Finite Group Databasebeta

← #12135665

Subgroups — #12135665

Order 1280.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#65376C211yesmaximal, characteristic, complemented
3#10614231C211yesmaximal, characteristic, complemented
4C2×C4×#686C211yesmaximal, characteristic
5#10602585C212yesmaximal, complemented
6#10602580C211yesmaximal, characteristic, complemented
7#10602575C212yesmaximal, complemented
8#10602427C211yesmaximal, characteristic, complemented
9#10602406C212yesmaximal
10#10602399C212yesmaximal, complemented
11#10602397C211yesmaximal, characteristic, complemented
12C23×C5×Q8—21no—
13C5×D4×Q8—24no—
14C5×D4×Q8—24no—
15C2×C5×C4⋊Q8C2212yes—
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×C22⋊Q8—22no—
18C2×C5×C22⋊Q8—21no—
19C2×C4×C5×Q8C2211yescharacteristic
20C2×C4×C5×D4C2212yes—
21C2×C4×C5×D4C2211yescharacteristic
22C2×#4092—21no—
23C22×#686C2211yescharacteristic
24C2×C4×D20C2211yescharacteristic
25C2×#3964C2211yescharacteristic
26C2×#3964C2212yes—
27#65376—24no—
28C4×#686—24no—
29#65283—22no—
30#65276—21no—
31C2×#3941C2211yescharacteristic
32C2×#3941C2212yes—
33C2×#3939C2211yescharacteristic
34C2×#3939C2212yes—
35#65228C2212yes—
36#65224C2211yescharacteristic
37#65217C2212yes—
38C2×C4⋊#168C2211yescharacteristic
39C2×C4×#168C2211yescharacteristic
40#64749C2212yes—
41#64749C2212yes—
42#64749C2212yes—
43#64744C2212yes—
44#64744C2212yes—
45#24326—51nomaximal, Sylow
46C22×C5×Q8—28no—
47C22×C5×Q8—22no—
48C22×C5×Q8—21no—
49C22×C5×Q8—44no—
50C22×C5×Q8—44no—
51C22×C5×Q8D411yescharacteristic
52C22×C5×D4C2311yescharacteristic
53C23×C4×C5—22no—
54C23×C4×C5—21no—
55C22×D20C2311yescharacteristic
56C22×C4×D5—21no—
57C5×C4⋊Q8—24no—
58C5×C4⋊Q8—24no—
59C5×C4⋊Q8—22no—
60C5×C4⋊Q8D414yes—
61C5×C22⋊Q8—28no—
62C5×C22⋊Q8—216no—
63C4×C5×Q8—22no—
64C4×C5×Q8—22no—
65C4×C5×D4—24no—
66C4×C5×D4—24no—
67C4×C5×D4—22no—
68C4×C5×D4D414yes—
69C2×C5×C4⋊C4—22no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4D412yes—
72C2×C5×C4⋊C4D411yescharacteristic
73C2×C5×C4⋊C4C2312yes—
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C22⋊C4—22no—
76C2×C5×C22⋊C4—21no—
77C2×C42×C5C2312yes—
78C2×C42×C5C2311yescharacteristic
79#4092—44no—
80C2×#686—22no—
81C2×#686—22no—
82C2×#686Q814yescomplemented
83C2×#685—21no—
84C2×#685—22no—
85C2×#685—22no—
86C2×C4⋊Dic5—21no—
87C2×C4⋊Dic5—21no—
88C2×C4⋊Dic5C2311yescharacteristic
89C2×C4⋊Dic5C2312yes—
90C2×#683—22no—
91C22×#168C2311yescharacteristic
92C22×#168C2312yes—
93C22×#168C2311yescharacteristic
94C4×D20—22no—
95C4×D20—22no—
96#3964—22no—
97#3964—24no—
98#3963—22no—
99#3941—22no—
100#3941—24no—
101#3939—22no—
102#3939—24no—
103C4⋊#168—22no—
104C4×#168—22no—
105C2×D4×Q8D511yescharacteristic, complemented
106C2×#474—51no—
107C2×C4×D8—51no—
108#2538—51no—
109#2057—52no—
110#2023—52no—
111#1990—51no—
112#1984—51no—
113#1879—52no—
114#1862—52no—
115#1802—51no—
116C24×C5—21no—
117C23×D5—21no—
118C2×C5×Q8—28no—
119C2×C5×Q8—22no—
120C2×C5×Q8—24no—
121C2×C5×Q8—22no—
122C2×C5×Q8—44no—
123C2×C5×Q8—432no—
124C2×C5×Q8—44no—
125C2×C5×Q8—42no—
