Finite Group Databasebeta

← #115491

Subgroups — #115491

Order 512.

1338 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
1C22×#474C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1981C211yesmaximal, characteristic, complemented
4C22×D4⋊C8C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6C2×#1296C211yesmaximal, characteristic, complemented
7C2×#1279C211yesmaximal, characteristic, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#474—22no—
11C2×#474—22no—
12C2×#474—22no—
13C2×#474—22no—
14C2×#474—22no—
15C2×#474C414yescomplemented
16C22×#107—21no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#103—22no—
20C22×#102—21no—
21C22×C4×C8C2211yescharacteristic
22C2×#390—21no—
23C2×#384C2211yescharacteristic
24C2×#380—22no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1990—22no—
27#1990—22no—
28#1981—22no—
29#1981—22no—
30C2×#340—22no—
31C2×D4⋊C8—22no—
32C2×D4⋊C8—22no—
33C2×D4⋊C8—22no—
34#1439—22no—
35C4⋊#95—21no—
36#1296—22no—
37#1279—22no—
38C23×Q8—21no—
39C23×D4D411yescharacteristic
40C24×C4—21no—
41C23×C8—21no—
42C23×C8—21no—
43C23×C8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2×C412yes—
54C2×C4⋊Q8D412yes—
55C2×C4⋊Q8D412yes—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4D412yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C4⋊C4D412yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—24no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#474—28no—
80#474—28no—
81#474—28no—
82#474—28no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—24no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C412yes—
95C2×C4⋊C8D412yes—
96C2×C4⋊C8D412yes—
97C2×C4⋊C8D412yes—
98C2×#103—28no—
99C2×#103—44no—
100C2×#103—44no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8D412yes—
110#390—42no—
111#390—42no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384—21no—
116#380—44no—
117#380—44no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C2×#95—24no—
123C2×#95—21no—
124#340—28no—
125D4⋊C8—28no—
126C25—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—41no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22⋊C412yes—
149C22×D4SD1612yes—
150C22×D4Q1612yes—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C22×C8—24no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—42no—
175C22×C8—41no—
176C22×C8—41no—
177C22×C8—41no—
178C22×C8—41no—
179C22×C8—41no—
180C22×C8—41no—
181C22×C8—41no—
182C22×C8—41no—
183C22×C8—42no—
184C22×C8—42no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C4⋊Q8—41no—
197C4⋊Q8—41no—
198C4⋊Q8—41no—
199C4⋊Q8—41no—
200C4⋊Q8C22⋊C414yescomplemented
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4C22⋊C412yes—
253C2×C4⋊C4D814yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—84no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42—41no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42#4111yescharacteristic
281#107—216no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—41no—
288C4⋊C8—41no—
289C4⋊C8C22⋊C414yescomplemented
290#103—416no—
291#102—216no—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8—41no—
295C4×C8—41no—
296#95—48no—
297#95—41no—
298#95—41no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—41no—
303C24—41no—
304C24—42no—
305C24—42no—
306C24—41no—
307C24—41no—
308C24—42no—
309C24—42no—
310C24#12011yescharacteristic
311C2×Q8—24no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—82no—
322C2×Q8—82no—
323C2×Q8—82no—
324C2×Q8—82no—
325C2×Q8—81no—
326C2×Q8—81no—
327C2×Q8—81no—
328C2×Q8—81no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4#10314yescomplemented
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C4×D411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4#12011yescharacteristic
403C22×C4#12011yescharacteristic
404C22×C4#12011yescharacteristic
405C22×C4#12011yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12311yescharacteristic
415C22×C4#12711yescharacteristic
416C22×C4#12711yescharacteristic
417C22×C4C2×D812yes—
418C22×C4C2×SD1611yescharacteristic
419C22×C4C2×Q1611yescharacteristic
420C22×C4#13511yescharacteristic
421C22×C4#13511yescharacteristic
422C22×C4#13512yes—
423C22×C4AGL(1,ℤ/8)11yescharacteristic
424C22×C4AGL(1,ℤ/8)12yes—
425C22×C4AGL(1,ℤ/8)12yes—
426C22×C4#13711yescharacteristic
427C22×C4#13711yescharacteristic
428C22×C4#13711yescharacteristic
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—44no—
432C2×C8—42no—
433C2×C8—44no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—44no—
443C2×C8—82no—
444C2×C8—81no—
445C2×C8—81no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—82no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—82no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—816no—
