Finite Group Databasebeta

← #12121147

Subgroups — #12121147

Order 1280.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65136C211yesmaximal, characteristic
2#10605501C211yesmaximal, characteristic
3#10605390C211yesmaximal, characteristic, complemented
4C5×C8×C4⋊C4C211yesmaximal, characteristic, complemented
5#10601248C211yesmaximal, characteristic, complemented
6#10600571C211yesmaximal, characteristic, complemented
7#10600563C211yesmaximal, characteristic, complemented
8#10600562C211yesmaximal, characteristic, complemented
9C4×#3953C211yesmaximal, characteristic
10C4×#3953C211yesmaximal, characteristic
11C4×#3953C211yesmaximal, characteristic
12#10599464C211yesmaximal, characteristic
13#10599463C211yesmaximal, characteristic
14#10599417C211yesmaximal, characteristic, complemented
15#10599416C211yesmaximal, characteristic, complemented
16C2×C4×D20C2211yescharacteristic
17C2×C4×D20C2211yescharacteristic
18C2×C4×D20C2211yescharacteristic
19C2×#4016C2211yescharacteristic
20C2×C5×C4⋊C8C2211yescharacteristic
21C2×C4×C5×C8C2211yescharacteristic
22C2×C4×C5×C8C2211yescharacteristic
23C2×C4×C5×C8C2211yescharacteristic
24C4×C5×C4⋊C4C2211yescharacteristic
25C2×#3953C2211yescharacteristic
26C2×#3953C2211yescharacteristic
27C2×#3953C2211yescharacteristic
28C2×#3953C2211yescharacteristic
29C2×#3950C2211yescharacteristic
30C2×#3950C2211yescharacteristic
31C2×#3949C2211yescharacteristic
32C2×#3949C2211yescharacteristic
33#65278—21no—
34#65278—21no—
35#65267—21no—
36#65261—21no—
37#65225C2211yescharacteristic
38C4×C4⋊Dic5C2211yescharacteristic
39C4×C4⋊Dic5C2211yescharacteristic
40C4×C4⋊Dic5C2211yescharacteristic
41#65217C2211yescharacteristic
42#65217C2211yescharacteristic
43#65217C2211yescharacteristic
44#65217C2211yescharacteristic
45#65136C418yescomplemented
46#64956—21no—
47D5×#95—21no—
48#64941—21no—
49#64938—21no—
50C5×#336C2211yescharacteristic
51C5×#336C2211yescharacteristic
52#64771C2211yescharacteristic
53#64771C2211yescharacteristic
54#64771C2211yescharacteristic
55#64771C2211yescharacteristic
56#64771C2211yescharacteristic
57#64771C2211yescharacteristic
58#64771C2211yescharacteristic
59#64771C2211yescharacteristic
60#22966—51nomaximal, Sylow
61C22×D20C2311yescharacteristic
62C22×C4×D5—21no—
63C22×C4×D5—21no—
64C22×C4×D5—21no—
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C2×C5×C4⋊C4D411yescharacteristic
70C2×C5×C4⋊C4D411yescharacteristic
71C2×C42×C5C2311yescharacteristic
72C2×C42×C5C2311yescharacteristic
73C2×C42×C5C2311yescharacteristic
74C2×#685—21no—
75C2×#685—21no—
76C2×#685—21no—
77C2×C4⋊Dic5C2311yescharacteristic
78C2×C4⋊Dic5C2311yescharacteristic
79C2×C4⋊Dic5C2311yescharacteristic
80C2×C4⋊Dic5C2311yescharacteristic
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4⋊Dic5C2311yescharacteristic
84C2×#683—21no—
85C2×#683—21no—
86C2×#683—21no—
87C2×#683—21no—
88C2×#683—21no—
89C2×#683—21no—
90C2×C4×Dic5—21no—
91C2×C4×Dic5—21no—
92C2×C4×Dic5—21no—
93C4×D20—22no—
94C4×D20—22no—
95C4×D20—22no—
96C4×D20—41no—
97C4×D20—41no—
98C4×D20C2×C414yes—
99#4016—22no—
100#4016C2×C414yescomplemented
101C5×C4⋊C8C2×C414yescomplemented
102C4×C5×C8—22no—
103C4×C5×C8—41no—
104C4×C5×C8—41no—
105C5×#95D411yescharacteristic
106C5×#95D411yescharacteristic
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113#3963—21no—
