Finite Group Databasebeta

← #12156432

Subgroups — #12156432

Order 1280.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×Q8×C22⋊C4C211yesmaximal, characteristic
2C22×#3967C211yesmaximal, characteristic
3C2×#65309C211yesmaximal, characteristic, complemented
4#10605224C211yesmaximal, characteristic
5#10602925C211yesmaximal, characteristic
6#10602422C211yesmaximal, characteristic, complemented
7#10602422C211yesmaximal, characteristic, complemented
8#10602419C211yesmaximal, characteristic, complemented
9#10600478C211yesmaximal, characteristic, complemented
10#10600475C211yesmaximal, characteristic, complemented
11#10600475C211yesmaximal, characteristic, complemented
12#10600461C211yesmaximal, characteristic, complemented
13#10600304C211yesmaximal, characteristic
14#10600298C211yesmaximal, characteristic
15#10600242C211yesmaximal, characteristic
16C23×C5×Q8C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×Q8C2211yescharacteristic
19C22×C4⋊Dic5C2211yescharacteristic
20C23×#168C2211yescharacteristic
21C2×#4024—21no—
22C2×#4015C2211yescharacteristic
23C5×#391—21no—
24C5×#391—21no—
25C5×#391C2211yescharacteristic
26C5×#391C2211yescharacteristic
27C5×#380—21no—
28C5×#380C2211yescharacteristic
29C4×C5×C22⋊C4—21no—
30C4×C5×C22⋊C4C2211yescharacteristic
31C2×#3967—24no—
32C2×#3967C2211yescharacteristic
33C2×#3967C2211yescharacteristic
34C2×#3967C2211yescharacteristic
35C2×#3967C2211yescharacteristic
36C2×#3967C2211yescharacteristic
37C2×#3967C2211yescharacteristic
38C2×#3967C2211yescharacteristic
39C2×#3967C2211yescharacteristic
40#65501C2211yescharacteristic
41C2×C22⋊#168C2211yescharacteristic
42#65309—24no—
43C2×#3940C2211yescharacteristic
44C2×#3940C2211yescharacteristic
45C2×#3940C2211yescharacteristic
46C2×#3940C2211yescharacteristic
47#65239—21no—
48#65224C2211yescharacteristic
49#65217C2211yescharacteristic
50#65217C2211yescharacteristic
51C2×C4⋊#168C2211yescharacteristic
52C2×C4⋊#168C2211yescharacteristic
53#64950—21no—
54#64945—21no—
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64744C2211yescharacteristic
60#64744C2211yescharacteristic
61#25955—51nomaximal, Sylow
62C22×C5×Q8—21no—
63C22×C5×Q8—22no—
64C22×C5×Q8—22no—
65C22×C5×Q8—22no—
66C22×C5×Q8—41no—
67C22×C5×Q8—41no—
68C22×C5×Q8Q814yescomplemented
69C22×C5×Q8C2311yescharacteristic
70C22×C5×Q8C2311yescharacteristic
71C23×C4×C5—21no—
72C23×C4×C5C2311yescharacteristic
73C23×Dic5—21no—
74C22×Dic10—21no—
75C4×C5×Q8—22no—
76C4×C5×Q8—22no—
77C4×C5×Q8—41no—
78C4×C5×Q8—41no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C5×C22⋊C4—21no—
84C2×C5×C22⋊C4D411yescharacteristic
85C2×C5×C22⋊C4D411yescharacteristic
86C2×C42×C5—21no—
87C2×C42×C5—21no—
88C2×C42×C5C2311yescharacteristic
89C2×C42×C5C2311yescharacteristic
90C2×C22⋊Dic5—21no—
91C2×C4⋊Dic5—22no—
92C2×C4⋊Dic5—21no—
93C2×C4⋊Dic5—21no—
94C2×C4⋊Dic5—21no—
95C2×C4⋊Dic5—21no—
96C2×C4⋊Dic5—41no—
97C2×C4⋊Dic5C2311yescharacteristic
