Finite Group Databasebeta

← #227973

Subgroups — #227973

Order 512.

745 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23572C211yesmaximal, characteristic
2#23055C211yesmaximal, characteristic, complemented
3#22531C211yesmaximal, characteristic
4C2×#1587C211yesmaximal, characteristic, complemented
5C2×#1494C211yesmaximal, characteristic
6#9666C211yesmaximal, characteristic, complemented
7#9560C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic, complemented
9#9071C211yesmaximal, characteristic, complemented
10#8952C211yesmaximal, characteristic, complemented
11#8950C211yesmaximal, characteristic, complemented
12#8844C211yesmaximal, characteristic, complemented
13#8834C211yesmaximal, characteristic, complemented
14#8213C211yesmaximal, characteristic, complemented
15#8144C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×#446C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25#2059C411yescharacteristic
26#2042C411yescharacteristic
27#2035C411yescharacteristic
28#2033C411yescharacteristic
29#2026C411yescharacteristic
30#2015C411yescharacteristic
31#2001C411yescharacteristic
32#2000C411yescharacteristic
33#1984C2211yescharacteristic
34#1955—21no—
35#1954—21no—
36#1954C2211yescharacteristic
37#1954C2211yescharacteristic
38#1939C2211yescharacteristic
39#1924—21no—
40#1923—21no—
41#1917—21no—
42#1798—21no—
43#1797C2211yescharacteristic
44#1770—21no—
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1587—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1494—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C23×C8—21no—
72#529—22no—
73#529—22no—
74#529—41no—
75#529—41no—
76C2×#126—21no—
77C2×#125C2×C411yescharacteristic
78C2×#124C2×C411yescharacteristic
79C2×#123—21no—
80C2×C22⋊Q8C2×C411yescharacteristic
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×#121C2×C411yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88C22×C42C2311yescharacteristic
89#446—22no—
90#446—22no—
91#446—41no—
92#446—41no—
93C8×Q8—22no—
94C8×Q8—41no—
95C4×Q16—22no—
96C4×Q16—22no—
97C4×SD16—41no—
98C4×SD16—41no—
99#432—22no—
100#432—41no—
101C2×#107C2×C411yescharacteristic
102C2×#106C2×C411yescharacteristic
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8C2×C411yescharacteristic
126C2×C22⋊C8C2×C411yescharacteristic
127C2×C22⋊C8D411yescharacteristic
128C2×C22⋊C8D411yescharacteristic
129C2×#97—21no—
130C2×#97—21no—
131C2×#97—21no—
132C2×C4×C8—22no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8—41no—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143#391C2×C411yescharacteristic
144#389C2×C411yescharacteristic
145#387—21no—
146#385—21no—
147#380C2×C411yescharacteristic
148C4×C4⋊C4—21no—
149C4×C4⋊C4C2311yescharacteristic
150C4×C22⋊C4—21no—
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336C2×C411yescharacteristic
161#336C2×C411yescharacteristic
162#336D411yescharacteristic
163#336D411yescharacteristic
164#335—41no—
165#334—22no—
166Q8⋊C8—22no—
167Q8⋊C8—22no—
168Q8⋊C8—41no—
169Q8⋊C8D414yes—
170D4⋊C8—22no—
171D4⋊C8—22no—
172#322—22no—
173#322—41no—
174C22×Q8C22×C411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—21no—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C2×Q16—22no—
183C2×Q16—41no—
184C2×Q16—41no—
185C2×SD16—41no—
186C2×SD16—41no—
187C2×SD16—41no—
188C22×C8—21no—
189C22×C8—24no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8C22×C411yescharacteristic
198C22×C8C22×C411yescharacteristic
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205#126—42no—
206#126—42no—
207#125—22no—
208#125—41no—
209#124—22no—
210#124—41no—
211#123—42no—
212#123—42no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—41no—
216C22⋊Q8—41no—
217#121—22no—
218#121—41no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8C2×D412yes—
226C4×Q8#4112yes—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—42no—
232C4×D4—42no—
233#117—42no—
234#117—42no—
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⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42C22×C411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42C2411yescharacteristic
265#107—22no—
266#106—41no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—42no—
277C4⋊C8—42no—
278C4⋊C8C2×D412yes—
279#103—22no—
280#103—22no—
281#103—22no—
282#103—41no—
283#103—41no—
284#102—22no—
285#102—41no—
286#102—41no—
287C22⋊C8—41no—
288C22⋊C8—41no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293C22⋊C8—42no—
294C22⋊C8—81no—
295C22⋊C8—81no—
296C22⋊C8—81no—
297#97—42no—
298#97—42no—
299#97—42no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—42no—
312C4×C8—42no—
313C4×C8—42no—
314C4×C8C2×D412yes—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320C24—21no—
321C24C4×D411yescharacteristic
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8C4×D412yes—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—81no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378Q16—44no—
379SD16—44no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—48no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—81no—
440C4⋊C4C4×D412yes—
441C4⋊C4#13112yes—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C42—21no—
451C42—21no—
452C42—22no—
453C42—21no—
454C42—21no—
455C42—22no—
456C42—21no—
457C42—21no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42C22×D411yescharacteristic
463C42C2×#4111yescharacteristic
464C42H(5,𝔽2)11yescharacteristic
465C42H(5,𝔽2)11yescharacteristic
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23#44312yes—
480C23C2×C4×D411yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
485Q8—41no—
486Q8—41no—
487Q8—41no—
488Q8—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—84no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4C2×C4×D411yescharacteristic
541C2×C4#51811yescharacteristic
542C2×C4D4211yescharacteristic
543C2×C4#54611yescharacteristic
544C2×C4#54611yescharacteristic
545C2×C4#55111yescharacteristic
546C2×C4C2×#13111yescharacteristic
547C2×C4#56811yescharacteristic
548C2×C4#56811yescharacteristic
549C2×C4#56811yescharacteristic
550C2×C4#57511yescharacteristic
551C2×C4#57511yescharacteristic
552C2×C4#57711yescharacteristic
553C2×C4#57811yescharacteristic
554C8—24no—
555C8—42no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—41no—
561C8—42no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—81no—
566C8—81no—
567C8—81no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22#235811yescharacteristic
583C22#293211yescharacteristic
584C22#293311yescharacteristic
585C22#294311yescharacteristic
586C22#294711yescharacteristic
587C22C2×#44311yescharacteristic
588C22#295611yescharacteristic
589C4—21no—
590C4—21no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4#293011yescharacteristic
602C4#293511yescharacteristic
603C4#297811yescharacteristic
604C4#298111yescharacteristic
605C4#326911yescharacteristic
606C4#327111yescharacteristic
607C4#331311yescharacteristic
608C4#331711yescharacteristic
609C2—42no—
610C2—161no—
611C2#1798211yescharacteristic
612C2#1798811yescharacteristic
613C2#1800211yescharacteristic
614C2#1801711yescharacteristic
615C2#2284911yescharacteristic
616C2#2286111yescharacteristic
617C2#2288811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23055C2semidirect
4. C2×#1587C2semidirect
6. #9666C2semidirect
7. #9560C2semidirect
8. #9542C2semidirect
9. #9071C2semidirect
10. #8952C2semidirect
11. #8950C2semidirect
12. #8844C2semidirect
13. #8834C2semidirect
14. #8213C2semidirect
15. #8144C2semidirect

Maximal subgroups

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

subgroupindex
#235722
#230552
#225312
C2×#15872
C2×#14942
#96662
#95602
#95422
#90712
#89522
#89502
#88442
#88342
#82132
#81442