—1222no—
416C22×Q8—1222no—
417C22×Q8—1221no—
418#131—6116no—
419C2×M16—614no—
420C2×M16C22×D6112yes—
421C22×C8—612no—
422C22×C8—614no—
423C4⋊Q8—21no—
424C4⋊Q8—21no—
425C4⋊Q8—614no—
426C4⋊Q8—614no—
427C4⋊Q8—611no—
428C4⋊Q8—612no—
429C4⋊Q8—612no—
430C4⋊Q8—611no—
431C4⋊Q8—1222no—
432C4⋊Q8—1222no—
433C4⋊Q8—1222no—
434C4⋊Q8—1222no—
435C4⋊Q8—1222no—
436C4⋊Q8—1224no—
437C4⋊Q8—1222no—
438C4⋊Q8—1224no—
439C4⋊Q8—1221no—
440C4⋊Q8—1221no—
441C4⋊Q8C22×D6111yescharacteristic
442C4⋊Q8C22×D6112yes—
443C4⋊Q8C22×D6112yes—
444C4⋊Q8C22×D6111yescharacteristic
445#125—614no—
446#125—614no—
447#125—612no—
448#125—1222no—
449#125—1224no—
450#125—1224no—
451#125—1222no—
452#125C22×D6112yes—
453C22⋊Q8—21no—
454C22⋊Q8—21no—
455C22⋊Q8—1224no—
456C22⋊Q8—1224no—
457C22⋊Q8—1224no—
458C22⋊Q8—1224no—
459C22⋊Q8—1224no—
460C22⋊Q8—1224no—
461C22⋊Q8—1222no—
462C22⋊Q8—1222no—
463C22⋊Q8—1222no—
464C22⋊Q8—1222no—
465C22⋊Q8—1221no—
466C22⋊Q8—1221no—
467C4×Q8—21no—
468C4×Q8—21no—
469C4×Q8—611no—
470C4×Q8—611no—
471C4×Q8—1222no—
472C4×Q8—1222no—
473C4×Q8—1222no—
474C4×Q8—1222no—
475C4×Q8—1222no—
476C4×Q8—1222no—
477C4×Q8—1221no—
478C4×Q8—1221no—
479C4×D4—612no—
480C4×D4—612no—
481C4×D4—611no—
482C4×D4—611no—
483C4×D4—1222no—
484C4×D4—1224no—
485C4×D4—1222no—
486C4×D4—1224no—
487#117—611no—
488#117—612no—
489#117—612no—
490#117—614no—
491#117—614no—
492#117C22×D6112yes—
493C2×C4⋊C4—21no—
494C2×C4⋊C4—612no—
495C2×C4⋊C4—614no—
496C2×C4⋊C4—614no—
497C2×C4⋊C4—1222no—
498C2×C4⋊C4—1222no—
499C2×C4⋊C4—1221no—
500C2×C4⋊C4C22×D6112yes—
501C2×C42—612no—
502C2×C42C22×D6111yescharacteristic
503#107—618no—
504#107—614no—
505#107—614no—
506#107C22×D6118yes—
507#106—618no—
508#106—618no—
509#106—614no—
510#106—618no—
511#106—614no—
512#106C22×D6118yes—
513#97—614no—
514#97C22×D6112yes—
515C4×C8—614no—
516C4×C8—611no—
517C4×C8—611no—
518C4×C8—612no—
519C4×C8C22×D6112yes—
520#41—618no—
521C2×Q8—21no—
522C2×Q8—21no—
523C2×Q8—22no—
524C2×Q8—22no—
525C2×Q8—82no—
526C2×Q8—611no—
527C2×Q8—1224no—
528C2×Q8—1224no—
529C2×Q8—1222no—
530C2×Q8—1221no—
531C2×Q8—1222no—
532C2×Q8—1221no—
533C2×Q8—4882no—
534C2×Q8—4884no—
535C2×Q8—4884no—
536C2×D4—612no—
537C2×D4—611no—
538C22×C4—22no—
539C22×C4—612no—
540C22×C4—612no—
541C22×C4—612no—
542C22×C4—1224no—
543C22×C4—1224no—
544C22×C4—1222no—
545C22×C4D4×D6111yescharacteristic
546C22×C4D4×D6111yescharacteristic
547C22×C4C23×D6111yescharacteristic
548M16—6116no—
549M16Q8×D6118yescomplemented
550C2×C8—618no—
551C2×C8—612no—
552C2×C8—612no—
553C2×C8—6116no—
554C2×C8C23×D6114yes—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—612no—
563C4⋊C4—612no—
564C4⋊C4—611no—
565C4⋊C4—611no—
566C4⋊C4—612no—
567C4⋊C4—614no—
568C4⋊C4—612no—
569C4⋊C4—1224no—
570C4⋊C4—1224no—
571C4⋊C4—1224no—
572C4⋊C4—1224no—
