Finite Group Databasebeta

← #29721365

Subgroups — #29721365

Order 15616.

794 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840062C211yesmaximal, characteristic, complemented
2C2×#16839823C211yesmaximal, characteristic, complemented
3#21451649C211yesmaximal, characteristic, complemented
4#21447255C211yesmaximal, characteristic, complemented
5#21447250C211yesmaximal, characteristic, complemented
6#21446998C211yesmaximal, characteristic
7#21446998C211yesmaximal, characteristic
8#21446776C211yesmaximal, characteristic, complemented
9#21446577C211yesmaximal, characteristic, complemented
10#21446554C211yesmaximal, characteristic, complemented
11#21446549C211yesmaximal, characteristic, complemented
12#21446519C211yesmaximal, characteristic, complemented
13#21446505C211yesmaximal, characteristic, complemented
14#21445819C211yesmaximal, characteristic, complemented
15C61×#2324C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17#16840062—24no—
18#16839823—24no—
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519688C2211yescharacteristic
32C2×C4×#92418C2211yescharacteristic
33#16839695—21no—
34#16839695—21no—
35#16839685—21no—
36#16839685—21no—
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839632C2211yescharacteristic
40#16839622C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50C2×#121×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C2×#117×C61C2211yescharacteristic
53C61×#390—21no—
54C61×#390C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C61×#388—21no—
57C61×#386—21no—
58C61×#385—21no—
59C61×#380C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519769—22no—
66#14519769—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79#14519710—22no—
80#14519710—22no—
81#14519710—22no—
82#14519708—22no—
83C4⋊#92418—22no—
84C4⋊#92418D414yescomplemented
85#14519698—22no—
86#14519698—22no—
87#14519698—22no—
88#14519698—22no—
89C22⋊#92418—22no—
90C22⋊#92418—22no—
91#14519688—22no—
92C4×#92418—22no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×D4×C61—21no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101#121×C61—22no—
102#121×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105#117×C61—22no—
106#117×C61—22no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61D411yescharacteristic
119C2×C22⋊C4×C61D411yescharacteristic
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126C24×C61—21no—
127C24×C61C2×D411yescharacteristic
128C2×D4×C61—24no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—81no—
135C2×D4×C61—81no—
136C2×D4×C61#4112yes—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—81no—
148C22×C4×C61—81no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—81no—
167C4⋊C4×C61#4112yes—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—42no—
172C22⋊C4×C61—81no—
173C22⋊C4×C61—81no—
174C22⋊C4×C61—81no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—81no—
178C42×C61C2×D412yes—
179C42×C61#4112yes—
180C22⋊Dic61—81no—
181C22⋊Dic61—81no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—81no—
191C4⋊Dic61—81no—
192#11758196—42no—
193#11758196—42no—
194#11758196—81no—
195#11758196—81no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418C2×D412yes—
207C2×#92418C2×D412yes—
208C23×C61—21no—
209C23×C61—22no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C22×D411yescharacteristic
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—84no—
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—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
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—41no—
254C2×C4×C61—84no—
255C2×C4×C61#12112yes—
256C2×C4×C61#12112yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×C4×C612−1+411yescharacteristic
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275#92418—22no—
276#92418—22no—
277#92418—41no—
278#92418—41no—
279#24713—611nomaximal, Sylow
280C22×C61—41no—
281C22×C61—42no—
282C22×C61—41no—
283C22×C61—42no—
284C22×C61—41no—
285C22×C61—42no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61C2×#12111yescharacteristic
295C22×C61#52511yescharacteristic
296C22×C61#53611yescharacteristic
297C22×C61#57611yescharacteristic
298C22×C61#57611yescharacteristic
299C22×C61#57711yescharacteristic
300C22×C61#57811yescharacteristic
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—42no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61#54711yescharacteristic
310C4×C61#54811yescharacteristic
311C4×C61#55211yescharacteristic
312C4×C61#55611yescharacteristic
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317C2×#478—611no—
318C2×#433—611no—
319#2629—611no—
320#2324D6111yescharacteristic
321#2077—611no—
322#2076—611no—
323#2033—611no—
324#2033—611no—
325#1989—611no—
326#1947—611no—
327#1943—611no—
328#1942—611no—
329#1936—611no—
330#1934—611no—
331#1802—611no—
332C2×C61—81no—
333C2×C61—82no—