126C2×C5×Q8—42no—
127C2×C5×Q8D814yescomplemented
128C2×C5×D4—21no—
129C2×C5×D4—21no—
130C2×C5×D4—21no—
131C2×C5×D4C2×D412yes—
132C2×C5×D4C2×Q812yes—
133C2×C5×D4#4112yes—
134C22×C4×C5—28no—
135C22×C4×C5—22no—
136C22×C4×C5—21no—
137C22×C4×C5—24no—
138C22×C4×C5—44no—
139C22×C4×C5—44no—
140C22×C4×C5—42no—
141C22×C4×C5—42no—
142C22×C4×C5C2×D412yes—
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D412yes—
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2411yescharacteristic
147C22×Dic5—21no—
148C22×Dic5—22no—
149C2×D20—21no—
150C2×D20—21no—
151C2×D20—21no—
152C2×D20—21no—
153C2×D20—21no—
154C2×D20C2×Q812yescomplemented
155C2×C4×D5—42no—
156C2×C4×D5—42no—
157C2×C4×D5—42no—
158C5×C4⋊C4—28no—
159C5×C4⋊C4—24no—
160C5×C4⋊C4—24no—
161C5×C4⋊C4—22no—
162C5×C4⋊C4—22no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4C2×D414yes—
165C5×C4⋊C4C2×D412yes—
166C5×C22⋊C4—28no—
167C5×C22⋊C4—24no—
168C42×C5—22no—
169C42×C5—22no—
170C42×C5C2×D412yes—
171C42×C5C2×D412yes—
172#686—28no—
173#685—42no—
174#685—44no—
175#685—82no—
176C4⋊Dic5—21no—
177C4⋊Dic5—21no—
178C4⋊Dic5—22no—
179C4⋊Dic5—22no—
180C4⋊Dic5—42no—
181C4⋊Dic5—81no—
182#683—44no—
183C2×#168—21no—
184C2×#168—21no—
185C2×#168—21no—
186C2×#168—22no—
187C2×#168—22no—
188C2×#168—21no—
189C2×#168—21no—
190C2×#168—22no—
191C2×#168C2×Q812yescomplemented
192C23×Q8—21no—
193C22×D8—51no—
194D4×Q8—24no—
195D4×Q8—24no—
196C2×C4⋊Q8C2×D512yes—
197C2×C4⋊Q8C2×D511yescharacteristic
198C2×C22⋊Q8—21no—
199C2×C22⋊Q8—22no—
200C2×C22⋊Q8—101no—
201C2×C4×Q8C2×D511yescharacteristic
202C2×C4×D4—51no—
203C2×C4×D4C2×D512yes—
204C2×C4×D4C2×D511yescharacteristic
205#474—104no—
206C4×D8—104no—
207C2×#107—51no—
208C2×#107—52no—
209C2×C4⋊C8—51no—
210C2×#102—52no—
211C2×#102—51no—
212C2×#102—52no—
213C2×#102—51no—
214C2×C4×C8—51no—
215#396—102no—
216#390—52no—
217#389—52no—
218#384—51no—
219#380—101no—
220#340—52no—
221#340—52no—
222#340—52no—
223#336—52no—
224#336—52no—
225C23×C5—22no—
226C23×C5—22no—
227C23×C5—22no—
228C23×C5—41no—
229C23×C5—41no—
230C23×C5—41no—
231C23×C5—41no—
232C23×C5C22×D411yescharacteristic
233C22×D5—22no—
234C22×D5—41no—
235C22×D5—41no—
236C22×D5—41no—
237C22×D5—41no—
238C22×D5—41no—
239C22×D5—41no—
240C5×Q8—28no—
241C5×Q8—416no—
242C5×Q8—48no—
243C5×D4—22no—
244C5×D4—22no—
245C5×D4—22no—
246C5×D4C22⋊Q814yescomplemented
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—21no—
250C2×C4×C5—21no—
251C2×C4×C5—22no—
252C2×C4×C5—24no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—416no—
257C2×C4×C5—42no—
258C2×C4×C5—41no—
259C2×C4×C5—42no—
260C2×C4×C5—41no—
261C2×C4×C5—48no—
262C2×C4×C5#12012yes—
263C2×C4×C5#12012yes—
264C2×C4×C5C2×D814yes—
265C2×C4×C5C2×D812yes—
266C2×C4×C5AGL(1,ℤ/8)12yes—
267C2×C4×C5C22×D411yescharacteristic
268C2×C4×C5C22×D411yescharacteristic
269C2×C4×C5C22×Q811yescharacteristic
270C2×C4×C5C2×#4111yescharacteristic
271C2×C4×C52−1+411yescharacteristic
272C2×C4×C52−1+411yescharacteristic
273C2×Dic5—41no—
274C2×Dic5—42no—
275C2×Dic5—41no—
276C2×Dic5—42no—
277C2×Dic5—41no—
278C2×Dic5—42no—
279D20—22no—
