Finite Group Databasebeta

← #705160

Subgroups — #705160

Order 512.

5056 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C2×#2486C211yesmaximal, characteristic, complemented
4C2×#2311C211yesmaximal, characteristic, complemented
5#16359C214yesmaximal, complemented
6#15982C214yesmaximal, complemented
7#15950C214yesmaximal, complemented
8#15188C212yesmaximal, complemented
9#15182C212yesmaximal, complemented
10#14821C214yesmaximal, complemented
11#14258C211yesmaximal, characteristic, complemented
12#13977C212yesmaximal, complemented
13#13937C212yesmaximal, complemented
14#13937C211yesmaximal, characteristic, complemented
15#13839C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C23×#41C2211yescharacteristic
18C2×#547—28no—
19C2×#547—28no—
20C2×#547—28no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#123C2212yes—
23C22×#123C2212yes—
24C22×#123C2212yes—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×#121C2212yes—
28C22×C4×D4C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32C23×C4⋊C4C2212yes—
33#2502—28no—
34#2486—28no—
35C2×#398C2212yes—
36C2×#398C2214yes—
37C2×#397C2212yes—
38#2311—28no—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#391C2212yes—
43C2×#390C2212yes—
44C2×#390C2212yes—
45C2×#389C2212yes—
46C2×#386C2214yes—
47C2×#385C2214yes—
48C2×#385C2218yes—
49C2×#384C2212yes—
50C2×#381C2212yes—
51C2×#380C2212yes—
52C2×#380C2211yescharacteristic
53C2×#380C2214yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56#1442C2218yes—
57#1441C2212yes—
58#1441C2218yes—
59#1441C2218yes—
60#1441C2214yes—
61#1440C2212yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2218yes—
65#1439C2218yes—
66#1438C2212yes—
67#1438C2214yes—
68#1438C2218yes—
69C4⋊#95C2212yes—
70C22⋊#95C2218yes—
71C22⋊#95C2214yes—
72C4×#95C2212yes—
73C22×#41—24no—
74C22×#41—24no—
75C22×#41—28no—
76C22×#41—28no—
77C22×#41—24no—
78C22×#41—24no—
79C22×#41—24no—
80C22×#41—44no—
81C22×#41—44no—
82C22×#41D418yes—
83C23×Q8C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C23×D4C2312yes—
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2312yes—
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C24×C4C2312yes—
95#547—2128no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×#123—28no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—28no—
103C2×#123—28no—
104C2×#123—28no—
105C2×#123—28no—
106C2×#123—48no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8D418yes—
113C2×#121—28no—
114C2×#121—28no—
115C2×#121—28no—
116C2×C4×D4—24no—
117C2×C4×D4—24no—
118C2×C4×D4—24no—
119C2×C4×D4—24no—
120C2×C4×D4—24no—
121C2×C4×D4D418yes—
122C2×#117—24no—
123C2×#117—44no—
124C22×C4⋊C4—28no—
125C22×C4⋊C4—28no—
126C22×C4⋊C4—28no—
127C22×C4⋊C4—24no—
128C22×C4⋊C4—24no—
129C22×C4⋊C4—24no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2318yes—
141C22×C4⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2318yes—
146C22×C22⋊C4C2311yescharacteristic
147C22×C22⋊C4C2311yescharacteristic
148C22×C22⋊C4C2312yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2312yes—
151C22×C22⋊C4C2312yes—
152C22×C22⋊C4C2318yes—
153C22×C22⋊C4C2314yes—
154C22×C42C2312yes—
155C22×C42C2312yes—
156C22×C42C2311yescharacteristic
157#398—28no—
158#398—216no—
159#397—28no—
160#391—28no—
161#391—28no—
162#391—44no—
163#391—44no—
164#390—28no—
165#390—28no—
166#389—28no—
167#386—216no—
168#385—216no—
169#385—416no—
170#384—28no—
171#381—28no—
172#380—216no—
173#380—28no—
174#380—24no—
175C4×C22⋊C4—24no—
176C4×C22⋊C4—44no—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2318yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2318yes—
185C2×#95C2314yes—
186C2×#95C2318yes—
187C2×#95C2318yes—
188C2×#95C2318yes—
189C25C2411yescharacteristic
190C25C2411yescharacteristic
191C25C2412yes—
192C2×#41—232no—
193C2×#41—232no—
194C2×#41—232no—
195C2×#41—416no—
196C2×#41—432no—
197C2×#41—432no—
198C2×#41—432no—
199C2×#41—432no—
200C2×#41—416no—
201C2×#41—416no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8C2×D414yes—
211C22×Q8C2×D414yes—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4—28no—
219C22×D4—28no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—24no—
225C22×D4—42no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4C2×D414yes—
229C22×D4C2×D418yes—
230C22×D4C2×D414yes—
231C22×D4#4118yes—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—28no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—28no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—28no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4C2×D414yes—
258C23×C4C2×D418yes—
259C23×C4C2×D414yes—
260C23×C4C2×D414yes—
261C23×C4C2×D414yes—
262C23×C4C2414yes—
263C23×C4C2414yes—
264C23×C4C2414yes—
265C23×C4C2411yescharacteristic
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C23×C4C2412yes—
269C23×C4C2412yes—
270C23×C4C2414yes—
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C4⋊Q8—232no—
275#123—264no—
276#123—264no—
277#123—432no—
278C22⋊Q8—232no—
279C22⋊Q8—232no—
280#121—264no—
281C4×D4—232no—
282C4×D4—232no—
283#117—416no—
284C2×C4⋊C4—264no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—216no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—216no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—416no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D418yes—
319C2×C4⋊C4C2×D414yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—216no—
338C2×C22⋊C4—216no—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—216no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—48no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4—48no—
