Finite Group Databasebeta

← #382552

Subgroups — #382552

Order 512.

2280 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2297C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×#1789C212yesmaximal
4C4×#380C211yesmaximal, characteristic, complemented
5#11075C214yesmaximal, complemented
6#11053C214yesmaximal, complemented
7#10733C212yesmaximal, complemented
8C23×#41C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#423C2212yescomplemented
12C22×#104C2218yes—
13C2×#407C2212yes—
14#2297—21no—
15#2297—21no—
16#2297—21no—
17#2297—21no—
18#2297C412yescomplemented
19#2297C412yescomplemented
20#2297C2212yes—
21#2297C2212yes—
22C2×#391C411yescharacteristic, complemented
23C2×#391C411yescharacteristic, complemented
24C2×#390C411yescharacteristic, complemented
25C2×#390C411yescharacteristic, complemented
26C2×#386—21no—
27C2×#386—21no—
28C2×#380—21no—
29C2×C4×C4⋊C4C2212yescomplemented
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4—21no—
32C2×C4×C22⋊C4C2212yes—
33#1797—21no—
34#1797—21no—
35#1797—21no—
36#1797—21no—
37#1789—28no—
38C2×#339C2218yescomplemented
39#1728C2212yescomplemented
40C4⋊#95—21no—
41#1434—21no—
42C22⋊#95—21no—
43C4×#95C2212yescomplemented
44C22×#41—21no—
45C22×#41—21no—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—44no—
49C22×#41—44no—
50C22×#41C2×C412yes—
51C22×#41D414yes—
52C22×#41Q814yes—
53C22×#41C2312yes—
54C23×Q8C2×C411yescharacteristic
55C23×D4—21no—
56C23×D4C2×C411yescharacteristic
57C24×C4—21no—
58C24×C4D411yescharacteristic
59C24×C4D411yescharacteristic
60C24×C4C2311yescharacteristic
61C22×M16C2314yes—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—44no—
65C2×#117C2×C412yes—
66C2×#117C2312yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—21no—
77C22×C42—44no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2314yes—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#423—44no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×#104—216no—
95C2×#104—48no—
96C2×#104—48no—
97#407—24no—
98#407—24no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101#391—21no—
102#391—21no—
103#391—21no—
104#391—21no—
105#391C2×C412yes—
106#391C2×C412yes—
107#391C2×C412yes—
108#391C2×C412yes—
109#390—21no—
110#390—21no—
111#390—21no—
112#390—21no—
113#390C2×C412yes—
114#390C2×C412yes—
115#390C2×C412yes—
116#390C2×C412yes—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#380—28no—
126C4×C4⋊C4—44no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—24no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145#339—48no—
146#336—22no—
147#336—22no—
148C25—21no—
149C25C2×D411yescharacteristic
150C2×#41—24no—
151C2×#41—24no—
152C2×#41—24no—
153C2×#41—24no—
154C2×#41—24no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—28no—
158C2×#41—28no—
159C2×#41—44no—
160C2×#41—44no—
161C2×#41—416no—
162C2×#41—416no—
163C2×#41—44no—
164C2×#41—44no—
165C2×#41—44no—
166C2×#41—44no—
167C2×#41—416no—
168C2×#41—416no—
169C2×#41C4⋊C418yescomplemented
170C2×#41C2×D414yes—
171C2×#41C2×Q814yes—
172C22×Q8—21no—
173C22×Q8—22no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8C4⋊C414yescomplemented
177C22×Q8C22×C412yes—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—22no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4C4⋊C414yescomplemented
189C22×D4C22×C412yes—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—21no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C22⋊C411yescharacteristic
209C23×C4C22⋊C411yescharacteristic
210C23×C4C22⋊C411yescharacteristic
211C23×C4C22⋊C411yescharacteristic
212C23×C4C22⋊C411yescharacteristic
213C23×C4C22⋊C411yescharacteristic
214C23×C4C22⋊C411yescharacteristic
215C23×C4C22⋊C411yescharacteristic
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×Q812yes—
227C23×C4#4112yes—
228C23×C4#4112yes—
229C23×C4C2411yescharacteristic
230C2×M16—24no—
231C2×M16—24no—
232C2×M16—28no—
233C2×M16—28no—
234C22×C8—24no—
235#117—28no—
236#117—44no—
237#117—44no—
238#117—44no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—28no—
273C2×C42—28no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—44no—
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C4⋊C8—44no—
302C4⋊C8—44no—
303#104—432no—
304C22⋊C8—44no—
305C22⋊C8—44no—
306#95—28no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—44no—
316#95—44no—
317#95—44no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—41no—
