Finite Group Databasebeta

← #12964314

Subgroups — #12964314

Order 1280.

1702 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
1Q8×#709C211yesmaximal, characteristic
2C5×#2982C211yesmaximal, characteristic, complemented
3#10614168C211yesmaximal, characteristic, complemented
4#10611654C213yesmaximal, complemented
5#10611653C213yesmaximal, complemented
6#10611652C211yesmaximal, characteristic, complemented
7#10611651C211yesmaximal, characteristic, complemented
8Q8×#676C211yesmaximal, characteristic, complemented
9#10611649C213yesmaximal, complemented
10#10611648C213yesmaximal, complemented
11C8×Q8×D5C211yesmaximal, characteristic, complemented
12#10611016C213yesmaximal
13#10611014C213yesmaximal
14#10610998C213yesmaximal, complemented
15#10610997C213yesmaximal, complemented
16C2×C4×C5×Q8C2211yescharacteristic
17Q8×C22⋊D5C411yescharacteristic
18Q8×C22⋊D5C411yescharacteristic
19#65913C413yes—
20#65912C2213yes—
21#65912C2213yes—
22Q8×D20C411yescharacteristic
23C4×Q8×D5C2211yescharacteristic
24C4×Q8×D5C2211yescharacteristic
25#65901C413yes—
26Q8×Dic10C411yescharacteristic
27#65845C413yes—
28#65845C413yes—
29C4×#709C2213yes—
30#65804C2213yes—
31C5×C8×Q8C2211yescharacteristic
32C5×C8×Q8C2213yes—
33C5×C8×Q8C2213yes—
34C5×C8×Q8C2211yescharacteristic
35C5×#431C2213yes—
36C5×#431C2213yes—
37#65313C2213yes—
38#65313C2213yes—
39Q8×#168C2211yescharacteristic
40Q8×#168C2211yescharacteristic
41#65215C2213yes—
42#65214C2213yes—
43#65129C2213yes—
44#65129C2213yes—
45#65128C2213yes—
46#65128C2213yes—
47#65126C2213yes—
48#65124C2213yes—
49#65122C2213yes—
50D5×C4⋊C8C2213yes—
51#65120C2213yes—
52#65120C2213yes—
53#65119C2213yes—
54#65119C2213yes—
55#65117C2216yes—
56#65117C2216yes—
57#65016C2211yescharacteristic
58#65016C2213yes—
59#65009C2213yes—
60#65009C2216yes—
61#65009C2213yes—
62#64979C2216yes—
63#64978C2216yes—
64C4×#676C2213yes—
65#64974C2213yes—
66#64974C2213yes—
67C4×C8×D5C2213yes—
68#64968C2213yes—
69#64968C2213yes—
70#64968C2216yes—
71C8×Dic10C2213yes—
72C8×Dic10C2213yes—
73C8×Dic10C2216yes—
74#60253—51nomaximal, Sylow
75C22×C5×Q8C2×C411yescharacteristic
76C2×Q8×D5C2×C411yescharacteristic
77C2×Q8×D5C2×C411yescharacteristic
78C2×#709C2311yescharacteristic
79C4×C5×Q8C2×C411yescharacteristic
80C4×C5×Q8C2×C413yes—
81C4×C5×Q8C2311yescharacteristic
82C4×C5×Q8C2313yes—
83C2×C5×C4⋊C4C2×C413yes—
84C2×C42×C5C2313yes—
85#4092C2×C413yes—
86#4092C2×C413yes—
87Q8×Dic5C2×C411yescharacteristic
88Q8×Dic5C2×C411yescharacteristic
89#4090C2×C413yes—
90#4090C2×C413yes—
91C4×C22⋊D5C2×C413yes—
92C4×C22⋊D5C2×C413yes—
93C4×C22⋊D5C2×C413yes—
94C4×C22⋊D5C2×C413yes—
95#4072C2313yes—