349C2×C22⋊C4C2×D414yes—
350C2×C22⋊C4C2×D414yes—
351C2×C22⋊C4C2×D418yes—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42C2×D414yes—
361#95—24no—
362#95—28no—
363#95—216no—
364#95—24no—
365#95—28no—
366#95—216no—
367#95—28no—
368#95—28no—
369#95—44no—
370#95—48no—
371#95—48no—
372#95—48no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—28no—
382C24—24no—
383C24—22no—
384C24—21no—
385C24—21no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—24no—
390C24—22no—
391C24C22×D412yes—
392C24C22×D412yes—
393C24C22×D412yes—
394C24C22×D412yes—
395C24C22×D414yes—
396C24C2×#4118yes—
397C24C2511yescharacteristic
398#41—4128no—
399#41—4128no—
400C2×Q8—28no—
401C2×Q8—28no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—216no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—48no—
408C2×Q8—48no—
409C2×Q8—48no—
410C2×D4—216no—
411C2×D4—28no—
412C2×D4—216no—
413C2×D4—216no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—44no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—416no—
424C2×D4—416no—
425C2×D4—416no—
426C2×D4—416no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4#123116yescomplemented
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—216no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—28no—
469C22×C4—216no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—28no—
481C22×C4—28no—
482C22×C4—24no—
483C22×C4—28no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—28no—
487C22×C4—48no—
488C22×C4—416no—
489C22×C4—416no—
490C22×C4—416no—
491C22×C4—416no—
492C22×C4—416no—
493C22×C4—416no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—88no—
505C22×C4—88no—
506C22×C4#120116yes—
507C22×C4C22×D412yes—
508C22×C4C22×D414yes—
509C22×C4C22×D412yes—
510C22×C4C22×D414yes—
511C22×C4C22×D414yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C2×#4114yes—
520C22×C4H(5,𝔽2)14yes—
521C22×C4H(5,𝔽2)14yes—
522C22×C42−1+418yes—
523C4⋊C4—216no—
524C4⋊C4—216no—
525C4⋊C4—216no—
526C4⋊C4—28no—
527C4⋊C4—216no—
528C4⋊C4—28no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—416no—
533C4⋊C4—48no—
534C4⋊C4—816no—
535C22⋊C4—216no—
536C22⋊C4—216no—
537C22⋊C4—216no—
538C22⋊C4—28no—
539C22⋊C4—216no—
540C22⋊C4—28no—
541C22⋊C4—432no—
542C22⋊C4—416no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—416no—
546C22⋊C4—816no—
547C42—28no—
548C42—48no—
549C42—48no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—28no—
559C23—28no—
560C23—24no—
561C23—28no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—28no—
567C23—216no—
568C23—216no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—24no—
574C23—42no—
575C23—42no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—48no—
583C23—48no—
584C23—44no—
585C23—41no—
586C23—41no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23C2×#12014yes—
597C23C2×#12318yes—
598C23C23×D411yescharacteristic
599C23C23×D411yescharacteristic
600C23C23×D411yescharacteristic
601C23C22×#4112yes—
602C23C2×H(5,𝔽2)12yes—
603C23C2×H(5,𝔽2)12yes—
604C23C2×2−1+414yes—
605C23#58512yes—
606Q8—416no—
607Q8—416no—
608D4—232no—
609D4—416no—
610D4—432no—
611D4—432no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—232no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—432no—
636C2×C4—432no—
637C2×C4—432no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—416no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—48no—
664C2×C4—48no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—48no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—48no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—816no—
674C2×C4C2×#12014yes—
675C2×C4C2×#12014yes—
676C2×C4C2×#12314yes—
677C2×C4#53414yes—
678C2×C4#53514yes—
679C2×C4#54618yes—
680C2×C4#54618yes—
681C22—24no—
682C22—24no—
683C22—28no—
684C22—216no—
685C22—24no—
686C22—44no—
687C22—48no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—48no—
692C22—44no—
693C22—48no—
694C22—416no—
695C22—44no—
696C22—48no—
697C22—44no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—44no—
702C22—82no—
703C22—82no—
704C22—84no—
705C22C22×#12011yescharacteristic
706C22C22×#12311yescharacteristic
707C22C2×#53411yescharacteristic
708C22C2×#53511yescharacteristic
709C22#344911yescharacteristic
710C22C2×#54612yes—
711C22C2×#54612yes—
712C22#348612yes—
713C22#348812yes—
714C22#348812yes—
715C22#349912yes—
716C22#349912yes—
717C22#351514yes—
718C22#352014yes—
719C22#354114yes—
720C22#354814yes—
721C4—24no—
722C4—24no—
723C4—42no—
724C4—42no—
725C4—44no—
726C4—44no—
727C4—44no—
728C4—48no—
729C4—84no—
730C4—84no—
731C4—84no—
732C4#240518yes—
733C2—44no—
734C2—48no—
735C2—82no—
736C2C2×#240511yescharacteristic
737C2#3525011yescharacteristic
738C2#3566611yescharacteristic
739C2#3599812yes—
740C2#3808812yes—
741C2#3895112yes—
742C2#4291612yes—
743C2#4499614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#547C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2502C2semidirect
3. C2×#2486C2semidirect
4. C2×#2311C2semidirect
5. #16359C2semidirect
6. #15982C2semidirect
7. #15950C2semidirect
8. #15188C2semidirect
9. #15182C2semidirect
10. #14821C2semidirect
11. #14258C2semidirect
12. #13977C2semidirect
13. #13937C2semidirect
14. #13937C2semidirect
15. #13839C2semidirect
429. C2×D4#123semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#25022
C2×#24862
C2×#23112
#163592
#159822
#159502
#151882
#151822
#148212
#142582
#139772
#139372
#138392