Finite Group Databasebeta

← #29647726

Subgroups — #29647726

Order 15616.

1070 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459615C211yesmaximal, characteristic
2#21454841C211yesmaximal, characteristic
3#21448723C211yesmaximal, characteristic
4#21444390C211yesmaximal, characteristic, complemented
5#21444389C212yesmaximal, complemented
6#21444381C212yesmaximal, complemented
7#21444316C212yesmaximal, complemented
8C2×#16839500C211yesmaximal, characteristic, complemented
9C2×#16839499C211yesmaximal, characteristic, complemented
10#328×D61C212yesmaximal, complemented
11C61×#1520C211yesmaximal, characteristic, complemented
12#16840470C412yescomplemented
13#16840467—22no—
14#16840467C412yescomplemented
15#16840466—22no—
16Q8×C22⋊D61C412yescomplemented
17#16840425C412yescomplemented
18#16840391—21no—
19#16840388—21no—
20#16840387—21no—
21#16840314C2212yes—
22C2×#14519756C2211yescharacteristic
23C22⋊Q8×D61C2212yes—
24#16840235—21no—
25C2×C4×C22⋊D61C2211yescharacteristic
26C2×#14519737—21no—
27C8×C22⋊D61—22no—
28#16839825—22no—
29#16839812C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C22⋊C8×D61C2212yes—
32#16839657—21no—
33#16839634—21no—
34#16839633—21no—
35#16839626C2211yescharacteristic
36#16839621C2211yescharacteristic
37#16839613C2211yescharacteristic
38C2×#14519684C2211yescharacteristic
39#95×D61C2211yescharacteristic
40#16839500C2212yes—
41#16839500C2212yes—
42#16839499C2212yes—
43#16839499C2212yes—
44#16839498C2212yes—
45#16839498C2212yes—
46C2×C22⋊Q8×C61C2211yescharacteristic
47C2×C22⋊C8×C61C2211yescharacteristic
48C61×#380C2211yescharacteristic
49C61×#328C2212yes—
50C61×#328C2212yes—
51C22×Dic122C2×C411yescharacteristic, complemented
52C2×Q8×D61C2×C412yescomplemented
53C22×C22⋊D61D411yescharacteristic
54C23×Dic61—21no—
55C22×C4×D61—21no—
56C22×C4×D61C2311yescharacteristic
57C22×#92418—21no—
58C2×C8×D61—22no—
59#14519776—22no—
60#14519776C2×C412yescomplemented
61#14519775—22no—
62#14519775C2×C412yescomplemented
63#14519767—22no—
64#14519767—22no—
65#14519767—22no—
66#14519757—22no—
67#14519757C2×C412yescomplemented
68#14519756—21no—
69#14519756—21no—
70#14519756—22no—
71#14519756—22no—
72#14519756C2×C412yescomplemented
73#14519756C2312yes—
74#14519755—22no—
75#14519755—22no—
76#14519755—22no—
77#14519754—22no—
78#14519754—22no—
79#14519754—22no—
80#14519754—22no—
81#14519753—22no—
82#14519753C2×C412yescomplemented
83Q8×Dic61C2×C412yescomplemented
84#14519742—22no—
85#14519742C2×C412yescomplemented
86C4×C22⋊D61—22no—
87C4×C22⋊D61—22no—
88C4×C22⋊D61C2×C412yes—
89C4×C22⋊D61C2×C412yes—
90C4×C22⋊D61D412yes—
91C4×C22⋊D61D412yes—
92#14519737—22no—
93#14519737—22no—
94#14519737—44no—
95#14519733—22no—
96C2×C4⋊Dic61C2×C411yescharacteristic, complemented
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196D411yescharacteristic
102C4⋊C4×D61—22no—
103C4⋊C4×D61C2×C412yescomplemented
104C2×C22⋊Dic61—21no—
105C2×C22⋊Dic61D411yescharacteristic
106C2×#11758198—21no—
107C2×#11758198D411yescharacteristic
108C22⋊C4×D61—22no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61C2311yescharacteristic
111#14519706—22no—
112C22⋊#92418—21no—
113C22⋊#92418C2312yes—
114#14519689—22no—
115C8×Dic61—22no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684C2311yescharacteristic
120#14519684C2311yescharacteristic
121C22×Q8×C61C2×C411yescharacteristic, complemented