96#4070C2×C416yes—
97#4070C2×C413yes—
98#4070C2×C413yes—
99C2×#676C2313yes—
100C2×#676C2311yescharacteristic
101C2×C8×D5C2313yes—
102C2×C8×D5C2311yescharacteristic
103#4043C2×C413yes—
104#4043C2×C413yes—
105#4042C2×C416yes—
106#4042C2×C416yes—
107D5×C4⋊C4C2×C413yes—
108D5×C4⋊C4C2×C413yes—
109#4035C2×C416yes—
110#4035C2×C416yes—
111#4034C2×C413yes—
112#4034C2×C413yes—
113#4033C2×C413yes—
114#4033C2×C413yes—
115C4×D20C2×C413yes—
116C4×D20C2×C413yes—
117#4018C2313yes—
118#4018C2313yes—
119C42×D5C2313yes—
120C42×D5C2313yes—
121#4016C2×C416yes—
122#4015C2×C413yes—
123#4015C2×C413yes—
124C4×Dic10C2×C413yes—
125C4×Dic10C2×C413yes—
126C4×Dic10C2316yes—
127C4×Dic10C2313yes—
128C4×Dic10C2313yes—
129C5×C4⋊C8C2313yes—
130C5×C4⋊C8C2313yes—
131C5×C4⋊C8C2313yes—
132C5×C4⋊C8C2313yes—
133C5×C22⋊C8C2311yescharacteristic
134C5×C22⋊C8C2313yes—
135C4×C5×C8C2313yes—
136C4×C5×C8C2316yes—
137C4×C5×C8C2313yes—
138C22⋊#168C2311yescharacteristic
139C22⋊#168C2313yes—
140#3952C2313yes—
141#3952C2313yes—
142#3952C2311yescharacteristic
143#3952C2311yescharacteristic
144#3947C2313yes—
145#3947C2316yes—
146#3947C2313yes—
147#3946C2313yes—
148#3946C2313yes—
149#3946C2316yes—
150#3946C2316yes—
151#3946C2313yes—
152#3946C2313yes—
153C8×Dic5C2313yes—
154C8×Dic5C2316yes—
155C8×Dic5C2313yes—
156C4⋊#168C2313yes—
157C4⋊#168C2313yes—
158C4⋊#168C2313yes—
159C4⋊#168C2313yes—
160#3935C2316yes—
161C4×#168C2313yes—
162C4×#168C2313yes—
163Q8×#41—51no—
164#2986—53no—
165#2986—53no—
166#2985—53no—
167#2985—53no—
168#2984—51no—
169#2984—51no—
170#2982—51no—
171#2982D511yescharacteristic, complemented
172Q8×M16—51no—
173#2962—53no—
174#2960—53no—
175C2×C8×Q8—51no—
176#2925—53no—
177#2922—53no—
178C2×C5×Q8—24no—
179C2×C5×Q8C22×C413yes—
180C2×C5×Q8C22×C411yescharacteristic
181C22×C4×C5C22×C413yes—
182C22×C4×C5C22×C413yes—
183C22×C4×C5C2411yescharacteristic
184C2×C22⋊D5C22×C411yescharacteristic
185C2×C22⋊D5C22×C411yescharacteristic
186Q8×D5—24no—
187Q8×D5—24no—
188#709—22no—
189#709—22no—
190C2×D20C22×C411yescharacteristic
191C2×C4×D5C22×C413yes—
192C2×C4×D5C22×C413yes—
193C2×C4×D5C22×C413yes—
194C2×C4×D5C22×C413yes—
195C2×C4×D5C2411yescharacteristic
196C2×C4×D5C2411yescharacteristic
197C2×Dic10C22×C411yescharacteristic
198C2×Dic10C22×C413yes—
199C2×Dic10C22×C413yes—
200C2×Dic10C22×C416yes—
201C2×C5×C8C2411yescharacteristic
202C2×C5×C8C2413yes—
203C2×C5×C8C2411yescharacteristic
204C2×C5×C8C2413yes—
205C5×C4⋊C4C22×C413yes—
206C5×C4⋊C4C22×C413yes—
207C5×C4⋊C4C22×C416yes—
208C42×C5C22×C413yes—
209C42×C5C22×C413yes—
210C42×C5C2413yes—
211C42×C5C2413yes—
212C22⋊Dic5C22×C413yes—
213C22⋊Dic5C22×C413yes—
