Finite Group Databasebeta

← #28748112

Subgroups — #28748112

Order 15104.

2434 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273692C211yesmaximal, characteristic
2#21427050C214yesmaximal, complemented
3#21427049C212yesmaximal, complemented
4#21427010C212yesmaximal, complemented
5#21426974C212yesmaximal
6#21426973C211yesmaximal, characteristic
7#21426968C212yesmaximal
8C22⋊C4×Dic118C211yesmaximal, characteristic
9#21426928C211yesmaximal, characteristic
10#21426569C214yesmaximal, complemented
11#21426567C212yesmaximal, complemented
12#21426498C212yesmaximal
13#21426498C211yesmaximal, characteristic
14C4×#14273692C212yesmaximal, complemented
15C4×#14273683C212yesmaximal
16C4×#14273683C211yesmaximal, characteristic
17C59×#2345C211yesmaximal, characteristic
18C23×Dic118C2211yescharacteristic
19C2×C4×Dic118C2212yes—
20C2×#14273692C4116yescomplemented
21C2×#14273692C2212yes—
22C2×#14273692C2211yescharacteristic
23C2×#14273692C2211yescharacteristic
24C2×#14273692C2211yescharacteristic
25C2×#14273692C2212yes—
26C2×#14273692C2211yescharacteristic
27C2×#14273692C2212yes—
28C2×#14273683C2212yes—
29C2×#14273683C2211yescharacteristic
30C2×#14273683C2211yescharacteristic
31C22×Dic59⋊C4C2212yes—
32C22×C4×Dic59C2211yescharacteristic
33C22×C4×Dic59C2212yes—
34#16837308C2212yes—
35#16837307C2212yes—
36#16837306C2214yes—
37#16837306C2214yes—
38#16837301C2212yes—
39#16837299C2214yes—
40#16837297C2214yes—
41#16837289C2214yes—
42#16837288C2214yes—
43#16837286C2214yes—
44#16837284C2214yes—
45#16837283C2214yes—
46#16837271C2212yes—
47#16837271C2214yes—
48#16837266C2214yes—
49#16837266C2212yes—
50#16837262C2212yes—
51#16837262C2211yescharacteristic
52#16837260C2212yes—
53#16837260C2212yes—
54#16837260C2212yes—
55C4⋊C4×Dic59C2212yes—
56C4⋊C4×Dic59C2212yes—
57C4⋊C4×Dic59C2214yes—
58C4×Dic59⋊C4C2212yes—
59C4×Dic59⋊C4C2214yes—
60C4×Dic59⋊C4C2214yes—
61#16837253C2214yes—
62#16837253C2212yes—
63#16837253C2214yes—
64#16837253C2214yes—
65#16837253C2212yes—
66C22⋊C4×Dic59C2211yescharacteristic
67C22⋊C4×Dic59C2212yes—
68C22⋊C4×Dic59C2211yescharacteristic
69C22⋊C4×Dic59C2212yes—
70C4×C22⋊Dic59C2212yes—
71C4×C22⋊Dic59C2211yescharacteristic
72#16837247C2214yes—
73#16837247C2212yes—
74C42×Dic59C2212yes—
75C42×Dic59C2214yes—
76C22×C4⋊C4×C59C2211yescharacteristic
77C59×#389C2212yes—
78C59×#387C2214yes—
79C59×#380C2212yes—
80C59×#380C2211yescharacteristic
81C4×C4⋊C4×C59C2212yes—
82C4×C22⋊C4×C59C2212yes—
83C4×C22⋊C4×C59C2211yescharacteristic
84C22×Dic118—28no—
85C22×Dic118C2×C418yes—
86C22×Dic118C2312yes—
