Finite Group Databasebeta

← #364864

Subgroups — #364864

Order 512.

1912 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1984C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5C2×C4×#102C211yesmaximal, characteristic
6#11207C212yesmaximal, complemented
7#11049C212yesmaximal
8#10931C212yesmaximal
9#10884C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8—21no—
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2538—24no—
22#2538C418yescomplemented
23C2×#396—22no—
24Q8×C4⋊C4—24no—
25Q8×C4⋊C4—24no—
26C2×#390C2212yes—
27C2×#389—22no—
28C2×#389—21no—
29C2×#389C2212yes—
30C2×#384C2212yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#380—21no—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1984—24no—
37#1802—24no—
38C4×#102—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C23×Q8D411yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C22⋊Q8—28no—
55C2×C22⋊Q8—44no—
56C2×C22⋊Q8—44no—
57C2×C4×Q8—28no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—44no—
63C2×C4×Q8—44no—
64C2×C4×Q8C2×C414yes—
65C2×C4×Q8D414yes—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4C2×C414yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D412yes—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C42—22no—
90C22×C42—21no—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—42no—
104C2×#102Q814yescomplemented
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108#396—216no—
109#390—24no—
110#390C2×C418yes—
111#389—28no—
112#389—216no—
113#389—24no—
114#389C2×C418yes—
115#384—24no—
116#384—24no—
117#384—22no—
118#384—22no—
119#384C2×C414yes—
120#384D414yes—
121#380—28no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4D414yes—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129#340—24no—
130#340—24no—
131#340—22no—
132#340—22no—
133#340—24no—
134#340—42no—
135#340—42no—
136#336—24no—
137#336—24no—
138C25—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8C22⋊C414yes—
144C22×Q8D814yes—
145C22×Q8SD1614yes—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C22×C412yes—
158C22×D4C2×Q812yes—
159C22×D4#4112yes—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8C2×Q812yes—
197C22×C8#4112yes—
198C22⋊Q8—432no—
199C4×Q8—28no—
200C4×Q8—28no—
201C4×Q8—28no—
202C4×Q8—28no—
203C4×Q8—432no—
204C4×D4—28no—
205C4×D4—28no—
206C4×D4—44no—
207C4×D4—44no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
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—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—81no—
251C2×C4⋊C4—81no—
252C2×C4⋊C4—81no—
253C2×C4⋊C4—81no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4—82no—
256C2×C4⋊C4C22⋊C414yes—
257C2×C4⋊C4C22⋊C418yes—
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×Q812yes—
264C2×C4⋊C4#4112yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42C22×C414yes—
285C2×C42C22×C412yes—
286C2×C42C2×D414yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42#4112yes—
291C2×C42#4112yes—
292C4⋊C8—44no—
293#102—28no—
294#102—28no—
295#102—48no—
296C4×C8—44no—
297#95—28no—
298#95—28no—
299#95—24no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24C22×D411yescharacteristic
316C2×Q8—28no—
317C2×Q8—28no—
318C2×Q8—28no—
319C2×Q8—28no—
320C2×Q8—28no—
321C2×Q8—28no—
322C2×Q8#102116yescomplemented
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
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—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4C4×Q814yescomplemented
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4C2×C22⋊C414yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C414yes—
410C22×C4C2×C22⋊C412yes—
411C22×C4#12012yes—
412C22×C4#12012yes—
413C22×C4#12312yes—
414C22×C4#12312yes—
415C22×C4C2×D814yes—
416C22×C4C2×D812yes—
417C22×C4C2×SD1614yes—
418C22×C4C2×SD1612yes—
419C22×C4AGL(1,ℤ/8)12yes—
420C22×C4#13712yes—
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×Q811yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C42−1+411yescharacteristic
433C22×C42−1+411yescharacteristic
434C22×C42−1+411yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
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—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—216no—
460C4⋊C4—28no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—416no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—88no—
484C4⋊C4C22⋊Q814yescomplemented
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—88no—
488C42—24no—
489C42—24no—
490C42—24no—
491C42—24no—
492C42—416no—
493C42—48no—
494C42C4×D414yes—