573C4⋊C4—1224no—
574C4⋊C4—1224no—
575C4⋊C4—1224no—
576C4⋊C4—1224no—
577C4⋊C4—1222no—
578C4⋊C4—1221no—
579C4⋊C4—1222no—
580C4⋊C4—1222no—
581C4⋊C4—1222no—
582C4⋊C4—1222no—
583C4⋊C4—1221no—
584C4⋊C4C23×D6114yes—
585C4⋊C4C23×D6112yes—
586C4⋊C4C23×D6112yes—
587C22⋊C4—22no—
588C22⋊C4—612no—
589C22⋊C4—612no—
590C22⋊C4—612no—
591C22⋊C4—1224no—
592C22⋊C4—1224no—
593C22⋊C4—1222no—
594C42—22no—
595C42—612no—
596C42—612no—
597C42—612no—
598C42—1224no—
599C42—1224no—
600C42—1222no—
601C42D4×D6111yescharacteristic
602C42D4×D6111yescharacteristic
603C42C23×D6111yescharacteristic
604C42C23×D6111yescharacteristic
605C23—612no—
606C23C2×D4×D6111yescharacteristic
607Q8—42no—
608Q8—42no—
609Q8—44no—
610Q8—82no—
611Q8—614no—
612Q8—4882no—
613Q8—4884no—
614Q8—4884no—
615Q8—2442no—
616Q8—2444no—
617Q8—2442no—
618D4—618no—
619D4—614no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—82no—
626C2×C4—612no—
627C2×C4—618no—
628C2×C4—612no—
629C2×C4—612no—
630C2×C4—1224no—
631C2×C4—1224no—
632C2×C4—1222no—
633C2×C4—1224no—
634C2×C4—1222no—
635C2×C4—4884no—
636C2×C4—4884no—
637C2×C4—4882no—
638C2×C4C2×D4×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4C2×D4×D6111yescharacteristic
642C2×C4C2×D4×D6111yescharacteristic
643C2×C4C2×Q8×D6114yes—
644C2×C4C24×D6111yescharacteristic
645C8—614no—
646C8—614no—
647C8C2×Q8×D6118yes—
648C22—21no—
649C22—21no—
650C22—614no—
651C22—1222no—
652C22—1222no—
653C22C4⋊Q8×D6112yes—
654C22C22×D4×D6111yescharacteristic
655C4—21no—
656C4—21no—
657C4—42no—
658C4—44no—
659C4—42no—
660C4—614no—
661C4—1222no—
662C4—1222no—
663C4—2444no—
664C4—2442no—
665C4—2442no—
666C4C4⋊Q8×D6112yes—
667C4#1684057112yes—
668C4#1684057212yes—
669C4C22×Q8×D6112yes—
670C2—22no—
671C2—1224no—
672C2C2×C4⋊Q8×D6111yescharacteristic
673C2C2×#1684057111yescharacteristic
674C2C2×#1684057211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3165C2
socleC22×C61C23×D4
prime radicalC2×#11758222C23
hypercenter#3165D61
hypocenterC61#60665
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
3. #21456365C2semidirect
4. #21456362C2semidirect
5. #21456360C2semidirect
6. #21456349C2semidirect
7. #21456346C2semidirect
8. #21456330C2semidirect
9. #21456329C2semidirect
10. C2×#16840158C2semidirect
11. #498×D61C2semidirect
14. C61×#3165C2semidirect
79. #14519785Q8semidirect
210. M16×C61C2×Q8semidirect
262. C2×#92418C2×Q8semidirect
265. #11758189C2×Q8semidirect
266. C8×D61C2×Q8semidirect
338. #3165D61semidirect
412. C61#60665semidirect
549. M16Q8×D61semidirect

Maximal subgroups

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

subgroupindex
#214596352
#214596312
#214563652
#214563622
#214563602
#214563492
#214563462
#214563302
#214563292
C2×#168401582
#498×D612
#214544132
C4×#145197852
C61×#31652
#6066561