114#3963—21no—
115#3963—21no—
116#3953—22no—
117#3953C2×C414yes—
118#3953C2×C414yes—
119#3953Q814yes—
120#3950—41no—
121#3950C2×C414yescomplemented
122#3949—41no—
123#3949C2×C414yescomplemented
124C2×#478—51no—
125#2370—51no—
126#2349—51no—
127#1991—51no—
128#1949—51no—
129#1942—51no—
130#1941—51no—
131#1921—51no—
132#1920—51no—
133#1864—51no—
134#1862—51no—
135C8×C4⋊C4D511yescharacteristic, complemented
136C4×#102—51no—
137C4×#102—51no—
138C4×#102—51no—
139C23×D5—21no—
140C22×C4×C5C2×D411yescharacteristic
141C22×C4×C5C2×D411yescharacteristic
142C22×C4×C5C2×D411yescharacteristic
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2411yescharacteristic
147C22×Dic5—21no—
148C22×Dic5—21no—
149C22×Dic5—21no—
150C22×Dic5—21no—
151C22×Dic5—21no—
152C22×Dic5—21no—
153C22×Dic5—21no—
154C2×D20—21no—
155C2×D20—21no—
156C2×D20—21no—
157C2×D20C22×C412yes—
158C2×D20C2×Q812yes—
159C2×D20#4112yes—
160C2×C4×D5—24no—
161C2×C4×D5—42no—
162C2×C4×D5—42no—
163C2×C4×D5—42no—
164C2×C4×D5—42no—
165C2×C4×D5—81no—
166C2×C4×D5—81no—
167C2×C4×D5—81no—
168C2×C4×D5—81no—
169C2×C5×C8—21no—
170C2×C5×C8—21no—
171C2×C5×C8—21no—
172C2×C5×C8—21no—
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8—21no—
176C2×C5×C8—21no—
177C2×C5×C8—21no—
178C2×C5×C8C22×C412yes—
179C2×C5×C8C22×C412yes—
180C2×C5×C8C2×Q812yes—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—22no—
183C42×C5—21no—
184C42×C5—21no—
185C42×C5—21no—
186C42×C5—21no—
187C42×C5C22×C412yes—
188C42×C5#4112yes—
189#685—24no—
190#685—42no—
191#685—42no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194C4⋊Dic5—21no—
195C4⋊Dic5—21no—
196C4⋊Dic5—21no—
197C4⋊Dic5—21no—
198C4⋊Dic5—21no—
199C4⋊Dic5—21no—
200C4⋊Dic5—21no—
201C4⋊Dic5—21no—
202C4⋊Dic5C22×C412yes—
203C4⋊Dic5C22×C412yes—
204C4⋊Dic5C22×C412yes—
205C4⋊Dic5C2×Q812yes—
206#683—24no—
207#683—24no—
208#683—42no—
209#683—81no—
210#683—81no—
211#683—81no—
212C4×Dic5—42no—
213C4×Dic5—81no—
214C4×Dic5—81no—
215C2×#125—51no—
216C2×C4×D4—51no—
217C2×C4×D4—51no—
218C2×C4×D4—51no—
219#478—58no—
220C2×#107—51no—
221C2×#107—51no—
222C2×#106—51no—
223C2×#106—51no—
224C2×C4⋊C8C2×D511yescharacteristic
225C2×#102—51no—
226C2×#102—51no—
227C2×#102—51no—
228C2×#102—51no—
229C2×C4×C8C2×D511yescharacteristic
230C2×C4×C8C2×D511yescharacteristic
231C2×C4×C8C2×D511yescharacteristic
232#389—51no—
233#389—51no—
234#389—51no—
235#389—51no—
236#389—101no—
237#387—101no—
238#387—101no—
239#385—101no—
240#385—101no—
241#382—51no—
242#380—101no—
243C4×C4⋊C4—51no—
244C4×C4⋊C4—51no—
245C4×C4⋊C4—51no—
246C4×C4⋊C4—101no—
247C4×C4⋊C4C2×D511yescharacteristic
248C2×#95—101no—
249#340—51no—
250#340—51no—
251#340—51no—
252#340—51no—
253#340—51no—
254#340—51no—
255#340—51no—
256#340—51no—
257#336C2×D511yescharacteristic
258#336C2×D511yescharacteristic
259C23×C5C22×D411yescharacteristic
260C22×D5—22no—
261C22×D5—22no—
262C22×D5—22no—
263C22×D5—41no—
264C22×D5—41no—
265C22×D5—41no—
266C22×D5—41no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5C4×D412yes—
282C2×C4×C5C4×D412yes—
283C2×C4×C5#13512yes—
284C2×C4×C5#13512yes—
285C2×C4×C5C23×C411yescharacteristic
286C2×C4×C5C22×Q811yescharacteristic
287C2×C4×C5C2×#4111yescharacteristic
288C2×C4×C5C2×#4111yescharacteristic
289C2×C4×C5H(5,𝔽2)11yescharacteristic
290C2×C4×C52−1+411yescharacteristic
