Finite Group Databasebeta

← #360091

Subgroups — #360091

Order 512.

2092 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3C2×#1899C212yesmaximal
4C2×#1799C212yesmaximal
5#14685C211yesmaximal, characteristic, complemented
6#11203C212yesmaximal, complemented
7#11043C212yesmaximal
8#10846C212yesmaximal, complemented
9#10736C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2212yes—
13C23×C42C2211yescharacteristic
14C22×#99C2214yes—
15C4×#123—22no—
16C4×#123—22no—
17C4×#123—21no—
18C4×#123—21no—
19C4×#123—21no—
20C4×#123—21no—
21C2×#388C412yescomplemented
22C2×#387C412yescomplemented
23C2×#386C2211yescharacteristic
24C2×#385C2211yescharacteristic
25#2285—28no—
26C2×#384—21no—
27C2×#382—21no—
28C2×#381C2212yes—
29C2×#381C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C412yescomplemented
36C2×C4×C22⋊C4C412yescomplemented
37C2×C4×C22⋊C4C2211yescharacteristic
38#1899—28no—
39#1799—28no—
40C2×#342C2214yes—
41C2×#342C2214yes—
42C2×#342C2214yes—
43C4⋊#95—22no—
44#1434—22no—
45C22⋊#95C2212yes—
46C23×D4C2311yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C2×#123—21no—
52C2×#123—22no—
53C2×#123—21no—
54C2×#123—22no—
55C2×#123—22no—
56C2×#123D414yes—
57C2×#123Q814yes—
58C2×C4×D4—24no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—24no—
64C2×#117—28no—
65C2×#117—24no—
66C2×#117—24no—
67C2×#117—44no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4C2×C412yes—
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2311yescharacteristic
80C22×C22⋊C4C2311yescharacteristic
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2311yescharacteristic
83C22×C42—24no—
84C22×C42—24no—
85C22×C42—22no—
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C412yes—
88C22×C42C2×C412yes—
89C22×C42C2×C412yes—
90C22×C42D412yes—
91C22×C42D412yes—
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C2×#99—28no—
97C2×#99—28no—
98C2×#99—44no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#386—21no—
108#386—22no—
109#386—21no—
110#385—21no—
111#385—22no—
112#385—21no—
113#384—28no—
114#382—28no—
115#381—21no—
116#381—22no—
117#381—21no—
118#381—24no—
119#381—24no—
120#380—24no—
121#380—28no—
122#380—24no—
123#380—24no—
124#380D418yes—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—24no—
142C2×#95—22no—
143C2×#95—24no—
144C2×#95C2×C412yes—
145C2×#95C2×C412yes—
146C2×#95C2×C412yes—
147C2×#95C2×C412yes—
148C2×#95C2×C412yes—
149#342—28no—
150#342—28no—
151#342—44no—
152C25C22×C411yescharacteristic
153C25C2411yescharacteristic
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4C2×D412yes—
163C22×D4C2×Q812yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C412yes—
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C412yes—
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D414yes—
199C23×C4C2×Q812yes—
200C23×C4#4112yes—
201C23×C4#4112yes—
202C23×C4#4114yes—
203#123—28no—
204#123—24no—
205#123—24no—
206#123—24no—
207#123—24no—
208#123—28no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—42no—
214C4×D4—42no—
215#117—416no—
216#117—416no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C4⋊C418yescomplemented
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C4⋊C418yescomplemented
269C2×C22⋊C4C2×D418yes—
270C2×C22⋊C4C2×D412yes—
271C2×C22⋊C4C2×Q812yes—
272C2×C42—28no—
273C2×C42—28no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—28no—
279C2×C42—28no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—28no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42C22⋊C412yes—
294C2×C42C22⋊C412yes—
295C2×C42C22⋊C412yes—
296C2×C42C22⋊C412yes—
297C2×C42C22⋊C412yes—
298C2×C42C22⋊C412yes—
299C2×C42C22⋊C412yes—
300C2×C42C22⋊C412yes—
301#99—416no—
302#95—28no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—28no—
314#95—48no—
315#95—44no—
316#95—48no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24C2×C4⋊C412yes—
326C24C23×C411yescharacteristic
327C24C22×D411yescharacteristic
328C24C22×D411yescharacteristic
329C24C22×D411yescharacteristic