280D20—22no—
281D20—41no—
282D20—41no—
283C4×D5—84no—
284#168—22no—
285#168—41no—
286#168—42no—
287C22×Q8—28no—
288C22×Q8—22no—
289C22×Q8—21no—
290C22×Q8—44no—
291C22×Q8—44no—
292C22×Q8C22⋊D511yescharacteristic
293C22×D4—51no—
294C22×D4C22×D511yescharacteristic
295C23×C4—22no—
296C23×C4—21no—
297C23×C4—101no—
298C2×D8—54no—
299C2×D8—102no—
300C2×D8—102no—
301C22×C8—51no—
302C22×C8—51no—
303C22×C8—52no—
304C4⋊Q8—24no—
305C4⋊Q8—24no—
306C4⋊Q8—22no—
307C4⋊Q8C22⋊D514yes—
308C22⋊Q8—28no—
309C22⋊Q8—216no—
310C22⋊Q8—204no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—22no—
316C4×D4—102no—
317C4×D4—102no—
318C4×D4C22⋊D514yes—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—51no—
322C2×C4⋊C4—52no—
323C2×C4⋊C4—101no—
324C2×C4⋊C4—102no—
325C2×C4⋊C4—101no—
326C2×C4⋊C4C22⋊D511yescharacteristic
327C2×C4⋊C4C22⋊D512yes—
328C2×C4⋊C4C22×D512yes—
329C2×C4⋊C4C22×D511yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—101no—
333C2×C22⋊C4—102no—
334C2×C22⋊C4—102no—
335C2×C42C22×D512yes—
336C2×C42C22×D511yescharacteristic
337#107—102no—
338#107—104no—
339C4⋊C8—102no—
340#102—104no—
341#102—102no—
342#102—102no—
343#102—104no—
344C4×C8—102no—
345#95—102no—
346C22×C5—24no—
347C22×C5—42no—
348C22×C5—42no—
349C22×C5—42no—
350C22×C5—42no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5C2×#12011yescharacteristic
354C22×C5#54811yescharacteristic
355C22×C5D4×Q811yescharacteristic
356C22×C5C22×D811yescharacteristic
357C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
358C22×C5#57711yescharacteristic
359C22×C5#57811yescharacteristic
360C2×D5—42no—
361C2×D5—42no—
362C2×D5—42no—
363C2×D5—81no—
364C2×D5—81no—
365C2×D5—81no—
366C2×D5—81no—
367C4×C5—24no—
368C4×C5—22no—
369C4×C5—44no—
370C4×C5—42no—
371C4×C5—42no—
372C4×C5#44714yes—
373C4×C5C2×C22⋊Q811yescharacteristic
374C4×C5#53611yescharacteristic
375C4×C5#54811yescharacteristic
376C4×C5D4×Q811yescharacteristic
377Dic5—82no—
378Dic5—81no—
379C24—21no—
380C24—101no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—44no—
386C2×Q8—432no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8#68614yescomplemented
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—52no—
395C2×D4—101no—
396C2×D4—101no—
397C2×D4—101no—
398C2×D4—101no—
399C2×D4—101no—
400C2×D4Q8×D512yes—
401C2×D4#71312yes—
402C2×D4C2×C22⋊D512yes—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—101no—
412C22×C4—102no—
413C22×C4—202no—
414C22×C4—202no—
415C22×C4—202no—
416C22×C4C2×C22⋊D512yes—
417C22×C4C2×C22⋊D511yescharacteristic
418C22×C4C2×C22⋊D512yes—
419C22×C4C2×C22⋊D511yescharacteristic
420C22×C4C23×D511yescharacteristic
421D8—108no—
422C2×C8—52no—
423C2×C8—101no—
424C2×C8—101no—
425C2×C8—102no—
426C2×C8—101no—
427C2×C8—101no—
428C2×C8—102no—
429C2×C8—101no—
430C2×C8—102no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—102no—
438C4⋊C4—102no—
439C4⋊C4—101no—
440C4⋊C4—101no—
441C4⋊C4—204no—
442C4⋊C4—202no—
443C4⋊C4—401no—
444C4⋊C4C2×C22⋊D514yes—
445C4⋊C4C2×C22⋊D512yes—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—202no—