214#685C22×C413yes—
215#685C22×C413yes—
216#685C22×C413yes—
217#685C22×C413yes—
218C4⋊Dic5C22×C413yes—
219C4⋊Dic5C22×C413yes—
220C4⋊Dic5C22×C413yes—
221C4⋊Dic5C22×C413yes—
222C4⋊Dic5C22×C416yes—
223#683C22×C413yes—
224#683C22×C413yes—
225#683C22×C413yes—
226#683C22×C413yes—
227#683C22×C416yes—
228#683C22×C416yes—
229#683C22×C416yes—
230#683C22×C416yes—
231C4×Dic5C22×C413yes—
232C4×Dic5C22×C413yes—
233C4×Dic5C22×C413yes—
234C4×Dic5C22×C413yes—
235C4×Dic5C2413yes—
236C4×Dic5C2413yes—
237C2×#168C2411yescharacteristic
238C2×#168C2411yescharacteristic
239C2×#168C2413yes—
240C2×#168C2413yes—
241#676—22no—
242#676—26no—
243C8×D5—26no—
244C8×D5—22no—
245Q82—51no—
246#557—53no—
247#554—53no—
248#554—53no—
249#554—53no—
250D4×Q8—51no—
251D4×Q8—51no—
252D4×Q8—51no—
253#533—53no—
254#533—53no—
255#533—53no—
256C4×#41—53no—
257C2×C4×Q8—51no—
258C2×C4×Q8—51no—
259C2×C4×Q8C2×D511yescharacteristic
260#446—53no—
261#446—53no—
262#446—56no—
263#446—56no—
264#446—53no—
265#446—53no—
266#446—53no—
267#446—56no—
268#446—53no—
269C8×Q8—51no—
270C8×Q8—51no—
271C8×Q8—53no—
272C8×Q8—53no—
273C8×Q8—53no—
274C8×Q8—53no—
275C8×Q8—56no—
276C8×Q8C2×D511yescharacteristic
277C8×Q8C2×D513yes—
278C8×Q8C2×D513yes—
279C8×Q8C2×D511yescharacteristic
280#433—53no—
281#433—53no—
282#433—53no—
283#433—53no—
284#433—56no—
285#433—56no—
286#432—53no—
287#432—53no—
288#431—53no—
289#431—53no—
290#431—53no—
291#431—53no—
292#431C2×D513yes—
293#431C2×D513yes—
294#424—53no—
295#424—53no—
296#423—53no—
297C2×C4⋊C8—53no—
298#408—51no—
299#408—53no—
300#405—56no—
301C4×M16—53no—
302C2×C4×C8—53no—
303C23×C5C23×C411yescharacteristic
304C22×D5C23×C411yescharacteristic
305C22×D5C23×C411yescharacteristic
306C5×Q8—26no—
307C5×Q8#13114yes—
308C2×C4×C5—26no—
309C2×C4×C5—21no—
310C2×C4×C5—21no—
311C2×C4×C5—43no—
312C2×C4×C5C23×C413yes—
313C2×C4×C5C23×C413yes—
314C2×C4×C5C23×C413yes—
315C2×C4×C5C23×C411yescharacteristic
316C2×C4×C5C23×C413yes—
317C2×C4×C5C2511yescharacteristic
318C22⋊D5—22no—
319C22⋊D5—22no—
320C2×Dic5C23×C411yescharacteristic
321C2×Dic5C23×C413yes—
322C2×Dic5C23×C413yes—
323C2×Dic5C23×C413yes—
324C2×Dic5C23×C413yes—
325C2×Dic5C23×C411yescharacteristic
326D20—22no—
327C4×D5—26no—
328C4×D5—26no—
329C4×D5—21no—
330C4×D5—21no—
331C4×D5—43no—
332C4×D5—43no—
333C4×D52−1+412yes—
334C4×D52−1+412yes—
335Dic10—26no—
336Dic10—26no—
337Dic10—22no—
338Dic10—46no—
339C5×C8—23no—
340C5×C8—21no—
341C5×C8—21no—
342C5×C8—23no—
343#168—21no—
344#168—23no—
345#168—21no—
346#168—23no—
347C2×#41—51no—
348C22×Q8—51no—
349C22×Q8—51no—
350C22×Q8C4×D511yescharacteristic
351C2×M16—51no—
352C2×M16—53no—
353C22×C8—51no—
