Finite Group Databasebeta

← #363236

Subgroups — #363236

Order 512.

3874 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1954C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic, complemented
7#13906C211yesmaximal, characteristic, complemented
8#10969C214yesmaximal, complemented
9#10730C214yesmaximal, complemented
10C23×#41—21no—
11C2×#547—24no—
12C2×#547—24no—
13C2×#547—24no—
14C2×#547—24no—
15C2×#547—24no—
16C2×#547C418yescomplemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#123—22no—
19C22×C22⋊Q8—21no—
20C22×#121C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2212yes—
27C22×C22⋊C8C2214yes—
28C22×C4×C8C2211yescharacteristic
29C2×#390—21no—
30C2×#384C2211yescharacteristic
31C2×#380C2214yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990—24no—
34#1988—24no—
35#1954—24no—
36#1802—24no—
37C4×#102—24no—
38C2×#340C2214yes—
39C2×#340C2214yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42#1439—22no—
43C4⋊#95—21no—
44C22×#41—28no—
45C22×#41—28no—
46C22×#41—28no—
47C22×#41—44no—
48C22×#41—44no—
49C22×#41—44no—
50C22×#41—44no—
51C23×Q8—21no—
52C23×D4—22no—
53C23×D4D411yescharacteristic
54C23×D4C2311yescharacteristic
55C24×C4—22no—
56C24×C4—21no—
57C24×C4C2312yes—
58C23×C8C2312yes—
59C23×C8C2312yes—
60#547—232no—
61#547—232no—
62#547—232no—
63#547—232no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8C2×C414yes—
68C2×C4⋊Q8D414yes—
69C2×C4⋊Q8D414yes—
70C2×#123—216no—
71C2×#123—216no—
72C2×#123—216no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—28no—
76C2×#121—24no—
77C2×#121—24no—
78C2×#121—24no—
79C2×#121—24no—
80C2×#121—24no—
81C2×#121C2×C418yes—
82C2×C4×D4—22no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4—22no—
86C2×C4×D4—22no—
87C2×C4×D4—22no—
88C2×C4×D4—22no—
89C2×C4×D4C2×C414yes—
90C2×C4×D4C2×C414yes—
91C2×C4×D4D414yes—
92C2×C4×D4D414yes—
93C2×C4×D4D414yes—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2×C418yes—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4D412yes—
113C22×C42—21no—
114C22×C42C2311yescharacteristic
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×#102—24no—
119C2×#102—24no—
120C2×#102—24no—
121C2×#102—24no—
122C2×#102—44no—
123C2×C22⋊C8—28no—
124C2×C22⋊C8—28no—
125C2×C22⋊C8—44no—
126C2×C4×C8—22no—
127C2×C4×C8—22no—
128C2×C4×C8—22no—
129#390—44no—
130#384—22no—
131#384—22no—
132#380—28no—
133#380—44no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C2×#95—24no—
137C2×#95—21no—
138#340—24no—
139#340—28no—
140#340—28no—
141#340—24no—
142C25—21no—
143C25C2×D412yes—
144C2×#41—264no—
145C2×#41—432no—
146C2×#41—432no—
147C2×#41—432no—
148C2×#41—432no—
149C2×#41—432no—
150C2×#41—432no—
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4C22⋊C414yes—
177C22×D4D814yes—
178C22×D4SD1614yes—
179C22×D4C22×C412yes—
180C22×D4C2×D412yes—
181C22×D4C2×D412yes—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4C22×C414yes—
209C23×C4C2×D412yes—
210C23×C4C2×D414yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2411yescharacteristic
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—24no—
224C22×C8—24no—
225C22×C8#4114yescomplemented
226C4⋊Q8—28no—
227C4⋊Q8—28no—
228C4⋊Q8—28no—
229C4⋊Q8C22⋊C4116yescomplemented
230#123—2128no—
231C22⋊Q8—264no—
232#121—216no—
233#121—216no—
234#121—216no—
235#121—216no—
236C4×D4—28no—
237C4×D4—28no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—28no—
241C4×D4—28no—
242C4×D4—28no—
243C4×D4C22⋊C4116yescomplemented
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—216no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4C22⋊C414yes—
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C22×C414yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4#4114yescomplemented
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—84no—
295C2×C22⋊C4C22⋊C418yes—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42C22×C412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C4⋊C8—44no—
306#102—216no—
307#102—48no—
308C22⋊C8—416no—
309C4×C8—44no—
310#95—48no—
311#95—42no—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—41no—
326C24—41no—
327C24—41no—
328C24—41no—
329C24—42no—
330C24—42no—
331C24C2×C22⋊C414yes—
332C24C2×D814yes—
333C24C2×SD1614yes—
334C24C22×D411yescharacteristic
335#41—4256no—
336C2×Q8—216no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×D4—232no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—24no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—416no—
