Finite Group Databasebeta

← #28651730

Subgroups — #28651730

Order 15104.

884 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837702C211yesmaximal, characteristic
2C2×C4×#11745670C211yesmaximal, characteristic
3#21427957C211yesmaximal, characteristic, complemented
4#21425635C211yesmaximal, characteristic, complemented
5#21425573C211yesmaximal, characteristic, complemented
6#21425488C211yesmaximal, characteristic, complemented
7#21425434C211yesmaximal, characteristic, complemented
8#21424826C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424430C211yesmaximal, characteristic
11#21424430C211yesmaximal, characteristic
12#21424363C211yesmaximal, characteristic, complemented
13#21424343C211yesmaximal, characteristic
14#21424343C211yesmaximal, characteristic
15D4×C4⋊C4×C59C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19#16837702—24no—
20C4×#11745670—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837343—21no—
35#16837333—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837297—21no—
41#16837284—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C2×C4×D4×C59C2211yescharacteristic
53C22×C4⋊C4×C59—21no—
54C59×#389C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#385—21no—
57C59×#385—21no—
58C59×#384C2211yescharacteristic
59C59×#380—21no—
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C2×#11745670—22no—
63C2×#11745670—22no—
64C2×#11745670Q814yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273719—22no—
70#14273719—41no—
71C4×Dic118—22no—
72C4×Dic118—41no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C4×Dic59—21no—
82#14273660—22no—
83#14273660—22no—
84#14273660—41no—
85#14273658—22no—
86#14273658—22no—
87C4⋊(C59⋊C8)—22no—
88#14273650—22no—
89#14273650—22no—
90#14273650—41no—
91#14273648—22no—
92#14273648—22no—
93#14273648D414yes—
94C4×C59⋊C8—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×D4×C59C2311yescharacteristic
102C23×C4×C59—21no—
103C23×C4×C59—21no—
104C23×C4×C59—21no—
105C4×D4×C59—22no—
106C4×D4×C59—22no—
107C4×D4×C59—22no—
108C4×D4×C59—22no—
109C4×D4×C59—41no—
110C4×D4×C59—41no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—44no—
116C2×C4⋊C4×C59D411yescharacteristic
117C2×C4⋊C4×C59D411yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59—21no—
123C2×C22⋊C4×C59—21no—
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126C2×C42×C59C2311yescharacteristic
127#95×C59—21no—
128#95×C59—21no—
129#95×C59D411yescharacteristic
130#95×C59D411yescharacteristic
131C24×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59C2×D412yes—
136C2×D4×C59C2×Q812yes—
137C2×D4×C59#4112yes—
138C22×C4×C59—24no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—42no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—42no—
146C22×C4×C59—42no—
147C22×C4×C59—42no—
148C22×C4×C59—81no—
149C22×C4×C59—81no—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2411yescharacteristic
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118C2×Q812yes—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59—42no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59—81no—
178C4⋊C4×C59#4112yes—
179C22⋊C4×C59—24no—
180C22⋊C4×C59—42no—
181C22⋊C4×C59—42no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188#11745670—28no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59C2×D412yes—
195Dic59⋊C4—42no—
196Dic59⋊C4—42no—
197Dic59⋊C4—42no—
198Dic59⋊C4—42no—
199Dic59⋊C4—42no—
200C4×Dic59—42no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8C2×D412yes—
212C2×C59⋊C8C2×Q812yes—
213C23×C59—22no—
214C23×C59—22no—
215C23×C59—22no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59C22×D411yescharacteristic
221D4×C59—22no—
222D4×C59—22no—
223D4×C59—22no—
224D4×C59C22⋊Q814yes—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—22no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—48no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—84no—
252C2×C4×C59—84no—
253C2×C4×C59#13512yes—
254C2×C4×C59AGL(1,ℤ/8)12yes—
255C2×C4×C59C22×D411yescharacteristic
256C2×C4×C59C22×Q811yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59H(5,𝔽2)11yescharacteristic
260C2×C4×C592−1+411yescharacteristic
261C2×Dic59—22no—
262C2×Dic59—22no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273Dic118—22no—
274Dic118—22no—
275Dic118—41no—
276Dic118—41no—
277C59⋊C8—22no—
278C59⋊C8—22no—
279C59⋊C8—41no—
280C59⋊C8—41no—
281#26374—591nomaximal, Sylow
282C22×C59—24no—
283C22×C59—42no—
284C22×C59—42no—
285C22×C59—42no—
286C22×C59—42no—
287C22×C59—42no—
288C22×C59—42no—
289C22×C59#52511yescharacteristic
290C22×C59D4211yescharacteristic
291C22×C59D4×Q811yescharacteristic
292C22×C59C2×#13511yescharacteristic
293C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C59#57711yescharacteristic
295C22×C59#57811yescharacteristic
296C4×C59—22no—
297C4×C59—22no—
298C4×C59—42no—
299C4×C59—41no—
300C4×C59—41no—
301C4×C59—41no—
302C4×C59—41no—
303C4×C59—81no—
304C4×C59—81no—
305C4×C59C2×C22⋊Q811yescharacteristic
306C4×C59#53911yescharacteristic
307C4×C59#54711yescharacteristic
308C4×C59#55411yescharacteristic
309Dic59—42no—
310Dic59—42no—
311Dic59—81no—
312Dic59—81no—
313C2×#478—591no—
314C2×C4×SD16—591no—
315#2453—591no—
316D4×C4⋊C4D5911yescharacteristic
317#2095—591no—
318#2084—591no—
319#2068—591no—
320#2059—591no—
321#1942—591no—
322#1916—591no—
