Finite Group Databasebeta

← #368368

Subgroups — #368368

Order 512.

1944 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic, complemented
4C2×#1853C212yesmaximal, complemented
5C2×#1776C211yesmaximal, characteristic, complemented
6#13905C211yesmaximal, characteristic
7#11096C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal, complemented
12C22×C4×D4—21no—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C22×#107C2211yescharacteristic
16C22×#106C2212yes—
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×#97C2211yescharacteristic
22D4×C4⋊C4—24no—
23D4×C4⋊C4—24no—
24C2×#389—21no—
25C2×#385—22no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1955C418yescomplemented
34#1948—24no—
35#1853—28no—
36#1776—24no—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2212yes—
44#1445—21no—
45#1438—21no—
46#1437—22no—
47C23×D4D411yescharacteristic
48C24×C4—22no—
49C24×C4C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C2×C4×D4—28no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—48no—
59C2×C4×D4C2×C414yes—
60C2×C4×D4Q814yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—42no—
69C22×C4⋊C4—42no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—42no—
92C2×#106D414yescomplemented
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8C2×C414yescomplemented
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×C22⋊C8C2×C418yescomplemented
101C2×C22⋊C8C2×C418yescomplemented
102C2×#97—22no—
103C2×#97—22no—
104C2×#97C2×C414yescomplemented
105#389—28no—
106#385—216no—
107#384—22no—
108#384D414yes—
109#382—22no—
110#382—22no—
111#380—24no—
112#380—24no—
113#380—24no—
114#380—24no—
115#380—42no—
116#380—42no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4D414yes—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—22no—
132#340—22no—
133#340—42no—
134#340—42no—
135C25C2×D412yes—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4C4⋊C414yes—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—28no—
159C23×C4—22no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—82no—
165C23×C4—82no—
166C23×C4C22×C414yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×Q814yes—
171C23×C4C2411yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8C22×C412yes—
189C22×C8C22×C414yes—
190C22×C8C22×C412yes—
191C22×C8C2×D412yes—
192C22×C8#4112yes—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—28no—
196C4×D4—28no—
197C4×D4—432no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C4⋊C414yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4#4112yes—
247C2×C4⋊C4#4112yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4C4⋊C418yes—
260C2×C42—24no—
261C2×C42—44no—
262C2×C42—42no—
263C2×C42—82no—
264C2×C42C22×C412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×Q812yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270#107—28no—
271#106—28no—
272#106—48no—
273C4⋊C8—28no—
274C22⋊C8—216no—
275C22⋊C8—48no—
276#97—28no—
277#95—28no—
278#95—24no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—42no—
283#95—42no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24C2×C4⋊C414yes—
297C24AGL(1,ℤ/8)14yes—
298C24#13714yes—
299C24C22×D411yescharacteristic
300C2×D4—28no—
301C2×D4—216no—
302C2×D4—28no—
303C2×D4—216no—
304C2×D4—28no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—416no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—416no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
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—42no—
379C22×C4—88no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4C2×C4⋊C414yes—
385C22×C4C2×C4⋊C412yes—
386C22×C4C2×C4⋊C412yes—
387C22×C4#12012yes—
388C22×C4#12012yes—
389C22×C4#12312yes—
390C22×C4#12312yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×Q811yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C42−1+411yescharacteristic
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8C4×D414yescomplemented
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—28no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—84no—
453C4⋊C4#12114yes—
454C22⋊C4—216no—
455C22⋊C4—416no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C42—48no—
459C42—84no—
460C42C4×D414yes—
461C42C4×D414yes—
462C42C22⋊Q814yes—
463C42C22⋊Q814yes—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23#42818yes—
499C23C22×C4⋊C411yescharacteristic
500C23C2×#12011yescharacteristic
501C23C2×#12311yescharacteristic
502C23D4211yescharacteristic
503C23#54611yescharacteristic
504C23#54611yescharacteristic
505C23#54711yescharacteristic
506C23C2×AGL(1,ℤ/8)12yes—
507C23C2×#13712yes—
508C23#57712yes—
509C23#57712yes—
510D4—232no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—832no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#38514yes—
570C2×C4#38514yes—
571C2×C4C2×C4×D411yescharacteristic
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C4×#4111yescharacteristic
575C2×C4#51811yescharacteristic
576C2×C4#52011yescharacteristic
577C2×C4C2×#12111yescharacteristic
578C2×C4C2×C22⋊Q811yescharacteristic
579C2×C4C2×C22⋊Q811yescharacteristic
580C2×C4#52511yescharacteristic
581C2×C4#52911yescharacteristic
582C2×C4#53211yescharacteristic
583C2×C4#53511yescharacteristic
584C2×C4#53711yescharacteristic
585C2×C4#54011yescharacteristic
586C2×C4#54211yescharacteristic
587C2×C4D4211yescharacteristic
588C2×C4#54611yescharacteristic
589C2×C4#54811yescharacteristic
590C2×C4D4×Q811yescharacteristic
591C2×C4#55011yescharacteristic
592C2×C4#55011yescharacteristic
593C2×C4#55411yescharacteristic
594C2×C4#55411yescharacteristic
595C2×C4#55411yescharacteristic
596C2×C4#55611yescharacteristic
597C2×C4#55911yescharacteristic
598C2×C4#55911yescharacteristic
599C8—24no—
600C8—42no—
601C8—44no—
602C22—28no—
603C22—28no—
604C22—28no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22C2×#38511yescharacteristic
618C22D4×C4⋊C411yescharacteristic
619C22#234911yescharacteristic
620C22#240711yescharacteristic
621C22#240811yescharacteristic
622C22#244511yescharacteristic
623C22#244711yescharacteristic
624C22C2×#42812yes—
625C22#291212yes—
626C22#300311yescharacteristic
627C22#300411yescharacteristic
628C22#300911yescharacteristic
629C22#301011yescharacteristic
630C22#309212yes—
631C22#309912yes—
632C22#326512yes—
633C22#326612yes—
634C22#330612yes—
635C22#331012yes—
636C22#331612yes—
637C22#332012yes—
638C22#334412yes—
639C22#334512yes—
640C4—24no—
641C4—24no—
642C4—44no—
643C4—44no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—84no—
648C4—162no—
649C4C4×#12111yescharacteristic
650C4#232411yescharacteristic
651C4#236611yescharacteristic
652C4#236711yescharacteristic
653C4#245211yescharacteristic
654C4#245411yescharacteristic
655C4#253811yescharacteristic
656C4#258011yescharacteristic
657C2—48no—
658C2#1401811yescharacteristic
659C2#2084411yescharacteristic
660C2#2084811yescharacteristic
661C2#2265912yes—
662C2#2271112yes—
663C2#2412811yescharacteristic
664C2#2413411yescharacteristic
665C2#2425111yescharacteristic
666C2#2425211yescharacteristic
667C2#2624212yes—
668C2#2631012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
3. C2×#1948C2semidirect
4. C2×#1853C2semidirect
5. C2×#1776C2semidirect
7. #11096C2semidirect
8. #11095C2semidirect
9. #10850C2semidirect
10. #10843C2semidirect
11. #10730C2semidirect
33. #1955C4semidirect
92. C2×#106D4semidirect
95. C2×C4⋊C8C2×C4semidirect
100. C2×C22⋊C8C2×C4semidirect
101. C2×C22⋊C8C2×C4semidirect
104. C2×#97C2×C4semidirect
425. C2×C8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#19552
C2×#19482
C2×#18532
C2×#17762
#139052
#110962
#110952
#108502
#108432
#107302