122C23×C4×C61C2311yescharacteristic
123C22×C8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61C2×C412yescomplemented
127C22⋊Q8×C61C2312yes—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
131C2×C22⋊C4×C61—21no—
132C2×C22⋊C4×C61—21no—
133C22⋊C8×C61—21no—
134C22⋊C8×C61C2312yes—
135#95×C61C2311yescharacteristic
136#95×C61C2311yescharacteristic
137C24×C61C2×D411yescharacteristic
138C23×D61C2×D411yescharacteristic
139C2×Q8×C61—21no—
140C2×Q8×C61—81no—
141C2×Q8×C61C22×C412yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—81no—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61C22⋊C412yes—
154C2×C22⋊D61C22⋊C412yes—
155C2×C22⋊D61SD1614yes—
156C2×C22⋊D61Q1614yes—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—81no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167Q8×D61—82no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—81no—
173C2×C4×D61—81no—
174C2×C4×D61C22×C412yes—
175C2×C4×D61C2×D412yes—
176C2×Dic122—21no—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122—81no—
180C2×Dic122C22×C412yes—
181C2×C8×C61—22no—
182C2×C8×C61—81no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—81no—
187C4⋊C4×C61C22×C412yes—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—42no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61C22⋊C412yes—
194C22⋊Dic61C22⋊C412yes—
195#11758198—22no—
196#11758198—22no—
197#11758198—42no—
198#11758198C22⋊C412yes—
199#11758198C22⋊C412yes—
200C4⋊Dic61—21no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61C22×C412yes—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—42no—
213#11758196—42no—
214#11758196—81no—
215#11758196C22⋊C412yes—
216#11758196C22⋊C412yes—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—42no—
220C4×Dic61C22×C412yes—
221C4×Dic61C2×D412yes—
222C2×#92418—22no—
223C2×#92418—81no—
224C8×D61—82no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61C2×C22⋊C411yescharacteristic
228C23×C61C2×C22⋊C411yescharacteristic
229C23×C61C2×SD1612yes—
230C23×C61C2×Q1612yes—
231C23×C61C22×D411yescharacteristic
232C22×D61—41no—
233C22×D61—41no—
234C22×D61C2×C22⋊C412yes—
235C22×D61C2×SD1612yes—
236C22×D61C2×Q1612yes—
237Q8×C61—42no—
238Q8×C61—81no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—81no—
254C2×C4×C61—81no—
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C22⋊D61#10318yes—
260C22⋊D61#10318yes—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—81no—
272C2×Dic61—81no—
273C2×Dic61—84no—
274C2×Dic61C2×C22⋊C412yes—
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C2×C22⋊C412yes—
277C2×Dic61C2×SD1612yes—
278C2×Dic61C2×Q1612yes—
279C4×D61—42no—
280C4×D61—42no—
281C4×D61—82no—
282C4×D61—82no—
283C4×D61—84no—
284Dic122—42no—
285Dic122—82no—
286Dic122—82no—
287Dic122—81no—
288C8×C61—42no—
289#92418—42no—
290#17614—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61C2×#10312yes—
300C22×C61C2×#10312yes—
301C22×C61C22×C22⋊C411yescharacteristic
302C22×C61C22×SD1611yescharacteristic
303C22×C61C22×Q1611yescharacteristic
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61C2×#10314yes—
309C4×C61—21no—
310C4×C61—21no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—42no—
314C4×C61—42no—
315C4×C61—81no—
316C4×C61—81no—
317Dic61—22no—
318Dic61—22no—
319Dic61—42no—
320Dic61—42no—
321Dic61—42no—
322Dic61—81no—
323Dic61C2×#10314yes—
324#3501—611no—
325#2972—611no—
326#2349—611no—
327#1520—611no—
328#1520—612no—
