Finite Group Databasebeta

← #189248

Subgroups — #189248

Order 512.

961 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60278C211yesmaximal, characteristic
2#21362C211yesmaximal, characteristic, complemented
3#20747C211yesmaximal, characteristic
4#20654C211yesmaximal, characteristic
5#17817C211yesmaximal, characteristic, complemented
6#8669C212yesmaximal, complemented
7#8668C212yesmaximal, complemented
8#8667C212yesmaximal, complemented
9#8666C212yesmaximal, complemented
10#8097C211yesmaximal, characteristic, complemented
11#8094C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13#2991C411yescharacteristic, complemented
14#2991C411yescharacteristic, complemented
15#2990—21no—
16#2990—21no—
17#2990C411yescharacteristic, complemented
18#2990C411yescharacteristic, complemented
19#2984C412yescomplemented
20#2974—22no—
21#2974C412yescomplemented
22#2965—22no—
23#2965C2212yes—
24C8×#41C2212yes—
25#2920—21no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#431—21no—
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C2×#421—21no—
32#2285C2211yescharacteristic
33C4×#117C2211yescharacteristic
34#1948—21no—
35#1947—21no—
36#1854—22no—
37#1826—21no—
38#1806—21no—
39#1806—21no—
40#1806C2211yescharacteristic
41#1806C2211yescharacteristic
42#1806C2211yescharacteristic
43#1806C2211yescharacteristic
44#1806C2212yes—
45#1806C2212yes—
46#1789—21no—
47C4×#107C2212yescomplemented
48C4×#106C2212yescomplemented
49C4×#104—21no—
50C4×#104—21no—
51#1733—21no—
52#1574—21no—
53#1574—21no—
54#1573—22no—
55#1567C2212yescomplemented
56#1566C2212yescomplemented
57#1303C2212yescomplemented
58#1303C2212yescomplemented
59#1303C2212yescomplemented
60#1303C2212yescomplemented
61C22×#41D411yescharacteristic
62C22×M16—21no—
63C4×#41—21no—
64C4×#41—21no—
65C4×#41—21no—
66C4×#41—21no—
67C4×#41C2×C411yescharacteristic
68C4×#41C2×C411yescharacteristic
69C4×#41D411yescharacteristic
70C4×#41D411yescharacteristic
71C4×#41Q811yescharacteristic
72C4×#41Q811yescharacteristic
73C4×#41C2311yescharacteristic
74C4×#41C2311yescharacteristic
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8Q811yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4Q811yescharacteristic
81C2×#117—21no—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117D411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#446—22no—
88#446—22no—
89#446C2×C412yes—
90#446C2×C412yes—
91C8×Q8C2×C412yes—
92C8×Q8C2×C412yes—
93#436—22no—
94#436—22no—
95#436C2×C412yes—
96#436C2×C412yes—
97#435—22no—
98#435—22no—
99#435—22no—
100#435—22no—
101#435—22no—
102#435—22no—
103C8×D4—22no—
104C8×D4—22no—
105C8×D4C2×C412yes—
106C8×D4C2×C412yes—
107#432—21no—
108#432—21no—
109#432—21no—
110#432—22no—
111#432—21no—
112#432—21no—
113#432C2×C412yes—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—22no—
118#431—22no—
119#431—22no—
120#431—21no—
121#431—21no—
122#431C2×C412yes—
123#431C2×C411yescharacteristic
124#431C2×C411yescharacteristic
125#431D412yes—
126#431D411yescharacteristic
127#431D411yescharacteristic
128#431C2312yes—
129C2×#107—22no—
130C2×#106—22no—
131#423—21no—
132#423—21no—
133#423—21no—
134#423—21no—
135#423—21no—
136#423C2×C411yescharacteristic
137#423C2×C411yescharacteristic
138#423C2311yescharacteristic
139#423C2312yes—
140#423C2311yescharacteristic
141C2×C4⋊C8—21no—
142C2×C4⋊C8—21no—
143C2×C4⋊C8—21no—
144C2×C4⋊C8—21no—
145C2×C4⋊C8D411yescharacteristic
146C2×C4⋊C8D411yescharacteristic
147#421—44no—
148C2×#104—22no—
149C2×#104—22no—
150C2×#104—21no—
151C2×#104—21no—
152C2×#104—21no—
153C2×#104—21no—
154#407—21no—
155#407—22no—
156#407—21no—
157C2×C22⋊C8—22no—
158C2×C22⋊C8D412yes—
159C4×M16—21no—
160C4×M16—22no—
161C4×M16—21no—
162C2×C4×C8—22no—
163C2×C4×C8C2312yes—
164#384—21no—
165#382—21no—
166#380—21no—
167#380—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4C2311yescharacteristic
171C4×C4⋊C4C2311yescharacteristic
172C4×C4⋊C4C2312yes—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175#376—21no—
176C43—21no—
177#340—22no—
