Finite Group Databasebeta

← #28648786

Subgroups — #28648786

Order 15104.

904 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837607C211yesmaximal, characteristic
2C2×C4×#11745670C211yesmaximal, characteristic
3#21427872C211yesmaximal, characteristic, complemented
4#21425640C211yesmaximal, characteristic, complemented
5#21425597C211yesmaximal, characteristic, complemented
6#21425505C211yesmaximal, characteristic, complemented
7#21425342C211yesmaximal, characteristic, complemented
8#21424771C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424430C211yesmaximal, characteristic
11#21424430C211yesmaximal, characteristic
12#21424430C211yesmaximal, characteristic
13#21424363C211yesmaximal, characteristic, complemented
14#21424357C211yesmaximal, characteristic
15D4×C4⋊C4×C59C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C22×#11745670C2211yescharacteristic
18C2×#14273727C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837607—24no—
21C4×#11745670—24no—
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837344—21no—
35#16837335—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837308C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837289—21no—
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×Dic118C2311yescharacteristic
62C22×Dic118C2311yescharacteristic
63C2×#11745670—22no—
64C2×#11745670—22no—
65C2×#11745670—22no—
66C2×#11745670—22no—
67C2×#11745670—22no—
68C2×#11745670D414yes—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273727—22no—
74#14273727—41no—
75C4×Dic118—22no—
76C4×Dic118—41no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×C4×Dic59—21no—
84#14273658—22no—
85#14273658—41no—
86C4⋊(C59⋊C8)—22no—
87#14273650—22no—
88#14273650—22no—
89#14273650—22no—
90#14273650—22no—
91#14273650—22no—
92#14273650—41no—
93#14273648—22no—
94#14273648—22no—
95C4×C59⋊C8—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×D4×C59C2311yescharacteristic
104C23×C4×C59—21no—
105C23×C4×C59—21no—
106C23×C4×C59—21no—
107C4×D4×C59—22no—
108C4×D4×C59—22no—
109C4×D4×C59—22no—
110C4×D4×C59—22no—
111C4×D4×C59—41no—
112C4×D4×C59—41no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59—44no—
118C2×C4⋊C4×C59D411yescharacteristic
119C2×C4⋊C4×C59D411yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C22⋊C4×C59—21no—
124C2×C22⋊C4×C59—21no—
125C2×C22⋊C4×C59—21no—
126C2×C42×C59C2311yescharacteristic
127C2×C42×C59C2311yescharacteristic
128C2×C42×C59C2311yescharacteristic
129#95×C59—21no—
130#95×C59—21no—
131#95×C59D411yescharacteristic
132#95×C59D411yescharacteristic
133C24×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59C2×D412yes—
138C2×D4×C59#4112yes—
139C2×D4×C59#4112yes—
140C22×C4×C59—24no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—42no—
145C22×C4×C59—42no—
146C22×C4×C59—42no—
147C22×C4×C59—42no—
148C22×C4×C59—42no—
149C22×C4×C59—42no—
150C22×C4×C59—81no—
151C22×C4×C59—81no—
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2411yescharacteristic
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118—21no—
174C2×Dic118C2×D412yes—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—41no—
181C4⋊C4×C59—42no—
182C4⋊C4×C59—42no—
183C4⋊C4×C59—42no—
184C4⋊C4×C59—81no—
185C4⋊C4×C59SD1614yescomplemented
186C4⋊C4×C59#4112yes—
187C22⋊C4×C59—24no—
188C22⋊C4×C59—42no—
189C22⋊C4×C59—42no—
190C42×C59—21no—
191C42×C59—21no—
192C42×C59—21no—
193C42×C59—21no—
194C42×C59—21no—
195C42×C59—21no—
196#11745670—28no—
197#11745670—44no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—42no—
203C4⋊Dic59—42no—
204Dic59⋊C4—42no—
205Dic59⋊C4—42no—
206C4×Dic59—42no—
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⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8C2×D412yes—
219C23×C59—22no—
220C23×C59—22no—
