Finite Group Databasebeta

← #227390

Subgroups — #227390

Order 512.

790 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23002C211yesmaximal, characteristic, complemented
2#22962C211yesmaximal, characteristic, complemented
3#22503C211yesmaximal, characteristic
4#22503C211yesmaximal, characteristic
5C2×#1646C211yesmaximal, characteristic
6#9833C211yesmaximal, characteristic, complemented
7#9576C211yesmaximal, characteristic, complemented
8#9568C211yesmaximal, characteristic, complemented
9#9227C211yesmaximal, characteristic, complemented
10#9225C211yesmaximal, characteristic, complemented
11#9065C211yesmaximal, characteristic, complemented
12#8882C211yesmaximal, characteristic, complemented
13#8882C211yesmaximal, characteristic, complemented
14#8210C211yesmaximal, characteristic, complemented
15#8208C211yesmaximal, characteristic, complemented
16C2×#555C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×#461C2211yescharacteristic
20C2×#440—21no—
21C2×#436C2211yescharacteristic
22C2×#436C2211yescharacteristic
23#2365C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25D4×C4⋊C4C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1987C2211yescharacteristic
30#1987C2211yescharacteristic
31#1956C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1942—21no—
35#1935—21no—
36#1918—21no—
37#1895—21no—
38#1895—21no—
39#1877—21no—
40#1874—21no—
41#1874—21no—
42#1863—21no—
43#1862—21no—
44#1845C2211yescharacteristic
45#1825—21no—
46#1825—21no—
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1770C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C4×#102—21no—
52C4×#102C2211yescharacteristic
53C4×#102C2211yescharacteristic
54#1736—21no—
55#1736—21no—
56#1646C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×M16—21no—
70C22×M16—21no—
71#555—22no—
72#555—22no—
73#555—41no—
74#555C2×C414yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×#124—41no—
78C2×#121—21no—
79C2×#121—21no—
80C2×C4×Q8—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4C2311yescharacteristic
84C2×C4×D4C2311yescharacteristic
85C2×#117—21no—
86C2×#117—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89#487—41no—
90#487C2×C414yescomplemented
91#474—22no—
92#474C2×C414yescomplemented
93#461—22no—
94#461C2×C414yescomplemented
95#440—44no—
96#436—22no—
97#436—22no—
98#436—41no—
99#436—41no—
100C2×#107—21no—
101C2×#107—21no—
102C2×#107—21no—
103C2×#107—41no—
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—41no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102D411yescharacteristic
116C2×#102D411yescharacteristic
117C2×#102D411yescharacteristic
118C2×#102D411yescharacteristic
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#387—41no—
131#385—41no—
132#385—41no—
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#380—21no—
136#380—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#336—21no—
159#336—21no—
160#333C2×C414yescomplemented
161#332—41no—
162D4⋊C8C2×C414yescomplemented
163D4⋊C8C2×C414yescomplemented
164#322—41no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×D4—41no—
168C22×D4C2×D411yescharacteristic
169C22×D4C2×D411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—41no—
173C23×C4—41no—
174C2×SD16—24no—
175C2×SD16—42no—
176C2×SD16—42no—
177C2×M16—42no—
178C2×M16—42no—
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—81no—
182C2×M16—81no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8C22×C412yes—
199C4⋊Q8C22×C412yes—
200#124—44no—
201#124—44no—
202#121—24no—
203#121—24no—
204#121—42no—
205#121—42no—
206C4×Q8—42no—
207C4×Q8—81no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—81no—
215C4×D4—81no—
216C4×D4C22×C412yes—
217C4×D4C22×C412yes—
218C4×D4#4112yes—
219C4×D4#4112yes—
220#117—24no—
221#117—24no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—81no—
240C2×C4⋊C4—81no—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42C2411yescharacteristic