334C2×C61#241511yescharacteristic
335C2×C61#306111yescharacteristic
336C2×C61#306211yescharacteristic
337C2×C61#312011yescharacteristic
338C2×C61#312211yescharacteristic
339C2×C61#324111yescharacteristic
340C2×C61#324311yescharacteristic
341C2×#125—611no—
342C2×#121C2×D6111yescharacteristic
343C2×C4×D4C2×D6111yescharacteristic
344C2×#117C2×D6111yescharacteristic
345#478—1224no—
346#433—1224no—
347C2×#107—611no—
348C2×#107—611no—
349C2×#107—611no—
350C2×#106—611no—
351C2×C4⋊C8—611no—
352C2×C4⋊C8—611no—
353C2×#102—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×C22⋊C8—611no—
358C2×C22⋊C8—611no—
359C2×#97—611no—
360C2×C4×C8—611no—
361#398—1221no—
362#398—1221no—
363#396—1221no—
364#396—1221no—
365#390—21no—
366#390C2×D6111yescharacteristic
367#389—611no—
368#389—611no—
369#389C2×D6111yescharacteristic
370#388—21no—
371#386—21no—
372#385—21no—
373#384—611no—
374#380—611no—
375#380—611no—
376#380C2×D6111yescharacteristic
377C4×C4⋊C4C2×D6111yescharacteristic
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386C61#2471311yescharacteristic, complemented, Sylow
387C22×D4—21no—
388C22×D4C22×D6111yescharacteristic
389C23×C4—21no—
390C23×C4C22×D6111yescharacteristic
391C22×C8—611no—
392C22×C8—611no—
393C22×C8—611no—
394C22×C8—611no—
395#125—1222no—
396#125—1222no—
397#121—22no—
398#121—22no—
399C4×D4—22no—
400C4×D4—22no—
401#117—22no—
402#117—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22⋊D6111yescharacteristic
415C2×C4⋊C4C22⋊D6111yescharacteristic
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4C22⋊D6111yescharacteristic
427C2×C22⋊C4C22⋊D6111yescharacteristic
428C2×C42—21no—
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431#107—1222no—
432#107—1222no—
433#107—1222no—
434#106—1222no—
435C4⋊C8—614no—
436C4⋊C8—1222no—
437#102—1222no—
438#102—1222no—
439#102—1222no—
440#102—1222no—
441C22⋊C8—1222no—
442C22⋊C8—1222no—
443#97—1222no—
444C4×C8—1222no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452C24—21no—
453C24C2×C22⋊D6111yescharacteristic
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—81no—
461C2×D4—81no—
462C2×D4#1175822212yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C23×D6111yescharacteristic
485C2×C8—612no—
486C2×C8—612no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—81no—
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—4881no—
514C4⋊C4—4881no—
515C4⋊C4—4881no—
516C4⋊C4—4881no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4#1175822212yes—
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—4881no—
528C22⋊C4—4881no—
529C42—21no—
530C42—21no—
531C42—81no—
532C42D4×D6112yes—
533C42#1175822412yes—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C22×C22⋊D6111yescharacteristic
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553D4—84no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
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#1451974912yes—
594C2×C4#1451974912yes—
595C2×C4C2×D4×D6111yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4C2×#1175822211yescharacteristic
599C2×C4#1451982411yescharacteristic
600C2×C4#1451982411yescharacteristic
601C8—1222no—
602C8—1222no—
603C8—2441no—
604C8—2441no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—41no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C2×#1451974911yescharacteristic
620C22#1684027311yescharacteristic
621C22#1684033311yescharacteristic
622C22#1684056511yescharacteristic
623C22#1684056511yescharacteristic
624C22#1684057311yescharacteristic
625C22#1684057711yescharacteristic
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—4881no—
635C4—4881no—
636C4—4881no—
637C4—4881no—
638C4#1684041711yescharacteristic
639C4#1684042111yescharacteristic
640C4#1684045211yescharacteristic
641C4#1684048211yescharacteristic
642C2—81no—
643C2—82no—
644C2#2144924811yescharacteristic
645C2#2145548611yescharacteristic
646C2#2145549511yescharacteristic
647C2#2145603311yescharacteristic
648C2#2145606211yescharacteristic
649C2#2145706611yescharacteristic
650C2#2145709611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2324C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2324D61
hypocenterC61#24713
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. C2×#16840062C2semidirect
2. C2×#16839823C2semidirect
3. #21451649C2semidirect
4. #21447255C2semidirect
5. #21447250C2semidirect
8. #21446776C2semidirect
9. #21446577C2semidirect
10. #21446554C2semidirect
11. #21446549C2semidirect
12. #21446519C2semidirect
13. #21446505C2semidirect
14. #21445819C2semidirect
84. C4⋊#92418D4semidirect
386. C61#24713semidirect

Maximal subgroups

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

subgroupindex
C2×#168400622
C2×#168398232
#214516492
#214472552
#214472502
#214469982
#214467762
#214465772
#214465542
#214465492
#214465192
#214465052
#214458192
C61×#23242
#2471361