98C2×C4⋊Dic5C2311yescharacteristic
99C2×C4⋊Dic5C2311yescharacteristic
100C2×C4⋊Dic5C2311yescharacteristic
101C2×#683—21no—
102C2×#683—21no—
103C22×#168—22no—
104C22×#168—22no—
105C22×#168—41no—
106C22×#168C2311yescharacteristic
107C22×#168C2311yescharacteristic
108C22×#168C2311yescharacteristic
109C22×#168C2311yescharacteristic
110#4024—44no—
111#4015—22no—
112#4015—41no—
113C5×#95—21no—
114C5×#95—21no—
115C5×#95—21no—
116C5×#95—21no—
117C5×#95—21no—
118C5×#95D411yescharacteristic
119C5×#95D411yescharacteristic
120#3967—22no—
121#3967—22no—
122#3967—22no—
123#3967—22no—
124#3967—22no—
125#3967—41no—
126#3967—41no—
127#3967D414yes—
128#3963—21no—
129#3963—21no—
130#3963—21no—
131C22⋊#168—22no—
132#3940—22no—
133#3940—22no—
134#3940—41no—
135#3940—41no—
136C4⋊#168—22no—
137C4⋊#168—41no—
138C2×#475—51no—
139C22×#103—51no—
140#2440—51no—
141Q8×C22⋊C4D511yescharacteristic
142#2096—51no—
143#2066—51no—
144#2066—51no—
145#2024—51no—
146#1865—51no—
147#1865—51no—
148#1853—51no—
149#1850—51no—
150#1807—51no—
151#1803—51no—
152#1787—51no—
153C24×C5C2×D411yescharacteristic
154C2×C5×Q8—24no—
155C2×C5×Q8—21no—
156C2×C5×Q8—21no—
157C2×C5×Q8—21no—
158C2×C5×Q8—21no—
159C2×C5×Q8—28no—
160C2×C5×Q8—21no—
161C2×C5×Q8—21no—
162C2×C5×Q8—21no—
163C2×C5×Q8—21no—
164C2×C5×Q8—42no—
165C2×C5×Q8—42no—
166C2×C5×Q8—44no—
167C2×C5×Q8—44no—
168C2×C5×Q8—44no—
169C2×C5×Q8C2×D412yes—
170C2×C5×Q8C2×Q812yes—
171C2×C5×Q8C2×Q812yes—
172C2×C5×Q8#4112yes—
173C22×C4×C5—21no—
174C22×C4×C5—21no—
175C22×C4×C5—21no—
176C22×C4×C5—21no—
177C22×C4×C5—21no—
178C22×C4×C5—24no—
179C22×C4×C5—21no—
180C22×C4×C5—21no—
181C22×C4×C5—21no—
182C22×C4×C5—42no—
183C22×C4×C5—42no—
184C22×C4×C5C2×D411yescharacteristic
185C22×C4×C5C2×D411yescharacteristic
186C22×C4×C5C2×D411yescharacteristic
187C22×C4×C5C2×D411yescharacteristic
188C22×C4×C5C2×D411yescharacteristic
189C22×C4×C5C2×Q812yes—
190C22×C4×C5#4112yes—
191C22×C4×C5C2411yescharacteristic
192C22×Dic5—21no—
193C22×Dic5—21no—
194C22×Dic5—21no—
195C22×Dic5—21no—
196C22×Dic5—42no—
197C22×Dic5—42no—
198C22×Dic5—81no—
199C2×Dic10—42no—
200C2×Dic10—42no—
201C2×Dic10—81no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4—42no—
207C5×C4⋊C4—42no—
208C5×C22⋊C4—22no—
209C5×C22⋊C4—42no—
210C5×C22⋊C4SD1614yes—
211C42×C5—21no—
212C42×C5—21no—
213C42×C5—21no—
214C42×C5—21no—
215C42×C5—42no—
216C42×C5—42no—
217C22⋊Dic5—42no—
218C4⋊Dic5—21no—
219C4⋊Dic5—21no—
220C4⋊Dic5—21no—
221C4⋊Dic5—21no—
222C4⋊Dic5—21no—
223C4⋊Dic5—21no—
224C4⋊Dic5—21no—
225C4⋊Dic5—42no—
226C4⋊Dic5—42no—
227C4⋊Dic5—42no—
228C4⋊Dic5—81no—
229C4⋊Dic5C2×D412yes—
230#683—42no—
231#683—42no—
232C2×#168—21no—
233C2×#168—21no—
234C2×#168—21no—
235C2×#168—21no—
236C2×#168—21no—
237C2×#168—21no—
238C2×#168—21no—
239C2×#168—21no—
240C2×#168—21no—
241C2×#168—21no—
242C2×#168—21no—
243C2×#168—44no—
244C2×#168C2×D412yes—
245C23×Q8C2×D511yescharacteristic
246C23×C8—51no—
247C2×C4⋊Q8—51no—
