Finite Group Databasebeta

← #223249

Subgroups — #223249

Order 512.

865 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36210C211yesmaximal, characteristic
2#22648C211yesmaximal, characteristic
3#22648C211yesmaximal, characteristic
4#9906C211yesmaximal, characteristic, complemented
5#9906C211yesmaximal, characteristic, complemented
6#9139C211yesmaximal, characteristic, complemented
7#9137C211yesmaximal, characteristic, complemented
8#9131C211yesmaximal, characteristic, complemented
9#9129C211yesmaximal, characteristic, complemented
10#8727C211yesmaximal, characteristic, complemented
11#8726C211yesmaximal, characteristic, complemented
12#8586C211yesmaximal, characteristic, complemented
13#8233C211yesmaximal, characteristic, complemented
14#8045C211yesmaximal, characteristic, complemented
15#8045C211yesmaximal, characteristic, complemented
16C2×#435—21no—
17C2×#435—21no—
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2859C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2562C412yes—
26#2514—21no—
27#2509—21no—
28#2495C412yes—
29#2494—21no—
30#2494C412yes—
31#2485—21no—
32#2421—21no—
33#2415—21no—
34#2370C412yes—
35#2362—21no—
36#2347—21no—
37C4×C4⋊Q8C2211yescharacteristic
38#2285C2211yescharacteristic
39#2285C2211yescharacteristic
40C42×D4C2211yescharacteristic
41#1940—21no—
42#1940—21no—
43#1851—21no—
44#1851—21no—
45#1850—21no—
46#1850—21no—
47#1842C2211yescharacteristic
48#1842C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1767C2211yescharacteristic
54#1767C2211yescharacteristic
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C22×M16—21no—
73C23×C8—21no—
74C23×C8—21no—
75C2×C4⋊Q8C2311yescharacteristic
76C2×#125C2×C412yes—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4D411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×#117—21no—
84C2×#117—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#435—44no—
88#435—44no—
89#423—22no—
90#423—22no—
91#423—41no—
92#423—41no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—41no—
104C2×C4⋊C8—41no—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×#97—21no—
118C2×#97—21no—
119#401—22no—
120#400—22no—
121#400—22no—
122#397—22no—
123#397—22no—
124#394—22no—
125#391—21no—
126#391—21no—
127#391—21no—
128#389—22no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389C2×C412yes—
137#389C2×C412yes—
138#389C2×C412yes—
139#388—22no—
140#388—22no—
141#388—22no—
142#388—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#383—22no—
150#382—21no—
151#382—21no—
152#382C2×C412yes—
153#380—21no—
154#380—21no—
155#380—21no—
156#380—21no—
157#380C2×C412yes—
158#380C2×C412yes—
159#380C2×C412yes—
160#380C2×C412yes—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4C2×C412yes—
166C4×C4⋊C4C2×C412yes—
167C4×C4⋊C4C2×C412yes—
168C4×C4⋊C4C2×C411yescharacteristic
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4D411yescharacteristic
171C4×C22⋊C4C2×C411yescharacteristic
172C4×C22⋊C4C2×C411yescharacteristic
173C4×C22⋊C4D411yescharacteristic
174C4×C22⋊C4D411yescharacteristic
175C4×C22⋊C4D411yescharacteristic
176C4×C22⋊C4D411yescharacteristic
177C43C2311yescharacteristic
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#332—22no—
185#332—22no—
186#332—41no—
187#332—41no—
188C22×Q8—21no—
189C22×D4C22⋊C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C2×M16—42no—
197C2×M16—42no—
198C2×M16—42no—
199C2×M16—42no—
200C2×M16—81no—
201C2×M16—81no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—81no—
215C22×C8—81no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220#125—44no—
221C22⋊Q8—42no—
222C22⋊Q8—42no—
223C22⋊Q8—81no—
224C22⋊Q8—81no—
225C4×Q8—81no—
226C4×Q8—81no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—42no—
232#117—42no—
233#117—42no—
234#117—81no—
235#117—81no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
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—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2411yescharacteristic
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—42no—
304C4⋊C8—42no—
305C4⋊C8—42no—
306C4⋊C8—42no—
307C4⋊C8—42no—
308C4⋊C8—42no—
309C4⋊C8—81no—
310C4⋊C8—81no—
311C4⋊C8#4112yes—
312C4⋊C8#4112yes—
313C22⋊C8—42no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317#97—42no—
318#97—42no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95C22⋊C411yescharacteristic
332#95C22⋊C411yescharacteristic
333#95C22⋊C411yescharacteristic
334#95C22⋊C411yescharacteristic
335#95C22⋊C411yescharacteristic
336#95C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—41no—
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—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4AGL(1,ℤ/8)12yes—
391C22×C4AGL(1,ℤ/8)12yes—
392C22×C4#13712yes—
393C22×C4#13712yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397M16—84no—
398M16—84no—
399C2×C8—22no—
400C2×C8—22no—
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—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—
423C2×C8—84no—
424C2×C8—84no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—22no—
483C42—41no—
484C42—44no—
485C42—82no—
486C42—82no—
487C42C2×#4111yescharacteristic
488C42C2×#4111yescharacteristic
489C42H(5,𝔽2)11yescharacteristic
490C422−1+411yescharacteristic
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23#41712yes—
504C23#41712yes—
505C23C22×C22⋊C411yescharacteristic
506Q8—81no—
507Q8—81no—
508Q8—81no—
509Q8—81no—
510D4—44no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4#41312yes—
573C2×C4#41712yes—
574C2×C4#41712yes—
575C2×C4C4×#4111yescharacteristic
576C2×C4#52011yescharacteristic
577C2×C4#54611yescharacteristic
578C2×C4#54611yescharacteristic
579C2×C4#54711yescharacteristic
580C2×C4#54711yescharacteristic
581C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
582C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
583C2×C4C2×#13711yescharacteristic
584C2×C4C2×#13711yescharacteristic
585C2×C4#57611yescharacteristic
586C2×C4#57711yescharacteristic
587C2×C4#57711yescharacteristic
588C2×C4#57811yescharacteristic
589C8—42no—
590C8—42no—
591C8—81no—
592C8—81no—
593C8—81no—
594C8—81no—
595C8—81no—
596C8—81no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22#233911yescharacteristic
612C22C2×#41311yescharacteristic
613C22#288911yescharacteristic
614C22C2×#41711yescharacteristic
615C22C2×#41711yescharacteristic
616C22#289711yescharacteristic
617C22#289811yescharacteristic
618C4—22no—
619C4—21no—
620C4—21no—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4#331011yescharacteristic
635C4#331211yescharacteristic
636C4#331811yescharacteristic
637C4#332011yescharacteristic
638C4#334311yescharacteristic
639C4#334411yescharacteristic
640C4#334511yescharacteristic
641C4#334611yescharacteristic
642C2—42no—
643C2—42no—
644C2#1775211yescharacteristic
645C2#1776311yescharacteristic
646C2#1776511yescharacteristic
647C2#1777511yescharacteristic
648C2#2248411yescharacteristic
649C2#2249111yescharacteristic
650C2#2249311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9906C2semidirect
5. #9906C2semidirect
6. #9139C2semidirect
7. #9137C2semidirect
8. #9131C2semidirect
9. #9129C2semidirect
10. #8727C2semidirect
11. #8726C2semidirect
12. #8586C2semidirect
13. #8233C2semidirect
14. #8045C2semidirect
15. #8045C2semidirect

Maximal subgroups

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

subgroupindex
#362102
#226482
#99062
#91392
#91372
#91312
#91292
#87272
#87262
#85862
#82332
#80452