Finite Group Databasebeta

← #28651772

Subgroups — #28651772

Order 15104.

856 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
1C2×#16837694C211yesmaximal, characteristic, complemented
2C2×C4×#11745670C211yesmaximal, characteristic
3#21427016C211yesmaximal, characteristic, complemented
4#21425617C211yesmaximal, characteristic
5#21425577C211yesmaximal, characteristic, complemented
6#21425484C211yesmaximal, characteristic
7#21425426C211yesmaximal, characteristic, complemented
8#21424826C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424454C211yesmaximal, characteristic
11#21424454C211yesmaximal, characteristic
12#21424363C211yesmaximal, characteristic, complemented
13#21424349C211yesmaximal, characteristic, complemented
14#21424343C211yesmaximal, characteristic
15C59×#2452C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837694—24no—
21C4×#11745670—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×#14273648C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837297—21no—
38#16837297—21no—
39#16837268C2211yescharacteristic
40#16837266—21no—
41#16837253—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4⋊Q8×C59C2211yescharacteristic
51C2×C22⋊Q8×C59—21no—
52C2×C4×D4×C59C2211yescharacteristic
53C59×#396—21no—
54C59×#393—21no—
55C59×#390C2211yescharacteristic
56C59×#390C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#389C2211yescharacteristic
59C59×#385—21no—
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C2×#11745670—22no—
63C2×#11745670—22no—
64C2×#11745670Q814yescomplemented
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C4×Dic118—22no—
70C4×Dic118—22no—
71C4×Dic118—22no—
72C4×Dic118—22no—
73C4×Dic118—41no—
74C4×Dic118—41no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86#14273660—22no—
87#14273660—22no—
88#14273660—22no—
89#14273658—22no—
90#14273658—41no—
91C4⋊(C59⋊C8)—22no—
92#14273650—22no—
93#14273650—22no—
94#14273650D414yes—
95#14273648—22no—
96#14273648—22no—
97#14273648—41no—
98C4×C59⋊C8—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59—21no—
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59—21no—
106C4⋊Q8×C59—22no—
107C4⋊Q8×C59—41no—
108C22⋊Q8×C59—44no—
109C4×D4×C59—22no—
110C4×D4×C59—41no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59D411yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
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×Q8×C59—42no—
133C2×Q8×C59—42no—
134C2×Q8×C59—81no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59C2×Q812yes—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—42no—
145C22×C4×C59—42no—
146C22×C4×C59—81no—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2411yescharacteristic
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118C2×D412yes—
162C2×Dic118C2×Q812yes—
163C2×Dic118#4112yes—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59—42no—
174C4⋊C4×C59C2×D412yes—
175C22⋊C4×C59—42no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—42no—
179C22⋊C4×C59—81no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186#11745670—28no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59#4112yes—
193Dic59⋊C4—24no—
194Dic59⋊C4—42no—
195Dic59⋊C4—42no—
196Dic59⋊C4—42no—
197Dic59⋊C4—42no—
198C4×Dic59—24no—
199C4×Dic59—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—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59C22×D411yescharacteristic
221Q8×C59—84no—
222D4×C59—22no—
223D4×C59—22no—
224D4×C59—41no—
225D4×C59—41no—
226C2×C4×C59—21no—
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—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#13512yes—
253C2×C4×C59AGL(1,ℤ/8)12yes—
254C2×C4×C59C22×D411yescharacteristic
255C2×C4×C59C22×Q811yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59H(5,𝔽2)11yescharacteristic
259C2×C4×C592−1+411yescharacteristic
260C2×Dic59—22no—
261C2×Dic59—22no—
262C2×Dic59—22no—
263C2×Dic59—22no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272Dic118—22no—
273Dic118—22no—
274Dic118—22no—
275Dic118C22⋊Q814yescomplemented
276C59⋊C8—22no—
277C59⋊C8—22no—
278C59⋊C8—41no—
279C59⋊C8—41no—
280#26377—591nomaximal, Sylow
281C22×C59—42no—
282C22×C59—42no—
283C22×C59—42no—
284C22×C59—81no—
285C22×C59—81no—
286C22×C59—81no—
287C22×C59—81no—
288C22×C59#52511yescharacteristic
289C22×C59D4211yescharacteristic
290C22×C59D4×Q811yescharacteristic
291C22×C59C2×#13511yescharacteristic
292C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C59#57711yescharacteristic
294C22×C59#57811yescharacteristic
295C4×C59—22no—
296C4×C59—22no—
297C4×C59—41no—
298C4×C59—41no—
299C4×C59—41no—
300C4×C59—42no—
301C4×C59—41no—
302C4×C59—81no—
303C4×C59—81no—
304C4×C59C2×C22⋊Q811yescharacteristic
305C4×C59#53911yescharacteristic
306C4×C59#54711yescharacteristic
307C4×C59#55411yescharacteristic
308Dic59—24no—
309Dic59—42no—
310Dic59—81no—
311Dic59—81no—
312C2×#477—591no—
313C2×C4×SD16—591no—
314#2452D5911yescharacteristic
315#2351—591no—
316#2092—591no—
317#2085—591no—
318#2067—591no—
319#2058—591no—
320#1942—591no—
321#1916—591no—
322#1880—591no—
323#1880—591no—
324#1865—591no—
325#1863—591no—
326#1862—591no—
327C2×C59—82no—
328C2×C59#244911yescharacteristic
329C2×C59#310611yescharacteristic
330C2×C59#312111yescharacteristic
331C2×C59#326511yescharacteristic
