Finite Group Databasebeta

← #189140

Subgroups — #189140

Order 512.

1014 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
1C2×#2988C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#20743C211yesmaximal, characteristic
4#20737C211yesmaximal, characteristic
5C2×#1572C211yesmaximal, characteristic, complemented
6#8649C211yesmaximal, characteristic, complemented
7#8648C211yesmaximal, characteristic, complemented
8#8647C211yesmaximal, characteristic, complemented
9#8646C211yesmaximal, characteristic, complemented
10#8645C211yesmaximal, characteristic, complemented
11#8644C211yesmaximal, characteristic, complemented
12#8643C211yesmaximal, characteristic, complemented
13#8642C211yesmaximal, characteristic, complemented
14#8044C211yesmaximal, characteristic, complemented
15#8039C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17#2988—24no—
18#2988C418yescomplemented
19C2×C8×Q8C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C8×D4—21no—
22C2×C8×D4—21no—
23C2×C8×D4C2211yescharacteristic
24C2×C8×D4C2211yescharacteristic
25C2×#432C2211yescharacteristic
26C2×#431—21no—
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C22×C4⋊C8—21no—
30C22×C4⋊C8C2211yescharacteristic
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#1949—21no—
34#1949—21no—
35#1854—21no—
36#1854—21no—
37#1845C2211yescharacteristic
38#1844C2211yescharacteristic
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1825—21no—
42#1825—21no—
43#1803C2211yescharacteristic
44#1803C2211yescharacteristic
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1788—21no—
52#1788—21no—
53#1763—21no—
54#1762—21no—
55C4×#103—21no—
56C4×#102—21no—
57#1572—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
71C23×C8—21no—
72C23×C8—21no—
73C23×C8—21no—
74C4×#41—22no—
75C4×#41—22no—
76C4×#41C2×C414yes—
77C4×#41Q814yes—
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4D411yescharacteristic
83C2×C4×D4C2311yescharacteristic
84C2×#117—21no—
85C2×#117—21no—
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C8×Q8—22no—
91C8×Q8—22no—
92C8×Q8C2×C414yes—
93C8×Q8C2×C414yes—
94C8×D4—28no—
95C8×D4—28no—
96C8×D4—22no—
97C8×D4—22no—
98C8×D4C2×C414yes—
99C8×D4C2×C414yes—
100#432—22no—
101#432—22no—
102#431—28no—
103#431—22no—
104#431—22no—
105#431C2×C414yes—
106#431D414yes—
107C2×#107—21no—
108C2×#107—21no—
109C2×#106—21no—
110C2×#106—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—28no—
116C2×C4⋊C8C2×C414yes—
117C2×C4⋊C8D411yescharacteristic
118C2×C4⋊C8D411yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—21no—
129C2×#102—21no—
130C2×#102—21no—
131C2×#102—21no—
132C2×#102—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8D411yescharacteristic
138C2×C22⋊C8D411yescharacteristic
139C2×#97—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#384C2311yescharacteristic
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#382—21no—
150#382—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—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—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41—22no—
180C2×#41C4⋊C414yes—
181C22×Q8C2×D411yescharacteristic
182C22×D4—21no—
183C22×D4C2×D411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4#4111yescharacteristic
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—24no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—81no—
200C22×C8—81no—
201C22×C8—81no—
202C22×C8—81no—
203C22×C8—81no—
204C22×C8—81no—
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C4×Q8—22no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8C4⋊C414yes—
213C4×Q8C22×C412yes—
214C4×Q8C2×Q812yes—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—22no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4C4⋊C414yes—
223C4×D4C22×C412yes—
224C4×D4C2×Q812yes—
225#117—22no—
226#117—24no—
227#117—24no—
228#117—81no—
229#117—81no—
230#117—82no—
231#117C4⋊C414yes—
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⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42C22×C412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×Q812yes—
265C2×C42#4111yescharacteristic
266C2×C42#4112yes—
267C2×C42C2411yescharacteristic
268#107—81no—
269#107—81no—
270#106—81no—
271#106—81no—
272C4⋊C8—24no—
273C4⋊C8—24no—
274C4⋊C8—24no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8C22⋊C414yes—
286C4⋊C8C22⋊C414yes—
287C4⋊C8C22×C412yes—
288C4⋊C8C22×C412yes—
289#103—42no—
290#103—42no—
291#103—42no—
292#103—42no—
293#102—42no—
294#102—42no—
295#102—42no—