330C24C22×D412yes—
331C24C22×Q811yescharacteristic
332C24C2×#4111yescharacteristic
333C24C2×#4111yescharacteristic
334C24C2×#4112yes—
335C24H(5,𝔽2)12yes—
336C242−1+412yes—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—41no—
346C2×D4—42no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—88no—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C412yes—
432C22×C4C2×C22⋊C412yes—
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C4⋊C412yes—
439C22×C4C2×C4⋊C414yes—
440C22×C4C2×C4⋊C414yes—
441C22×C4#11714yes—
442C22×C4C4×D414yes—
443C22×C4C4×D414yes—
444C22×C4#12012yes—
445C22×C4#12012yes—
446C22×C4#12114yes—
447C22×C4#12112yes—
448C22×C4#12112yes—
449C22×C4#12114yes—
450C22×C4#12114yes—
451C22×C4C22⋊Q812yes—
452C22×C4C22⋊Q812yes—
453C22×C4C22⋊Q814yes—
454C22×C4#12312yes—
455C22×C4#12312yes—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—44no—
461C4⋊C4—416no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—88no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—44no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C42—24no—
491C42—24no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42—24no—
496C42—24no—
497C42—24no—
498C42—44no—
499C42—416no—
500C42—42no—
501C42—42no—
502C42—44no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—82no—
529C23—84no—
530C23—82no—
531C23C22×C22⋊C411yescharacteristic
532C23C22×C4⋊C411yescharacteristic
533C23C2×#11711yescharacteristic
534C23C2×C4×D412yes—
535C23#52012yes—
536C23C2×#12011yescharacteristic
537C23C2×#12111yescharacteristic
538C23C2×#12111yescharacteristic
539C23C2×#12112yes—
540C23C2×C22⋊Q811yescharacteristic
541C23C2×C22⋊Q811yescharacteristic
542C23C2×#12311yescharacteristic
543C23#53511yescharacteristic
544C23#53611yescharacteristic
545C23#53712yes—
546C23D4212yes—
547C23#54612yes—
548C23#54712yes—
549C23#54812yes—
550C23D4×Q812yes—
551C23#55212yes—
552C23#55312yes—
553C23#55412yes—
554D4—42no—
555D4—42no—
556D4—44no—
557D4—44no—
558D4—42no—
559D4—42no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—48no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—84no—
619C2×C4—88no—
620C2×C4—88no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—84no—
626C2×C4—88no—
627C2×C4—88no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4#38014yes—
631C2×C4#38014yes—
632C2×C4#38612yes—
633C2×C4#38612yes—
634C2×C4#38612yes—
635C2×C4#38612yes—
636C2×C4#41014yes—
637C2×C4#41014yes—
638C22—22no—
639C22—24no—
640C22—24no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—21no—
649C22—22no—
650C22—22no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—22no—
655C22—22no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—44no—
666C22—84no—
667C22—84no—
668C22—82no—
669C22—81no—
670C22—81no—
671C22C2×#38011yescharacteristic
672C22C2×#38611yescharacteristic
673C22#231111yescharacteristic
674C22D4×C4⋊C412yes—
675C22#235212yes—
676C22#240811yescharacteristic
677C22#240911yescharacteristic
678C22#241612yes—
679C22#241712yes—
680C22#248611yescharacteristic
681C22#248711yescharacteristic
682C22C2×#41012yes—
683C22C2×#41012yes—
684C4—24no—
685C4—24no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—24no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—88no—
694C4—82no—
695C4—81no—
696C4—81no—
697C4—164no—
698C4#189914yescomplemented
699C2—21no—
700C2—22no—
701C2—22no—
702C2—22no—
703C2—21no—
704C2—22no—
705C2—21no—
706C2—21no—
707C2—82no—
708C2#1387211yescharacteristic
709C2#2063212yes—
710C2C2×#189911yescharacteristic
711C2#2098311yescharacteristic
712C2#2187512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
5. #14685C2semidirect
6. #11203C2semidirect
8. #10846C2semidirect
9. #10736C2semidirect
21. C2×#388C4semidirect
22. C2×#387C4semidirect
35. C2×C4×C4⋊C4C4semidirect
36. C2×C4×C22⋊C4C4semidirect
235. C2×C4⋊C4C4⋊C4semidirect
268. C2×C22⋊C4C4⋊C4semidirect
698. C4#1899semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C2×#22852
C2×#18992
C2×#17992
#146852
#112032
#110432
#108462
#107362