323#1877—591no—
324#1877—591no—
325#1865—591no—
326#1862—591no—
327#1862—591no—
328C2×C59—44no—
329C2×C59#244911yescharacteristic
330C2×C59#310511yescharacteristic
331C2×C59#312011yescharacteristic
332C2×C59#326511yescharacteristic
333C2×C59#328311yescharacteristic
334C2×C59#338111yescharacteristic
335C2×C59#338611yescharacteristic
336C22×SD16—591no—
337C2×#125—591no—
338C2×C4×Q8—591no—
339C2×C4×D4C2×D5911yescharacteristic
340C2×C4×D4C2×D5911yescharacteristic
341C2×C4×D4C2×D5911yescharacteristic
342C22×C4⋊C4—21no—
343#478—1184no—
344C4×SD16—1184no—
345C2×#107—591no—
346C2×#107—591no—
347C2×#107—591no—
348C2×#106—591no—
349C2×#106—591no—
350C2×C4⋊C8—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×C4×C8—591no—
358#400—1181no—
359#398—1181no—
360#391—591no—
361#391—591no—
362#389—591no—
363#389—591no—
364#389—1181no—
365#389C2×D5911yescharacteristic
366#389C2×D5911yescharacteristic
367#387—1181no—
368#385—21no—
369#385—21no—
370#384C2×D5911yescharacteristic
371#380—21no—
372C4×C4⋊C4C2×D5911yescharacteristic
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#336—591no—
378#336—591no—
379#336—591no—
380#336—591no—
381C59#2637411yescharacteristic, complemented, Sylow
382C22×Q8—591no—
383C22×D4C22×D5911yescharacteristic
384C23×C4—21no—
385C23×C4—21no—
386C23×C4—21no—
387C2×SD16—594no—
388C2×SD16—1182no—
389C2×SD16—1182no—
390C22×C8—591no—
391C22×C8—591no—
392C22×C8—591no—
393C22×C8—591no—
394#125—1182no—
395#125—2361no—
396C4×Q8—1182no—
397C4×Q8—2361no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—41no—
403C4×D4—41no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—44no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1181no—
417C2×C4⋊C4C22⋊D5911yescharacteristic
418C2×C4⋊C4C22⋊D5911yescharacteristic
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C4⋊C4C22×D5911yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C42—1181no—
426C2×C42C22×D5911yescharacteristic
427C2×C42C22×D5911yescharacteristic
428C2×C42C22×D5911yescharacteristic
429#107—1182no—
430#107—1182no—
431#107—2361no—
432#106—1182no—
433#106—1182no—
434C4⋊C8—1182no—
435#103—1182no—
436#103—1182no—
437#103—2361no—
438#102—594no—
439#102—1182no—
440#102—1182no—
441C4×C8—1182no—
442#95—21no—
443#95—21no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95C22⋊D5911yescharacteristic
451#95C22⋊D5911yescharacteristic
452C24—21no—
453C2×Q8—592no—
454C2×Q8—1181no—
455C2×Q8—1181no—
456C2×Q8—1181no—
457C2×Q8—1181no—
458C2×Q8—1181no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4#1174568512yes—
463C2×D4D4×D5912yes—
464C2×D4Q8×D5912yes—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C23×D5911yescharacteristic
488SD16—1188no—
489C2×C8—592no—
490C2×C8—592no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—592no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—2362no—
519C4⋊C4—2362no—
520C4⋊C4—2362no—
521C4⋊C4—2362no—
522C4⋊C4—2362no—
523C4⋊C4#1174568512yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—2362no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C22×C22⋊D5911yescharacteristic
542Q8—1182no—
543Q8—1182no—
544Q8—2361no—
545Q8—2361no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4#1427370514yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
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—48no—
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—84no—
577C2×C4—84no—
578C2×C4—1182no—
579C2×C4—1182no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4#1427374712yes—
591C2×C4#1427375312yes—
592C2×C4C2×D4×D5911yescharacteristic
593C2×C4C2×Q8×D5911yescharacteristic
594C2×C4C2×#1174568511yescharacteristic
595C2×C4C2×#1174568511yescharacteristic
596C2×C4#1427377211yescharacteristic
597C2×C4#1427377511yescharacteristic
598C8—1182no—
599C8—1182no—
600C8—2361no—
601C8—2361no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#1683791311yescharacteristic
610C22D4×C22⋊D5911yescharacteristic
611C22Q8×C22⋊D5911yescharacteristic
612C22C2×#1427374711yescharacteristic
613C22C2×#1427375311yescharacteristic
614C22#1683821311yescharacteristic
615C22#1683821711yescharacteristic
616C4—22no—
617C4—22no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—4721no—
626C4—4721no—
627C4—2362no—
628C4—2362no—
629C4C2×#1427370511yescharacteristic
630C4#1683799911yescharacteristic
631C4#1683805311yescharacteristic
632C4#1683810711yescharacteristic
633C2—44no—
634C2#2142790511yescharacteristic
635C2#2143414311yescharacteristic
636C2#2143431011yescharacteristic
637C2#2143558711yescharacteristic
638C2#2143581411yescharacteristic
639C2#2143697111yescharacteristic
640C2#2143701711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingD4×C4⋊C4×C59C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenterD4×C4⋊C4D59
hypocenterC59#26374
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21427957C2semidirect
4. #21425635C2semidirect
5. #21425573C2semidirect
6. #21425488C2semidirect
7. #21425434C2semidirect
8. #21424826C2semidirect
9. #21424655C2semidirect
12. #21424363C2semidirect
381. C59#26374semidirect

Maximal subgroups

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

subgroupindex
C2×#168377022
C2×C4×#117456702
#214279572
#214256352
#214255732
#214254882
#214254342
#214248262
#214246552
#214244302
#214243632
#214243432
D4×C4⋊C4×C592
#2637459