178#340—22no—
179#339—22no—
180#339—22no—
181#339—21no—
182#339—21no—
183#339—21no—
184#339—21no—
185#335—22no—
186#335—22no—
187#334—22no—
188#334—22no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—22no—
192C2×#41—22no—
193C2×#41—41no—
194C2×#41—41no—
195C2×#41C4⋊C411yescharacteristic
196C2×#41C4⋊C411yescharacteristic
197C2×#41C2×D411yescharacteristic
198C2×#41C2×D411yescharacteristic
199C22×Q8C4⋊C411yescharacteristic
200C22×D4—21no—
201C22×D4C4⋊C411yescharacteristic
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4#4111yescharacteristic
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—41no—
216C2×M16—42no—
217C2×M16—42no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—41no—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8C4⋊C411yescharacteristic
228C4×Q8C4⋊C411yescharacteristic
229C4×Q8C4⋊C411yescharacteristic
230C4×Q8C4⋊C411yescharacteristic
231C4×Q8C22×C411yescharacteristic
232C4×Q8C22×C411yescharacteristic
233C4×Q8C2×Q811yescharacteristic
234C4×Q8C2×Q811yescharacteristic
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4C4⋊C411yescharacteristic
256C4×D4C4⋊C411yescharacteristic
257C4×D4C4⋊C411yescharacteristic
258C4×D4C4⋊C411yescharacteristic
259C4×D4C22×C411yescharacteristic
260C4×D4C22×C411yescharacteristic
261C4×D4C2×Q811yescharacteristic
262C4×D4C2×Q811yescharacteristic
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—42no—
274#117—42no—
275#117—42no—
276#117—42no—
277#117C4⋊C411yescharacteristic
278#117C4⋊C411yescharacteristic
279#117C2×D411yescharacteristic
280#117C2×D411yescharacteristic
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—41no—
287C2×C4⋊C4—41no—
288C2×C4⋊C4C4⋊C411yescharacteristic
289C2×C4⋊C4C4⋊C411yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—41no—
310C2×C42—41no—
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×Q811yescharacteristic
318C2×C42C2×Q811yescharacteristic
319C2×C42#4111yescharacteristic
320C2×C42#4111yescharacteristic
321C2×C42#4111yescharacteristic
322C2×C42C2411yescharacteristic
323#107—44no—
324#106—44no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—22no—
333C4⋊C8—21no—
334C4⋊C8—22no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—41no—
338C4⋊C8—41no—
339C4⋊C8C22⋊C411yescharacteristic
340C4⋊C8C22⋊C411yescharacteristic
341C4⋊C8C22⋊C411yescharacteristic
342C4⋊C8C22⋊C411yescharacteristic
343C4⋊C8C22⋊C412yes—
344C4⋊C8C22⋊C412yes—
345C4⋊C8C22×C412yes—
346C4⋊C8C22×C412yes—
347#104—42no—
348#104—42no—
349#104—44no—
350#104—42no—
351#104—44no—
352#104—42no—
353C22⋊C8—22no—
354C22⋊C8—22no—
355C22⋊C8—22no—
356C22⋊C8—22no—
357C22⋊C8—22no—
358C22⋊C8—22no—
359C22⋊C8—41no—
360C22⋊C8—41no—
361C22⋊C8C22⋊C412yes—
362C22⋊C8C22⋊C412yes—
363C22⋊C8C22⋊C412yes—
364C22⋊C8C22⋊C412yes—
365#97—22no—
366#97—22no—
367#97—22no—
368#97—41no—
369C4×C8—22no—
370C4×C8—22no—
371C4×C8—22no—
372C4×C8—41no—
373C4×C8C22×C412yes—
374C4×C8C22×C412yes—
375C4×C8C2×D412yes—
376C4×C8C2×D412yes—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—41no—
381#95—41no—
382C24—21no—
383C24#12111yescharacteristic
384#41—21no—
385#41—21no—
386#41—22no—
387#41—22no—
388#41—21no—
389#41—21no—
390#41—21no—
391#41—21no—
392#41—42no—
393#41—42no—
394#41—41no—
395#41—41no—
396#41—42no—
397#41—42no—
398#41AGL(1,ℤ/8)12yes—
399#41#13712yes—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—41no—
403C2×Q8C2×C4⋊C411yescharacteristic
404C2×Q8C2×C4⋊C411yescharacteristic
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—41no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×C4⋊C411yescharacteristic
415C2×D4C2×C4⋊C411yescharacteristic
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C4⋊C411yescharacteristic
449C22×C4C2×C4⋊C411yescharacteristic
450C22×C4#11711yescharacteristic
451C22×C4#11711yescharacteristic
452C22×C4#12111yescharacteristic
453C22×C4#12111yescharacteristic
454C22×C4#12111yescharacteristic
455C22×C4#12111yescharacteristic
456C22×C4#12111yescharacteristic
457C22×C4C22⋊Q811yescharacteristic
