Finite Group Databasebeta

← #356958

Subgroups — #356958

Order 512.

2202 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2361C211yesmaximal, characteristic
2C2×#1964C212yesmaximal
3C2×C4×#99C212yesmaximal, complemented
4C22⋊C4×C4⋊C4C212yesmaximal, complemented
5#10960C214yesmaximal, complemented
6#10846C211yesmaximal, characteristic, complemented
7#10761C212yesmaximal, complemented
8C4×#342C211yesmaximal, characteristic, complemented
9C22×#121C2211yescharacteristic
10C22×C4×D4C411yescharacteristic, complemented
11C22×C4×D4C2212yes—
12C23×C4⋊C4C2211yescharacteristic
13C22×#99C2214yes—
14C22×#99C2214yes—
15#2361—22no—
16#2361—22no—
17#2361—24no—
18C2×#390C411yescharacteristic, complemented
19C2×#389C411yescharacteristic, complemented
20C2×#388C412yescomplemented
21C2×#386C2212yes—
22C2×#385C412yescomplemented
23C2×#380C2212yes—
24C2×C4×C4⋊C4—22no—
25C2×C4×C4⋊C4C411yescharacteristic, complemented
26C2×C4×C22⋊C4C2214yes—
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1964C4116yescomplemented
30C4×#99—28no—
31C2×#342C2211yescharacteristic
32C2×#342C2212yes—
33C2×#342C2212yes—
34C2×#342C2212yes—
35C2×#342C2211yescharacteristic
36#1436—24no—
37C4⋊#95—22no—
38C22⋊#95C2214yes—
39C23×D4C2×C411yescharacteristic
40C23×D4C2312yes—
41C24×C4C2×C412yes—
42C24×C4C2312yes—
43C24×C4C2311yescharacteristic
44C2×#121—22no—
45C2×#121—22no—
46C2×#121—22no—
47C2×#121—22no—
48C2×#121C2×C418yes—
49C2×C4×D4—24no—
50C2×C4×D4—24no—
51C2×C4×D4—22no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—44no—
57C2×C4×D4—44no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4D412yes—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D414yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2311yescharacteristic
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2311yescharacteristic
76C22×C22⋊C4C2312yes—
77C22×C42—22no—
78C22×C42—24no—
79C22×C42C2×C412yes—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C2×#99—28no—
83C2×#99—28no—
84C2×#99—44no—
85C2×#99—44no—
86C2×#99C2×C4116yescomplemented
87C2×#99C2×C4116yescomplemented
88#390—22no—
89#390—22no—
90#389—22no—
91#389—22no—
92#388—24no—
93#388—24no—
94#386—24no—
95#386—24no—
96#385—24no—
97#385—24no—
98#380—24no—
99#380C2×C418yescomplemented
100C4×C4⋊C4—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—48no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—28no—
108C4×C22⋊C4—44no—
109C4×C22⋊C4C2×C418yescomplemented
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—24no—
113C2×#95—24no—
114C2×#95—24no—
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C412yes—
118C2×#95C2×C412yes—
119#342—24no—
120#342—22no—
121#342—24no—
122#342—42no—
123#342D414yescomplemented
124C25C22×C412yes—
125C25C2411yescharacteristic
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4C4214yescomplemented
141C22×D4C22×C418yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—44no—
156C23×C4—82no—
157C23×C4—82no—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C414yes—
168C23×C4C2×D414yes—
169C23×C4C2×D412yes—
170C23×C4C2×D414yes—
171C23×C4C2×D412yes—
172C23×C4#4112yes—
173C23×C4#4112yes—
174#121—28no—
175#121—28no—
176#121—28no—
177#121—28no—
178C4×D4—48no—
179C4×D4—48no—
180C4×D4—48no—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—44no—
184C4×D4—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C4214yescomplemented
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C414yes—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4C4218yescomplemented
238C2×C22⋊C4C22×C414yes—
239C2×C22⋊C4C22×C418yes—
240C2×C22⋊C4C22×C414yes—
241C2×C22⋊C4C2×D412yes—
242C2×C22⋊C4C2×D414yes—
243C2×C22⋊C4#4112yes—
244C2×C22⋊C4#4114yes—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—24no—
252C2×C42—28no—
253C2×C42—42no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—48no—
259C2×C42—48no—
260C2×C42—84no—
261#99—232no—
262#99—416no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—24no—
268#95—28no—
269#95—28no—
270#95—48no—
271#95—48no—
272#95—84no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24C2×C4214yes—
288C24C23×C411yescharacteristic
289C24C23×C412yes—
290C24C22×D412yes—
291C24C22×D411yescharacteristic
292C24C2×#4111yescharacteristic
293C24C2×#4111yescharacteristic
294C24C2×#4111yescharacteristic
295C24C2×#4112yes—
296C24H(5,𝔽2)12yes—
297C24H(5,𝔽2)11yescharacteristic
298C242−1+411yescharacteristic
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—28no—
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—42no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—44no—
