Finite Group Databasebeta

← #223920

Subgroups — #223920

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36182C211yesmaximal, characteristic
2#22750C211yesmaximal, characteristic, complemented
3#22748C211yesmaximal, characteristic, complemented
4#9907C211yesmaximal, characteristic, complemented
5#9897C211yesmaximal, characteristic, complemented
6#9142C211yesmaximal, characteristic, complemented
7#9134C211yesmaximal, characteristic, complemented
8#9133C211yesmaximal, characteristic, complemented
9#9133C211yesmaximal, characteristic, complemented
10#8727C211yesmaximal, characteristic, complemented
11#8724C211yesmaximal, characteristic, complemented
12#8592C211yesmaximal, characteristic, complemented
13#8234C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic, complemented
15C2×#1296C211yesmaximal, characteristic, complemented
16C2×#553—21no—
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2859C2211yescharacteristic
22#2856C2211yescharacteristic
23#2856C2211yescharacteristic
24#2566—21no—
25#2512—21no—
26#2512C412yes—
27#2504—21no—
28#2500—21no—
29#2422—21no—
30#2422C412yes—
31#2416—21no—
32#2416C412yes—
33#2361—21no—
34#2361C412yes—
35C4×#127C2211yescharacteristic
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1924—21no—
40#1923—21no—
41#1917—21no—
42#1916—21no—
43#1869—21no—
44#1869—21no—
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53#1752—21no—
54C8×C22⋊C4—21no—
55C2×#331C2211yescharacteristic
56C2×#331C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C23×C8—21no—
72C23×C8—21no—
73#553—44no—
74#553—44no—
75C2×#127C2311yescharacteristic
76C2×#126—22no—
77C2×#125—21no—
78C2×#123—22no—
79C2×#121—22no—
80C2×#121—21no—
81C2×#121—21no—
82C2×#121—21no—
83C2×#121—21no—
84C2×#121C2×C412yes—
85C2×#121C2×C412yes—
86C2×C4×D4—22no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4C2×C412yes—
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C2×C4×D4D411yescharacteristic
93C2×#117—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4C2×C412yes—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#432—22no—
100#432—41no—
101#431—22no—
102#431—41no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—41no—
108C2×C4⋊C8—41no—
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×#97—21no—
124C2×#97—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127#397—22no—
128#397—22no—
129#396—22no—
130#396—22no—
131#394—22no—
132#394—22no—
133#390—21no—
134#390—21no—
135#390—21no—
136#390—21no—
137#390—21no—
138#390—21no—
139#390C2×C412yes—
140#390C2×C412yes—
141#390C2×C412yes—
142#389—21no—
143#389—21no—
144#389C2×C412yes—
145#389C2×C412yes—
146#388—22no—
147#388—22no—
148#386—22no—
149#386—22no—
150#385—22no—
151#385—22no—
152#384—21no—
153#384—21no—
154#384C2×C412yes—
155#384C2311yescharacteristic
156#384C2311yescharacteristic
157#384C2311yescharacteristic
158#382—21no—
159#382—21no—
160#380—21no—
161#380C2×C412yes—
162C4×C4⋊C4—21no—
163C4×C4⋊C4C2×C412yes—
164C4×C4⋊C4C2×C411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4D411yescharacteristic
167C4×C22⋊C4C2×C411yescharacteristic
168C4×C22⋊C4C2×C411yescharacteristic
169C4×C22⋊C4D411yescharacteristic
170C4×C22⋊C4D411yescharacteristic
171C4×C22⋊C4D411yescharacteristic
172C4×C22⋊C4D411yescharacteristic
173C43C2311yescharacteristic
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#331—22no—
185#331—22no—
186#331—41no—
187#331—41no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4C22⋊C411yescharacteristic
193C22×D4C22×C412yes—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
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—
216#127—21no—
217#127—21no—
218#127—21no—
219#127—21no—
220#126—48no—
221#125—42no—
222#125—42no—
223#123—48no—
224#121—44no—
225#121—42no—
226#121—42no—
227#121—48no—
228#121—42no—
229#121—42no—
230#121—42no—
231#121—42no—
232#121—81no—
233#121—81no—
234#121—82no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4—42no—