364C2×D4—416no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4#102116yescomplemented
374C2×D4C2×C22⋊C414yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—232no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—24no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—416no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—416no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4C2×C22⋊C414yes—
440C22×C4C2×C22⋊C412yes—
441C22×C4C2×C22⋊C412yes—
442C22×C4#12012yes—
443C22×C4#12012yes—
444C22×C4#12012yes—
445C22×C4#12012yes—
446C22×C4C2×D812yes—
447C22×C4C2×SD1612yes—
448C22×C4AGL(1,ℤ/8)12yes—
449C22×C4AGL(1,ℤ/8)12yes—
450C22×C4C23×C411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+412yes—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—44no—
471C2×C8—44no—
472C4⋊C4—232no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—216no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4C2×C22⋊C414yes—
491C4⋊C4C2×C22⋊C418yes—
492C22⋊C4—232no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—816no—
496C42—84no—
497C42C2×C22⋊C414yes—
498C42C2×C22⋊C414yes—
499C42#12014yes—
500C42#12014yes—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—28no—
512C23—24no—
513C23—24no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23C2×#10218yes—
547C23C22×C22⋊C411yescharacteristic
548C23C2×#12011yescharacteristic
549C23C2×#12011yescharacteristic
550C23#54612yes—
551C23#54712yes—
552C23C22×D811yescharacteristic
553C23C22×SD1611yescharacteristic
554C23C2×AGL(1,ℤ/8)11yescharacteristic
555C23C2×AGL(1,ℤ/8)11yescharacteristic
556C23#57712yes—
557C23#57812yes—
558Q8—432no—
559D4—24no—
560D4—24no—
561D4—232no—
562D4—464no—
563D4—44no—
564D4—44no—
565D4—42no—
566D4—42no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—464no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—432no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—84no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—84no—
610C2×C4#37914yes—
611C2×C4#37914yes—
612C2×C4C2×#10214yes—
613C2×C4#41814yes—
614C2×C4#44714yes—
615C2×C4#44714yes—
616C2×C4#45014yes—
617C2×C4#45014yes—
618C2×C4C22×C22⋊C411yescharacteristic
619C2×C4C22×C22⋊C411yescharacteristic
620C2×C4C4×#4112yes—
621C2×C4#52012yes—
622C2×C4C2×#12011yescharacteristic
623C2×C4C2×#12011yescharacteristic
624C2×C4#52512yes—
625C2×C4#53211yescharacteristic
626C2×C4#53211yescharacteristic
627C2×C4#53511yescharacteristic
628C2×C4#53611yescharacteristic
629C2×C4#53712yes—
630C2×C4#54612yes—
631C2×C4#54712yes—
632C2×C4#54812yes—
633C2×C4#54812yes—
634C2×C4#55012yes—
635C2×C4#55012yes—
636C8—44no—
637C8—82no—
638C22—28no—
639C22—216no—
640C22—48no—
641C22—48no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—48no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—84no—
653C22—84no—
654C22—82no—
655C22—82no—
656C22—84no—
657C22—82no—
658C22—82no—
659C22—84no—
660C22C2×#37911yescharacteristic
661C22#233912yes—
662C22#240712yes—
663C22#243111yescharacteristic
664C22#243211yescharacteristic
665C22C22×#10211yescharacteristic
666C22C2×#41811yescharacteristic
667C22#289812yes—
668C22C2×#44711yescharacteristic
669C22C2×#44711yescharacteristic
670C22C2×#45011yescharacteristic
671C22C2×#45011yescharacteristic
672C22#301011yescharacteristic
673C22#301111yescharacteristic
674C22#301312yes—
675C22#329412yes—
676C22#329812yes—
677C22#333012yes—
678C22#333212yes—
679C22#333412yes—
680C22#333512yes—
681C22#335312yes—
682C22#335412yes—
683C4—24no—
684C4—44no—
685C4—44no—
686C4—48no—
687C4—84no—
688C4—162no—
689C4#178618yes—
690C4#229711yescharacteristic
691C4#231111yescharacteristic
692C4#234112yes—
693C4#240712yes—
694C4#243111yescharacteristic
695C4#243211yescharacteristic
696C2—48no—
697C2—84no—
698C2#1393711yescharacteristic
699C2C2×#178611yescharacteristic
700C2#2060311yescharacteristic
701C2#2248512yes—
702C2#2250012yes—
703C2#2411612yes—
704C2#2412012yes—
705C2#2556711yescharacteristic
706C2#2557111yescharacteristic
707C2#2569911yescharacteristic
708C2#2570311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1990C2semidirect
3. C2×#1988C2semidirect
4. C2×#1954C2semidirect
5. C2×#1802C2semidirect
6. C2×C4×#102C2semidirect
7. #13906C2semidirect
8. #10969C2semidirect
9. #10730C2semidirect
16. C2×#547C4semidirect
225. C22×C8#41semidirect
229. C4⋊Q8C22⋊C4semidirect
243. C4×D4C22⋊C4semidirect
283. C2×C4⋊C4#41semidirect
373. C2×D4#102semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#19902
C2×#19882
C2×#19542
C2×#18022
C2×C4×#1022
#139062
#109692
#107302