291C2×Dic5—22no—
292C2×Dic5—22no—
293C2×Dic5—22no—
294C2×Dic5—22no—
295C2×Dic5—41no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5—41no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312D20—22no—
313D20—22no—
314D20—22no—
315D20C4×Q814yes—
316C4×D5—84no—
317C4×D5—84no—
318C4×D5—84no—
319C5×C8—22no—
320C5×C8—22no—
321C5×C8—22no—
322C5×C8—41no—
323C22×D4—51no—
324C23×C4—101no—
325C23×C4—101no—
326C23×C4—101no—
327C22×C8C22×D511yescharacteristic
328C22×C8C22×D511yescharacteristic
329C22×C8C22×D511yescharacteristic
330C22×C8C22×D511yescharacteristic
331#125—54no—
332#125—102no—
333C4×D4—54no—
334C4×D4—102no—
335C4×D4—102no—
336C4×D4—102no—
337C4×D4—201no—
338C4×D4—201no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4—51no—
342C2×C4⋊C4—51no—
343C2×C4⋊C4—51no—
344C2×C4⋊C4—51no—
345C2×C4⋊C4—51no—
346C2×C4⋊C4—101no—
347C2×C4⋊C4—101no—
348C2×C4⋊C4—101no—
349C2×C4⋊C4—101no—
350C2×C4⋊C4—101no—
351C2×C4⋊C4—101no—
352C2×C4⋊C4D2011yescharacteristic
353C2×C4⋊C4D2011yescharacteristic
354C2×C22⋊C4—101no—
355C2×C22⋊C4—101no—
356C2×C22⋊C4—101no—
357C2×C42—101no—
358C2×C42—101no—
359C2×C42—101no—
360C2×C42C22×D511yescharacteristic
361C2×C42C22×D511yescharacteristic
362C2×C42C22×D511yescharacteristic
363#107—54no—
364#107—201no—
365#106—54no—
366#106—201no—
367C4⋊C8C4×D514yescomplemented
368#102—54no—
369#102—54no—
370#102—54no—
371#102—102no—
372C4×C8—22no—
373C4×C8—41no—
374C4×C8—41no—
375#95—101no—
376#95—101no—
377#95—101no—
378#95—101no—
379#95—101no—
380#95—101no—
381#95—101no—
382#95—101no—
383#95—101no—
384#95D2011yescharacteristic
385#95D2011yescharacteristic
386C22×C5C2×C4×D411yescharacteristic
387C22×C5#54611yescharacteristic
388C22×C5D4×Q811yescharacteristic
389C22×C5C2×#13511yescharacteristic
390C22×C5C2×#13511yescharacteristic
391C22×C5#57611yescharacteristic
392C22×C5#57711yescharacteristic
393C2×D5—24no—
394C2×D5—42no—
395C2×D5—42no—
396C2×D5—42no—
397C2×D5—42no—
398C2×D5—42no—
399C2×D5—42no—
400C4×C5—22no—
401C4×C5—22no—
402C4×C5—22no—
403C4×C5—22no—
404C4×C5—41no—
405C4×C5—41no—
406C4×C5C2×C4×Q811yescharacteristic
407C4×C5#52011yescharacteristic
408C4×C5#55211yescharacteristic
409C4×C5#55411yescharacteristic
410Dic5—42no—
411Dic5—42no—
412Dic5—42no—
413Dic5—42no—
414Dic5—81no—
415Dic5—81no—
416Dic5—81no—
417C24—101no—
418C2×D4—52no—
419C2×D4—52no—
420C2×D4—52no—
421C2×D4—101no—
422C2×D4—101no—
423C2×D4—101no—
424C22×C4—101no—
425C22×C4—101no—
426C22×C4—101no—
427C22×C4—101no—
428C22×C4—104no—
429C22×C4—101no—
430C22×C4—101no—
431C22×C4—101no—
432C22×C4—202no—
433C22×C4—202no—
434C22×C4—202no—
435C22×C4—202no—
436C22×C4—401no—
437C22×C4—401no—
438C22×C4—401no—
439C22×C4—401no—
440C22×C4C2×D2011yescharacteristic
441C22×C4C2×D2011yescharacteristic
442C22×C4C2×D2011yescharacteristic
443C22×C4C2×D2011yescharacteristic
444C22×C4C2×D2011yescharacteristic
445C22×C4C2×D2011yescharacteristic
446C22×C4C23×D511yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8C2×C4×D512yes—
457C2×C8C2×C4×D512yes—
458C2×C8Q8×D512yes—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—52no—
462C4⋊C4—52no—
463C4⋊C4—52no—
464C4⋊C4—52no—
465C4⋊C4—101no—
466C4⋊C4—101no—
467C4⋊C4—101no—
468C4⋊C4—101no—
469C4⋊C4—104no—
470C4⋊C4—104no—
471C4⋊C4—101no—
472C4⋊C4—101no—
473C4⋊C4—101no—