354C22×C8—53no—
355C4⋊Q8—53no—
356C4⋊Q8—53no—
357C4⋊Q8—53no—
358C4⋊Q8—53no—
359C4⋊Q8—53no—
360C4⋊Q8—53no—
361#125—56no—
362#125—56no—
363#125—56no—
364C22⋊Q8—53no—
365C22⋊Q8—53no—
366C22⋊Q8—53no—
367C22⋊Q8—53no—
368C22⋊Q8—56no—
369C22⋊Q8—53no—
370C22⋊Q8—53no—
371C22⋊Q8—56no—
372C22⋊Q8—56no—
373C4×Q8—51no—
374C4×Q8—51no—
375C4×Q8—53no—
376C4×Q8—53no—
377C4×Q8—53no—
378C4×Q8—53no—
379C4×Q8—53no—
380C4×Q8—53no—
381C4×Q8—56no—
382C4×Q8C4×D511yescharacteristic
383C4×Q8C4×D513yes—
384C4×Q8C22×D511yescharacteristic
385C4×Q8C22×D513yes—
386C4×D4—53no—
387C4×D4—53no—
388C4×D4—53no—
389C4×D4—53no—
390C4×D4—53no—
391C4×D4—53no—
392#117—53no—
393#117—53no—
394#117—53no—
395C2×C4⋊C4—53no—
396C2×C4⋊C4—53no—
397C2×C4⋊C4C4×D513yes—
398C2×C42—53no—
399C2×C42—53no—
400C2×C42C22×D513yes—
401C4⋊C8—56no—
402C4⋊C8—56no—
403C4⋊C8—53no—
404C4⋊C8—53no—
405C4⋊C8—53no—
406C4⋊C8—53no—
407C4⋊C8—53no—
408C4⋊C8—53no—
409C4⋊C8—53no—
410C4⋊C8—53no—
411C4⋊C8C22×D513yes—
412C4⋊C8C22×D513yes—
413C4⋊C8C22×D513yes—
414C4⋊C8C22×D513yes—
415C22⋊C8—51no—
416C22⋊C8—53no—
417C22⋊C8—51no—
418C22⋊C8—53no—
419C22⋊C8—51no—
420C22⋊C8—53no—
421C22⋊C8C22×D511yescharacteristic
422C22⋊C8C22×D513yes—
423#97—53no—
424#97—53no—
425#97—56no—
426#97—56no—
427C4×C8—53no—
428C4×C8—53no—
429C4×C8—56no—
430C4×C8—53no—
431C4×C8—53no—
432C4×C8C22×D513yes—
433C4×C8C22×D513yes—
434C4×C8C22×D516yes—
435C22×C5—21no—
436C22×C5—21no—
437C22×C5—21no—
438C22×C5C24×C411yescharacteristic
439C2×D5—21no—
440C2×D5—21no—
441C2×D5—21no—
442C2×D5—21no—
443C2×D5—21no—
444C2×D5#51912yes—
445C4×C5—21no—
446C4×C5—23no—
447C4×C5—23no—
448C4×C5—23no—
449C4×C5C2×#13116yes—
450C4×C5C2×2−1+411yescharacteristic
451C4×C5C2×2−1+411yescharacteristic
452Dic5—21no—
453Dic5—23no—
454Dic5—23no—
455Dic5—23no—
456Dic5—23no—
457Dic5#51912yes—
458#41—102no—
459#41—102no—
460C2×Q8—24no—
461C2×Q8—51no—
462C2×Q8—53no—
463C2×Q8—53no—
464C2×Q8—56no—
465C2×Q8—104no—
466C2×Q8—104no—
467C2×Q8C2×C4×D513yes—
468C2×Q8C2×C4×D511yescharacteristic
469C2×D4—51no—
470C2×D4—51no—
471C2×D4—51no—
472C22×C4—51no—
473C22×C4—53no—
474C22×C4—53no—
475C22×C4—51no—
476C22×C4—53no—
477C22×C4—53no—
478C22×C4C2×C4×D513yes—
479C22×C4C2×C4×D513yes—
480C22×C4C23×D511yescharacteristic
481M16—102no—
482M16—106no—
483C2×C8—51no—
484C2×C8—53no—
485C2×C8—51no—
486C2×C8—53no—
487C2×C8—106no—
488C2×C8—102no—
489C2×C8C23×D513yes—
490C2×C8C23×D511yescharacteristic
491C2×C8C23×D511yescharacteristic
492C2×C8C23×D513yes—
493C4⋊C4—56no—
494C4⋊C4—56no—
495C4⋊C4—53no—
496C4⋊C4—53no—
497C4⋊C4—53no—
498C4⋊C4—53no—
499C4⋊C4—53no—
500C4⋊C4—53no—
501C4⋊C4—53no—
502C4⋊C4—53no—
503C4⋊C4—56no—
504C4⋊C4—56no—