332C2×C59#328511yescharacteristic
333C2×C59#338111yescharacteristic
334C2×C59#338611yescharacteristic
335C22×SD16—591no—
336C2×C4⋊Q8C2×D5911yescharacteristic
337C2×C22⋊Q8—21no—
338C2×C4×Q8—591no—
339C2×C4×Q8—591no—
340C2×C4×Q8—591no—
341C2×C4×D4C2×D5911yescharacteristic
342#477—1184no—
343C4×SD16—1184no—
344C2×#107—591no—
345C2×#107—591no—
346C2×#107—591no—
347C2×#106—591no—
348C2×#106—591no—
349C2×C4⋊C8—591no—
350C2×#103—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×C4×C8—591no—
357#396—21no—
358#393—21no—
359#390C2×D5911yescharacteristic
360#390C2×D5911yescharacteristic
361#389—591no—
362#389—591no—
363#389—1181no—
364#389—1181no—
365#389C2×D5911yescharacteristic
366#389C2×D5911yescharacteristic
367#385—21no—
368#382—591no—
369#382—1181no—
370C4×C4⋊C4—1181no—
371C4×C4⋊C4C2×D5911yescharacteristic
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#336—591no—
377#336—591no—
378#336—591no—
379#336—591no—
380C59#2637711yescharacteristic, complemented, Sylow
381C22×Q8—21no—
382C22×Q8—591no—
383C22×D4C22×D5911yescharacteristic
384C23×C4—21no—
385C2×SD16—594no—
386C2×SD16—1182no—
387C2×SD16—1182no—
388C22×C8—591no—
389C22×C8—591no—
390C22×C8—591no—
391C22×C8—591no—
392C4⋊Q8—22no—
393C4⋊Q8—41no—
394C22⋊Q8—44no—
395C4×Q8—1182no—
396C4×Q8—1182no—
397C4×Q8—1182no—
398C4×Q8—1182no—
399C4×Q8—2361no—
400C4×Q8—2361no—
401C4×D4—22no—
402C4×D4—41no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4C22⋊D5911yescharacteristic
415C2×C4⋊C4C22⋊D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C42—1181no—
425C2×C42—1181no—
426C2×C42—1181no—
427C2×C42C22×D5911yescharacteristic
428C2×C42C22×D5911yescharacteristic
429C2×C42C22×D5911yescharacteristic
430#107—1182no—
431#107—1182no—
432#107—1182no—
433#106—1182no—
434#106—2361no—
435C4⋊C8—1182no—
436#103—594no—
437#103—1182no—
438#103—1182no—
439#102—1182no—
440#102—1182no—
441#102—2361no—
442C4×C8—1182no—
443#95—21no—
444#95—21no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449#95C22⋊D5911yescharacteristic
450#95C22⋊D5911yescharacteristic
451C24—21no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×Q8—592no—
456C2×Q8—592no—
457C2×Q8—592no—
458C2×Q8—1181no—
459C2×Q8—1181no—
460C2×Q8—1181no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4Q8×D5912yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—81no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C2×C22⋊D5911yescharacteristic
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C23×D5911yescharacteristic
484SD16—1188no—
485C2×C8—592no—
486C2×C8—592no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—592no—
508C4⋊C4—1184no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—2362no—
515C4⋊C4—2362no—
516C4⋊C4—2362no—
517C4⋊C4—2362no—
518C4⋊C4D4×D5912yes—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—1184no—
531C42—2362no—
532C42—2362no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C22×C22⋊D5911yescharacteristic
541Q8—84no—
542Q8—594no—
543Q8—1182no—
544Q8—1182no—
545Q8—1182no—
546D4—22no—
547D4—22no—
548D4—41no—
549D4—41no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
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—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—1182no—
577C2×C4—1182no—
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#1427374712yes—
589C2×C4#1427375312yes—
590C2×C4C2×D4×D5911yescharacteristic
591C2×C4C2×Q8×D5911yescharacteristic
592C2×C4#41×D5911yescharacteristic
593C2×C4#41×D5911yescharacteristic
594C2×C4#1427377211yescharacteristic
595C2×C4#1427377511yescharacteristic
596C8—1182no—
597C8—1182no—
598C8—2361no—
599C8—2361no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#1683790911yescharacteristic
608C22D4×C22⋊D5911yescharacteristic
609C22Q8×C22⋊D5911yescharacteristic
610C22C2×#1427374711yescharacteristic
611C22C2×#1427375311yescharacteristic
612C22#1683821311yescharacteristic
613C22#1683821711yescharacteristic
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—1184no—
624C4—4721no—
625C4—4721no—
626C4—2362no—
627C4C22⋊Q8×D5911yescharacteristic
628C4#1683799811yescharacteristic
629C4#1683805411yescharacteristic
630C4#1683810611yescharacteristic
631C2—82no—
632C2#2142790211yescharacteristic
633C2#2143415011yescharacteristic
634C2#2143431911yescharacteristic
635C2#2143558711yescharacteristic
636C2#2143583311yescharacteristic
637C2#2143697111yescharacteristic
638C2#2143701711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2452C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2452D59
hypocenterC59#26377
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2452

Complements

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

NH ≅ G/Ntype
1. C2×#16837694C2semidirect
3. #21427016C2semidirect
5. #21425577C2semidirect
7. #21425426C2semidirect
8. #21424826C2semidirect
9. #21424655C2semidirect
12. #21424363C2semidirect
13. #21424349C2semidirect
64. C2×#11745670Q8semidirect
275. Dic118C22⋊Q8semidirect
380. C59#26377semidirect

Maximal subgroups

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

subgroupindex
C2×#168376942
C2×C4×#117456702
#214270162
#214256172
#214255772
#214254842
#214254262
#214248262
#214246552
#214244542
#214243632
#214243492
#214243432
C59×#24522
#2637759