248C2×C22⋊Q8—101no—
249C2×C4×Q8C2×D511yescharacteristic
250C2×C4×Q8C2×D511yescharacteristic
251C22×C4⋊C4—51no—
252#475—104no—
253C2×#106—51no—
254C2×#106—51no—
255C2×#106—51no—
256C2×#106—51no—
257C2×C4⋊C8—51no—
258C2×C4⋊C8—51no—
259C2×#103—51no—
260C2×#103—51no—
261C2×#103—51no—
262C2×#103—51no—
263C2×#103—51no—
264C2×#103—51no—
265C2×#103—51no—
266C2×#103—51no—
267C2×#103—104no—
268C2×C22⋊C8—51no—
269#398—101no—
270#395—101no—
271#391—21no—
272#391—21no—
273#391C2×D511yescharacteristic
274#391C2×D511yescharacteristic
275#389—51no—
276#389—51no—
277#385—101no—
278#384—51no—
279#380—21no—
280#380—51no—
281#380C2×D511yescharacteristic
282C4×C22⋊C4—21no—
283C4×C22⋊C4C2×D511yescharacteristic
284#340—51no—
285#340—51no—
286#340—51no—
287#340—51no—
288#336—51no—
289#336—51no—
290C23×C5—21no—
291C23×C5—21no—
292C23×C5—21no—
293C23×C5—21no—
294C23×C5C22⋊Q812yes—
295C23×C5C2×SD1612yes—
296C23×C5#13712yes—
297C23×C5C22×D411yescharacteristic
298C5×Q8—22no—
299C5×Q8—22no—
300C5×Q8—22no—
301C5×Q8—22no—
302C5×Q8—22no—
303C5×Q8—48no—
304C5×Q8—41no—
305C5×Q8—41no—
306C5×Q8C22⋊Q814yescomplemented
307C2×C4×C5—22no—
308C2×C4×C5—21no—
309C2×C4×C5—21no—
310C2×C4×C5—22no—
311C2×C4×C5—21no—
312C2×C4×C5—22no—
313C2×C4×C5—21no—
314C2×C4×C5—21no—
315C2×C4×C5—21no—
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5—22no—
320C2×C4×C5—22no—
321C2×C4×C5—21no—
322C2×C4×C5—21no—
323C2×C4×C5—21no—
324C2×C4×C5—21no—
325C2×C4×C5—21no—
326C2×C4×C5—21no—
327C2×C4×C5—48no—
328C2×C4×C5—41no—
329C2×C4×C5—41no—
330C2×C4×C5—41no—
331C2×C4×C5—41no—
332C2×C4×C5—41no—
333C2×C4×C5—41no—
334C2×C4×C5—41no—
335C2×C4×C5—41no—
336C2×C4×C5C22⋊Q812yes—
337C2×C4×C5C2×SD1612yes—
338C2×C4×C5C2×SD1612yes—
339C2×C4×C5C22×D411yescharacteristic
340C2×C4×C5C22×Q811yescharacteristic
341C2×C4×C5C2×#4111yescharacteristic
342C2×C4×C5C2×#4111yescharacteristic
343C2×C4×C5H(5,𝔽2)11yescharacteristic
344C2×C4×C5H(5,𝔽2)11yescharacteristic
345C2×Dic5—22no—
346C2×Dic5—41no—
347C2×Dic5—41no—
348C2×Dic5—41no—
349C2×Dic5—41no—
350C2×Dic5—41no—
351C2×Dic5—41no—
352C2×Dic5—41no—
353C2×Dic5—41no—
354C2×Dic5—41no—
355C2×Dic5—41no—
356C2×Dic5—41no—
357C2×Dic5—84no—
358Dic10—84no—
359#168—22no—
360#168—41no—
361#168—41no—
362#168—41no—
363C22×Q8—22no—
364C22×Q8—21no—
365C22×Q8—22no—
366C22×Q8—22no—
367C22×Q8—41no—
368C22×Q8—41no—
369C22×Q8—101no—
370C22×Q8Dic1014yescomplemented
371C22×Q8C22×D511yescharacteristic
372C22×Q8C22×D511yescharacteristic
373C23×C4—21no—
374C23×C4—101no—
375C23×C4C22×D511yescharacteristic
376C22×C8—51no—
377C22×C8—51no—
378C22×C8—51no—
379C22×C8—51no—
380C22×C8—102no—
381C22×C8—102no—
382C22×C8—201no—
383C4⋊Q8—102no—
384C4⋊Q8—201no—
385C22⋊Q8—204no—
386C4×Q8—22no—
387C4×Q8—22no—
388C4×Q8—41no—
389C4×Q8—41no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—51no—
393C2×C4⋊C4—51no—
394C2×C4⋊C4—51no—
395C2×C4⋊C4—51no—
396C2×C4⋊C4—102no—
397C2×C4⋊C4—101no—
398C2×C4⋊C4—101no—