296#102—42no—
297C22⋊C8—24no—
298C22⋊C8—24no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301C22⋊C8C22⋊C414yes—
302C22⋊C8C22⋊C414yes—
303#97—42no—
304C4×C8—24no—
305C4×C8—24no—
306C4×C8—42no—
307C4×C8C22×C412yes—
308C4×C8C22×C412yes—
309C4×C8C2×D412yes—
310C4×C8C2×D412yes—
311#95—41no—
312#95—41no—
313#95—41no—
314#95—41no—
315C24—21no—
316C24#12111yescharacteristic
317#41—28no—
318#41—28no—
319#41—28no—
320#41—28no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8C2×C4⋊C412yes—
325C2×Q8#13512yes—
326C2×Q8#13512yes—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4C2×C4⋊C412yes—
334C2×D4#13512yes—
335C2×D4#13512yes—
336C22×C4—24no—
337C22×C4—24no—
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—21no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C4⋊C412yes—
367C22×C4#11712yes—
368C22×C4#12112yes—
369C22×C4#12111yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4C22⋊Q812yes—
373C22×C4C22⋊Q812yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—22no—
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—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—84no—
404C2×C8—84no—
405C2×C8—84no—
406C2×C8C2×C22⋊C412yes—
407C2×C8C2×C22⋊C412yes—
408C2×C8C2×C22⋊C412yes—
409C2×C8C2×C22⋊C412yes—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—24no—
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—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4C2×C4⋊C412yes—
436C4⋊C4C2×C4⋊C412yes—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—24no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C42—22no—
444C42—22no—
445C42—24no—
446C42—24no—
447C42—82no—
448C42—82no—
449C42C2×C22⋊C412yes—
450C42C2×C4⋊C412yes—
451C42#11712yes—
452C42#12112yes—
453C42C22⋊Q812yes—
454C42C22⋊Q812yes—
455C42C23×C411yescharacteristic
456C42C22×D411yescharacteristic
457C42C22×Q811yescharacteristic
458C42C2×#4111yescharacteristic
459C23—21no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23#38012yes—
473C23C2×#12111yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—22no—
477Q8#42714yes—
478D4—216no—
479D4—22no—
480D4—22no—
481D4—22no—
482D4#42714yes—
483C2×C4—216no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—84no—
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—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4#38012yes—
529C2×C4#38012yes—
530C2×C4#38012yes—
531C2×C4C22×C22⋊C411yescharacteristic
532C2×C4C22×C4⋊C411yescharacteristic
533C2×C4C2×#11711yescharacteristic
534C2×C4C2×#12111yescharacteristic
535C2×C4C2×C22⋊Q811yescharacteristic
536C2×C4C2×C22⋊Q811yescharacteristic
537C2×C4C2×#13511yescharacteristic
538C2×C4C2×#13511yescharacteristic
539C2×C4C2×#13511yescharacteristic
540C2×C4C2×#13511yescharacteristic
541C2×C4#57611yescharacteristic
542C2×C4#57711yescharacteristic
543C2×C4#57711yescharacteristic
544C2×C4#57811yescharacteristic
545C8—22no—
546C8—22no—
547C8—22no—
548C8—22no—
549C8—42no—
550C8—42no—
551C8—81no—
552C8—81no—
553C8—81no—
554C8—81no—
555C22—22no—
556C22—22no—
557C22—24no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—42no—
563C22—41no—
564C22—42no—
565C22—41no—
566C22—42no—
567C22—41no—
568C22—42no—
569C22C2×#38011yescharacteristic
570C22#304511yescharacteristic
571C22#304611yescharacteristic
572C22#305011yescharacteristic
573C22#305011yescharacteristic
574C22#306811yescharacteristic
575C22#306911yescharacteristic
576C4—22no—
577C4—22no—
578C4—24no—
579C4—22no—
580C4—21no—
581C4—21no—
582C4—42no—
583C4—42no—
584C4—161no—
585C4—161no—
586C4—161no—
587C4—161no—
588C4#289711yescharacteristic
589C4#289811yescharacteristic
590C4C2×#42711yescharacteristic
591C4C2×#42711yescharacteristic
592C4#305011yescharacteristic
593C4#305011yescharacteristic
594C4#308311yescharacteristic
595C4#308411yescharacteristic
596C2—42no—
597C2—44no—
598C2#1941211yescharacteristic
599C2#1941311yescharacteristic
600C2#1941811yescharacteristic
601C2#1941911yescharacteristic
602C2#2065211yescharacteristic
603C2#2065811yescharacteristic
604C2#2067911yescharacteristic
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×#1572C2semidirect
6. #8649C2semidirect
7. #8648C2semidirect
8. #8647C2semidirect
9. #8646C2semidirect
10. #8645C2semidirect
11. #8644C2semidirect
12. #8643C2semidirect
13. #8642C2semidirect
14. #8044C2semidirect
15. #8039C2semidirect
18. #2988C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29882
#219842
#207432
#207372
C2×#15722
#86492
#86482
#86472
#86462
#86452
#86442
#86432
#86422
#80442
#80392