Finite Group Databasebeta

← #189159

Subgroups — #189159

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2991C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#20743C211yesmaximal, characteristic
4#20645C211yesmaximal, characteristic
5C2×#1573C211yesmaximal, characteristic, complemented
6#8656C211yesmaximal, characteristic, complemented
7#8654C211yesmaximal, characteristic, complemented
8#8652C211yesmaximal, characteristic, complemented
9#8650C211yesmaximal, characteristic, complemented
10#8649C211yesmaximal, characteristic, complemented
11#8647C211yesmaximal, characteristic, complemented
12#8645C211yesmaximal, characteristic, complemented
13#8643C211yesmaximal, characteristic, complemented
14#8045C211yesmaximal, characteristic, complemented
15#8042C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17#2991—24no—
18#2991C418yescomplemented
19C2×C8×Q8C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#436—21no—
22C2×#436—21no—
23C2×#436C2211yescharacteristic
24C2×#436C2211yescharacteristic
25C2×#432C2211yescharacteristic
26C2×#431—21no—
27C2×#431C2211yescharacteristic
28C2×#423—21no—
29C2×#423C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C2×#416—21no—
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34#1949—21no—
35#1949—21no—
36#1854—21no—
37#1854—21no—
38#1845C2211yescharacteristic
39#1844C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1825—21no—
43#1805—21no—
44#1804—21no—
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1802C2211yescharacteristic
53#1788—21no—
54#1763—21no—
55#1762—21no—
56#1735—21no—
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×#41D411yescharacteristic
71C22×M16—21no—
72C22×M16—21no—
73C22×M16—21no—
74C23×C8—21no—
75C4×#41—22no—
76C4×#41—22no—
77C4×#41C2×C414yes—
78C4×#41Q814yes—
79C2×C4×Q8D411yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4D411yescharacteristic
84C2×C4×D4C2311yescharacteristic
85C2×#117—21no—
86C2×#117—21no—
87C2×#117—21no—
88C2×#117D411yescharacteristic
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C8×Q8—22no—
92C8×Q8—22no—
93C8×Q8C2×C414yes—
94C8×Q8C2×C414yes—
95#436—28no—
96#436—28no—
97#436—22no—
98#436—22no—
99#436C2×C414yes—
100#436C2×C414yes—
101#432—22no—
102#432—22no—
103#431—28no—
104#431C2×C414yes—
105#431D414yes—
106C2×#107—21no—
107C2×#107—21no—
108C2×#106—21no—
109C2×#106—21no—
110#423—22no—
111#423—28no—
112#423C2×C414yes—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8D411yescharacteristic
120C2×C4⋊C8D411yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127#416—44no—
128C2×#103—21no—
129C2×#103—21no—
130C2×#103—21no—
131C2×#103—21no—
132C2×#102—21no—
133C2×#102—21no—
134C2×#102—21no—
135C2×#102—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8D411yescharacteristic
140C2×C22⋊C8D411yescharacteristic
141C2×#97—21no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#382—21no—
151#382—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—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#340—21no—
172#335—22no—
173#335—22no—
174#334—22no—
175#334—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41—41no—
180C2×#41—41no—
181C2×#41C4⋊C414yes—
182C22×Q8C2×D411yescharacteristic
183C22×D4—21no—
184C22×D4C2×D411yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4#4111yescharacteristic
191C2×M16—24no—
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—81no—
195C2×M16—81no—
196C2×M16—81no—
197C2×M16—81no—
198C2×M16—81no—
199C2×M16—81no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C4×Q8—22no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8C4⋊C414yes—
217C4×Q8C22×C412yes—
218C4×Q8C2×Q812yes—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—22no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4C4⋊C414yes—
227C4×D4C22×C412yes—
228C4×D4C2×Q812yes—
229#117—22no—
230#117—24no—
231#117—24no—
232#117—42no—
233#117—42no—
234#117—81no—
235#117—81no—
236#117C4⋊C414yes—
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—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42C22×C412yes—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D412yes—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×Q812yes—
270C2×C42#4111yescharacteristic
271C2×C42#4112yes—
272C2×C42C2411yescharacteristic
273#107—81no—
274#107—81no—
275#106—81no—
276#106—81no—