221C23×C59—22no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59C22×D411yescharacteristic
227D4×C59—22no—
228D4×C59—22no—
229D4×C59—22no—
230D4×C59#12314yes—
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—22no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—48no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—84no—
257C2×C4×C59—84no—
258C2×C4×C59C2×SD1612yes—
259C2×C4×C59C2×SD1612yes—
260C2×C4×C59C2×SD1612yes—
261C2×C4×C59C22×D411yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59H(5,𝔽2)11yescharacteristic
266C2×C4×C592−1+411yescharacteristic
267C2×Dic59—22no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279Dic118—22no—
280Dic118—22no—
281Dic118—41no—
282Dic118—41no—
283Dic118—41no—
284Dic118—41no—
285Dic118—41no—
286Dic118—41no—
287C59⋊C8—22no—
288C59⋊C8—41no—
289C59⋊C8—41no—
290C59⋊C8—41no—
291#26166—591nomaximal, Sylow
292C22×C59—24no—
293C22×C59—42no—
294C22×C59—42no—
295C22×C59—42no—
296C22×C59—42no—
297C22×C59—42no—
298C22×C59—42no—
299C22×C59#52511yescharacteristic
300C22×C59D4211yescharacteristic
301C22×C59#54811yescharacteristic
302C22×C59C22×SD1611yescharacteristic
303C22×C59#57511yescharacteristic
304C22×C59#57711yescharacteristic
305C22×C59#57811yescharacteristic
306C4×C59—22no—
307C4×C59—22no—
308C4×C59—22no—
309C4×C59—42no—
310C4×C59—41no—
311C4×C59—41no—
312C4×C59—41no—
313C4×C59—81no—
314C4×C59—81no—
315C4×C59C2×#12311yescharacteristic
316C4×C59#53911yescharacteristic
317C4×C59#54711yescharacteristic
318C4×C59#55311yescharacteristic
319Dic59—42no—
320Dic59—81no—
321Dic59—81no—
322Dic59—81no—
323C2×#461—591no—
324C2×C4×SD16—591no—
325#2446—591no—
326D4×C4⋊C4D5911yescharacteristic
327#2096—591no—
328#2088—591no—
329#2072—591no—
330#2043—591no—
331#1933—591no—
332#1916—591no—
333#1877—591no—
334#1877—591no—
335#1877—591no—
336#1865—591no—
337#1864—591no—
338C2×C59—44no—
339C2×C59#244311yescharacteristic
340C2×C59#310111yescharacteristic
341C2×C59#312011yescharacteristic
342C2×C59#326911yescharacteristic
343C2×C59D4×SD1611yescharacteristic
344C2×C59#335911yescharacteristic
345C2×C59#337611yescharacteristic
346C22×SD16—591no—
347C22×SD16—591no—
348C2×C4⋊Q8—591no—
349C2×C4×Q8—591no—
350C2×C4×D4C2×D5911yescharacteristic
351C2×C4×D4C2×D5911yescharacteristic
352C2×C4×D4C2×D5911yescharacteristic
353C22×C4⋊C4—21no—
354#461—1184no—
355C4×SD16—1184no—
356C2×#106—591no—
357C2×#106—591no—
358C2×C4⋊C8—591no—
359C2×#103—591no—
360C2×#103—591no—
361C2×#103—591no—
362C2×#103—591no—
363C2×#103—591no—
364C2×#103—591no—
365C2×#102—591no—
366C2×#102—591no—
367C2×C4×C8—591no—
368#400—1181no—
369#398—1181no—
370#391—591no—
371#391—591no—
372#391—591no—
373#389—591no—
374#389C2×D5911yescharacteristic
375#389C2×D5911yescharacteristic
376#387—1181no—
377#387—1181no—
378#385—21no—
379#385—21no—
380#384C2×D5911yescharacteristic
381#380—21no—
382C4×C4⋊C4C2×D5911yescharacteristic
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#336—591no—
388#336—591no—
389#336—591no—
390#336—591no—
391C59#2616611yescharacteristic, complemented, Sylow
392C22×Q8—591no—
393C22×Q8—591no—
394C22×D4C22×D5911yescharacteristic
395C23×C4—21no—
396C23×C4—21no—
397C23×C4—21no—
398C2×SD16—594no—
399C2×SD16—1182no—
400C2×SD16—1182no—
401C2×SD16—1182no—
402C2×SD16—1182no—
403C2×SD16—1182no—
404C22×C8—591no—
405C22×C8—591no—
406C22×C8—591no—
407C22×C8—591no—
408C4⋊Q8—1182no—
409C4⋊Q8—2361no—
410C4×Q8—1182no—
411C4×Q8—2361no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—41no—
417C4×D4—41no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—44no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4C22⋊D5911yescharacteristic
430C2×C4⋊C4C22⋊D5911yescharacteristic
431C2×C4⋊C4C22×D5911yescharacteristic
432C2×C4⋊C4C22×D5911yescharacteristic
433C2×C4⋊C4C22×D5911yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C42—1181no—
