Finite Group Databasebeta

← #29752709

Subgroups — #29752709

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450219C211yesmaximal, characteristic, complemented
2#21447232C211yesmaximal, characteristic, complemented
3#21447207C211yesmaximal, characteristic, complemented
4#21447005C211yesmaximal, characteristic
5#21446998C211yesmaximal, characteristic
6#21446587C211yesmaximal, characteristic, complemented
7#21446549C211yesmaximal, characteristic, complemented
8#21446427C211yesmaximal, characteristic, complemented
9#21446411C211yesmaximal, characteristic, complemented
10#21446394C211yesmaximal, characteristic, complemented
11#21446066C211yesmaximal, characteristic, complemented
12#21446007C211yesmaximal, characteristic, complemented
13#21445909C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic, complemented
15C61×#2361C211yesmaximal, characteristic
16C2×#14519710C2211yescharacteristic
17C2×#14519710C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26C2×C22⋊#92418C2211yescharacteristic
27C2×C4×#92418C2211yescharacteristic
28C61⋊#401—21no—
29#16839690—21no—
30#16839660C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839656—21no—
33#16839644—21no—
34#16839628C2211yescharacteristic
35#16839622C2211yescharacteristic
36#16839622C2211yescharacteristic
37C4×C4⋊Dic61C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×#121×C61C2211yescharacteristic
49C2×C4×D4×C61—21no—
50C2×C4×D4×C61C2211yescharacteristic
51C22×C4⋊C4×C61C2211yescharacteristic
52C61×#390C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#388—21no—
55C61×#386—21no—
56C61×#385—21no—
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—21no—
69C2×C22⋊Dic61—21no—
70C2×C22⋊Dic61—21no—
71C2×C4×Dic61—21no—
72#14519710—22no—
73#14519710—22no—
74#14519710—41no—
75#14519708—22no—
76C4⋊#92418—22no—
77#14519698—22no—
78#14519698—22no—
79#14519698—22no—
80#14519698—41no—
81C22⋊#92418—22no—
82C22⋊#92418—41no—
83C4×#92418—41no—
84#14519684—21no—
85#14519684—21no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C22×D4×C61—21no—
92C22×D4×C61C2311yescharacteristic
93C23×C4×C61—21no—
94C23×C4×C61—21no—
95C23×C4×C61C2311yescharacteristic
96#121×C61—22no—
97#121×C61—22no—
98C4×D4×C61—22no—
99C4×D4×C61—41no—
100C4×D4×C61—44no—
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—41no—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#95×C61—21no—
118#95×C61—21no—
119#95×C61D411yescharacteristic
120#95×C61D411yescharacteristic
121C24×C61—21no—
122C24×C61C2×D411yescharacteristic
123C2×D4×C61—24no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—42no—
130C2×D4×C61—42no—
131C2×D4×C61#4112yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—81no—
146C22×C4×C61—81no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61#4112yes—
164C4⋊C4×C61#4112yes—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—42no—
170C22⋊C4×C61—81no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—42no—
176C22⋊Dic61—42no—
177C22⋊Dic61—81no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186#11758196—42no—
187#11758196—81no—
188C4×Dic61—81no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418#4112yes—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—22no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61#13512yes—
215C23×C61C22×D411yescharacteristic
216D4×C61—22no—
217D4×C61—22no—
218D4×C61—48no—
219D4×C61—41no—
220D4×C61—41no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
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—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
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—84no—
249C2×C4×C61—84no—
250C2×C4×C61AGL(1,ℤ/8)12yes—
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C612−1+411yescharacteristic
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—
269#92418—22no—
270#92418—41no—
271#92418—41no—
272#92418—41no—
273#27033—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61#52511yescharacteristic
289C22×C61#54611yescharacteristic
290C22×C61#54711yescharacteristic
291C22×C61C2×#13511yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C61#57611yescharacteristic
294C22×C61#57811yescharacteristic
295C4×C61—22no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—42no—
302C4×C61—81no—
303C4×C61#52911yescharacteristic
304C4×C61#53911yescharacteristic
305C4×C61#55311yescharacteristic
306C4×C61#55611yescharacteristic
307Dic61—81no—
308Dic61—81no—
309Dic61—81no—
310Dic61—81no—
311#2494—611no—
312#2361D6111yescharacteristic
313#2073—611no—
314#2067—611no—
315#2034—611no—
316#2033—611no—
317#1949—611no—
318#1942—611no—
319#1923—611no—
320#1921—611no—
321#1919—611no—
