Finite Group Databasebeta

← #189170

Subgroups — #189170

Order 512.

936 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2994C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3#21887C211yesmaximal, characteristic
4#21867C211yesmaximal, characteristic
5C2×#1573C211yesmaximal, characteristic, complemented
6#8657C211yesmaximal, characteristic, complemented
7#8656C211yesmaximal, characteristic, complemented
8#8651C211yesmaximal, characteristic, complemented
9#8650C211yesmaximal, characteristic, complemented
10#8649C211yesmaximal, characteristic, complemented
11#8648C211yesmaximal, characteristic, complemented
12#8643C211yesmaximal, characteristic, complemented
13#8642C211yesmaximal, characteristic, complemented
14#8044C211yesmaximal, characteristic, complemented
15C2×#1296C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2994—24no—
18#2994C418yescomplemented
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#435—21no—
23C2×C8×D4—21no—
24C2×C8×D4C2211yescharacteristic
25C2×#433—21no—
26C2×#432C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#424—21no—
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31#2285C2211yescharacteristic
32C2×#384C2211yescharacteristic
33#1992—21no—
34#1989—21no—
35#1983—21no—
36#1959—21no—
37#1948—21no—
38#1948—21no—
39#1893—21no—
40#1892—21no—
41#1874—21no—
42#1872—21no—
43#1853—21no—
44#1853—21no—
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1802C2211yescharacteristic
54#1802C2211yescharacteristic
55#1802C2211yescharacteristic
56#1802C2211yescharacteristic
57#1573—24no—
58C2×#335C2211yescharacteristic
59C2×#335C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#334C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70C22×M16—21no—
71C23×C8—21no—
72C23×C8—21no—
73C23×C8—21no—
74#529—22no—
75#529—22no—
76#529C2×C414yes—
77#529Q814yes—
78C2×#126—21no—
79C2×#125D411yescharacteristic
80C2×#124D411yescharacteristic
81C2×#123—21no—
82C2×C22⋊Q8D411yescharacteristic
83C2×#121D411yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88C2×#117—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C42C2311yescharacteristic
93#446—22no—
94#446C2×C414yescomplemented
95C8×Q8—22no—
96C8×Q8C2×C414yescomplemented
97#436—22no—
98#436C2×C414yescomplemented
99#435—28no—
100C8×D4—28no—
101C8×D4—22no—
102C8×D4C2×C414yescomplemented
103#433—28no—
104#432—22no—
105#432—22no—
106#431C2×C414yes—
107#431D414yes—
108C2×#106—21no—
109C2×#106—21no—
110C2×#106—21no—
111C2×#106—21no—
112#424—28no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8C2×C414yes—
121C2×C4⋊C8D411yescharacteristic
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×#103—21no—
130C2×#103—21no—
131C2×#103—21no—
132C2×#103—21no—
133C2×#102—21no—
134C2×#102—21no—
135C2×#102—21no—
136C2×#102—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8D411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×#97—21no—
143C2×#97—21no—
144C2×C4×C8—21no—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#384—21no—
148#384—21no—
149#384—21no—
150#384C2311yescharacteristic
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#384C2311yescharacteristic
154#382—21no—
155#380—21no—
156#380—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159#340—21no—
160#340—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#340—21no—
171#335—22no—
172#335—22no—
173#334—22no—
174#334—22no—
175C22×Q8C2×D411yescharacteristic
176C22×D4—21no—
177C22×D4C2×D411yescharacteristic
178C23×C4—21no—
179C23×C4—41no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4111yescharacteristic
183C2×M16—24no—
184C2×M16—81no—
185C2×M16—81no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—42no—
195C22×C8—81no—
196C22×C8—81no—
197C22×C8—81no—
198C22×C8—81no—
199C22×C8—81no—
200C22×C8—81no—
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205#126—24no—
206#126—24no—
207#125—22no—
208#125C4⋊C414yes—
209#124—22no—
210#124C4⋊C414yes—
211#123—24no—
212#123—24no—
213C22⋊Q8—22no—
214C22⋊Q8C4⋊C414yes—
215#121—22no—
216#121C4⋊C414yes—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8C22×C412yes—
220C4×Q8C2×Q812yes—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4C22×C412yes—
226C4×D4C2×Q812yes—
227#117—24no—
228#117—24no—
229#117—82no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—81no—
250C2×C4⋊C4—81no—
251C2×C4⋊C4—81no—