87C22×Dic118C2311yescharacteristic
88C22×Dic118C2311yescharacteristic
89C23×Dic59C2312yes—
90C23×Dic59C2311yescharacteristic
91C23×Dic59C2311yescharacteristic
92C4×Dic118—28no—
93#14273692—28no—
94#14273692C2×C418yes—
95#14273692C2×C4116yes—
96#14273692C2×C418yes—
97#14273692C2×C4116yes—
98#14273692C2×C418yes—
99#14273692C2×C418yes—
100#14273683—24no—
101#14273683—28no—
102#14273683Q818yescomplemented
103C2×C4⋊Dic59C2312yes—
104C2×C4⋊Dic59C2314yes—
105C2×Dic59⋊C4C2×C4116yes—
106C2×Dic59⋊C4C2314yes—
107C2×Dic59⋊C4C2314yes—
108C2×Dic59⋊C4C2314yes—
109C2×Dic59⋊C4C2312yes—
110C2×Dic59⋊C4C2314yes—
111C2×Dic59⋊C4C2312yes—
112C2×Dic59⋊C4C2312yes—
113C2×Dic59⋊C4C2312yes—
114C2×Dic59⋊C4C2312yes—
115C2×Dic59⋊C4C2312yes—
116C2×C22⋊Dic59C2312yes—
117C2×C22⋊Dic59C2311yescharacteristic
118C2×C22⋊Dic59C2311yescharacteristic
119C2×C4×Dic59C2×C4116yes—
120C2×C4×Dic59C2×C418yes—
121C2×C4×Dic59C2312yes—
122C2×C4×Dic59C2314yes—
123C2×C4×Dic59C2312yes—
124C2×C4×Dic59C2314yes—
125C2×C4×Dic59C2314yes—
126C2×C4×Dic59C2314yes—
127C2×C4×Dic59C2312yes—
128C2×C4×Dic59C2312yes—
129C2×C4×Dic59C2312yes—
130C2×C4×Dic59C2311yescharacteristic
131C2×C4×Dic59C2311yescharacteristic
132C2×C4×Dic59C2312yes—
133C2×C4×Dic59C2311yescharacteristic
134C2×C4×Dic59C2312yes—
135C2×C4×Dic59C2312yes—
136C2×C4×Dic59C2311yescharacteristic
137C2×C4×Dic59C2312yes—
138C59⋊#95C2314yes—
139C59⋊#95C2312yes—
140C59⋊#95C2314yes—
141C59⋊#95C2312yes—
142C59⋊#95C2312yes—
143C59⋊#95C2314yes—
144C59⋊#95C2314yes—
145C59⋊#95C2314yes—
146C59⋊#95C2312yes—
147C59⋊#95C2314yes—
148C59⋊#95C2314yes—
149C23×C4×C59C2311yescharacteristic
150C23×C4×C59C2312yes—
151C2×C4⋊C4×C59C2×C418yes—
152C2×C4⋊C4×C59C2312yes—
153C2×C4⋊C4×C59C2314yes—
154C2×C4⋊C4×C59C2311yescharacteristic
155C2×C4⋊C4×C59C2312yes—
156C2×C4⋊C4×C59C2311yescharacteristic
157C2×C22⋊C4×C59C2312yes—
158C2×C22⋊C4×C59C2311yescharacteristic
159C2×C22⋊C4×C59C2311yescharacteristic
160C2×C42×C59C2312yes—
161C2×C42×C59C2314yes—
162#95×C59C2312yes—
163#95×C59C2314yes—
164#95×C59C2314yes—
165#95×C59C2312yes—
166C24×C59C2411yescharacteristic
167C22×C4×C59—24no—
168C22×C4×C59—44no—
169C22×C4×C59C22×C414yes—
170C22×C4×C59C22×C418yes—
171C22×C4×C59C2414yes—
172C22×C4×C59C2412yes—
173C22×C4×C59C2411yescharacteristic
174C22×C4×C59C2412yes—
175C22×C4×C59C2412yes—
176C22×C4×C59C2412yes—
177C22×C4×C59C2411yescharacteristic
178C22×Dic59—28no—
179C22×Dic59—42no—
180C22×Dic59C22×C418yes—
181C22×Dic59C22×C414yes—
182C22×Dic59C22×C414yes—
183C22×Dic59C2×Q814yes—
184C22×Dic59#4114yes—
185C22×Dic59C2412yes—
