Finite Group Databasebeta

← #370121

Subgroups — #370121

Order 512.

1364 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2905C211yesmaximal, characteristic
2C2×#1946C211yesmaximal, characteristic, complemented
3C2×#1945C211yesmaximal, characteristic, complemented
4C2×#1855C211yesmaximal, characteristic
5#20843C211yesmaximal, characteristic
6#20835C211yesmaximal, characteristic
7C2×#1753C211yesmaximal, characteristic
8#11329C211yesmaximal, characteristic, complemented
9#11329C211yesmaximal, characteristic, complemented
10#11221C211yesmaximal, characteristic, complemented
11#11221C211yesmaximal, characteristic, complemented
12#10883C211yesmaximal, characteristic, complemented
13#10882C211yesmaximal, characteristic, complemented
14#10733C211yesmaximal, characteristic
15#10732C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C23×M16C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C2×#427C2211yescharacteristic
23#2905—24no—
24#2905—24no—
25C2×#424—22no—
26C2×#424C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C2×#424C2211yescharacteristic
31C2×#424C2211yescharacteristic
32C2×#423—22no—
33C2×#407C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35C22×#97C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37#2280C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39#1946—24no—
40#1945—24no—
41#1855—24no—
42#1854C412yescomplemented
43#1854C412yescomplemented
44#1853C412yescomplemented
45#1852C412yescomplemented
46#1851—22no—
47#1851—22no—
48#1753—24no—
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51#1735C2211yescharacteristic
52#1735C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61#1729C2211yescharacteristic
62#1729C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C24×C4C2311yescharacteristic
66C22×M16—21no—
67C22×M16—22no—
68C22×M16—22no—
69C22×M16—22no—
70C22×M16—22no—
71C22×M16—22no—
72C22×M16—22no—
73C22×M16—22no—
74C22×M16—22no—
75C22×M16—41no—
76C22×M16—41no—
77C22×M16—41no—
78C22×M16—41no—
79C22×M16D412yes—
80C22×M16D412yes—
81C22×M16C2311yescharacteristic
82C22×M16C2311yescharacteristic
83C22×M16C2311yescharacteristic
84C22×M16C2311yescharacteristic
85C23×C8—21no—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×#117—21no—
91C2×#117—21no—
92C2×#117—22no—
93C2×#117—22no—
94C2×#117—22no—
95C2×#117—22no—
96C2×#117C2×C412yes—
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C4⋊C4C2×C412yes—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C42—21no—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C2×#108—22no—
108C2×#108—22no—
109C2×#108—22no—
110C2×#108—22no—
111C2×#108—41no—
112C2×#108—41no—
113#427—22no—
114#427—22no—
115#427—22no—
116#427—22no—
117C2×#107C2×C412yes—
118C2×#107C2×C412yes—
119C2×#106C2×C412yes—
120C2×#106C2×C412yes—
121#424—216no—
122#424—22no—
123#424—22no—
124#424—22no—
125#424—22no—
126#424—22no—
127#424—22no—
128#424—22no—
129#424—22no—
130#424—41no—
131#424—41no—
132#424D414yes—
133#424D414yes—
134#423—216no—
135C2×C4⋊C8—22no—
136C2×C4⋊C8—22no—
137C2×C4⋊C8—22no—
138C2×C4⋊C8—22no—
139C2×C4⋊C8—22no—
140C2×C4⋊C8—22no—
141#407—22no—
142#407—22no—
143C2×C22⋊C8—21no—
144C2×C22⋊C8—21no—
145C2×C22⋊C8C2×C411yescharacteristic
146C2×C22⋊C8C2×C411yescharacteristic
147C2×C22⋊C8C2×C411yescharacteristic
148C2×C22⋊C8C2×C411yescharacteristic
149C2×C22⋊C8D414yes—
150C2×C22⋊C8Q814yes—
151C2×#97—22no—
152C2×#97—22no—
153C2×#97—41no—
154C2×C4×C8—22no—
155C2×C4×C8—22no—
156C2×C4×C8—41no—
157#381—21no—
158#380C2×C412yes—
159#380C2×C412yes—
160#379—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—22no—