329#1520D6111yescharacteristic, complemented
330#1518—612no—
331#1503—612no—
332C2×#328—611no—
333C2×#328—612no—
334C2×#328—611no—
335C2×C61—21no—
336C2×C61—21no—
337C2×C61—22no—
338C2×C61—22no—
339C2×C61#288311yescharacteristic
340C2×C61C22×#10311yescharacteristic
341C2×C61#290111yescharacteristic
342D61—24no—
343#554—612no—
344#554—612no—
345#554—1222no—
346#554—1222no—
347D4×Q8—612no—
348D4×Q8—612no—
349#544—1221no—
350#543—1221no—
351#543—1221no—
352#533—612no—
353C2×C22⋊Q8—611no—
354C2×C22⋊Q8—612no—
355C2×C22⋊Q8C2×D6111yescharacteristic
356#518—1221no—
357C2×C4×D4—611no—
358C2×C4×D4—1221no—
359C8×D4—1222no—
360C8×D4—1222no—
361#431—612no—
362C2×C22⋊C8—611no—
363C2×C22⋊C8—612no—
364C2×C22⋊C8C2×D6111yescharacteristic
365#389—1221no—
366#385—1221no—
367#385—1221no—
368#382—611no—
369#380—611no—
370#380C2×D6111yescharacteristic
371C4×C4⋊C4—611no—
372C2×#95—611no—
373C2×#95—611no—
374#328—612no—
375#328—612no—
376#328—612no—
377#328—612no—
378#328—612no—
379#328—612no—
380#328C2×D6112yes—
381#328C2×D6112yes—
382C61#1761411yescharacteristic, complemented, Sylow
383C22×Q8—612no—
384C22×Q8—611no—
385C22×Q8C4×D6111yescharacteristic, complemented
386C22×D4—611no—
387C23×C4—611no—
388C23×C4—1221no—
389C23×C4—1221no—
390C23×C4C22×D6111yescharacteristic
391C22×C8—21no—
392C22×C8—1222no—
393C22×C8—1221no—
394C4⋊Q8—612no—
395C4⋊Q8—612no—
396C4⋊Q8—1222no—
397C4⋊Q8—1222no—
398#125—1222no—
399#125—1222no—
400#125—1222no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8—612no—
404C22⋊Q8—612no—
405C22⋊Q8—612no—
406C22⋊Q8—612no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409C22⋊Q8—1222no—
410C22⋊Q8—1222no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1222no—
414C22⋊Q8—1222no—
415C22⋊Q8—1222no—
416C22⋊Q8—1222no—
417C22⋊Q8—1222no—
418C22⋊Q8—1221no—
419C22⋊Q8—1221no—
420C22⋊Q8C4×D6112yescomplemented
421C22⋊Q8C22×D6112yes—
422C4×Q8—612no—
423C4×Q8—612no—
424C4×Q8—1222no—
425C4×D4—612no—
426C4×D4—612no—
427C4×D4—612no—
428C4×D4—612no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—2444no—
434#117—1222no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—612no—
440C2×C4⋊C4—1222no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4C4×D6111yescharacteristic, complemented
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—611no—
449C2×C22⋊C4—611no—
450C2×C22⋊C4—1222no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C42—611no—
454C2×C42—1221no—
455C4⋊C8—1222no—
456C22⋊C8—21no—
457C22⋊C8—612no—
458C22⋊C8—1222no—
459C22⋊C8—1221no—
460C22⋊C8C22×D6112yes—
461C4×C8—1222no—
462#95—611no—
463#95—611no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95C22×D6111yescharacteristic
468#95C22×D6111yescharacteristic
469C24—611no—
470C24D4×D6111yescharacteristic
471C2×Q8—21no—
472C2×Q8—81no—
473C2×Q8—612no—
474C2×Q8—1222no—
475C2×Q8—1222no—
476C2×Q8—1221no—
477C2×Q8—4881no—
478C2×Q8—4882no—
479C2×Q8C2×C4×D6112yes—
480C2×D4—612no—
481C2×D4—612no—
482C2×D4—614no—
483C2×D4—614no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—81no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—612no—
493C22×C4—612no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—4881no—
501C22×C4—4881no—
502C22×C4—4881no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4C2×C4×D6111yescharacteristic
508C22×C4C2×C4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4D4×D6111yescharacteristic