186C22×Dic59C2414yes—
187C22×Dic59C2412yes—
188C22×Dic59C2411yescharacteristic
189C22×Dic59C2411yescharacteristic
190C22×Dic59C2412yes—
191C22×Dic59C2411yescharacteristic
192C22×Dic59C2411yescharacteristic
193C22×Dic59C2412yes—
194C2×Dic118—24no—
195C2×Dic118—28no—
196C2×Dic118—232no—
197C2×Dic118C22×C414yes—
198C2×Dic118C22×C418yes—
199C2×Dic118C22×C414yes—
200C2×Dic118#4118yes—
201C4⋊C4×C59—24no—
202C4⋊C4×C59—44no—
203C4⋊C4×C59C22×C418yes—
204C4⋊C4×C59C22×C414yes—
205C4⋊C4×C59C22×C414yes—
206C22⋊C4×C59—24no—
207C22⋊C4×C59—22no—
208C22⋊C4×C59C2×Q814yes—
209C42×C59—24no—
210C42×C59—44no—
211C22⋊Dic59—24no—
212C22⋊Dic59—22no—
213C22⋊Dic59C2×Q814yes—
214C4⋊Dic59—24no—
215C4⋊Dic59—44no—
216Dic59⋊C4—28no—
217Dic59⋊C4—24no—
218Dic59⋊C4—28no—
219Dic59⋊C4—28no—
220Dic59⋊C4—44no—
221Dic59⋊C4C22×C418yes—
222Dic59⋊C4C22×C418yes—
223Dic59⋊C4C22×C418yes—
224Dic59⋊C4C22×C418yes—
225Dic59⋊C4C2×Q818yescomplemented
226C4×Dic59—24no—
227C4×Dic59—24no—
228C4×Dic59—28no—
229C4×Dic59—28no—
230C4×Dic59—28no—
231C4×Dic59—42no—
232C4×Dic59—44no—
233C4×Dic59—42no—
234C4×Dic59C22×C414yes—
235C4×Dic59C22×C414yes—
236C4×Dic59C22×C414yes—
237C4×Dic59C22×C414yes—
238C4×Dic59C22×C418yes—
239C4×Dic59C22×C418yes—
240C4×Dic59C22×C418yes—
241C4×Dic59C22×C418yes—
242C4×Dic59C2×Q818yescomplemented
243C23×C59—22no—
244C23×C59—22no—
245C23×C59C23×C412yes—
246C23×C59C22×Q812yes—
247C23×C59C2×#4112yes—
248C23×C59C2511yescharacteristic
249C2×C4×C59—22no—
250C2×C4×C59—22no—
251C2×C4×C59—22no—
252C2×C4×C59—24no—
253C2×C4×C59—24no—
254C2×C4×C59—28no—
255C2×C4×C59—24no—
256C2×C4×C59—22no—
257C2×C4×C59—24no—
258C2×C4×C59—22no—
259C2×C4×C59—22no—
260C2×C4×C59—24no—
261C2×C4×C59—48no—
262C2×C4×C59C23×C414yes—
263C2×C4×C59C23×C414yes—
264C2×C4×C59C23×C412yes—
265C2×C4×C59C23×C412yes—
266C2×C4×C59C22×Q814yes—
267C2×C4×C59C2×#4114yes—
268C2×Dic59—22no—
269C2×Dic59—24no—
270C2×Dic59—22no—
271C2×Dic59—22no—
272C2×Dic59—22no—
273C2×Dic59—216no—
274C2×Dic59—22no—
275C2×Dic59—24no—
276C2×Dic59—22no—
277C2×Dic59—24no—
278C2×Dic59—24no—
279C2×Dic59—22no—
280C2×Dic59—44no—
281C2×Dic59C4×Q8116yescomplemented
282C2×Dic59C23×C412yes—
283C2×Dic59C23×C412yes—
284C2×Dic59C23×C414yes—
285C2×Dic59C23×C414yes—
286C2×Dic59C23×C412yes—
287C2×Dic59C23×C412yes—
288C2×Dic59C22×Q814yes—
289C2×Dic59C22×Q812yes—
290C2×Dic59C22×Q812yes—
291C2×Dic59C2×#4118yes—
292C2×Dic59C2×#4112yes—
293C2×Dic59C2×#4112yes—
294Dic118—28no—
295Dic118—216no—
296Dic118#117116yescomplemented
297Q8×#117—591nomaximal, Sylow
298C22×C59—22no—
299C22×C59—22no—
