Finite Group Databasebeta

← #29724364

Subgroups — #29724364

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#21448748C211yesmaximal, characteristic
2#21447387C211yesmaximal, characteristic, complemented
3#21447300C211yesmaximal, characteristic, complemented
4#21447256C211yesmaximal, characteristic, complemented
5#21447245C211yesmaximal, characteristic, complemented
6#21446408C211yesmaximal, characteristic, complemented
7#21446387C211yesmaximal, characteristic, complemented
8#21446112C211yesmaximal, characteristic
9#21446016C211yesmaximal, characteristic, complemented
10#21446006C211yesmaximal, characteristic, complemented
11#21445914C211yesmaximal, characteristic, complemented
12#21445907C211yesmaximal, characteristic, complemented
13#21445846C211yesmaximal, characteristic, complemented
14#21445535C211yesmaximal, characteristic, complemented
15C61×#2485C211yesmaximal, characteristic
16C23×#92418C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26#16839655—21no—
27#16839648—21no—
28#16839642—21no—
29#16839628C2211yescharacteristic
30#16839622C2211yescharacteristic
31#16839622C2211yescharacteristic
32C4×C4⋊Dic61C2211yescharacteristic
33C22⋊C4×Dic61—21no—
34#16839524C2211yescharacteristic
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C4×D4×C61C2211yescharacteristic
49C2×#117×C61C2211yescharacteristic
50C61×#397—21no—
51C61×#394—21no—
52C61×#389C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#388—21no—
55C61×#387—21no—
56C61×#380C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#380C2211yescharacteristic
59C23×Dic61—21no—
60C22×#92418—22no—
61C22×#92418—22no—
62C22×#92418—41no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519736—22no—
68#14519736—41no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—41no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81#14519710—22no—
82#14519710—22no—
83#14519710—41no—
84#14519708—41no—
85C22⋊#92418—22no—
86C22⋊#92418—22no—
87C22⋊#92418—41no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93C22×D4×C61D411yescharacteristic
94C23×C4×C61C2311yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98#117×C61—22no—
99#117×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C42×C61—21no—
114C2×C42×C61C2311yescharacteristic
115#95×C61—21no—
116#95×C61—21no—
117#95×C61—21no—
118#95×C61—21no—
119#95×C61—21no—
120C24×C61C2×D411yescharacteristic
121C24×C61C2×D411yescharacteristic
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—41no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—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×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61#4112yes—
144C22×C4×C61#4112yes—
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61—81no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61—81no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—42no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—81no—
169C22⋊C4×C61—81no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—42no—
173C22⋊Dic61—42no—
174C22⋊Dic61—42no—
175C22⋊Dic61—81no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—42no—
184C4⋊Dic61#4112yes—
185C4×Dic61—42no—
186C4×Dic61—81no—
187C2×#92418—21no—
188C2×#92418—21no—
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—44no—
199C2×#92418#4112yes—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61#12312yes—
213C23×C61#13512yes—
214C23×C61C22×D411yescharacteristic
215D4×C61—44no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
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—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
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#12312yes—
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×C4×C612−1+411yescharacteristic
254C2×Dic61—22no—
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—84no—
267#92418—22no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#24934—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61C2×#12311yescharacteristic
287C22×C61#54611yescharacteristic
288C22×C61#54711yescharacteristic
289C22×C61C2×#13511yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57611yescharacteristic
292C22×C61#57811yescharacteristic
293C4×C61—22no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—81no—
298C4×C61—81no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61#52911yescharacteristic
302C4×C61#53911yescharacteristic
303C4×C61#55111yescharacteristic
304C4×C61#55211yescharacteristic
305Dic61—42no—
306Dic61—81no—
307Dic61—81no—
308Dic61—81no—
309#2485D6111yescharacteristic
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#1735—611no—
324C2×C61—42no—
325C2×C61—42no—
326C2×C61#241911yescharacteristic
327C2×C61#309111yescharacteristic
