Finite Group Databasebeta

← #29724368

Subgroups — #29724368

Order 15616.

754 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450110C211yesmaximal, characteristic
2#21448749C211yesmaximal, characteristic
3#21447389C211yesmaximal, characteristic, complemented
4#21447306C211yesmaximal, characteristic, complemented
5#21447257C211yesmaximal, characteristic, complemented
6#21447244C211yesmaximal, characteristic, complemented
7#21446409C211yesmaximal, characteristic, complemented
8#21446389C211yesmaximal, characteristic, complemented
9#21446111C211yesmaximal, characteristic
10#21446018C211yesmaximal, characteristic, complemented
11#21446008C211yesmaximal, characteristic, complemented
12#21445914C211yesmaximal, characteristic, complemented
13#21445908C211yesmaximal, characteristic, complemented
14#21445848C211yesmaximal, characteristic, complemented
15C61×#1735C211yesmaximal, characteristic, complemented
16C2×C4×C22⋊D61C2211yescharacteristic
17C2×C4⋊C4×D61C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519711C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23C2×#14519689C2211yescharacteristic
24C2×#14519689C2211yescharacteristic
25#16839690—21no—
26#16839677—21no—
27#16839658C2211yescharacteristic
28#16839658C2211yescharacteristic
29#16839656—21no—
30#16839653—21no—
31#16839646—21no—
32#16839646—21no—
33#16839638—21no—
34#16839627C2211yescharacteristic
35#16839622C2211yescharacteristic
36#16839622C2211yescharacteristic
37#16839619C2211yescharacteristic
38#16839619C2211yescharacteristic
39#16839619C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758198—21no—
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C23×C8×C61C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C2×#117×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×C22⋊D61D411yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C4×C22⋊D61—22no—
65C4×C22⋊D61—22no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196D411yescharacteristic
74C4⋊C4×D61—22no—
75C4⋊C4×D61—41no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61D411yescharacteristic
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198D411yescharacteristic
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85#14519711—22no—
86#14519711—41no—
87#14519710—22no—
88#14519708—41no—
89C22⋊#92418—22no—
90#14519689—22no—
91#14519689—41no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102C23×C4×C61C2311yescharacteristic
103C22×C8×C61—22no—
104C22×C8×C61—22no—
105C22×C8×C61—41no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108#117×C61—22no—
109#117×C61—22no—
110#117×C61—22no—
111#117×C61—41no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C42×C61—21no—
119C2×C42×C61—21no—
120C24×C61C2×D411yescharacteristic
121C23×D61C2×D411yescharacteristic
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61#4112yes—
132C22×C4×C61#4112yes—
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—22no—
135C2×C22⋊D61—22no—
136C2×C22⋊D61—41no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—81no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—44no—
159C2×C8×C61#4112yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61#4112yes—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—42no—
170C42×C61—42no—
171C42×C61—42no—
172C22⋊Dic61—22no—
173C22⋊Dic61—81no—
174C22⋊Dic61—81no—
175#11758198—22no—
176#11758198—42no—
177#11758198—42no—
178#11758198—81no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187#11758196—22no—
188#11758196—42no—
189#11758196—42no—
190#11758196—81no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—81no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61#12312yes—
207C23×C61C22×D411yescharacteristic
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61#13512yes—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61#12312yes—
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C22⋊D61—44no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
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—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—84no—
267C8×C61—22no—
268C8×C61—41no—
269#92418—41no—
270#92418—41no—
271#24934—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61C2×#12311yescharacteristic
280C22×C61#54611yescharacteristic
281C22×C61#54711yescharacteristic
282C22×C61C2×#13511yescharacteristic
283C22×C61#57511yescharacteristic
284C22×C61#57611yescharacteristic
285C22×C61#57811yescharacteristic
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—41no—
293C4×C61—22no—
294C4×C61—42no—
295C4×C61—81no—
296C4×C61—81no—
297C4×C61—81no—
298C4×C61#52911yescharacteristic
299C4×C61#53911yescharacteristic
300C4×C61#55111yescharacteristic
301C4×C61#55211yescharacteristic
302Dic61—41no—
303Dic61—41no—
304Dic61—41no—
305Dic61—81no—
306Dic61—81no—
307Dic61—81no—
308Dic61—81no—
309#2485—611no—
310#2352—611no—
311#2101—611no—
312#2085—611no—
313#2077—611no—
314#2075—611no—
315#1921—611no—
316#1918—611no—
317#1869—611no—
318#1854—611no—
319#1852—611no—
320#1825—611no—
321#1824—611no—
322#1808—611no—
323#1735D6111yescharacteristic, complemented