458C22×C4C22⋊Q811yescharacteristic
459C22×C4C22⋊Q811yescharacteristic
460C22×C4C22⋊Q811yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4111yescharacteristic
464M16—42no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—42no—
470M16—82no—
471M16—82no—
472M16—81no—
473M16—81no—
474M16—82no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—42no—
482C2×C8—82no—
483C2×C8—82no—
484C2×C8—82no—
485C2×C8C2×C22⋊C412yes—
486C2×C8C2×C22⋊C412yes—
487C2×C8C2×C22⋊C412yes—
488C2×C8C2×C22⋊C412yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4C2×C4⋊C411yescharacteristic
508C4⋊C4C2×C4⋊C411yescharacteristic
509C4⋊C4C2×C4⋊C411yescharacteristic
510C4⋊C4C2×C4⋊C411yescharacteristic
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—44no—
538C42—42no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—42no—
544C42—82no—
545C42C2×C22⋊C411yescharacteristic
546C42C2×C22⋊C411yescharacteristic
547C42C2×C4⋊C411yescharacteristic
548C42C2×C4⋊C411yescharacteristic
549C42#11711yescharacteristic
550C42#11711yescharacteristic
551C42#12111yescharacteristic
552C42#12111yescharacteristic
553C42C22⋊Q811yescharacteristic
554C42C22⋊Q811yescharacteristic
555C42C22⋊Q811yescharacteristic
556C42C22⋊Q811yescharacteristic
557C42C23×C411yescharacteristic
558C42C22×D411yescharacteristic
559C42C22×Q811yescharacteristic
560C42C2×#4111yescharacteristic
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—81no—
570C23#38011yescharacteristic
571C23#38011yescharacteristic
572C23C2×#12111yescharacteristic
573Q8—21no—
574Q8—21no—
575Q8—21no—
576Q8—42no—
577Q8#42812yes—
578D4—22no—
579D4—22no—
580D4—21no—
581D4—21no—
582D4—21no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—41no—
587D4—41no—
588D4#42812yes—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4#38011yescharacteristic
649C2×C4#38011yescharacteristic
650C2×C4#38011yescharacteristic
651C2×C4#38011yescharacteristic
652C2×C4#38011yescharacteristic
653C2×C4#38011yescharacteristic
654C2×C4C22×C22⋊C411yescharacteristic
655C2×C4C22×C4⋊C411yescharacteristic
656C2×C4C2×#11711yescharacteristic
657C2×C4C2×#12111yescharacteristic
658C2×C4C2×C22⋊Q811yescharacteristic
659C2×C4C2×C22⋊Q811yescharacteristic
660C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
661C2×C4C2×#13711yescharacteristic
662C2×C4#57511yescharacteristic
663C2×C4#57511yescharacteristic
664C8—22no—
665C8—22no—
666C8—22no—
667C8—22no—
668C8—42no—
669C8—42no—
670C8—82no—
671C8—81no—
672C8—81no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—22no—
680C22—21no—
681C22—21no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—81no—
688C22—81no—
689C22—84no—
690C22C2×#38011yescharacteristic
691C22#305511yescharacteristic
692C22#305611yescharacteristic
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—22no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—82no—
708C4—82no—
709C4—162no—
710C4#181412yes—
711C4#290111yescharacteristic
712C4#290111yescharacteristic
713C4C2×#42811yescharacteristic
714C4#291111yescharacteristic
715C4#305511yescharacteristic
716C4#305611yescharacteristic
717C2—21no—
718C2—21no—
719C2—41no—
720C2—41no—
721C2—42no—
722C2—81no—
723C2#2065511yescharacteristic
724C2#2066111yescharacteristic
725C2C2×#181411yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#380
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21362C2semidirect
5. #17817C2semidirect
6. #8669C2semidirect
7. #8668C2semidirect
8. #8667C2semidirect
9. #8666C2semidirect
10. #8097C2semidirect
11. #8094C2semidirect
13. #2991C4semidirect
14. #2991C4semidirect
17. #2990C4semidirect
18. #2990C4semidirect
19. #2984C4semidirect
21. #2974C4semidirect
47. C4×#107C22semidirect
48. C4×#106C22semidirect
55. #1567C22semidirect
56. #1566C22semidirect
57. #1303C22semidirect
58. #1303C22semidirect
59. #1303C22semidirect
60. #1303C22semidirect

Maximal subgroups

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

subgroupindex
#602782
#213622
#207472
#206542
#178172
#86692
#86682
#86672
#86662
#80972
#80942