328C2×C61#309811yescharacteristic
329C2×C61#330011yescharacteristic
330C2×C61#331911yescharacteristic
331C2×C61#333911yescharacteristic
332C2×C61#334911yescharacteristic
333C23×C8—611no—
334C2×C4×D4C2×D6111yescharacteristic
335C2×#117—611no—
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—21no—
346#394—21no—
347#389C2×D6111yescharacteristic
348#389C2×D6111yescharacteristic
349#388—21no—
350#388—1221no—
351#387—21no—
352#387—1221no—
353#386—1221no—
354#382—611no—
355#380—611no—
356#380—611no—
357#380C2×D6111yescharacteristic
358#380C2×D6111yescharacteristic
359#380C2×D6111yescharacteristic
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#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#336—611no—
375#336—611no—
376C61#2493411yescharacteristic, complemented, Sylow
377C22×D4C22⋊D6111yescharacteristic
378C23×C4—1221no—
379C23×C4C22×D6111yescharacteristic
380C23×C4C22×D6111yescharacteristic
381C22×C8—611no—
382C22×C8—611no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8—1222no—
386C22×C8—1222no—
387C22×C8—2441no—
388C4×D4—22no—
389C4×D4—22no—
390#117—22no—
391#117—22no—
392#117—1222no—
393#117—2441no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—2441no—
403C2×C4⋊C4C22⋊D6111yescharacteristic
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—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4C22⋊D6111yescharacteristic
416C2×C22⋊C4C22⋊D6111yescharacteristic
417C2×C42—21no—
418C2×C42—1221no—
419C2×C42—1221no—
420C2×C42C22×D6111yescharacteristic
421#107—1222no—
422#107—1222no—
423#107—2441no—
424#106—2441no—
425C22⋊C8—1222no—
426C22⋊C8—1222no—
427C22⋊C8—2441no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438C24C2×C22⋊D6111yescharacteristic
439C24C2×C22⋊D6111yescharacteristic
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—41no—
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—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—4881no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4#1175822412yes—
465C22×C4#1175822412yes—
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C23×D6111yescharacteristic
471C2×C8—612no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—2444no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—612no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—2442no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—4881no—
511C22⋊C4—2442no—
512C22⋊C4—2442no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—4881no—
517C42—2442no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1451975812yes—
531C23#1451979712yes—
532C23C22×C22⋊D6111yescharacteristic
533D4—44no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
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—1222no—
566C2×C4—4884no—
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#1451975812yes—
579C2×C4#41×D6111yescharacteristic
580C2×C4#41×D6111yescharacteristic
581C2×C4C2×#1175822411yescharacteristic
582C2×C4C2×#1175822411yescharacteristic
583C2×C4#1451982311yescharacteristic
584C2×C4#1451982411yescharacteristic
585C8—1222no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22C2×#1451975811yescharacteristic
604C22#1684040011yescharacteristic
605C22#1684041111yescharacteristic
606C22C2×#1451979711yescharacteristic
607C22#1684056011yescharacteristic
608C22#1684056511yescharacteristic
609C22#1684057711yescharacteristic
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—4881no—
619C4—4881no—
620C4—4881no—
621C4—2442no—
622C4#1684028911yescharacteristic
623C4#1684035611yescharacteristic
624C4#1684043511yescharacteristic
625C4#1684044911yescharacteristic
626C2—42no—
627C2—42no—
628C2#2144929211yescharacteristic
629C2#2145572811yescharacteristic
630C2#2145579311yescharacteristic
631C2#2145770011yescharacteristic
632C2#2145796911yescharacteristic
633C2#2145821511yescharacteristic
634C2#2145834311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2485C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#2485D61
hypocenterC61#24934
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2485

Complements

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

NH ≅ G/Ntype
2. #21447387C2semidirect
3. #21447300C2semidirect
4. #21447256C2semidirect
5. #21447245C2semidirect
6. #21446408C2semidirect
7. #21446387C2semidirect
9. #21446016C2semidirect
10. #21446006C2semidirect
11. #21445914C2semidirect
12. #21445907C2semidirect
13. #21445846C2semidirect
14. #21445535C2semidirect
376. C61#24934semidirect

Maximal subgroups

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

subgroupindex
#214487482
#214473872
#214473002
#214472562
#214472452
#214464082
#214463872
#214461122
#214460162
#214460062
#214459142
#214459072
#214458462
#214455352
C61×#24852
#2493461