324C2×C61—42no—
325C2×C61#241911yescharacteristic
326C2×C61#309111yescharacteristic
327C2×C61#309811yescharacteristic
328C2×C61#330011yescharacteristic
329C2×C61#331911yescharacteristic
330C2×C61#333911yescharacteristic
331C2×C61#334911yescharacteristic
332D61—42no—
333C23×C8C2×D6111yescharacteristic
334C2×C4×D4—611no—
335C2×#117C2×D6111yescharacteristic
336C2×#117C2×D6111yescharacteristic
337C22×C4⋊C4—611no—
338C2×#107—611no—
339C2×#107—611no—
340C2×#107—611no—
341C2×#106—611no—
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
344C2×C22⋊C8—611no—
345#397—1221no—
346#394—1221no—
347#389—611no—
348#389—611no—
349#388—1221no—
350#388—1221no—
351#387—1221no—
352#387—1221no—
353#386—1221no—
354#382—611no—
355#380—611no—
356#380—611no—
357#380—611no—
358#380—611no—
359#380—611no—
360C4×C4⋊C4—611no—
361C4×C22⋊C4—1221no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340C2×D6111yescharacteristic
371#340C2×D6111yescharacteristic
372#340C2×D6111yescharacteristic
373#340C2×D6111yescharacteristic
374#336—611no—
375#336—611no—
376C61#2493411yescharacteristic, complemented, Sylow
377C22×D4—611no—
378C23×C4—611no—
379C23×C4—1221no—
380C23×C4C22×D6111yescharacteristic
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—41no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388C4×D4—1222no—
389C4×D4—1222no—
390#117—22no—
391#117—22no—
392#117—22no—
393#117—41no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—2441no—
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—611no—
411C2×C22⋊C4—611no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—611no—
420C2×C42—1221no—
421#107—1222no—
422#107—1222no—
423#107—2441no—
424#106—2441no—
425C22⋊C8—1222no—
426C22⋊C8—1222no—
427C22⋊C8—2441no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438C24—611no—
439C24D4×D6111yescharacteristic
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—2441no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—4881no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4#1175822212yes—
468C22×C4D4×D6111yescharacteristic
469C22×C4#1175822412yes—
470C22×C4C23×D6111yescharacteristic
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—44no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8#1175822212yes—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—1222no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—4881no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C4⋊C4#1175822212yes—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—4881no—
509C22⋊C4—4881no—
510C22⋊C4—4881no—
511C22⋊C4—2442no—
512C22⋊C4—2442no—
513C42—42no—
514C42—42no—
515C42—1221no—
516C42—1221no—
517C42—4881no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—612no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23#1451975912yes—
532C23C2×D4×D6111yescharacteristic
533D4—2444no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—4884no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
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#1451975912yes—
579C2×C4C2×#1175822411yescharacteristic
580C2×C4C2×#1175822211yescharacteristic
581C2×C4C2×#1175822211yescharacteristic
582C2×C4C2×#1175822211yescharacteristic
583C2×C4#1451982211yescharacteristic
584C2×C4#1451982511yescharacteristic
585C8—22no—
586C8—41no—
587C8—2441no—
588C8—2441no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—1222no—
597C22—1222no—
598C22—1222no—
599C22—2441no—
600C22—2441no—
601C22—2441no—
602C22—2441no—
603C22C2×#1451975911yescharacteristic
604C22#1684040211yescharacteristic
605C22#1684041311yescharacteristic
606C22#135×D6111yescharacteristic
607C22#1684055911yescharacteristic
608C22#1684056411yescharacteristic
609C22#1684057611yescharacteristic
610C4—22no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—4881no—
616C4—4881no—
617C4—4881no—
618C4—4881no—
619C4—2441no—
620C4—2441no—
621C4—2441no—
622C4#1684029611yescharacteristic
623C4#1684035911yescharacteristic
624C4#1684043811yescharacteristic
625C4#1684044711yescharacteristic
626C2—42no—
627C2—2442no—
628C2#2144929611yescharacteristic
629C2#2145573011yescharacteristic
630C2#2145579511yescharacteristic
631C2#2145770711yescharacteristic
632C2#2145797111yescharacteristic
633C2#2145822111yescharacteristic
634C2#2145834511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1735C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenter#1735D61
hypocenterC61#24934
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
3. #21447389C2semidirect
4. #21447306C2semidirect
5. #21447257C2semidirect
6. #21447244C2semidirect
7. #21446409C2semidirect
8. #21446389C2semidirect
10. #21446018C2semidirect
11. #21446008C2semidirect
12. #21445914C2semidirect
13. #21445908C2semidirect
14. #21445848C2semidirect
15. C61×#1735C2semidirect
323. #1735D61semidirect
376. C61#24934semidirect

Maximal subgroups

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

subgroupindex
#214501102
#214487492
#214473892
#214473062
#214472572
#214472442
#214464092
#214463892
#214461112
#214460182
#214460082
#214459142
#214459082
#214458482
C61×#17352
#2493461