Finite Group Databasebeta

← #29736084

Subgroups — #29736084

Order 15616.

1040 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840471C211yesmaximal, characteristic
2C2×#16840170C211yesmaximal, characteristic, complemented
3C2×#16840060C211yesmaximal, characteristic
4#21451206C211yesmaximal, characteristic, complemented
5#21447160C211yesmaximal, characteristic, complemented
6#21447158C211yesmaximal, characteristic, complemented
7#21447157C211yesmaximal, characteristic, complemented
8#21447155C211yesmaximal, characteristic, complemented
9#21446769C211yesmaximal, characteristic, complemented
10#21446757C211yesmaximal, characteristic, complemented
11#21446082C211yesmaximal, characteristic
12#21446080C211yesmaximal, characteristic
13#21446064C211yesmaximal, characteristic
14#21445646C211yesmaximal, characteristic
15C61×#1862C211yesmaximal, characteristic
16#16840471—24no—
17#16840471—24no—
18C2×#14519776C2211yescharacteristic
19C2×#14519769C2211yescharacteristic
20C2×#14519768C2211yescharacteristic
21C2×#14519768C2211yescharacteristic
22C2×#14519756—21no—
23C2×#14519754—21no—
24C2×#14519754—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×D4×Dic61C2211yescharacteristic
27C2×D4×Dic61C2211yescharacteristic
28C22×#11758196—21no—
29#16840170—24no—
30#16840060—24no—
31C2×#14519711C2211yescharacteristic
32C2×#14519710C2211yescharacteristic
33C2×#14519709C2211yescharacteristic
34C2×#14519708C2211yescharacteristic
35C2×#14519708C2211yescharacteristic
36C2×#14519706C2211yescharacteristic
37C2×#14519698C2211yescharacteristic
38C2×#14519698C2211yescharacteristic
39C2×#14519687C2211yescharacteristic
40#16839695—21no—
41#16839683—21no—
42#16839661C2211yescharacteristic
43#16839660C2211yescharacteristic
44#16839659—21no—
45#16839658—21no—
46#16839658C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C61×#389C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122D411yescharacteristic
65C23×Dic61—21no—
66C23×Dic61—21no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519776—22no—
70#14519776D414yes—
71#14519769—22no—
72#14519769—22no—
73#14519768—22no—
74#14519768—22no—
75#14519768—22no—
76#14519768—22no—
77#14519756—28no—
78#14519754—28no—
79#14519754—28no—
80C4×Dic122—22no—
81C4×Dic122—22no—
82D4×Dic61—22no—
83D4×Dic61—22no—
84D4×Dic61—22no—
85D4×Dic61D414yes—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61D411yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×#11758196—28no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196D411yescharacteristic
103C2×#11758196D411yescharacteristic
104C2×C22⋊Dic61—21no—
105C2×C22⋊Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109#14519711—22no—
110#14519710—22no—
111#14519709—22no—
112#14519708—22no—
113#14519708—22no—
114#14519706—22no—
115#14519698—22no—
116#14519698Q814yescomplemented
117#14519687—22no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122C22×D4×C61C2311yescharacteristic
123C23×C4×C61—21no—
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C4×D4×C61—22no—
127C4×D4×C61—22no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C22⋊C4×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#107×C61—22no—
135#102×C61—22no—
136#102×C61Q814yescomplemented
137#95×C61—21no—
138C24×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61C2×D412yes—
143C2×D4×C61C2×Q812yes—
144C2×D4×C61C2×Q812yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—24no—
148C22×C4×C61—21no—
149C22×C4×C61—81no—
150C22×C4×C61—81no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—24no—
158C22×Dic61—24no—
159C22×Dic61—21no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—81no—
163C22×Dic61—81no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×Dic122—22no—
169C2×Dic122—24no—
170C2×Dic122—41no—
171C2×Dic122—41no—
172C2×Dic122—81no—
