Finite Group Databasebeta

← #29721404

Subgroups — #29721404

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840053C211yesmaximal, characteristic, complemented
2#21451640C211yesmaximal, characteristic, complemented
3#21448512C211yesmaximal, characteristic
4#21447252C211yesmaximal, characteristic, complemented
5#21447252C211yesmaximal, characteristic, complemented
6#21447009C211yesmaximal, characteristic
7#21447002C211yesmaximal, characteristic
8#21446770C211yesmaximal, characteristic, complemented
9#21446589C211yesmaximal, characteristic, complemented
10#21446557C211yesmaximal, characteristic, complemented
11#21446547C211yesmaximal, characteristic, complemented
12#21446516C211yesmaximal, characteristic, complemented
13#21446502C211yesmaximal, characteristic, complemented
14#21445830C211yesmaximal, characteristic, complemented
15C2×C61×#433C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×#14519751C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19#16840053—24no—
20C2×#14519711C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28#16839701—21no—
29#16839698—21no—
30#16839682—21no—
31#16839674—21no—
32#16839664C2211yescharacteristic
33#16839662—21no—
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839652—21no—
38#16839638—21no—
39#16839633—21no—
40#16839628C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839622C2211yescharacteristic
43#16839622C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C61×#433—24no—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244C2311yescharacteristic
63C22×C22⋊D61—21no—
64C22×C4×D61—21no—
65#14519777—22no—
66#14519777—22no—
67#14519751—22no—
68#14519751—22no—
69C4×D244—22no—
70C4×D244—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—21no—
90#14519711—22no—
91#14519709—22no—
92#14519709—22no—
93#14519709—22no—
94#14519699—22no—
95#14519699—22no—
96#14519699—22no—
97#14519699—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107C23×C4×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112#117×C61—22no—
113#117×C61—22no—
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C22⋊C4×C61D411yescharacteristic
117C2×C22⋊C4×C61D411yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C4⋊C8×C61—22no—
121C4⋊C8×C61D414yescomplemented
122C22⋊C8×C61—22no—
123C22⋊C8×C61—22no—
124#97×C61—22no—
125C4×C8×C61—22no—
126C24×C61C2×D411yescharacteristic
127C23×D61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—24no—
141C2×C22⋊D61—81no—
142C2×C22⋊D61—81no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244#4112yes—
156C2×C4×D61—42no—
157C2×C4×D61—81no—
158C2×C4×D61—81no—
159C2×Dic122—42no—
160C2×Dic122—81no—
161C2×Dic122—81no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D412yes—
173C2×C8×C61C2×D412yes—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61C2×D412yes—
181C42×C61#4112yes—
182C22⋊Dic61—81no—
183C22⋊Dic61—81no—
184C22⋊Dic61—81no—
185#11758198—24no—
186#11758198—42no—
187#11758198—81no—
188#11758198—81no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—42no—
203C4⋊Dic61#4112yes—
204#11758196—81no—
205#11758196—81no—
206#11758196—81no—
207C4×Dic61—81no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61#12112yes—
241C2×C4×C61#12112yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×C4×C612−1+411yescharacteristic
248C22⋊D61—84no—
249C2×Dic61—22no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270D244—22no—
271D244—22no—
272D244—41no—
273D244—41no—
274C4×D61—84no—
275Dic122—84no—
276C8×C61—22no—
277C8×C61—22no—
278C8×C61—41no—
279C8×C61—41no—
280#24716—611nomaximal, Sylow
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61C2×#12111yescharacteristic
289C22×C61#52511yescharacteristic
290C22×C61#53611yescharacteristic
291C22×C61#57611yescharacteristic
292C22×C61#57711yescharacteristic
293C22×C61#57711yescharacteristic
294C22×C61#57711yescharacteristic
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—81no—
302C4×C61—22no—
303C4×C61—22no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61#54711yescharacteristic
308C4×C61#54811yescharacteristic
309C4×C61#55211yescharacteristic
310C4×C61#55611yescharacteristic
311Dic61—42no—
312Dic61—81no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317Dic61—81no—
318C2×#476—611no—
319C2×#433D6111yescharacteristic, complemented
320#2628—611no—
321#2324—611no—
322#2077—611no—
323#2077—611no—
324#2034—611no—
325#2033—611no—
326#1988—611no—
327#1949—611no—
328#1943—611no—
329#1941—611no—
330#1936—611no—
331#1934—611no—
332#1804—611no—
333C2×C61—81no—
334C2×C61#241511yescharacteristic
335C2×C61#305711yescharacteristic
336C2×C61#306211yescharacteristic
337C2×C61#311811yescharacteristic
338C2×C61#312511yescharacteristic
339C2×C61#324111yescharacteristic
340C2×C61#324411yescharacteristic
341D61—82no—
342C2×C4⋊Q8—611no—
343C2×#121—611no—
344C2×C4×D4—611no—
345C2×#117C2×D6111yescharacteristic
346#476—1224no—
347#433—24no—
348C2×#107—611no—
349C2×#106—611no—
350C2×#106—611no—
351C2×#106—611no—
352C2×C4⋊C8C2×D6111yescharacteristic