277C4⋊C8—24no—
278C4⋊C8—24no—
279C4⋊C8—24no—
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—21no—
290C4⋊C8—42no—
291C4⋊C8C22⋊C414yes—
292C4⋊C8C22⋊C414yes—
293C4⋊C8C22×C412yes—
294C4⋊C8C22×C412yes—
295#103—42no—
296#103—42no—
297#103—42no—
298#103—42no—
299#102—42no—
300#102—42no—
301#102—42no—
302#102—42no—
303C22⋊C8—24no—
304C22⋊C8—24no—
305C22⋊C8—42no—
306C22⋊C8C22⋊C414yes—
307C22⋊C8C22⋊C414yes—
308#97—42no—
309C4×C8—24no—
310C4×C8—24no—
311C4×C8C22×C412yes—
312C4×C8C22×C412yes—
313C4×C8C2×D412yes—
314C4×C8C2×D412yes—
315#95—41no—
316#95—41no—
317#95—41no—
318#95—41no—
319C24—21no—
320C24#12111yescharacteristic
321#41—28no—
322#41—44no—
323#41—44no—
324#41—44no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8C2×C4⋊C412yes—
329C2×Q8AGL(1,ℤ/8)12yes—
330C2×Q8#13712yes—
331C2×D4—24no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C2×C4⋊C412yes—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C4⋊C412yes—
371C22×C4#11712yes—
372C22×C4#12112yes—
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4C22⋊Q812yes—
377C22×C4C22⋊Q812yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2×#4111yescharacteristic
381M16—84no—
382M16—84no—
383M16—84no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—84no—
411C2×C8C2×C22⋊C412yes—
412C2×C8C2×C22⋊C412yes—
413C2×C8C2×C22⋊C412yes—
414C2×C8C2×C22⋊C412yes—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—24no—
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—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4C2×C4⋊C412yes—
441C4⋊C4C2×C4⋊C412yes—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—24no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C42—22no—
449C42—22no—
450C42—24no—
451C42—24no—
452C42—82no—
453C42—82no—
454C42C2×C22⋊C412yes—
455C42C2×C4⋊C412yes—
456C42#11712yes—
457C42#12112yes—
458C42C22⋊Q812yes—
459C42C22⋊Q812yes—
460C42C23×C411yescharacteristic
461C42C22×D411yescharacteristic
462C42C22×Q811yescharacteristic
463C42C2×#4111yescharacteristic
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#38012yes—
478C23C2×#12111yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—22no—
482Q8#42814yes—
483D4—22no—
484D4—22no—
485D4—48no—
486D4—41no—
487D4—41no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4#38012yes—
534C2×C4#38012yes—
535C2×C4#38012yes—
536C2×C4C22×C22⋊C411yescharacteristic
537C2×C4C22×C4⋊C411yescharacteristic
538C2×C4C2×#11711yescharacteristic
539C2×C4C2×#12111yescharacteristic
540C2×C4C2×C22⋊Q811yescharacteristic
541C2×C4C2×C22⋊Q811yescharacteristic
542C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
543C2×C4C2×#13711yescharacteristic
544C2×C4#57511yescharacteristic
545C2×C4#57511yescharacteristic
546C2×C4#57611yescharacteristic
547C2×C4#57711yescharacteristic
548C2×C4#57711yescharacteristic
549C2×C4#57811yescharacteristic
550C8—22no—
551C8—22no—
552C8—22no—
553C8—22no—
554C8—42no—
555C8—42no—
556C8—81no—
557C8—81no—
558C8—81no—
559C8—81no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22C2×#38011yescharacteristic
575C22#304511yescharacteristic
576C22#304611yescharacteristic
577C22#305511yescharacteristic
578C22#305611yescharacteristic
579C22#306711yescharacteristic
580C22#306711yescharacteristic
581C4—22no—
582C4—22no—
583C4—24no—
584C4—22no—
585C4—21no—
586C4—21no—
587C4—42no—
588C4—42no—
589C4—161no—
590C4—161no—
591C4—161no—
592C4—161no—
593C4#289711yescharacteristic
594C4#289811yescharacteristic
595C4C2×#42811yescharacteristic
596C4#291111yescharacteristic
597C4#305511yescharacteristic
598C4#305611yescharacteristic
599C4#308311yescharacteristic
600C4#308411yescharacteristic
601C2—42no—
602C2—82no—
603C2#1949211yescharacteristic
604C2#1949311yescharacteristic
605C2#1949811yescharacteristic
606C2#1949911yescharacteristic
607C2#2065211yescharacteristic
608C2#2066111yescharacteristic
609C2#2067811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. #21984C2semidirect
5. C2×#1573C2semidirect
6. #8656C2semidirect
7. #8654C2semidirect
8. #8652C2semidirect
9. #8650C2semidirect
10. #8649C2semidirect
11. #8647C2semidirect
12. #8645C2semidirect
13. #8643C2semidirect
14. #8045C2semidirect
15. #8042C2semidirect
18. #2991C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29912
#219842
#207432
#206452
C2×#15732
#86562
#86542
#86522
#86502
#86492
#86472
#86452
#86432
#80452
#80422