173C2×Dic122—81no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C2×Q812yes—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—81no—
185C4⋊C4×C61C2×D412yes—
186C4⋊C4×C61C2×Q812yes—
187C22⋊C4×C61—24no—
188C42×C61—21no—
189C42×C61—21no—
190C22⋊Dic61—24no—
191C22⋊Dic61—24no—
192C4⋊Dic61—24no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4⋊Dic61C2×D412yes—
202C4⋊Dic61C2×D412yes—
203C4⋊Dic61C2×Q812yes—
204#11758196—24no—
205#11758196—24no—
206#11758196—24no—
207#11758196—22no—
208#11758196—24no—
209#11758196—22no—
210#11758196—42no—
211#11758196—42no—
212#11758196—81no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—81no—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D412yes—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418C2×Q812yes—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61C22×D411yescharacteristic
232D4×C61—22no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61C4⋊Q814yes—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
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—84no—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×Q811yescharacteristic
255C2×C4×C61C22×Q811yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—48no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—84no—
282C2×Dic61#12012yes—
283C2×Dic61#12012yes—
284Dic122—44no—
285Dic122—84no—
286C8×C61—22no—
287C8×C61—41no—
288#92418—22no—
289#92418—41no—
290#25845—611nomaximal, Sylow
291C22×C61—24no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61C2×#12011yescharacteristic
299C22×C61D4×Q811yescharacteristic
300C22×C61D4×Q811yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57511yescharacteristic
303C22×C61#57611yescharacteristic
304C22×C61#57711yescharacteristic
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—81no—
310C4×C61C2×C4⋊Q811yescharacteristic
311C4×C61#53611yescharacteristic
312C4×C61D4×Q811yescharacteristic
313C4×C61D4×Q811yescharacteristic
314Dic61—22no—
315Dic61—22no—
316Dic61—42no—
317Dic61—41no—
318Dic61—42no—
319Dic61—42no—
320Dic61—41no—
321Dic61—81no—
322C2×#554—611no—
323C2×#501—611no—
324C2×#478—611no—
325#2584—611no—
326#2059—611no—
327#2059—611no—
328#2059—611no—
329#2059—611no—
330#1988—611no—
331#1985—611no—
332#1864—611no—
333#1864—611no—
334#1862—611no—
335#1862D6111yescharacteristic
336#1762—611no—
337C2×C61—44no—
338C2×C61#243411yescharacteristic
339C2×C61#300511yescharacteristic
340C2×C61#301211yescharacteristic
341C2×C61#338811yescharacteristic
342C2×C61#338811yescharacteristic
343C2×C61#338911yescharacteristic
344C2×C61#338911yescharacteristic
345#554—1224no—
346#554—1224no—
347C2×C4⋊Q8—611no—
348C2×#125—611no—
349C2×#125—611no—
350C2×#125—611no—
351C2×C22⋊Q8—1221no—
352C2×C22⋊Q8—1221no—
353C2×C22⋊Q8—1221no—
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356C2×C4×D4—611no—
357C2×C4×D4C2×D6111yescharacteristic
358C22×C4⋊C4—1221no—
359#501—1224no—
360#478—1224no—
361C2×#107—611no—
362C2×#107—611no—
363C2×#107C2×D6111yescharacteristic
364C2×#106—611no—
365C2×#106—611no—
366C2×#106—611no—
367C2×C4⋊C8—611no—
368C2×#102—611no—
369C2×#102—611no—
370C2×#102C2×D6111yescharacteristic
371C2×#102C2×D6111yescharacteristic
372C2×#97—611no—
373#398—1221no—
374#395—1221no—
375#389—611no—
376#389—611no—
377#389—611no—
378#389—1221no—
379#389—1221no—
380#389C2×D6111yescharacteristic
381C4×C4⋊C4—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340C2×D6111yescharacteristic
388#336—611no—
389#336—611no—
390#336—611no—
391#336C2×D6111yescharacteristic
392C61#2584511yescharacteristic, complemented, Sylow
393C22×Q8—611no—
394C22×Q8—1221no—
395C22×D4C22×D6111yescharacteristic
396C23×C4—21no—
397C23×C4—1221no—
398C23×C4—1221no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8C22×D6111yescharacteristic
402C22×C8C22×D6111yescharacteristic
403C4⋊Q8—614no—
404C4⋊Q8—1222no—
405#125—1222no—
406#125—1222no—
407#125—1222no—
408#125—1222no—