164C4×C22⋊C4Q814yes—
165C2×#95—21no—
166#341—22no—
167#341—22no—
168#340C2×C412yes—
169#340C2×C412yes—
170#340C2×C412yes—
171#340C2×C412yes—
172#339—22no—
173#339—22no—
174#339—22no—
175#339—22no—
176#339—22no—
177#339—22no—
178#339—41no—
179#339—41no—
180#336—22no—
181#336—22no—
182#336—22no—
183#336—22no—
184C25#4111yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—22no—
195C23×C4—21no—
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×Q812yes—
202C23×C4C2×Q812yes—
203C23×C4#4111yescharacteristic
204C23×C4#4112yes—
205C23×C4C2411yescharacteristic
206C2×M16—24no—
207C2×M16—24no—
208C2×M16—24no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—24no—
212C2×M16—24no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—42no—
234C2×M16—42no—
235C2×M16—44no—
236C2×M16—44no—
237C2×M16—44no—
238C2×M16—44no—
239C2×M16—44no—
240C2×M16—44no—
241C2×M16—44no—
242C2×M16—44no—
243C2×M16—42no—
244C2×M16—42no—
245C2×M16C2×D412yes—
246C2×M16C2×D412yes—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—24no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—42no—
271C22×C8—42no—
272C22×C8C22×C411yescharacteristic
273C22×C8C22×C411yescharacteristic
274C22×C8C22×C411yescharacteristic
275C22×C8C22×C411yescharacteristic
276C22×C8C2×D412yes—
277C22×C8C2×D412yes—
278C22×C8C2×D412yes—
279C22×C8C2×D412yes—
280C22×C8C2×D412yes—
281C22×C8C2×D412yes—
282C22×C8C2×Q812yes—
283C22×C8C2×Q812yes—
284#117—24no—
285#117—24no—
286#117—24no—
287#117—24no—
288#117—21no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117—24no—
293#117—24no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299#117—21no—
300#117—21no—
301#117—21no—
302#117C2×D412yes—
303#117C2×D412yes—
304#117#4112yes—
305#117#4112yes—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4C22×C412yes—
310C2×C4⋊C4C22×C412yes—
311C2×C4⋊C4C22×C412yes—
312C2×C4⋊C4C22×C412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—44no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42C2×Q812yes—
342C2×C42C2×Q812yes—
343#108—44no—
344#108—44no—
345#107—24no—
346#107—24no—
347#106—24no—
348#106—24no—
349C4⋊C8—28no—
350C4⋊C8—28no—
351C4⋊C8—28no—
352C4⋊C8—28no—
353C4⋊C8—44no—
354C4⋊C8—44no—
355C22⋊C8—42no—
356C22⋊C8—42no—
357C22⋊C8C4⋊C414yes—
358C22⋊C8C4⋊C414yes—
359C22⋊C8C4⋊C414yes—
360C22⋊C8C4⋊C414yes—
361#97—44no—
362C4×C8—44no—
363#95—22no—
364#95—22no—
365#95—24no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—42no—
371C24—42no—
372C24C4×D411yescharacteristic
373C24C4×D411yescharacteristic
374C24C22⋊Q812yes—
375C24#12512yes—
376C24C2×#4111yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
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—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4C2×C4⋊C412yes—
431C22×C4C2×C4⋊C412yes—
432C22×C4C4×D411yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4C4×Q812yes—
435C22×C4C4×Q812yes—
436C22×C4#12112yes—
437C22×C4#12112yes—
438C22×C4C22⋊Q812yes—
439C22×C4#12512yes—
440C22×C4C4⋊Q812yes—
441C22×C4C4⋊Q812yes—
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×Q811yescharacteristic
448C22×C4C22×Q811yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4H(5,𝔽2)11yescharacteristic
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C42−1+411yescharacteristic
455C22×C42−1+411yescharacteristic
456M16—22no—
457M16—22no—
458M16—22no—
459M16—22no—
460M16—48no—
461M16—48no—
462M16—48no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—41no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—48no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—41no—
517C2×C8C2×C4⋊C412yes—