252C2×C4⋊C4—81no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—41no—
262C2×C42—21no—
263C2×C42—41no—
264C2×C42C22×C412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×Q812yes—
269C2×C42#4111yescharacteristic
270C2×C42#4112yes—
271C2×C42C2411yescharacteristic
272#106—81no—
273#106—81no—
274#106—81no—
275#106—81no—
276C4⋊C8—24no—
277C4⋊C8—24no—
278C4⋊C8—24no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—42no—
290C4⋊C8C22⋊C414yes—
291C4⋊C8C22⋊C414yes—
292C4⋊C8C22×C412yes—
293C4⋊C8C22×C412yes—
294#103—81no—
295#103—81no—
296#103—81no—
297#103—81no—
298#102—81no—
299#102—81no—
300#102—81no—
301#102—81no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304C22⋊C8—42no—
305C22⋊C8C22⋊C414yes—
306C22⋊C8C22⋊C414yes—
307#97—24no—
308#97—42no—
309C4×C8—24no—
310C4×C8C22×C412yes—
311C4×C8C22×C412yes—
312C4×C8C2×D412yes—
313C4×C8C2×D412yes—
314#95—41no—
315#95—41no—
316C24—21no—
317C24#12111yescharacteristic
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8C2×C4⋊C412yes—
324C2×D4—24no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4C2×C4⋊C412yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C4⋊C412yes—
361C22×C4#11712yes—
362C22×C4#12112yes—
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4C22⋊Q812yes—
367C22×C4C22⋊Q812yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371M16—84no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—84no—
399C2×C8—84no—
400C2×C8—84no—
401C2×C8C2×C22⋊C412yes—
402C2×C8C2×C22⋊C412yes—
403C2×C8C2×C22⋊C412yes—
404C2×C8C2×C22⋊C412yes—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4C2×C4⋊C412yes—
436C4⋊C4C2×C4⋊C412yes—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—82no—
444C42—22no—
445C42—22no—
446C42—82no—
447C42C2×C22⋊C412yes—
448C42C2×C4⋊C412yes—
449C42#11712yes—
450C42#12112yes—
451C42C22⋊Q812yes—
452C42C22⋊Q812yes—
453C42C23×C411yescharacteristic
454C42C22×D411yescharacteristic
455C42C22×Q811yescharacteristic
456C42C2×#4111yescharacteristic
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23#38012yes—
469C23#47112yes—
470C23#47112yes—
471C23C2×#12111yescharacteristic
472Q8—22no—
473Q8—22no—
474Q8—41no—
475Q8—41no—
476D4—22no—
477D4—22no—
478D4—48no—
479D4—41no—
480D4—41no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—164no—
525C2×C4#38012yes—
526C2×C4#38012yes—
527C2×C4#38012yes—
528C2×C4C22×C22⋊C411yescharacteristic
529C2×C4C22×C4⋊C411yescharacteristic
530C2×C4C2×#11711yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4C2×C22⋊Q811yescharacteristic
534C2×C4#57611yescharacteristic
535C2×C4#57611yescharacteristic
536C2×C4#57711yescharacteristic
537C2×C4#57711yescharacteristic
538C2×C4#57711yescharacteristic
539C2×C4#57711yescharacteristic
540C2×C4#57811yescharacteristic
541C2×C4#57811yescharacteristic
542C8—22no—
543C8—22no—
544C8—22no—
545C8—22no—
546C8—42no—
547C8—42no—
548C8—81no—
549C8—81no—
550C8—81no—
551C8—81no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22#181514yes—
566C22C2×#38011yescharacteristic
567C22#304511yescharacteristic
568C22#304611yescharacteristic
569C22#306211yescharacteristic
570C22#306311yescharacteristic
571C22C2×#47111yescharacteristic
572C22C2×#47111yescharacteristic
573C4—22no—
574C4—22no—
575C4—21no—
576C4—21no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—161no—
582C4—161no—
583C4—161no—
584C4—161no—
585C4#289711yescharacteristic
586C4#289811yescharacteristic
587C4#291211yescharacteristic
588C4#291311yescharacteristic
589C4#306211yescharacteristic
590C4#306311yescharacteristic
591C4#308311yescharacteristic
592C4#308411yescharacteristic
593C2—24no—
594C2—82no—
595C2#1958011yescharacteristic
596C2#1958111yescharacteristic
597C2#1958811yescharacteristic
598C2#1958911yescharacteristic
599C2#2065211yescharacteristic
600C2#2066411yescharacteristic
601C2C2×#181511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21981C2semidirect
5. C2×#1573C2semidirect
6. #8657C2semidirect
7. #8656C2semidirect
8. #8651C2semidirect
9. #8650C2semidirect
10. #8649C2semidirect
11. #8648C2semidirect
12. #8643C2semidirect
13. #8642C2semidirect
14. #8044C2semidirect
15. C2×#1296C2semidirect
18. #2994C4semidirect
94. #446C2×C4semidirect
96. C8×Q8C2×C4semidirect
98. #436C2×C4semidirect
102. C8×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29942
#219812
#218872
#218672
C2×#15732
#86572
#86562
#86512
#86502
#86492
#86482
#86432
#86422
#80442
C2×#12962