300C22×C59—24no—
301C22×C59—24no—
302C22×C59C2×C4×Q814yes—
303C22×C59C24×C411yescharacteristic
304C22×C59C23×Q811yescharacteristic
305C22×C59C22×#4112yes—
306C22×C59C22×#4111yescharacteristic
307C22×C59#58512yes—
308C4×C59—22no—
309C4×C59—24no—
310C4×C59—24no—
311C4×C59—24no—
312C4×C59—44no—
313C4×C59—42no—
314C4×C59C2×#11714yes—
315Dic59—24no—
316Dic59—22no—
317Dic59—22no—
318Dic59—24no—
319Dic59—42no—
320Dic59—44no—
321Dic59C2×#11718yes—
322Dic59C2×C4×Q814yes—
323Dic59C2×C4×Q814yes—
324C22×C4×Q8—591no—
325#2355—594no—
326#2355—592no—
327Q8×C4⋊C4—592no—
328#2345—592no—
329#2345—592no—
330#2345—591no—
331#2345D5911yescharacteristic
332Q8×C22⋊C4—591no—
333Q8×C22⋊C4—591no—
334#2295—592no—
335#2295—594no—
336#2285—591no—
337#2285—592no—
338C42×Q8—592no—
339C4×#117—591no—
340C4×#117—592no—
341C2×C59—24no—
342C2×C59C22×#11711yescharacteristic
343C2×C59C22×C4×Q811yescharacteristic
344C2×C59#342811yescharacteristic
345C2×C59Q8×#4112yes—
346C2×C59#348012yes—
347C23×Q8—591no—
348C2×C4×Q8—5916no—
349C2×C4×Q8—591no—
350C2×C4×Q8—592no—
351C2×C4×Q8—591no—
352C2×C4×Q8—591no—
353C2×C4×Q8—591no—
354C2×C4×Q8—592no—
355C2×C4×Q8—592no—
356C2×C4×Q8—592no—
357C2×#117—591no—
358C2×#117—592no—
359C2×#117—591no—
360C22×C4⋊C4—592no—
361C22×C4⋊C4C2×D5911yescharacteristic
362C22×C42—592no—
363C22×C42—591no—
364#391—592no—
365#391—592no—
366#391—594no—
367#391—594no—
368#389—594no—
369#389—592no—
370#389—594no—
371#389C2×D5912yes—
372#387—594no—
373#387—594no—
374#387—594no—
375#387—594no—
376#387—594no—
377#387C2×D5914yes—
378#384—594no—
379#384—592no—
380#382—592no—
381#382—594no—
382#380—591no—
383#380—592no—
384#380—592no—
385#380—592no—
386#380—592no—
387#380C2×D5911yescharacteristic
388#380C2×D5912yes—
389C4×C4⋊C4—592no—
390C4×C4⋊C4—592no—
391C4×C4⋊C4—594no—
392C4×C4⋊C4—594no—
393C4×C4⋊C4—592no—
394C4×C4⋊C4—594no—
395C4×C4⋊C4—594no—
396C4×C4⋊C4—594no—
397C4×C4⋊C4—592no—
398C4×C4⋊C4—594no—
399C4×C4⋊C4—592no—
400C4×C4⋊C4C2×D5912yes—
401C4×C22⋊C4—591no—
402C4×C22⋊C4—592no—
403C4×C22⋊C4—592no—
404C4×C22⋊C4—592no—
405C4×C22⋊C4—591no—
406C4×C22⋊C4—591no—
407C4×C22⋊C4C2×D5911yescharacteristic
408C4×C22⋊C4C2×D5912yes—
409#376—592no—
410#376—594no—
411C43—594no—
412C43—592no—
413C59Q8×#11711yescharacteristic, complemented, Sylow
414C22×Q8—598no—
415C22×Q8—591no—
416C22×Q8—592no—
417C22×Q8—591no—
418C22×Q8—1188no—
419C23×C4—591no—
420C23×C4—592no—
421C23×C4—591no—
422C23×C4C22×D5911yescharacteristic
423C23×C4C22×D5912yes—
424C4×Q8—598no—
425C4×Q8—598no—
426C4×Q8—5916no—
427C4×Q8—598no—