438C2×C42C22×D5911yescharacteristic
439C2×C42C22×D5911yescharacteristic
440C2×C42C22×D5911yescharacteristic
441#106—1182no—
442#106—2361no—
443C4⋊C8—1182no—
444#103—1182no—
445#103—1182no—
446#103—1182no—
447#103—1182no—
448#103—1182no—
449#103—2361no—
450#102—1182no—
451#102—1182no—
452C4×C8—1182no—
453#95—21no—
454#95—21no—
455#95—1181no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462#95C22⋊D5911yescharacteristic
463#95C22⋊D5911yescharacteristic
464C24—21no—
465C2×Q8—592no—
466C2×Q8—1181no—
467C2×Q8—1181no—
468C2×Q8—1181no—
469C2×Q8—1181no—
470C2×Q8—1181no—
471C2×Q8—1181no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4#1174568512yes—
481C2×D4D4×D5912yes—
482C2×D4#1174568912yes—
483C22×C4—24no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—1181no—
498C22×C4—1181no—
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C2×C22⋊D5911yescharacteristic
502C22×C4C2×C22⋊D5911yescharacteristic
503C22×C4C2×C22⋊D5911yescharacteristic
504C22×C4C2×C22⋊D5911yescharacteristic
505C22×C4C23×D5911yescharacteristic
506SD16—1188no—
507SD16—2364no—
508C2×C8—592no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—41no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—81no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—1181no—
534C4⋊C4—2362no—
535C4⋊C4—2362no—
536C4⋊C4—2362no—
537C4⋊C4—2362no—
538C4⋊C4#1174567014yescomplemented
539C4⋊C4#1174568512yes—
540C22⋊C4—24no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—2362no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C22×C22⋊D5911yescharacteristic
558Q8—1182no—
559Q8—1182no—
560Q8—2361no—
561Q8—2361no—
562Q8—2361no—
563Q8—2361no—
564Q8—2361no—
565Q8—2361no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4#1427371214yes—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—48no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—1182no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4C2×#1174567012yes—
610C2×C4C2×#1174567012yes—
611C2×C4C2×#1174567012yes—
612C2×C4C2×D4×D5911yescharacteristic
613C2×C4C2×#1174568511yescharacteristic
614C2×C4C2×#1174568511yescharacteristic
615C2×C4C2×#1174568911yescharacteristic
616C2×C4#1427377211yescharacteristic
617C2×C4#1427377511yescharacteristic
618C8—1182no—
619C8—2361no—
620C8—2361no—
621C8—2361no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22#1683791311yescharacteristic
630C22D4×C22⋊D5911yescharacteristic
631C22#1683806311yescharacteristic
632C22C22×#1174567011yescharacteristic
633C22#1683820011yescharacteristic
634C22#1683821311yescharacteristic
635C22#1683821711yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—81no—
644C4—81no—
645C4—4721no—
646C4—4721no—
647C4—4721no—
648C4—2362no—
649C4C2×#1427371211yescharacteristic
650C4#1683799911yescharacteristic
651C4#1683805311yescharacteristic
652C4#1683810011yescharacteristic
653C2—44no—
654C2#2142783111yescharacteristic
655C2#2143410411yescharacteristic
656C2#2143431011yescharacteristic
657C2#2143564711yescharacteristic
658C2D4×#1174567011yescharacteristic
659C2#2143673811yescharacteristic
660C2#2143692311yescharacteristic
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#26166
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. #21427872C2semidirect
4. #21425640C2semidirect
5. #21425597C2semidirect
6. #21425505C2semidirect
7. #21425342C2semidirect
8. #21424771C2semidirect
9. #21424655C2semidirect
13. #21424363C2semidirect
185. C4⋊C4×C59SD16semidirect
391. C59#26166semidirect
538. C4⋊C4#11745670semidirect

Maximal subgroups

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

subgroupindex
C2×#168376072
C2×C4×#117456702
#214278722
#214256402
#214255972
#214255052
#214253422
#214247712
#214246552
#214244302
#214243632
#214243572
D4×C4⋊C4×C592
#2616659