322#1862—611no—
323#1852—611no—
324#1824—611no—
325C4×#102—611no—
326C2×C61—42no—
327C2×C61—82no—
328C2×C61#246611yescharacteristic
329C2×C61#310511yescharacteristic
330C2×C61#312411yescharacteristic
331C2×C61#330011yescharacteristic
332C2×C61#331811yescharacteristic
333C2×C61#333911yescharacteristic
334C2×C61#335311yescharacteristic
335C2×#121C2×D6111yescharacteristic
336C2×C4×D4—21no—
337C2×C4×D4C2×D6111yescharacteristic
338C22×C4⋊C4C2×D6111yescharacteristic
339C2×#107—611no—
340C2×#107—611no—
341C2×#107—611no—
342C2×#106—611no—
343C2×C4⋊C8—611no—
344C2×#102—611no—
345C2×#102—611no—
346C2×#102—611no—
347C2×#102—611no—
348C2×C22⋊C8—611no—
349C2×C22⋊C8—611no—
350C2×C4×C8—611no—
351#401—1221no—
352#397—1221no—
353#390C2×D6111yescharacteristic
354#389—611no—
355#389—611no—
356#389C2×D6111yescharacteristic
357#388—21no—
358#388—1221no—
359#387—1221no—
360#386—21no—
361#385—21no—
362#382—611no—
363#380—611no—
364#380—611no—
365C4×C4⋊C4—611no—
366C4×C4⋊C4C2×D6111yescharacteristic
367C4×C22⋊C4C2×D6111yescharacteristic
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#336—611no—
377#336—611no—
378C61#2703311yescharacteristic, complemented, Sylow
379C22×D4—21no—
380C22×D4C22×D6111yescharacteristic
381C23×C4—21no—
382C23×C4—21no—
383C23×C4C22×D6111yescharacteristic
384C22×C8—611no—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8—611no—
388#121—22no—
389#121—22no—
390C4×D4—22no—
391C4×D4—41no—
392C4×D4—44no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4C22⋊D6111yescharacteristic
413C2×C22⋊C4C22⋊D6111yescharacteristic
414C2×C42—21no—
415C2×C42—1221no—
416C2×C42C22×D6111yescharacteristic
417C2×C42C22×D6111yescharacteristic
418#107—1222no—
419#107—1222no—
420#107—2441no—
421#106—1222no—
422C4⋊C8—1222no—
423#102—1222no—
424#102—1222no—
425#102—1222no—
426#102—2441no—
427C22⋊C8—1222no—
428C22⋊C8—2441no—
429C4×C8—2441no—
430#95—21no—
431#95—21no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95C22⋊D6111yescharacteristic
440#95C22⋊D6111yescharacteristic
441C24—21no—
442C24C2×C22⋊D6111yescharacteristic
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4#1175822212yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C2×C22⋊D6111yescharacteristic
476C22×C4C23×D6111yescharacteristic
477C2×C8—612no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—4881no—
504C4⋊C4—2442no—
505C4⋊C4#1175822212yes—
506C4⋊C4#1175822412yes—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C22⋊C4—4881no—
514C22⋊C4—2442no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—4881no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23#1451979712yes—
535C23C22×C22⋊D6111yescharacteristic
536D4—22no—
537D4—22no—
538D4—48no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4#1451980312yes—
583C2×C4#41×D6111yescharacteristic
584C2×C4C2×#1175822411yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4C2×#1175822211yescharacteristic
587C2×C4#1451982211yescharacteristic
588C2×C4#1451982411yescharacteristic
589C8—1222no—
590C8—2441no—
591C8—2441no—
592C8—2441no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#1684027311yescharacteristic
608C22#1684040611yescharacteristic
609C22#1684041111yescharacteristic
610C22C2×#1451979711yescharacteristic
611C22C2×#1451980311yescharacteristic
612C22#1684056511yescharacteristic
613C22#1684057711yescharacteristic
614C4—22no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—42no—
621C4—81no—
622C4—4881no—
623C4—4881no—
624C4—4881no—
625C4—4881no—
626C4#1684029211yescharacteristic
627C4#1684035411yescharacteristic
628C4#1684046211yescharacteristic
629C4#1684048211yescharacteristic
630C2—42no—
631C2—82no—
632C2#2144984911yescharacteristic
633C2#2145586611yescharacteristic
634C2#2145608111yescharacteristic
635C2#2145770611yescharacteristic
636C2#2145796011yescharacteristic
637C2#2145821511yescharacteristic
638C2#2145840311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2361C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2361D61
hypocenterC61#27033
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #21450219C2semidirect
2. #21447232C2semidirect
3. #21447207C2semidirect
6. #21446587C2semidirect
7. #21446549C2semidirect
8. #21446427C2semidirect
9. #21446411C2semidirect
10. #21446394C2semidirect
11. #21446066C2semidirect
12. #21446007C2semidirect
13. #21445909C2semidirect
14. C4×#14519698C2semidirect
378. C61#27033semidirect

Maximal subgroups

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

subgroupindex
#214502192
#214472322
#214472072
#214470052
#214469982
#214465872
#214465492
#214464272
#214464112
#214463942
#214460662
#214460072
#214459092
C4×#145196982
C61×#23612
#2703361