Finite Group Databasebeta

← #28651760

Subgroups — #28651760

Order 15104.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837692C211yesmaximal, characteristic
2C2×C4×#11745670C211yesmaximal, characteristic
3#21427935C211yesmaximal, characteristic, complemented
4#21425640C211yesmaximal, characteristic, complemented
5#21425573C211yesmaximal, characteristic, complemented
6#21425474C211yesmaximal, characteristic, complemented
7#21425429C211yesmaximal, characteristic, complemented
8#21424821C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424430C211yesmaximal, characteristic
11#21424430C211yesmaximal, characteristic
12#21424363C211yesmaximal, characteristic, complemented
13#21424357C211yesmaximal, characteristic
14#21424357C211yesmaximal, characteristic
15D4×C4⋊C4×C59C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×#14273727C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19#16837692—24no—
20C4×#11745670—24no—
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
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#16837336—21no—
35#16837322—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837306—21no—
39#16837300C2211yescharacteristic
40#16837300C2211yescharacteristic
41#16837288—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×Dic118—21no—
62C22×Dic118C2311yescharacteristic
63C2×#11745670—22no—
64C2×#11745670—22no—
65C2×#11745670Q814yescomplemented
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70#14273727—22no—
71#14273727—41no—
72C4×Dic118—22no—
73C4×Dic118—41no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C4×Dic59—21no—
82#14273658—22no—
83#14273658—22no—
84#14273658—22no—
85#14273658—22no—
86#14273658—41no—
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×Dic118—42no—
167C2×Dic118—42no—
168C2×Dic118—81no—
169C2×Dic118C2×Q812yes—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—41no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59—81no—
180C4⋊C4×C59SD1614yescomplemented
181C4⋊C4×C59#4112yes—
182C22⋊C4×C59—24no—
183C22⋊C4×C59—42no—
184C22⋊C4×C59—42no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—21no—
191#11745670—28no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—42no—
198C4⋊Dic59C2×D412yes—
199Dic59⋊C4—42no—
200Dic59⋊C4—81no—
201Dic59⋊C4—81no—
202C4×Dic59—42no—
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⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8C2×D412yes—
214C2×C59⋊C8C2×Q812yes—
215C23×C59—22no—
216C23×C59—22no—
217C23×C59—22no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59C22×D411yescharacteristic
223D4×C59—22no—
224D4×C59—22no—
225D4×C59—22no—
226D4×C59C22⋊Q814yes—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—22no—
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—21no—
243C2×C4×C59—48no—
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—41no—
252C2×C4×C59—84no—
253C2×C4×C59—84no—
254C2×C4×C59C2×SD1612yes—
255C2×C4×C59C2×SD1612yes—
256C2×C4×C59C2×SD1612yes—
257C2×C4×C59C22×D411yescharacteristic
258C2×C4×C59C22×Q811yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59H(5,𝔽2)11yescharacteristic
262C2×C4×C592−1+411yescharacteristic
263C2×Dic59—22no—
264C2×Dic59—22no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275Dic118—22no—
276Dic118—22no—
277Dic118—41no—
278Dic118—41no—
279Dic118—84no—
280C59⋊C8—22no—
281C59⋊C8—22no—
282C59⋊C8—41no—
283C59⋊C8—41no—
284#26376—591nomaximal, Sylow
285C22×C59—24no—
286C22×C59—42no—
287C22×C59—42no—
288C22×C59—42no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—42no—
292C22×C59#52511yescharacteristic
293C22×C59D4211yescharacteristic
294C22×C59D4×Q811yescharacteristic
295C22×C59C22×SD1611yescharacteristic
296C22×C59#57511yescharacteristic
297C22×C59#57711yescharacteristic
298C22×C59#57711yescharacteristic
299C4×C59—22no—
300C4×C59—22no—
301C4×C59—22no—
302C4×C59—42no—
303C4×C59—41no—
304C4×C59—41no—
305C4×C59—41no—
306C4×C59—81no—
307C4×C59—81no—
308C4×C59C2×C22⋊Q811yescharacteristic
309C4×C59#53911yescharacteristic
310C4×C59#54711yescharacteristic
311C4×C59#55411yescharacteristic
312Dic59—42no—
313Dic59—42no—
314Dic59—81no—
315Dic59—81no—
316C2×#476—591no—
317C2×C4×SD16—591no—
318#2451—591no—
319D4×C4⋊C4D5911yescharacteristic
320#2096—591no—
321#2084—591no—
322#2066—591no—
323#2058—591no—
324#1941—591no—
325#1916—591no—
326#1877—591no—
327#1877—591no—
328#1865—591no—
329#1864—591no—
330#1864—591no—
331C2×C59—44no—
332C2×C59#244911yescharacteristic
333C2×C59#310111yescharacteristic
334C2×C59#312511yescharacteristic
335C2×C59#326711yescharacteristic
336C2×C59D4×SD1611yescharacteristic
337C2×C59Q8×SD1611yescharacteristic
338C2×C59#338811yescharacteristic
339C22×SD16—591no—
340C2×C4⋊Q8—591no—
341C2×C4×Q8—591no—
342C2×C4×D4C2×D5911yescharacteristic
343C2×C4×D4C2×D5911yescharacteristic
344C2×C4×D4C2×D5911yescharacteristic
345C22×C4⋊C4—21no—