321C2×D4—42no—
322C2×D4—42no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—48no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—84no—
402C22×C4—88no—
403C22×C4—84no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4C2×C4214yes—
407C22×C4C2×C4212yes—
408C22×C4C2×C4212yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C414yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4#11712yes—
416C22×C4#11712yes—
417C22×C4C4×D414yes—
418C22×C4C4×D414yes—
419C22×C4C4×D412yes—
420C22×C4C4×D412yes—
421C22×C4C4×D414yes—
422C22×C4C4×D414yes—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—416no—
433C4⋊C4#9914yescomplemented
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—216no—
440C22⋊C4—28no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—28no—
444C22⋊C4—48no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—44no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—84no—
454C22⋊C4C4×D418yescomplemented
455C42—44no—
456C42—44no—
457C42—42no—
458C42—416no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—816no—
463C23—22no—
464C23—21no—
465C23—22no—
466C23—21no—
467C23—22no—
468C23—21no—
469C23—22no—
470C23—21no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23C22×C4211yescharacteristic
513C23C22×C22⋊C411yescharacteristic
514C23C2×#11711yescharacteristic
515C23C2×C4×D411yescharacteristic
516C23C2×C4×D412yes—
517C23C2×C4×D412yes—
518C23C2×C4×D412yes—
519C23C4×#4111yescharacteristic
520C23C4×#4112yes—
521C23#51811yescharacteristic
522C23#51812yes—
523C23#52011yescharacteristic
524C23#52012yes—
525C23D4212yes—
526C23#54612yes—
527C23#54711yescharacteristic
528C23#54812yes—
529C23#55012yes—
530C23#55011yescharacteristic
531C23#55312yes—
532C23#55511yescharacteristic
533C23#55611yescharacteristic
534C23#55912yes—
535D4—48no—
536D4—48no—
537D4—48no—
538D4—48no—
539D4—48no—
540D4—48no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—88no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—88no—
607C2×C4—84no—
608C2×C4—82no—
609C2×C4—84no—
610C2×C4—82no—
611C2×C4—84no—
612C2×C4—82no—
613C2×C4—84no—
614C2×C4—82no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4C4×C22⋊C414yes—
620C2×C4C4×C22⋊C412yes—
621C2×C4C4×C22⋊C412yes—
622C2×C4C2×#9914yes—
623C2×C4C2×#9912yes—
624C2×C4#41012yes—
625C22—22no—
626C22—22no—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—22no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—21no—
635C22—21no—
636C22—22no—
637C22—22no—
638C22—21no—
639C22—21no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—22no—
647C22—21no—
648C22—21no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—41no—
654C22—41no—
655C22—42no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22C2×C4×C22⋊C411yescharacteristic
671C22C42×D411yescharacteristic
672C22#227511yescharacteristic
673C22D4×C22⋊C411yescharacteristic
674C22#234111yescharacteristic
675C22#235411yescharacteristic
676C22#235711yescharacteristic
677C22#236112yes—
678C22#236312yes—
679C22#236612yes—
680C22#236712yes—
681C22C22×#9911yescharacteristic
682C22C2×#41011yescharacteristic
683C22#288311yescharacteristic
684C22#288411yescharacteristic
685C4—24no—
686C4—24no—
687C4—22no—
688C4—48no—
689C4—44no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—88no—
695C4—84no—
696C4—81no—
697C4—81no—
698C4C4×#9914yescomplemented
699C2—22no—
700C2—22no—
701C2—22no—
702C2—21no—
703C2—21no—
704C2—21no—
705C2—21no—
706C2—22no—
707C2—84no—
708C2#1382011yescharacteristic
709C2C2×C4×#9911yescharacteristic
710C2#2039311yescharacteristic
711C2D4×#9911yescharacteristic
712C2#2240611yescharacteristic
713C2#2253411yescharacteristic
714C2#2253511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×#99C2semidirect
4. C22⋊C4×C4⋊C4C2semidirect
5. #10960C2semidirect
6. #10846C2semidirect
7. #10761C2semidirect
8. C4×#342C2semidirect
10. C22×C4×D4C4semidirect
18. C2×#390C4semidirect
19. C2×#389C4semidirect
20. C2×#388C4semidirect
22. C2×#385C4semidirect
25. C2×C4×C4⋊C4C4semidirect
29. #1964C4semidirect
86. C2×#99C2×C4semidirect
87. C2×#99C2×C4semidirect
99. #380C2×C4semidirect
109. C4×C22⋊C4C2×C4semidirect
123. #342D4semidirect
140. C22×D4C42semidirect
200. C2×C4⋊C4C42semidirect
237. C2×C22⋊C4C42semidirect
433. C4⋊C4#99semidirect
454. C22⋊C4C4×D4semidirect
698. C4C4×#99semidirect

Maximal subgroups

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

subgroupindex
C2×#23612
C2×#19642
C2×C4×#992
C22⋊C4×C4⋊C42
#109602
#108462
#107612
C4×#3422