Finite Group Databasebeta

← #209198

Subgroups — #209198

Order 512.

1107 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2988C211yesmaximal, characteristic, complemented
2#22133C211yesmaximal, characteristic, complemented
3C4×#449C211yesmaximal, characteristic
4C2×#1580C211yesmaximal, characteristic, complemented
5C2×C8×SD16C211yesmaximal, characteristic, complemented
6#8948C211yesmaximal, characteristic, complemented
7#8947C211yesmaximal, characteristic, complemented
8#8939C211yesmaximal, characteristic, complemented
9#8846C211yesmaximal, characteristic, complemented
10#8845C211yesmaximal, characteristic, complemented
11#8830C211yesmaximal, characteristic, complemented
12#8312C211yesmaximal, characteristic, complemented
13#8201C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#449C411yescharacteristic
18#2988—24no—
19#2988—24no—
20C2×C8×Q8C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C2×C8×D4—21no—
25C2×C8×D4—21no—
26C2×C8×D4C2211yescharacteristic
27C2×C8×D4C2211yescharacteristic
28C2×C8×D4C2211yescharacteristic
29C2×#431—21no—
30C2×#431C2211yescharacteristic
31C22×C4⋊C8—21no—
32C22×C4⋊C8C2211yescharacteristic
33C4×#121C2211yescharacteristic
34#1954—21no—
35#1954C2211yescharacteristic
36#1939C2211yescharacteristic
37#1923—21no—
38#1880C411yescharacteristic
39#1879C411yescharacteristic
40#1875C411yescharacteristic
41#1869—21no—
42#1864C411yescharacteristic
43#1863C411yescharacteristic
44#1854C411yescharacteristic
45#1850—21no—
46#1797C2211yescharacteristic
47#1788C411yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×#103C2211yescharacteristic
50C4×#102C2211yescharacteristic
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1580—24no—
54C8×SD16—24no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×#41C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×D8C2×C411yescharacteristic
74C23×C8—21no—
75C23×C8—21no—
76C23×C8—21no—
77C23×C8—21no—
78C2×#121C2×C411yescharacteristic
79C4×#41—22no—
80C4×#41—22no—
81C4×#41—22no—
82C4×#41—22no—
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4C2×C411yescharacteristic
89C2×C4×D4C2311yescharacteristic
90C2×C4×D4C2311yescharacteristic
91C2×#117—21no—
92C2×#117C2×C411yescharacteristic
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95#449C818yescomplemented
96C8×Q8—22no—
97C8×Q8—22no—
98C8×Q8—22no—
99C8×Q8D414yescomplemented
100C4×SD16—22no—
101C4×SD16—22no—
102C4×D8—22no—
103C4×D8—22no—
104C8×D4—28no—
105C8×D4—28no—
106C8×D4—22no—
107C8×D4—22no—
108C8×D4—22no—
109C8×D4—22no—
110C8×D4—22no—
111C8×D4D414yescomplemented
112#431—28no—
113#431—22no—
114#431—22no—
115C2×#107C2×C411yescharacteristic
116C2×#106C2×C411yescharacteristic
117C2×C4⋊C8—28no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—22no—
122C2×C4⋊C8—22no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8D411yescharacteristic
125C2×C4⋊C8D411yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×#102C2×C411yescharacteristic
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8—21no—
143C2×C22⋊C8—21no—
144C2×C22⋊C8—21no—
145C2×C22⋊C8C2×C411yescharacteristic
146C2×C22⋊C8C2×C411yescharacteristic
147C2×C22⋊C8D411yescharacteristic
148C2×C22⋊C8D411yescharacteristic
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158#390C2×C411yescharacteristic
159#389C2×C411yescharacteristic
160#388—21no—
161#386—21no—
162#380C2×C411yescharacteristic
163C4×C4⋊C4C2311yescharacteristic
164C4×C22⋊C4—21no—
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336C2×C411yescharacteristic
174#336C2×C411yescharacteristic
175#334—22no—
176#334—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179D4⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183#322—22no—
184C82—22no—
185C2×#41—22no—
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41C2×C814yescomplemented
191C22×Q8C22×C411yescharacteristic
192C22×D4—21no—
193C22×D4—21no—
194C22×D4C22×C411yescharacteristic
195C22×D4C22×C411yescharacteristic
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4C22×C411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4#4111yescharacteristic
203C2×SD16—22no—
204C2×SD16—22no—
205C2×SD16C2×C814yescomplemented
206C2×D8—22no—
207C2×D8—22no—
208C2×D8C2×C814yescomplemented
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—24no—
213C22×C8—24no—
214C22×C8—24no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—42no—
219C22×C8—81no—
220C22×C8—81no—
221C22×C8—81no—
222C22×C8—81no—
223C22×C8—81no—
224C22×C8—81no—
225C22×C8—81no—
226C22×C8—81no—
227C22×C8C22×C411yescharacteristic
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235#121—22no—
236#121C2×C814yescomplemented
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8C2×D412yes—
242C4×Q8#4112yes—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—81no—
256C4×D4—81no—
257C4×D4C2×D412yes—
258C4×D4#4112yes—
259#117—24no—
260#117—24no—
261#117—22no—
262#117—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42C22×C411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42C2411yescharacteristic
293#107—22no—