242C4×D4—48no—
243C4×D4—81no—
244C4×D4—81no—
245C4×D4—82no—
246#117—42no—
247#117—42no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—81no—
262C2×C4⋊C4—81no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22⋊C411yescharacteristic
285C2×C22⋊C4C22⋊C411yescharacteristic
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—22no—
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2411yescharacteristic
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—42no—
323C4⋊C8—42no—
324C4⋊C8#4112yes—
325C4⋊C8#4112yes—
326C22⋊C8—42no—
327C22⋊C8—42no—
328C22⋊C8—42no—
329C22⋊C8—42no—
330C22⋊C8—81no—
331C22⋊C8—81no—
332#97—42no—
333#97—81no—
334C4×C8—42no—
335C4×C8—81no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346C24—22no—
347C24C2×C22⋊C411yescharacteristic
348C24C2×C22⋊C411yescharacteristic
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—82no—
373C2×D4—82no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4#13512yes—
422C22×C4#13512yes—
423C22×C4AGL(1,ℤ/8)12yes—
424C22×C4AGL(1,ℤ/8)12yes—
425C22×C4C23×C411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—84no—
453C2×C8—84no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—44no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—41no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—41no—
514C42—41no—
515C42—44no—
516C42C2×#4111yescharacteristic
517C42C2×#4111yescharacteristic
518C42H(5,𝔽2)11yescharacteristic
519C422−1+411yescharacteristic
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23#41312yes—
539C23#41612yes—
540C23#41812yes—
541C23C22×C22⋊C411yescharacteristic
542D4—44no—
543D4—82no—
544D4—82no—
545D4—88no—
546D4—81no—
547D4—81no—
548D4—81no—
549D4—81no—
550D4—81no—
551D4—81no—
552D4—81no—
553D4—81no—
554D4—164no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—164no—
615C2×C4—164no—
616C2×C4#41612yes—
617C2×C4#41812yes—
618C2×C4C4×#4111yescharacteristic
619C2×C4#52011yescharacteristic
620C2×C4#54611yescharacteristic
621C2×C4#54611yescharacteristic
622C2×C4#54711yescharacteristic
623C2×C4#54711yescharacteristic
624C2×C4C2×#13511yescharacteristic
625C2×C4C2×#13511yescharacteristic
626C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
627C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
628C2×C4#57611yescharacteristic
629C2×C4#57711yescharacteristic
630C2×C4#57711yescharacteristic
631C2×C4#57811yescharacteristic
632C8—42no—
633C8—42no—
634C8—81no—
635C8—81no—
636C8—81no—
637C8—81no—
638C8—81no—
639C8—81no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22#155314yes—
661C22#233911yescharacteristic
662C22C2×#41311yescharacteristic
663C22#288811yescharacteristic
664C22C2×#41611yescharacteristic
665C22C2×#41811yescharacteristic
666C22#289711yescharacteristic
667C22#289811yescharacteristic
668C4—21no—
669C4—21no—
670C4—22no—
671C4—22no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#329711yescharacteristic
684C4#330111yescharacteristic
685C4#330611yescharacteristic
686C4#330811yescharacteristic
687C4#333711yescharacteristic
688C4#333911yescharacteristic
689C4#334311yescharacteristic
690C4#334511yescharacteristic
691C2—24no—
692C2—81no—
693C2—162no—
694C2C2×#155311yescharacteristic
695C2#1772611yescharacteristic
696C2#1775811yescharacteristic
697C2#1777511yescharacteristic
698C2#2248211yescharacteristic
699C2#2248811yescharacteristic
700C2#2249511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36182C2
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
2. #22750C2semidirect
3. #22748C2semidirect
4. #9907C2semidirect
5. #9897C2semidirect
6. #9142C2semidirect
7. #9134C2semidirect
8. #9133C2semidirect
9. #9133C2semidirect
10. #8727C2semidirect
11. #8724C2semidirect
12. #8592C2semidirect
13. #8234C2semidirect
14. #8046C2semidirect
15. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
#361822
#227502
#227482
#99072
#98972
#91422
#91342
#91332
#87272
#87242
#85922
#82342
#80462
C2×#12962