449C22⋊C4—204no—
450C22⋊C4—402no—
451C42—22no—
452C42—22no—
453C42C2×C22⋊D512yes—
454C42C2×C22⋊D512yes—
455C2×C5—44no—
456C2×C5#240911yescharacteristic
457C2×C5C2×#44711yescharacteristic
458C2×C5#301311yescharacteristic
459C2×C5#335811yescharacteristic
460C2×C5#336711yescharacteristic
461C2×C5Q8×D811yescharacteristic
462C2×C5#338411yescharacteristic
463D5—82no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—102no—
472C23—201no—
473C23—201no—
474C23—201no—
475C23—201no—
476C23—201no—
477C23—201no—
478C23C22×C22⋊D511yescharacteristic
479Q8—28no—
480Q8—416no—
481Q8—48no—
482D4—22no—
483D4—22no—
484D4—22no—
485D4—102no—
486D4—102no—
487D4—201no—
488D4—201no—
489D4#409214yescomplemented
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—416no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—48no—
505C2×C4—202no—
506C2×C4—201no—
507C2×C4—201no—
508C2×C4—202no—
509C2×C4—202no—
510C2×C4—201no—
511C2×C4—404no—
512C2×C4C2×#68614yes—
513C2×C4C2×#68612yes—
514C2×C4#407812yes—
515C2×C4#409912yes—
516C2×C4#409912yes—
517C2×C4C2×Q8×D511yescharacteristic
518C2×C4C2×#71311yescharacteristic
519C2×C4#414711yescharacteristic
520C2×C4#414711yescharacteristic
521C2×C4C22×C22⋊D511yescharacteristic
522C2×C4C22×C22⋊D511yescharacteristic
523C8—102no—
524C8—201no—
525C8—202no—
526C5#2432611yescharacteristic, complemented, Sylow
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—202no—
535C22—202no—
536C22—202no—
537C22—401no—
538C22—401no—
539C22—401no—
540C22—401no—
541C22C22×#68611yescharacteristic
542C22C2×#407811yescharacteristic
543C22Q8×C22⋊D511yescharacteristic
544C22#6615111yescharacteristic
545C22#6617111yescharacteristic
546C22#6617211yescharacteristic
547C22C2×#409911yescharacteristic
548C4—22no—
549C4—24no—
550C4—42no—
551C4—44no—
552C4—42no—
553C4—401no—
554C4—402no—
555C4#6550614yes—
556C4C2×#409211yescharacteristic
557C4Q8×C22⋊D511yescharacteristic
558C4#6615011yescharacteristic
559C4#6615111yescharacteristic
560C2—44no—
561C2—402no—
562C2Q8×#68611yescharacteristic
563C2#1061529311yescharacteristic
564C2#1061529411yescharacteristic
565C2#1061529511yescharacteristic
566C2C2×#6550611yescharacteristic
567C2#1061579911yescharacteristic
568C2#1061584511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×D4×Q8C2
socleC23×C5C22×D4
prime radicalC22×#686C22
hypercenterC2×D4×Q8D5
hypocenterC5#24326
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×D4×Q8C2semidirect
2. C2×#65376C2semidirect
3. #10614231C2semidirect
5. #10602585C2semidirect
6. #10602580C2semidirect
7. #10602575C2semidirect
8. #10602427C2semidirect
10. #10602399C2semidirect
11. #10602397C2semidirect
82. C2×#686Q8semidirect
105. C2×D4×Q8D5semidirect
127. C2×C5×Q8D8semidirect
154. C2×D20C2×Q8semidirect
191. C2×#168C2×Q8semidirect
246. C5×D4C22⋊Q8semidirect
390. C2×Q8#686semidirect
489. D4#4092semidirect
526. C5#24326semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D4×Q82
C2×#653762
#106142312
C2×C4×#6862
#106025852
#106025802
#106025752
#106024272
#106024062
#106023992
#106023972
#243265