428C4×Q8—5916no—
429C4×Q8—598no—
430C4×Q8—1188no—
431C4×Q8—1188no—
432#117—598no—
433#117—1188no—
434#117—1184no—
435C2×C4⋊C4—5916no—
436C2×C4⋊C4—592no—
437C2×C4⋊C4—592no—
438C2×C4⋊C4—592no—
439C2×C4⋊C4—592no—
440C2×C4⋊C4—592no—
441C2×C4⋊C4—592no—
442C2×C4⋊C4—594no—
443C2×C4⋊C4—594no—
444C2×C4⋊C4—594no—
445C2×C4⋊C4—592no—
446C2×C4⋊C4—594no—
447C2×C4⋊C4—594no—
448C2×C4⋊C4C4×D5918yes—
449C2×C4⋊C4C22×D5912yes—
450C2×C4⋊C4C22×D5914yes—
451C2×C4⋊C4C22×D5911yescharacteristic
452C2×C4⋊C4C22×D5912yes—
453C2×C4⋊C4C22×D5911yescharacteristic
454C2×C22⋊C4—591no—
455C2×C22⋊C4—591no—
456C2×C22⋊C4—592no—
457C2×C22⋊C4C22×D5912yes—
458C2×C22⋊C4C22×D5911yescharacteristic
459C2×C22⋊C4C22×D5911yescharacteristic
460C2×C42—5916no—
461C2×C42—598no—
462C2×C42—592no—
463C2×C42—592no—
464C2×C42—591no—
465C2×C42—592no—
466C2×C42—591no—
467C2×C42—592no—
468C2×C42—591no—
469C2×C42—592no—
470C2×C42—591no—
471C2×C42—594no—
472C2×C42—592no—
473C2×C42—594no—
474C2×C42—594no—
475C2×C42—592no—
476C2×C42—592no—
477C2×C42—594no—
478C2×C42—592no—
479C2×C42C22×D5912yes—
480C2×C42C22×D5914yes—
481#95—592no—
482#95—594no—
483#95—592no—
484#95—594no—
485#95—594no—
486#95—592no—
487#95—594no—
488#95—594no—
489#95—594no—
490#95—594no—
491#95—592no—
492#95C22×D5914yes—
493#95C22×D5912yes—
494#95C22×D5914yes—
495#95C22×D5912yes—
496C24C23×D5911yescharacteristic
497C2×Q8—594no—
498C2×Q8—598no—
499C2×Q8—594no—
500C2×Q8—598no—
501C2×Q8—1188no—
502C2×Q8—11832no—
503C2×Q8—1184no—
504C22×C4—24no—
505C22×C4—44no—
506C22×C4—594no—
507C22×C4—594no—
508C22×C4—594no—
509C22×C4—591no—
510C22×C4—591no—
511C22×C4—592no—
512C22×C4—598no—
513C22×C4—592no—
514C22×C4—592no—
515C22×C4—591no—
516C22×C4—591no—
517C22×C4—594no—
518C22×C4—592no—
519C22×C4—594no—
520C22×C4—1188no—
521C22×C4—2362no—
522C22×C4C2×C4×D5914yes—
523C22×C4C2×C4×D5918yes—
524C22×C4C23×D5912yes—
525C22×C4C23×D5914yes—
526C22×C4C23×D5911yescharacteristic
527C22×C4C23×D5912yes—
528C22×C4C23×D5912yes—
529C22×C4C23×D5912yes—
530C22×C4C23×D5911yescharacteristic
531C4⋊C4—24no—
532C4⋊C4—44no—
533C4⋊C4—598no—
534C4⋊C4—598no—
535C4⋊C4—598no—
536C4⋊C4—598no—
537C4⋊C4—598no—
538C4⋊C4—1184no—
539C4⋊C4—1188no—
540C4⋊C4—1188no—
541C4⋊C4—1188no—
542C4⋊C4—1184no—
543C4⋊C4—2364no—
544C4⋊C4—2364no—
545C4⋊C4C2×C4×D5918yes—
546C4⋊C4C2×C4×D5914yes—
547C4⋊C4C2×C4×D5914yes—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—594no—
551C22⋊C4—1182no—
552C22⋊C4—1184no—
553C22⋊C4Q8×D5914yes—
554C42—24no—
555C42—44no—
556C42—598no—
557C42—598no—
558C42—594no—
559C42—594no—
560C42—594no—