399C2×C4⋊C4—101no—
400C2×C4⋊C4—101no—
401C2×C4⋊C4—101no—
402C2×C4⋊C4—101no—
403C2×C4⋊C4—201no—
404C2×C4⋊C4C22×D511yescharacteristic
405C2×C4⋊C4C22×D511yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—101no—
408C2×C22⋊C4C22⋊D511yescharacteristic
409C2×C22⋊C4C22⋊D511yescharacteristic
410C2×C42—21no—
411C2×C42—21no—
412C2×C42C22×D511yescharacteristic
413C2×C42C22×D511yescharacteristic
414#106—102no—
415#106—102no—
416#106—201no—
417#106—201no—
418C4⋊C8—102no—
419C4⋊C8—201no—
420#103—54no—
421#103—102no—
422#103—102no—
423#103—102no—
424#103—102no—
425#103—102no—
426#103—201no—
427#103—201no—
428C22⋊C8—102no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—101no—
435#95—101no—
436#95—101no—
437#95C22⋊D511yescharacteristic
438#95C22⋊D511yescharacteristic
439C22×C5—22no—
440C22×C5—22no—
441C22×C5—22no—
442C22×C5—22no—
443C22×C5—22no—
444C22×C5—22no—
445C22×C5#47514yescomplemented
446C22×C5C2×C22⋊Q811yescharacteristic
447C22×C5D4211yescharacteristic
448C22×C5#54611yescharacteristic
449C22×C5C22×SD1611yescharacteristic
450C22×C5C2×#13711yescharacteristic
451C22×C5#57711yescharacteristic
452C22×C5#57811yescharacteristic
453C4×C5—22no—
454C4×C5—24no—
455C4×C5—22no—
456C4×C5—22no—
457C4×C5—42no—
458C4×C5—41no—
459C4×C5—41no—
460C4×C5—81no—
461C4×C5—81no—
462C4×C5C2×C22⋊Q811yescharacteristic
463C4×C5#54311yescharacteristic
464C4×C5#54611yescharacteristic
465C4×C5#55111yescharacteristic
466Dic5—42no—
467Dic5—81no—
468Dic5—81no—
469Dic5—81no—
470C24C2×C22⋊D511yescharacteristic
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—24no—
476C2×Q8—28no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—42no—
482C2×Q8—42no—
483C2×Q8—44no—
484C2×Q8—44no—
485C2×Q8—44no—
486C2×Q8—202no—
487C2×Q8—202no—
488C2×Q8—401no—
489C2×Q8C2×Dic1012yes—
490C2×Q8C2×Dic1012yes—
491C2×Q8D4×D512yes—
492C2×Q8#71112yes—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—24no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—101no—
505C22×C4—101no—
506C22×C4—101no—
507C22×C4—101no—
508C22×C4—202no—
509C22×C4—202no—
510C22×C4—401no—
511C22×C4C2×Dic1012yes—
512C22×C4#70912yes—
513C22×C4C2×C22⋊D511yescharacteristic
514C22×C4C2×C22⋊D511yescharacteristic
515C22×C4C2×C22⋊D511yescharacteristic
516C22×C4C2×C22⋊D511yescharacteristic
517C22×C4C2×C22⋊D511yescharacteristic
518C22×C4C23×D511yescharacteristic
519C2×C8—52no—
520C2×C8—101no—
521C2×C8—101no—
522C2×C8—101no—
523C2×C8—101no—
524C2×C8—101no—
525C2×C8—101no—
526C2×C8—101no—
527C2×C8—101no—
528C2×C8—101no—
529C2×C8—101no—
530C2×C8—101no—
531C2×C8—204no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—52no—
539C4⋊C4—101no—
540C4⋊C4—101no—
541C4⋊C4—101no—
542C4⋊C4—101no—
543C4⋊C4—101no—
544C4⋊C4—101no—
545C4⋊C4—101no—
546C4⋊C4—202no—
547C4⋊C4—202no—
548C4⋊C4—202no—
549C4⋊C4—202no—
550C4⋊C4—202no—
551C4⋊C4—401no—
552C22⋊C4—22no—
553C22⋊C4—42no—
554C22⋊C4—202no—
555C22⋊C4#68814yes—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—42no—