409#125—1222no—
410#125—1222no—
411C22⋊Q8—1228no—
412C22⋊Q8—1228no—
413C22⋊Q8—1228no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—614no—
419C4×D4—1222no—
420C4×D4—1222no—
421C4×D4—1222no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1228no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451#107—22no—
452#107—1222no—
453#107—1222no—
454#106—1222no—
455#106—1222no—
456#106—1222no—
457C4⋊C8—1222no—
458#102—22no—
459#102—614no—
460#102—1222no—
461#102Dic12214yescomplemented
462#97—1222no—
463#95—21no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468C24—21no—
469C2×Q8—1224no—
470C2×Q8—1222no—
471C2×Q8—4881no—
472C2×Q8—4881no—
473C2×Q8—2441no—
474C2×Q8—2441no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C2×Dic12212yes—
479C2×D4D4×D6112yes—
480C2×D4Q8×D6112yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—21no—
485C22×C4—81no—
486C22×C4—81no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—1224no—
492C22×C4—1224no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—4881no—
498C22×C4—4881no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—612no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8C2×Dic12212yes—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—81no—
521C4⋊C4—612no—
522C4⋊C4—612no—
523C4⋊C4—612no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1224no—
528C4⋊C4—1224no—
529C4⋊C4—1224no—
530C4⋊C4—1224no—
531C4⋊C4—1224no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—4881no—
538C4⋊C4—2442no—
539C4⋊C4—2442no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C4⋊C4C2×Dic12212yes—
543C4⋊C4D4×D6112yes—
544C22⋊C4—24no—
545C22⋊C4—1224no—
546C22⋊C4—1224no—
547C42—21no—
548C42—21no—
549C42—612no—
550C42—612no—
551C42—1221no—
552C42—1221no—
553C42—4881no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C2×D4×D6111yescharacteristic
562Q8—4884no—
563Q8—2444no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4#1451977614yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1222no—
599C2×C4—4884no—
600C2×C4—2448no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C2×Q8×D6111yescharacteristic
613C2×C4C22×Dic12211yescharacteristic
614C2×C4#1451982411yescharacteristic
615C2×C4#1451982411yescharacteristic
616C8—22no—
617C8—41no—
618C8—1222no—
619C8—2441no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22#120×D6111yescharacteristic
628C22#1684042511yescharacteristic
629C22D4×Dic12211yescharacteristic
630C22#1684055911yescharacteristic
631C22#1684056111yescharacteristic
632C22#1684056311yescharacteristic
633C22#1684057211yescharacteristic
634C4—41no—
635C4—42no—
636C4—42no—
637C4—41no—
638C4—81no—
639C4—1222no—
640C4—1222no—
641C4—4881no—
642C4—2441no—
643C4—2442no—
644C4—2442no—
645C4—2441no—
646C4—2442no—
647C4C2×#1451977611yescharacteristic
648C4#1684033411yescharacteristic
649C4#1684042511yescharacteristic
650C4D4×Dic12211yescharacteristic
651C2—44no—
652C2#2144946111yescharacteristic
653C2#2145506211yescharacteristic
654C2#2145511811yescharacteristic
655C2#2145876511yescharacteristic
656C2#2145876811yescharacteristic
657C2#2145877211yescharacteristic
658C2#2145877511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1862C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1862D61
hypocenterC61#25845
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. C2×#16840170C2semidirect
4. #21451206C2semidirect
5. #21447160C2semidirect
6. #21447158C2semidirect
7. #21447157C2semidirect
8. #21447155C2semidirect
9. #21446769C2semidirect
10. #21446757C2semidirect
116. #14519698Q8semidirect
136. #102×C61Q8semidirect
392. C61#25845semidirect
461. #102Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168404712
C2×#168401702
C2×#168400602
#214512062
#214471602
#214471582
#214471572
#214471552
#214467692
#214467572
#214460822
#214460802
#214460642
#214456462
C61×#18622
#2584561