561C42—594no—
562C42—598no—
563C42—598no—
564C42—598no—
565C42—1188no—
566C42—1184no—
567C42—1188no—
568C42—1188no—
569C42—1184no—
570C42—2362no—
571C42—2364no—
572C42—2362no—
573C23—22no—
574C23—22no—
575C23C22×C4×D5912yes—
576C23C2×Q8×D5912yes—
577C23#41×D5912yes—
578C23C24×D5911yescharacteristic
579Q8—5916no—
580Q8—11816no—
581Q8—1188no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—592no—
596C2×C4—592no—
597C2×C4—594no—
598C2×C4—592no—
599C2×C4—592no—
600C2×C4—592no—
601C2×C4—598no—
602C2×C4—592no—
603C2×C4—5916no—
604C2×C4—592no—
605C2×C4—592no—
606C2×C4—594no—
607C2×C4—594no—
608C2×C4—11816no—
609C2×C4—1182no—
610C2×C4—1184no—
611C2×C4—1182no—
612C2×C4—1182no—
613C2×C4—1182no—
614C2×C4—1182no—
615C2×C4—1184no—
616C2×C4—1182no—
617C2×C4—1184no—
618C2×C4—1184no—
619C2×C4—1182no—
620C2×C4—2364no—
621C2×C4C22×C4×D5914yes—
622C2×C4C22×C4×D5914yes—
623C2×C4C22×C4×D5912yes—
624C2×C4C22×C4×D5912yes—
625C2×C4C2×Q8×D5914yes—
626C2×C4#41×D5914yes—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—24no—
631C22C4×Q8×D5914yes—
632C22C23×C4×D5911yescharacteristic
633C22C22×Q8×D5911yescharacteristic
634C22C2×#41×D5912yes—
635C22C2×#41×D5911yescharacteristic
636C22#1683823712yes—
637C4—22no—
638C4—24no—
639C4—24no—
640C4—24no—
641C4—42no—
642C4—44no—
643C4—598no—
644C4—594no—
645C4—594no—
646C4—1184no—
647C4—1182no—
648C4—1182no—
649C4—1184no—
650C4—2362no—
651C4—2364no—
652C4#117×D5914yes—
653C2—24no—
654C2C2×#117×D5911yescharacteristic
655C2C2×C4×Q8×D5911yescharacteristic
656C2#2143729411yescharacteristic
657C2#2143767512yes—
658C2#2143769012yes—
subgroupG/H
center, FrattiniC23C24×D59
commutatorC22×C59C24×C4
FittingC59×#2345C2
socle, cosocleC23×C59C25
prime radicalC24×C59C24
hypercenter#2345D59
hypocenterC59Q8×#117
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ #2345

Complements

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

NH ≅ G/Ntype
2. #21427050C2semidirect
3. #21427049C2semidirect
4. #21427010C2semidirect
10. #21426569C2semidirect
11. #21426567C2semidirect
14. C4×#14273692C2semidirect
20. C2×#14273692C4semidirect
102. #14273683Q8semidirect
225. Dic59⋊C4C2×Q8semidirect
242. C4×Dic59C2×Q8semidirect
281. C2×Dic59C4×Q8semidirect
296. Dic118#117semidirect
413. C59Q8×#117semidirect

Maximal subgroups

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

subgroupindex
C22×#142736922
#214270502
#214270492
#214270102
#214269742
#214269732
#214269682
C22⋊C4×Dic1182
#214269282
#214265692
#214265672
#214264982
C4×#142736922
C4×#142736832
C59×#23452
Q8×#11759