561C42—42no—
562C2×C5—24no—
563C2×C5#243711yescharacteristic
564C2×C5C2×#47511yescharacteristic
565C2×C5#308211yescharacteristic
566C2×C5#327011yescharacteristic
567C2×C5D4×SD1611yescharacteristic
568C2×C5#329811yescharacteristic
569C2×C5#331611yescharacteristic
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23#407012yes—
575C23C2×#68812yes—
576C23#409712yes—
577C23C22×C22⋊D511yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—48no—
584Q8—41no—
585Q8—41no—
586Q8—404no—
587Q8#402414yescomplemented
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—48no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—102no—
618C2×C4—201no—
619C2×C4—201no—
620C2×C4—201no—
621C2×C4—201no—
622C2×C4—201no—
623C2×C4—201no—
624C2×C4—201no—
625C2×C4—201no—
626C2×C4—201no—
627C2×C4—201no—
628C2×C4—201no—
629C2×C4—404no—
630C2×C4#407012yes—
631C2×C4C2×#68812yes—
632C2×C4C2×#68812yes—
633C2×C4C22×Dic1011yescharacteristic
634C2×C4C2×#70911yescharacteristic
635C2×C4C2×D4×D511yescharacteristic
636C2×C4C2×#71111yescharacteristic
637C2×C4#414411yescharacteristic
638C2×C4#414411yescharacteristic
639C8—102no—
640C8—201no—
641C8—201no—
642C8—201no—
643C5#2595511yescharacteristic, complemented, Sylow
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#6530914yescomplemented
651C22C2×#407011yescharacteristic
652C22D4×C22⋊D511yescharacteristic
653C22#6614011yescharacteristic
654C22C22×#68811yescharacteristic
655C22C2×#409711yescharacteristic
656C22#6617111yescharacteristic
657C22#6617211yescharacteristic
658C4—22no—
659C4—24no—
660C4—22no—
661C4—22no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—81no—
666C4—81no—
667C4—202no—
668C4—401no—
669C4—401no—
670C4—401no—
671C4C2×#402411yescharacteristic
672C4#6581711yescharacteristic
673C4#6582411yescharacteristic
674C4#6582911yescharacteristic
675C2—24no—
676C2#1061384111yescharacteristic
677C2C2×#6530911yescharacteristic
678C2#1061428111yescharacteristic
679C2#1061526911yescharacteristic
680C2#1061527711yescharacteristic
681C2#1061528111yescharacteristic
682C2D4×#68811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×Q8×C22⋊C4C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenterQ8×C22⋊C4D5
hypocenterC5#25955
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#65309C2semidirect
6. #10602422C2semidirect
7. #10602422C2semidirect
8. #10602419C2semidirect
9. #10600478C2semidirect
10. #10600475C2semidirect
11. #10600475C2semidirect
12. #10600461C2semidirect
68. C22×C5×Q8Q8semidirect
306. C5×Q8C22⋊Q8semidirect
370. C22×Q8Dic10semidirect
445. C22×C5#475semidirect
587. Q8#4024semidirect
643. C5#25955semidirect
650. C22#65309semidirect

Maximal subgroups

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

subgroupindex
C5×Q8×C22⋊C42
C22×#39672
C2×#653092
#106052242
#106029252
#106024222
#106024192
#106004782
#106004752
#106004612
#106003042
#106002982
#106002422
#259555