476C42—81no—
477C42—81no—
478C42—81no—
479C42—81no—
480C42C2×C22⋊C411yescharacteristic
481C42C2×C22⋊C411yescharacteristic
482C42C4×D411yescharacteristic
483C42C4×D411yescharacteristic
484C42#12011yescharacteristic
485C42#12011yescharacteristic
486C42#12011yescharacteristic
487C42#12111yescharacteristic
488C42#12111yescharacteristic
489C42#12111yescharacteristic
490C42#12111yescharacteristic
491C42#12311yescharacteristic
492C42#12311yescharacteristic
493C42#12311yescharacteristic
494C42#12411yescharacteristic
495C42#12411yescharacteristic
496C23—24no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—81no—
515C23—81no—
516C23—82no—
517C23—81no—
518C23—81no—
519C23#38611yescharacteristic
520C23#39211yescharacteristic
521C23#39211yescharacteristic
522C23#44712yes—
523C23#44912yes—
524C23#45011yescharacteristic
525C23#45111yescharacteristic
526C23#45211yescharacteristic
527C23#45211yescharacteristic
528C23#45512yes—
529C23#45612yes—
530Q8—48no—
531Q8—84no—
532Q8—82no—
533Q8—82no—
534Q8—82no—
535Q8—82no—
536D4—24no—
537D4—24no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—84no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—84no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4—82no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4#37911yescharacteristic
581C2×C4#38611yescharacteristic
582C2×C4#38611yescharacteristic
583C2×C4#38611yescharacteristic
584C2×C4#39011yescharacteristic
585C2×C4#39011yescharacteristic
586C2×C4#39211yescharacteristic
587C2×C4#39211yescharacteristic
588C2×C4#39411yescharacteristic
589C2×C4#39411yescharacteristic
590C2×C4#39411yescharacteristic
591C2×C4#39411yescharacteristic
592C2×C4C2×#10311yescharacteristic
593C2×C4#41611yescharacteristic
594C2×C4#41711yescharacteristic
595C2×C4#41911yescharacteristic
596C2×C4C4×D812yes—
597C2×C4#44212yes—
598C2×C4#44712yes—
599C2×C4#44811yescharacteristic
600C2×C4#44811yescharacteristic
601C2×C4#44912yes—
602C2×C4#44911yescharacteristic
603C2×C4#44911yescharacteristic
604C2×C4#45111yescharacteristic
605C2×C4#45111yescharacteristic
606C2×C4#45211yescharacteristic
607C2×C4#45211yescharacteristic
608C2×C4#45912yes—
609C2×C4#45912yes—
610C2×C4#46312yes—
611C2×C4#46312yes—
612C2×C4#46511yescharacteristic
613C2×C4#46511yescharacteristic
614C2×C4#46612yes—
615C2×C4#46711yescharacteristic
616C2×C4#46711yescharacteristic
617C2×C4#46811yescharacteristic
618C2×C4#46811yescharacteristic
619C2×C4#46912yes—
620C2×C4#47011yescharacteristic
621C2×C4#47011yescharacteristic
622C2×C4#48311yescharacteristic
623C2×C4#48411yescharacteristic
624C2×C4#48511yescharacteristic
625C2×C4#48511yescharacteristic
626C2×C4#49311yescharacteristic
627C2×C4#49511yescharacteristic
628C2×C4#49611yescharacteristic
629C2×C4#49611yescharacteristic
630C8—44no—
631C8—44no—
632C8—82no—
633C22—48no—
634C22—44no—
635C22—82no—
636C22—82no—
637C22—84no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—84no—
644C22#143811yescharacteristic
645C22#163912yes—
646C22#164112yes—
647C22#164412yes—
648C22#164612yes—
649C22#164712yes—
650C22#164912yes—
651C22#165212yes—
652C22#165412yes—
653C22#187211yescharacteristic
654C22#187311yescharacteristic
655C22#187912yes—
656C22#188212yes—
657C22#199912yes—
658C22#200011yescharacteristic
659C22#200011yescharacteristic
660C22#200111yescharacteristic
661C22#200111yescharacteristic
662C22#200411yescharacteristic
663C22#200511yescharacteristic
664C22#200712yes—
665C22#201111yescharacteristic
666C22#201211yescharacteristic
667C4—21no—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—82no—
672C4—84no—
673C4—84no—
674C4—162no—
675C4#178711yescharacteristic
676C4#178811yescharacteristic
677C4#187912yes—
678C4#189211yescharacteristic
679C4#189411yescharacteristic
680C4#197011yescharacteristic
681C4#197211yescharacteristic
682C4#201111yescharacteristic
683C4#201211yescharacteristic
684C4#203511yescharacteristic
685C4#203511yescharacteristic
686C4#203611yescharacteristic
687C4#203611yescharacteristic
688C4#205412yes—
689C2—84no—
690C2#898112yes—
691C2#898412yes—
692C2#1007011yescharacteristic
693C2#1007211yescharacteristic
694C2#1007411yescharacteristic
695C2#1007611yescharacteristic
696C2#1023112yes—
697C2#1023312yes—
698C2#1097011yescharacteristic
699C2#1101711yescharacteristic
700C2#1102211yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1990C2semidirect
3. C2×#1981C2semidirect
6. C2×#1296C2semidirect
7. C2×#1279C2semidirect
15. C2×#474C4semidirect
200. C4⋊Q8C22⋊C4semidirect
289. C4⋊C8C22⋊C4semidirect
356. C2×D4#103semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#19902
C2×#19812
C22×D4⋊C82
#139062
C2×#12962
C2×#12792