495C42C4×D414yes—
496C42#12114yes—
497C42#12114yes—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23—81no—
523C23—81no—
524C23—81no—
525C23—81no—
526C23—81no—
527C23—81no—
528C23—81no—
529C23—81no—
530C23—81no—
531C23—81no—
532C23—81no—
533C23C22×C22⋊C411yescharacteristic
534C23C2×#12011yescharacteristic
535C23C2×#12311yescharacteristic
536C23#54811yescharacteristic
537C23#54811yescharacteristic
538C23D4×Q811yescharacteristic
539C23#55011yescharacteristic
540C23C22×D811yescharacteristic
541C23C22×SD1611yescharacteristic
542C23C2×AGL(1,ℤ/8)11yescharacteristic
543C23C2×#13711yescharacteristic
544C23#57611yescharacteristic
545C23#57711yescharacteristic
546C23#57711yescharacteristic
547C23#57811yescharacteristic
548Q8—232no—
549D4—24no—
550D4—24no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—81no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—81no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—816no—
612C2×C4#38614yes—
613C2×C4#38614yes—
614C2×C4C2×#10218yes—
615C2×C4C2×#10214yes—
616C2×C4#41714yes—
617C2×C4#44714yes—
618C2×C4#44814yes—
619C2×C4#48014yes—
620C2×C4#48114yes—
621C2×C4C2×C4×D411yescharacteristic
622C2×C4C2×C4×D411yescharacteristic
623C2×C4C2×C4×Q811yescharacteristic
624C2×C4C4×#4111yescharacteristic
625C2×C4#51911yescharacteristic
626C2×C4#52011yescharacteristic
627C2×C4C2×#12111yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4C2×C22⋊Q811yescharacteristic
630C2×C4#52511yescharacteristic
631C2×C4#52911yescharacteristic
632C2×C4#53311yescharacteristic
633C2×C4#53611yescharacteristic
634C2×C4#53711yescharacteristic
635C2×C4#54111yescharacteristic
636C2×C4#54211yescharacteristic
637C2×C4#54611yescharacteristic
638C2×C4#54711yescharacteristic
639C2×C4#54811yescharacteristic
640C2×C4#54811yescharacteristic
641C2×C4#54811yescharacteristic
642C2×C4D4×Q811yescharacteristic
643C2×C4D4×Q811yescharacteristic
644C2×C4#55011yescharacteristic
645C2×C4#55211yescharacteristic
646C2×C4#55211yescharacteristic
647C2×C4#55411yescharacteristic
648C2×C4#55611yescharacteristic
649C8—42no—
650C8—42no—
651C8—82no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22C2×#38611yescharacteristic
668C22Q8×C22⋊C411yescharacteristic
669C22#234111yescharacteristic
670C22#240711yescharacteristic
671C22#240911yescharacteristic
672C22#242111yescharacteristic
673C22#242311yescharacteristic
674C22C22×#10211yescharacteristic
675C22C2×#41711yescharacteristic
676C22#289711yescharacteristic
677C22#289811yescharacteristic
678C22C2×#44711yescharacteristic
679C22C2×#44811yescharacteristic
680C22#301211yescharacteristic
681C22#301311yescharacteristic
682C22C2×#48011yescharacteristic
683C22C2×#48111yescharacteristic
684C22#309811yescharacteristic
685C22#309911yescharacteristic
686C22#335811yescharacteristic
687C22#335811yescharacteristic
688C22#335911yescharacteristic
689C22#336211yescharacteristic
690C22#336611yescharacteristic
691C22#336711yescharacteristic
692C22#336911yescharacteristic
693C22#337211yescharacteristic
694C22Q8×D811yescharacteristic
695C22Q8×SD1611yescharacteristic
696C22#338411yescharacteristic
697C22#338511yescharacteristic
698C22#339111yescharacteristic
699C22#339211yescharacteristic
700C22#339711yescharacteristic
701C22#339811yescharacteristic
702C4—28no—
703C4—24no—
704C4—44no—
705C4—44no—
706C4—84no—
707C4—82no—
708C4—84no—
709C4—82no—
710C4#187418yes—
711C4C4×C22⋊Q811yescharacteristic
712C4#232311yescharacteristic
713C4#236011yescharacteristic
714C4#236211yescharacteristic
715C4#242611yescharacteristic
716C4#242911yescharacteristic
717C4#253611yescharacteristic
718C4#253811yescharacteristic
719C2—84no—
720C2#1396911yescharacteristic
721C2C2×#187411yescharacteristic
722C2#2091511yescharacteristic
723C2Q8×#10211yescharacteristic
724C2#2252211yescharacteristic
725C2#2254011yescharacteristic
726C2#2254411yescharacteristic
727C2#2418411yescharacteristic
728C2#2418711yescharacteristic
729C2#2432611yescharacteristic
730C2#2432911yescharacteristic
731C2#2499011yescharacteristic
732C2#2499211yescharacteristic
733C2#2505611yescharacteristic
734C2#2505811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
3. C2×#1984C2semidirect
6. #11207C2semidirect
9. #10884C2semidirect
10. #10849C2semidirect
11. #10841C2semidirect
22. #2538C4semidirect
104. C2×#102Q8semidirect
322. C2×Q8#102semidirect
350. C2×D4C4×Q8semidirect
484. C4⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×Q8×C4⋊C42
C2×#19842
C2×#18022
C2×C4×#1022
#112072
#110492
#109312
#108842
#108492
#108412