346#476—1184no—
347C4×SD16—1184no—
348C2×#106—591no—
349C2×#106—591no—
350C2×#106—591no—
351C2×#106—591no—
352C2×#106—591no—
353C2×C4⋊C8—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×C4×C8—591no—
361#398—1181no—
362#395—1181no—
363#391—591no—
364#391—591no—
365#391—1181no—
366#389—591no—
367#389—591no—
368#389C2×D5911yescharacteristic
369#389C2×D5911yescharacteristic
370#387—1181no—
371#385—21no—
372#385—21no—
373#384C2×D5911yescharacteristic
374#380—21no—
375C4×C4⋊C4C2×D5911yescharacteristic
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#336—591no—
381#336—591no—
382#336—591no—
383#336—591no—
384C59#2637611yescharacteristic, complemented, Sylow
385C22×Q8—591no—
386C22×Q8—1181no—
387C22×D4C22×D5911yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—21no—
391C2×SD16—594no—
392C2×SD16—1182no—
393C2×SD16—1182no—
394C22×C8—591no—
395C22×C8—591no—
396C22×C8—591no—
397C22×C8—591no—
398C4⋊Q8—1182no—
399C4⋊Q8—2361no—
400C4×Q8—1182no—
401C4×Q8—2361no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—41no—
407C4×D4—41no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—44no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—1181no—
417C2×C4⋊C4—1181no—
418C2×C4⋊C4—1181no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4C22⋊D5911yescharacteristic
421C2×C4⋊C4C22⋊D5911yescharacteristic
422C2×C4⋊C4C22×D5911yescharacteristic
423C2×C4⋊C4C22×D5911yescharacteristic
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C42—1181no—
429C2×C42C22×D5911yescharacteristic
430C2×C42C22×D5911yescharacteristic
431C2×C42C22×D5911yescharacteristic
432#106—1182no—
433#106—1182no—
434#106—1182no—
435#106—1182no—
436#106—2361no—
437C4⋊C8—1182no—
438#103—1182no—
439#103—1182no—
440#103—2361no—
441#102—594no—
442#102—1182no—
443#102—1182no—
444C4×C8—1182no—
445#95—21no—
446#95—21no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453#95C22⋊D5911yescharacteristic
454#95C22⋊D5911yescharacteristic
455C24—21no—
456C2×Q8—592no—
457C2×Q8—1181no—
458C2×Q8—1181no—
459C2×Q8—1181no—
460C2×Q8—1181no—
461C2×Q8—1181no—
462C2×Q8—4721no—
463C2×Q8—2362no—
464C2×Q8—2362no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4#1174568512yes—
469C2×D4D4×D5912yes—
470C2×D4Q8×D5912yes—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C23×D5911yescharacteristic
494SD16—1188no—
495C2×C8—592no—
496C2×C8—592no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—592no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—4721no—
524C4⋊C4—4721no—
525C4⋊C4—2362no—
526C4⋊C4—2362no—
527C4⋊C4#1174567014yescomplemented
528C4⋊C4#1174568512yes—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—2362no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C22×C22⋊D5911yescharacteristic
547Q8—1182no—
548Q8—1182no—
549Q8—4724no—
550Q8—2361no—
551Q8—2361no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4#1427370514yes—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—1182no—
584C2×C4—1182no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4C2×#1174567012yes—
596C2×C4C2×#1174567012yes—
597C2×C4C2×#1174567012yes—
598C2×C4C2×D4×D5911yescharacteristic
599C2×C4C2×Q8×D5911yescharacteristic
600C2×C4C2×#1174568511yescharacteristic
601C2×C4C2×#1174568511yescharacteristic
602C2×C4#1427377211yescharacteristic
603C2×C4#1427377511yescharacteristic
604C8—1182no—
605C8—1182no—
606C8—2361no—
607C8—2361no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#1683791311yescharacteristic
616C22D4×C22⋊D5911yescharacteristic
617C22Q8×C22⋊D5911yescharacteristic
618C22C22×#1174567011yescharacteristic
619C22#1683820011yescharacteristic
620C22#1683821011yescharacteristic
621C22#1683821311yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—81no—
630C4—81no—
631C4—4721no—
632C4—4721no—
633C4—2362no—
634C4—2362no—
635C4C2×#1427370511yescharacteristic
636C4#1683799911yescharacteristic
637C4#1683805311yescharacteristic
638C4#1683810711yescharacteristic
639C2—44no—
640C2#2142790511yescharacteristic
641C2#2143410411yescharacteristic
642C2#2143437311yescharacteristic
643C2#2143562211yescharacteristic
644C2D4×#1174567011yescharacteristic
645C2Q8×#1174567011yescharacteristic
646C2#2143704111yescharacteristic
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#26376
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. #21427935C2semidirect
4. #21425640C2semidirect
5. #21425573C2semidirect
6. #21425474C2semidirect
7. #21425429C2semidirect
8. #21424821C2semidirect
9. #21424655C2semidirect
12. #21424363C2semidirect
65. C2×#11745670Q8semidirect
180. C4⋊C4×C59SD16semidirect
384. C59#26376semidirect
527. C4⋊C4#11745670semidirect

Maximal subgroups

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

subgroupindex
C2×#168376922
C2×C4×#117456702
#214279352
#214256402
#214255732
#214254742
#214254292
#214248212
#214246552
#214244302
#214243632
#214243572
D4×C4⋊C4×C592
#2637659