512C22×C4C23×D6111yescharacteristic
513C2×C8—22no—
514C2×C8—81no—
515C2×C8—1222no—
516C2×C8—4882no—
517C2×C8—4881no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—81no—
522C4⋊C4—612no—
523C4⋊C4—612no—
524C4⋊C4—612no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1221no—
535C4⋊C4—1222no—
536C4⋊C4—4881no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4C2×C4×D6112yes—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—42no—
543C22⋊C4—612no—
544C22⋊C4—612no—
545C22⋊C4—612no—
546C22⋊C4—612no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—1222no—
551C22⋊C4—2442no—
552C42—612no—
553C42—612no—
554C42—1222no—
555C42—1222no—
556C42—2442no—
557C23—41no—
558C23—41no—
559C23—612no—
560C23—612no—
561C23—612no—
562C23—2441no—
563C23—2441no—
564C23C22⋊C4×D6111yescharacteristic
565C23C22⋊C4×D6111yescharacteristic
566C23SD16×D6112yes—
567C23Q16×D6112yes—
568C23C2×D4×D6111yescharacteristic
569Q8—42no—
570Q8—81no—
571Q8—4881no—
572Q8—4882no—
573Q8—4882no—
574Q8—2442no—
575D4—618no—
576D4—618no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—612no—
597C2×C4—612no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—4882no—
602C2×C4—4882no—
603C2×C4—4884no—
604C2×C4—4884no—
605C2×C4—4881no—
606C2×C4—4881no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4C22⋊C4×D6111yescharacteristic
617C2×C4C22⋊C4×D6111yescharacteristic
618C2×C4C22×C4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C8—42no—
621C8—2442no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—42no—
629C22—42no—
630C22—614no—
631C22—1222no—
632C22—1222no—
633C22—2442no—
634C22—2442no—
635C22#103×D6112yes—
636C22#103×D6112yes—
637C22C2×C22⋊C4×D6111yescharacteristic
638C22C2×SD16×D6111yescharacteristic
639C22C2×Q16×D6111yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—42no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—614no—
649C4—1222no—
650C4—1222no—
651C4—4881no—
652C4—2442no—
653C4—2442no—
654C4—2442no—
655C2—21no—
656C2—21no—
657C2—22no—
658C2—22no—
659C2—1224no—
660C2#2145427711yescharacteristic
661C2C2×#103×D6111yescharacteristic
662C2#2145436511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1520C2
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61D4
cosocleC22×C4×C61C24
hypercenter#1520D61
hypocenterC61#17614
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
4. #21444390C2semidirect
5. #21444389C2semidirect
6. #21444381C2semidirect
7. #21444316C2semidirect
8. C2×#16839500C2semidirect
9. C2×#16839499C2semidirect
10. #328×D61C2semidirect
11. C61×#1520C2semidirect
12. #16840470C4semidirect
14. #16840467C4semidirect
16. Q8×C22⋊D61C4semidirect
17. #16840425C4semidirect
51. C22×Dic122C2×C4semidirect
52. C2×Q8×D61C2×C4semidirect
60. #14519776C2×C4semidirect
62. #14519775C2×C4semidirect
67. #14519757C2×C4semidirect
72. #14519756C2×C4semidirect
82. #14519753C2×C4semidirect
83. Q8×Dic61C2×C4semidirect
85. #14519742C2×C4semidirect
96. C2×C4⋊Dic61C2×C4semidirect
103. C4⋊C4×D61C2×C4semidirect
121. C22×Q8×C61C2×C4semidirect
126. C22⋊Q8×C61C2×C4semidirect
130. C2×C4⋊C4×C61C2×C4semidirect
329. #1520D61semidirect
382. C61#17614semidirect
385. C22×Q8C4×D61semidirect
420. C22⋊Q8C4×D61semidirect
445. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214596152
#214548412
#214487232
#214443902
#214443892
#214443812
#214443162
C2×#168395002
C2×#168394992
#328×D612
C61×#15202
#1761461