518C2×C8C2×C4⋊C412yes—
519C2×C8C2×C4⋊C412yes—
520C2×C8C2×C4⋊C412yes—
521C2×C8#12014yes—
522C4⋊C4—28no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4C4×D414yes—
530C4⋊C4C4×D414yes—
531C22⋊C4—28no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—48no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C42—28no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C23—22no—
561C23—22no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—21no—
569C23—21no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—82no—
579C23—82no—
580C23—82no—
581C23—82no—
582C23#38912yes—
583C23#38912yes—
584C23C2×C4×D411yescharacteristic
585C23C2×C22⋊Q811yescharacteristic
586C23#52511yescharacteristic
587C23#52511yescharacteristic
588C23C2×#12511yescharacteristic
589C23#53911yescharacteristic
590C23#53911yescharacteristic
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
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—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—88no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4#38912yes—
639C2×C4#38912yes—
640C2×C4C22×C4⋊C411yescharacteristic
641C2×C4C2×C4×D411yescharacteristic
642C2×C4C2×C4×D411yescharacteristic
643C2×C4C2×C4×Q811yescharacteristic
644C2×C4#52011yescharacteristic
645C2×C4#52011yescharacteristic
646C2×C4C2×#12011yescharacteristic
647C2×C4C2×#12111yescharacteristic
648C2×C4#52511yescharacteristic
649C2×C4C2×C4⋊Q811yescharacteristic
650C2×C4#53411yescharacteristic
651C2×C4#53611yescharacteristic
652C2×C4D4211yescharacteristic
653C2×C4D4211yescharacteristic
654C2×C4#54611yescharacteristic
655C2×C4#54611yescharacteristic
656C2×C4#54711yescharacteristic
657C2×C4#54711yescharacteristic
658C2×C4D4×Q811yescharacteristic
659C2×C4D4×Q811yescharacteristic
660C2×C4D4×Q811yescharacteristic
661C2×C4D4×Q811yescharacteristic
662C2×C4#55211yescharacteristic
663C2×C4#55211yescharacteristic
664C2×C4#55411yescharacteristic
665C2×C4#55411yescharacteristic
666C2×C4#55411yescharacteristic
667C2×C4#55411yescharacteristic
668C8—22no—
669C8—22no—
670C8—22no—
671C8—22no—
672C8—24no—
673C8—42no—
674C8—42no—
675C8—42no—
676C8—81no—
677C8—81no—
678C22—22no—
679C22—22no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22C2×#38911yescharacteristic
701C22C4×#12011yescharacteristic
702C22#232211yescharacteristic
703C22#244911yescharacteristic
704C22#244911yescharacteristic
705C22#247311yescharacteristic
706C22#247311yescharacteristic
707C22#295612yes—
708C22#295712yes—
709C4—22no—
710C4—22no—
711C4—44no—
712C4—44no—
713C4—44no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4D4×C4⋊C411yescharacteristic
718C4D4×C4⋊C411yescharacteristic
719C4#236011yescharacteristic
720C4#236011yescharacteristic
721C4#241011yescharacteristic
722C4#241311yescharacteristic
723C4#243411yescharacteristic
724C4#258311yescharacteristic
725C2—21no—
726C2—21no—
727C2—21no—
728C2—21no—
729C2—42no—
730C2—42no—
731C2#1404311yescharacteristic
732C2#2116311yescharacteristic
733C2#2116511yescharacteristic
734C2#2146812yes—
735C2#2209712yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×M16 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1946C2semidirect
3. C2×#1945C2semidirect
8. #11329C2semidirect
9. #11329C2semidirect
10. #11221C2semidirect
11. #11221C2semidirect
12. #10883C2semidirect
13. #10882C2semidirect
42. #1854C4semidirect
43. #1854C4semidirect
44. #1853C4semidirect
45. #1852C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29052
C2×#19462
C2×#19452
C2×#18552
#208432
#208352
C2×#17532
#113292
#112212
#108832
#108822
#107332
#107322