325C24—41no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334C24#12112yes—
335C24C22⋊Q812yes—
336C24C22×D411yescharacteristic
337#41—216no—
338#41—216no—
339#41—416no—
340#41—416no—
341#41—416no—
342#41—416no—
343#41—416no—
344#41—416no—
345#41—48no—
346#41—48no—
347C2×Q8—22no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—48no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—48no—
355C2×Q8C2×C4⋊C414yes—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—28no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—48no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4C2×C4⋊C414yes—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
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—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—416no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—416no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4#10414yescomplemented
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C4⋊C412yes—
465C22×C4C2×C4⋊C412yes—
466C22×C4#11712yes—
467C22×C4#11712yes—
468C22×C4#12112yes—
469C22×C4#12112yes—
470C22×C4#12112yes—
471C22×C4C22⋊Q812yes—
472C22×C4C22⋊Q812yes—
473C22×C4C22⋊Q812yes—
474C22×C4C23×C411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×Q811yescharacteristic
478C22×C4C2×#4111yescharacteristic
479C22×C4C2×#4111yescharacteristic
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)14yes—
482M16—48no—
483M16—48no—
484C2×C8—44no—
485C2×C8—44no—
486C2×C8—44no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—416no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—416no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—44no—
520C42—44no—
521C42—48no—
522C42—48no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—48no—
549C23—48no—
550C23—42no—
551C23—42no—
552C23—48no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23#38012yes—
562C23#38012yes—
563C23C2×#10412yes—
564C23#42112yes—
565C23C22×C22⋊C411yescharacteristic
566C23C2×#12111yescharacteristic
567C23C2×C22⋊Q811yescharacteristic
568C23#53412yes—
569C23#53512yes—
570Q8—28no—
571Q8—48no—
572Q8—44no—
573D4—28no—
574D4—48no—
575D4—416no—
576D4—48no—
577D4—48no—
578D4—44no—
579C2×C4—28no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—24no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—416no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4#38012yes—
648C2×C4#38012yes—
649C2×C4C2×#10412yes—
650C2×C4C2×#10412yes—
651C2×C4#42112yes—
652C2×C4#42112yes—
653C2×C4C22×C4⋊C411yescharacteristic
654C2×C4C2×#11711yescharacteristic
655C2×C4#51812yes—
656C2×C4#51812yes—
657C2×C4C2×#12111yescharacteristic
658C2×C4C2×C22⋊Q811yescharacteristic
659C2×C4#53414yes—
660C2×C4#53412yes—
661C2×C4#53512yes—
662C2×C4#53814yes—
663C2×C4#53814yes—
664C2×C4#54314yes—
665C8—84no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—22no—
679C22—21no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—48no—
684C22—48no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—816no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22#180614yes—
698C22C2×#38011yescharacteristic
699C22#230811yescharacteristic
700C22#230911yescharacteristic
701C22#261712yes—
702C22#262612yes—
703C22C22×#10411yescharacteristic
704C22C2×#42111yescharacteristic
705C22C2×#42111yescharacteristic
706C22C2×#42111yescharacteristic
707C4—22no—
708C4—22no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—44no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—88no—
719C4—82no—
720C4—82no—
721C4#231212yes—
722C4#231512yes—
723C4#261712yes—
724C4#262612yes—
725C2—22no—
726C2—24no—
727C2—22no—
728C2—42no—
729C2—84no—
730C2#1433811yescharacteristic
731C2C2×#180611yescharacteristic
732C2#2065411yescharacteristic
733C2#2177311yescharacteristic
734C2#2177411yescharacteristic
735C2#2181511yescharacteristic
736C2#2181611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
4. C4×#380C2semidirect
5. #11075C2semidirect
6. #11053C2semidirect
7. #10733C2semidirect
11. C2×#423C22semidirect
18. #2297C4semidirect
19. #2297C4semidirect
22. C2×#391C4semidirect
23. C2×#391C4semidirect
24. C2×#390C4semidirect
25. C2×#390C4semidirect
29. C2×C4×C4⋊C4C22semidirect
38. C2×#339C22semidirect
39. #1728C22semidirect
43. C4×#95C22semidirect
169. C2×#41C4⋊C4semidirect
176. C22×Q8C4⋊C4semidirect
188. C22×D4C4⋊C4semidirect
453. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
C2×#22972
#206262
C2×#17892
C4×#3802
#110752
#110532
#107332