505C4⋊C4—56no—
506C4⋊C4C2×C4×D513yes—
507C4⋊C4C2×C4×D516yes—
508C4⋊C4C2×C4×D513yes—
509C22⋊C4—53no—
510C22⋊C4—53no—
511C22⋊C4—53no—
512C22⋊C4—53no—
513C22⋊C4—53no—
514C22⋊C4—53no—
515C42—53no—
516C42—53no—
517C42—53no—
518C42—53no—
519C42—53no—
520C42—53no—
521C42C2×C4×D513yes—
522C42C2×C4×D513yes—
523C42C23×D513yes—
524C42C23×D513yes—
525C2×C5—41no—
526C2×C5C2×#51911yescharacteristic
527C2×C5C22×#13111yescharacteristic
528C2×C5#357311yescharacteristic
529D5—22no—
530D5—41no—
531C23—51no—
532C23—51no—
533C23C22×C4×D511yescharacteristic
534Q8—26no—
535Q8—102no—
536Q8—106no—
537Q8—106no—
538Q8—206no—
539Q8#404714yes—
540D4—102no—
541D4—102no—
542D4—102no—
543C2×C4—26no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—43no—
547C2×C4—51no—
548C2×C4—52no—
549C2×C4—52no—
550C2×C4—51no—
551C2×C4—53no—
552C2×C4—53no—
553C2×C4—53no—
554C2×C4—53no—
555C2×C4—106no—
556C2×C4—106no—
557C2×C4—101no—
558C2×C4—101no—
559C2×C4—203no—
560C2×C4—203no—
561C2×C4C22×C4×D513yes—
562C2×C4C22×C4×D511yescharacteristic
563C2×C4C22×C4×D513yes—
564C2×C4C22×C4×D513yes—
565C2×C4C22×C4×D513yes—
566C2×C4C24×D511yescharacteristic
567C8—23no—
568C8—21no—
569C8—21no—
570C8—23no—
571C8—101no—
572C8—103no—
573C8—101no—
574C8—103no—
575C5#6025311yescharacteristic, complemented, Sylow
576C22—21no—
577C22—21no—
578C22—21no—
579C22—52no—
580C22—101no—
581C22—101no—
582C22—101no—
583C22—101no—
584C22—101no—
585C22C23×C4×D511yescharacteristic
586C4—23no—
587C4—21no—
588C4—23no—
589C4—23no—
590C4—52no—
591C4—103no—
592C4—103no—
593C4—101no—
594C4—103no—
595C4—103no—
596C4C2×#404716yes—
597C4D5×2−1+411yescharacteristic
598C4D5×2−1+411yescharacteristic
599C2—41no—
600C2—102no—
601C2—201no—
602C2D5×#51911yescharacteristic
603C2C22×#404711yescharacteristic
604C2#1061906711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D5
commutator, socleC22×C5C24×C4
FittingC5×#2982C2
prime radicalC2×#709C23
cosocleC2×C4×C5C25
hypercenter#2982D5
hypocenterC5#60253
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C5×#2982C2semidirect
3. #10614168C2semidirect
4. #10611654C2semidirect
5. #10611653C2semidirect
6. #10611652C2semidirect
7. #10611651C2semidirect
8. Q8×#676C2semidirect
9. #10611649C2semidirect
10. #10611648C2semidirect
11. C8×Q8×D5C2semidirect
14. #10610998C2semidirect
15. #10610997C2semidirect
171. #2982D5semidirect
575. C5#60253semidirect

Maximal subgroups

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

subgroupindex
Q8×#7092
C5×#29822
#106141682
#106116542
#106116532
#106116522
#106116512
Q8×#6762
#106116492
#106116482
C8×Q8×D52
#106110162
#106110142
#106109982
#106109972
#602535