294#106—22no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—22no—
298C4⋊C8—24no—
299C4⋊C8—22no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—81no—
307C4⋊C8C2×D412yes—
308C4⋊C8C2×D412yes—
309#103—22no—
310#103—22no—
311#103C2×C814yescomplemented
312#102—22no—
313#102—22no—
314#102—22no—
315#102—22no—
316#102C2×C814yescomplemented
317C22⋊C8—24no—
318C22⋊C8—22no—
319C22⋊C8—24no—
320C22⋊C8—22no—
321C22⋊C8—22no—
322C22⋊C8—42no—
323C22⋊C8—42no—
324C22⋊C8—81no—
325C22⋊C8—81no—
326C22⋊C8—81no—
327C22⋊C8C2×C814yescomplemented
328C4×C8—24no—
329C4×C8—24no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8—21no—
337C4×C8—21no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—81no—
341C4×C8C2×D412yes—
342C4×C8C2×D412yes—
343#95—21no—
344#95—21no—
345#95—21no—
346C24—21no—
347C24—21no—
348C24C4×D411yescharacteristic
349#41—28no—
350#41—28no—
351#41—28no—
352#41—28no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8C4×D412yes—
357C2×Q8C22×C812yes—
358C2×Q8#13112yes—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4C4×D412yes—
374C2×D4C22×C812yes—
375C2×D4C22×C812yes—
376C2×D4#13112yes—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4C4×D411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4C22×C812yes—
419C22×C4#13112yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2×#4111yescharacteristic
423SD16—28no—
424D8—28no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—22no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—84no—
464C2×C8—84no—
465C2×C8—84no—
466C2×C8—84no—
467C2×C8#12012yes—
468C2×C8#12012yes—
469C2×C8C22×C812yes—
470C2×C8C22×C812yes—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—81no—
482C4⋊C4C4×D412yes—
483C4⋊C4C4×D412yes—
484C4⋊C4C22×C812yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—24no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C42—24no—
496C42—24no—
497C42—21no—
498C42—21no—
499C42—22no—
500C42—21no—
501C42—21no—
502C42—22no—
503C42—21no—
504C42—21no—
505C42—41no—
506C42—41no—
507C42C22×D411yescharacteristic
508C42C22×D411yescharacteristic
509C42C2×#4111yescharacteristic
510C42C2×#4111yescharacteristic
511C23—22no—
512C23—21no—
513C23—22no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C8×D412yes—
532C23C2×C4×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8C8×D414yes—
537D4—216no—
538D4—22no—
539D4—22no—
540D4—22no—
541D4—22no—
542D4—22no—
543D4—41no—
544D4—41no—
545D4—84no—
546D4C8×D414yes—
547C2×C4—216no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—84no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4C8×D412yes—
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×#12011yescharacteristic
599C2×C4#52511yescharacteristic
600C2×C4#52511yescharacteristic
601C2×C4#52511yescharacteristic
602C2×C4C23×C811yescharacteristic
603C2×C4C2×#13111yescharacteristic
604C2×C4C2×#13111yescharacteristic
605C2×C4C2×#13111yescharacteristic
606C2×C4#57611yescharacteristic
607C2×C4#57711yescharacteristic
608C2×C4#57711yescharacteristic
609C2×C4#57711yescharacteristic
610C8—22no—
611C8—22no—
612C8—22no—
613C8—22no—
614C8—42no—
615C8—41no—
616C8—41no—
617C8—41no—
618C8—41no—
619C8—42no—
620C8—42no—
621C8—81no—
622C8—81no—
623C8—81no—
624C22—22no—
625C22—22no—
626C22—24no—
627C22—22no—
628C22—41no—
629C22—42no—
630C22—41no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—41no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22C4×#12011yescharacteristic
646C22C2×C8×D411yescharacteristic
647C22#292911yescharacteristic
648C22#294611yescharacteristic
649C22#295211yescharacteristic
650C22#295611yescharacteristic
651C22#295711yescharacteristic
652C4—22no—
653C4—24no—
654C4—21no—
655C4—21no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—41no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4C2×C8×D411yescharacteristic
664C4C2×C8×D411yescharacteristic
665C4#292911yescharacteristic
666C4#292911yescharacteristic
667C4#301011yescharacteristic
668C4#301211yescharacteristic
669C4#311911yescharacteristic
670C4#312511yescharacteristic
671C2—42no—
672C2—44no—
673C2—82no—
674C2#1785211yescharacteristic
675C2#1785811yescharacteristic
676C2#1792011yescharacteristic
677C2#1792711yescharacteristic
678C2C8×#12011yescharacteristic
679C2#2146411yescharacteristic
680C2#2146811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#449C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2988C2semidirect
2. #22133C2semidirect
4. C2×#1580C2semidirect
5. C2×C8×SD16C2semidirect
6. #8948C2semidirect
7. #8947C2semidirect
8. #8939C2semidirect
9. #8846C2semidirect
10. #8845C2semidirect
11. #8830C2semidirect
12. #8312C2semidirect
13. #8201C2semidirect
14. C8×#102C2semidirect
15. #8148C2semidirect
95. #449C8semidirect
99. C8×Q8D4semidirect
111. C8×D4D4semidirect
190. C2×#41C2×C8semidirect
205. C2×SD16C2×C8semidirect
208. C2×D8C2×C8semidirect
236. #121C2×C8semidirect
311. #103C2×C8semidirect
316. #102C2×C8semidirect
327. C22⋊C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#29882
#221332
C4×#4492
C2×#15802
C2×C8×SD162
#89482
#89472
#89392
#88462
#88452
#88302
#83122
#82012
C8×#1022
#81482