474C4⋊C4—101no—
475C4⋊C4—101no—
476C4⋊C4—101no—
477C4⋊C4—202no—
478C4⋊C4—401no—
479C4⋊C4—401no—
480C4⋊C4—401no—
481C22⋊C4—104no—
482C22⋊C4—202no—
483C22⋊C4—202no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—202no—
489C42—401no—
490C42—401no—
491C42C2×C4×D512yes—
492C42#70912yes—
493C2×C5#236011yescharacteristic
494C2×C5C4×#13511yescharacteristic
495C2×C5#294611yescharacteristic
496C2×C5#329611yescharacteristic
497C2×C5#329711yescharacteristic
498C2×C5#338011yescharacteristic
499C2×C5#338111yescharacteristic
500D5—44no—
501C23—102no—
502C23—102no—
503C23—102no—
504C23—201no—
505C23—201no—
506C23—201no—
507C23—201no—
508C23C22×D2011yescharacteristic
509D4—54no—
510D4—102no—
511D4—102no—
512D4—102no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—102no—
528C2×C4—102no—
529C2×C4—102no—
530C2×C4—102no—
531C2×C4—201no—
532C2×C4—201no—
533C2×C4—201no—
534C2×C4—201no—
535C2×C4—201no—
536C2×C4—201no—
537C2×C4—201no—
538C2×C4—201no—
539C2×C4—201no—
540C2×C4—201no—
541C2×C4—201no—
542C2×C4—201no—
543C2×C4—201no—
544C2×C4—201no—
545C2×C4—201no—
546C2×C4—201no—
547C2×C4—201no—
548C2×C4—404no—
549C2×C4—404no—
550C2×C4—404no—
551C2×C4C4×D2012yes—
552C2×C4C4×D2012yes—
553C2×C4#405012yes—
554C2×C4#405012yes—
555C2×C4C22×C4×D511yescharacteristic
556C2×C4C2×#70911yescharacteristic
557C2×C4#414411yescharacteristic
558C2×C4C2×Q8×D511yescharacteristic
559C2×C4D5×#4111yescharacteristic
560C2×C4#415011yescharacteristic
561C8—22no—
562C8—22no—
563C8—22no—
564C8—41no—
565C5#2296611yescharacteristic, complemented, Sylow
566C22—104no—
567C22—202no—
568C22—202no—
569C22—202no—
570C22—202no—
571C22—202no—
572C22—202no—
573C22C2×C4×D2011yescharacteristic
574C22#6588811yescharacteristic
575C22Q8×D2011yescharacteristic
576C22C2×#405011yescharacteristic
577C22C2×#405011yescharacteristic
578C22#6608511yescharacteristic
579C22#6608611yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—202no—
587C4—202no—
588C4—202no—
589C4—202no—
590C4—401no—
591C4—401no—
592C4—401no—
593C4#6584511yescharacteristic
594C4#6586311yescharacteristic
595C4#6588011yescharacteristic
596C4C4×Q8×D511yescharacteristic
597C2—204no—
598C2#1060550411yescharacteristic
599C2C4×#405011yescharacteristic
600C2#1061029111yescharacteristic
601C2#1061117111yescharacteristic
602C2#1061117211yescharacteristic
603C2#1061166111yescharacteristic
604C2#1061166211yescharacteristic
subgroupG/H
centerC23C22×D20
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×C8×C4⋊C4C2
socleC23×C5C22×D4
prime radicalC22×D20C23
cosocleC22×C4×C5C24
hypercenterC8×C4⋊C4D5
hypocenterC5#22966
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #10605390C2semidirect
4. C5×C8×C4⋊C4C2semidirect
5. #10601248C2semidirect
6. #10600571C2semidirect
7. #10600563C2semidirect
8. #10600562C2semidirect
14. #10599417C2semidirect
15. #10599416C2semidirect
45. #65136C4semidirect
100. #4016C2×C4semidirect
101. C5×C4⋊C8C2×C4semidirect
121. #3950C2×C4semidirect
123. #3949C2×C4semidirect
135. C8×C4⋊C4D5semidirect
367. C4⋊C8C4×D5semidirect
565. C5#22966semidirect

Maximal subgroups

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

subgroupindex
C2×#651362
#106055012
#106053902
C5×C8×C4⋊C42
#106012482
#106005712
#106005632
#106005622
C4×#39532
#105994642
#105994632
#105994172
#105994162
#229665