353C2×C4⋊C8C2×D6111yescharacteristic
354C2×#102—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×C22⋊C8C2×D6111yescharacteristic
359C2×C22⋊C8C2×D6111yescharacteristic
360C2×#97C2×D6111yescharacteristic
361C2×C4×C8C2×D6111yescharacteristic
362#400—1221no—
363#399—1221no—
364#395—1221no—
365#393—1221no—
366#390—611no—
367#390—1221no—
368#389—611no—
369#389—611no—
370#389—611no—
371#388—1221no—
372#386—1221no—
373#385—1221no—
374#382—611no—
375#380—611no—
376#380—611no—
377#380—611no—
378C4×C4⋊C4—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387C61#2471611yescharacteristic, complemented, Sylow
388C22×Q8—1221no—
389C22×D4—611no—
390C22×D4—1221no—
391C23×C4—1221no—
392C23×C4C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394C22×C8C22×D6111yescharacteristic
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397C4⋊Q8—1222no—
398C4⋊Q8—1222no—
399#121—1222no—
400#121—1222no—
401C4×D4—1222no—
402C4×D4—1222no—
403#117—22no—
404#117—22no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4D24411yescharacteristic
417C2×C4⋊C4D24411yescharacteristic
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4D24411yescharacteristic
426C2×C22⋊C4D24411yescharacteristic
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430#107—1222no—
431#106—1222no—
432#106—1222no—
433#106—1222no—
434C4⋊C8—22no—
435C4⋊C8C22⋊D6114yescomplemented
436#102—1222no—
437#102—1222no—
438#102—1222no—
439#102—1222no—
440C22⋊C8—22no—
441C22⋊C8—22no—
442#97—22no—
443C4×C8—22no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453C24—1221no—
454C24C2×D24411yescharacteristic
455C2×Q8—4881no—
456C2×Q8—4881no—
457C2×Q8—2442no—
458C2×D4—612no—
459C2×D4—1224no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—4881no—
466C2×D4—4881no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—4881no—
481C22×C4—4881no—
482C22×C4—2442no—
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×D24411yescharacteristic
485C22×C4C2×D24411yescharacteristic
486C22×C4C2×D24411yescharacteristic
487C22×C4C2×D24411yescharacteristic
488C22×C4C23×D6111yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8C2×C22⋊D6112yes—
500C2×C8C2×C22⋊D6112yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—612no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—4881no—
518C4⋊C4—4881no—
519C4⋊C4—4881no—
520C4⋊C4—2442no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—1224no—
524C22⋊C4—4881no—
525C22⋊C4—4881no—
526C22⋊C4—4881no—
527C22⋊C4—4881no—
528C22⋊C4—4881no—
529C22⋊C4—2442no—
530C42—21no—
531C42—21no—
532C42—4881no—
533C42#1175822212yes—
534C42C2×C22⋊D6112yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—1222no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23—2441no—
549C23C22×D24411yescharacteristic
550Q8—4884no—
551D4—1222no—
552D4—1222no—
553D4—4884no—
554D4—2441no—
555D4—2441no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1222no—
574C2×C4—4884no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4#1451975112yes—
596C2×C4#1451975112yes—
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4#41×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4C2×#1175822211yescharacteristic
601C2×C4#1451982411yescharacteristic
602C2×C4#1451982411yescharacteristic
603C8—22no—
604C8—22no—
605C8—41no—
606C8—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—4881no—
615C22—4881no—
616C22—4881no—
617C22—4881no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22C2×#1451975111yescharacteristic
622C22#1684027411yescharacteristic
623C22#1684033811yescharacteristic
624C22#1684056711yescharacteristic
625C22#1684057411yescharacteristic
626C22#1684057411yescharacteristic
627C22#1684057411yescharacteristic
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—4881no—
634C4—4881no—
635C4—4881no—
636C4—4881no—
637C4—4881no—
638C4—4881no—
639C4—2442no—
640C4#1684041111yescharacteristic
641C4#1684041911yescharacteristic
642C4#1684045111yescharacteristic
643C4#1684048011yescharacteristic
644C2—81no—
645C2—4882no—
646C2#2144924211yescharacteristic
647C2#2145545811yescharacteristic
648C2#2145549911yescharacteristic
649C2#2145600511yescharacteristic
650C2#2145609811yescharacteristic
651C2#2145707411yescharacteristic
652C2#2145711911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#433C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC2×#433D61
hypocenterC61#24716
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
1. C2×#16840053C2semidirect
2. #21451640C2semidirect
4. #21447252C2semidirect
5. #21447252C2semidirect
8. #21446770C2semidirect
9. #21446589C2semidirect
10. #21446557C2semidirect
11. #21446547C2semidirect
12. #21446516C2semidirect
13. #21446502C2semidirect
14. #21445830C2semidirect
15. C2×C61×#433C2semidirect
121. C4⋊C8×C61D4semidirect
319. C2×#433D61semidirect
387. C61#24716semidirect
435. C4⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400532
#214516402
#214485122
#214472522
#214470092
#214470022
#214467702
#214465892
#214465572
#214465472
#214465162
#214465022
#214458302
C2×C61×#4332
#2471661