258#107—24no—
259#107—42no—
260#107—81no—
261#107—82no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8C22×C412yes—
269C4⋊C8C22×C412yes—
270#103—24no—
271#103—44no—
272#103—42no—
273#103—81no—
274#102—24no—
275#102—22no—
276#102—22no—
277#102—41no—
278#102—41no—
279#102—81no—
280#102—81no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283#97—42no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—81no—
290C4×C8C22×C412yes—
291#95—21no—
292#95—21no—
293#95—41no—
294#95—41no—
295#95—41no—
296#95—41no—
297C24—21no—
298C24—21no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—81no—
308C2×Q8—81no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—44no—
320C2×D4—161no—
321C2×D4—161no—
322C2×D4C4×D412yes—
323C2×D4C4×D412yes—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—161no—
351C22×C4—161no—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355SD16—48no—
356M16—84no—
357M16—84no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8#13712yes—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C4⋊C4—82no—
415C4⋊C4C4×D412yes—
416C4⋊C4C4×D412yes—
417C4⋊C4AGL(1,ℤ/8)12yes—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C42—44no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—82no—
432C42C23×C411yescharacteristic
433C42C2×#4111yescharacteristic
434C42C2×#4111yescharacteristic
435C42H(5,𝔽2)11yescharacteristic
436C23—22no—
437C23—22no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23D4211yescharacteristic
451Q8—42no—
452Q8—42no—
453Q8—82no—
454Q8—81no—
455Q8—81no—
456Q8—161no—
457D4—22no—
458D4—22no—
459D4—22no—
460D4—22no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—164no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—164no—
507C2×C4—164no—
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4#51811yescharacteristic
511C2×C4#54611yescharacteristic
512C2×C4#54611yescharacteristic
513C2×C4#55111yescharacteristic
514C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
515C2×C4C2×#13711yescharacteristic
516C2×C4#57511yescharacteristic
517C2×C4#57511yescharacteristic
518C2×C4#57611yescharacteristic
519C2×C4#57611yescharacteristic
520C2×C4#57811yescharacteristic
521C2×C4#57811yescharacteristic
522C8—22no—
523C8—42no—
524C8—41no—
525C8—42no—
526C8—42no—
527C8—41no—
528C8—41no—
529C8—81no—
530C8—81no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22#235811yescharacteristic
546C22#326411yescharacteristic
547C22#326911yescharacteristic
548C22#327311yescharacteristic
549C22#327611yescharacteristic
550C22#328811yescharacteristic
551C22#328911yescharacteristic
552C4—21no—
553C4—21no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—82no—
558C4—82no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—161no—
563C4—161no—
564C4C4×AGL(1,ℤ/8)11yescharacteristic
565C4C4×#13711yescharacteristic
566C4#295111yescharacteristic
567C4#295111yescharacteristic
568C4#331411yescharacteristic
569C4#331511yescharacteristic
570C4#332211yescharacteristic
571C4#333011yescharacteristic
572C2—82no—
573C2—82no—
574C2#2015911yescharacteristic
575C2#2016711yescharacteristic
576C2#2022311yescharacteristic
577C2#2023111yescharacteristic
578C2#2286211yescharacteristic
579C2#2287811yescharacteristic
580C2#2289011yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#555C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23002C2semidirect
2. #22962C2semidirect
6. #9833C2semidirect
7. #9576C2semidirect
8. #9568C2semidirect
9. #9227C2semidirect
10. #9225C2semidirect
11. #9065C2semidirect
12. #8882C2semidirect
13. #8882C2semidirect
14. #8210C2semidirect
15. #8208C2semidirect
56. #1646C4semidirect
90. #487C2×C4semidirect
92. #474C2×C4semidirect
94. #461C2×C4semidirect
160. #333C2×C4semidirect
162. D4⋊C8C2×C4semidirect
163. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#230022
#229622
#225032
C2×#16462
#98332
#95762
#95682
#92272
#92252
#90652
#88822
#82102
#82082