Finite Group Databasebeta

← #6446236

Subgroups — #6446236

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#26295C214yesmaximal, complemented
4#26293C214yesmaximal, complemented
5#26282C212yesmaximal, complemented
6#26280C212yesmaximal, complemented
7#26279C212yesmaximal, complemented
8#26277C212yesmaximal, complemented
9#23747C212yesmaximal
10#23657C212yesmaximal
11#21324C211yesmaximal, characteristic
12#21313C211yesmaximal, characteristic
13#21303C211yesmaximal, characteristic
14#21170C211yesmaximal, characteristic
15#21034C212yesmaximal
16#21034C212yesmaximal
17C4×#428C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#428C2212yes—
21C2×#428C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2567—21no—
26#2567—22no—
27#2567—21no—
28#2567C2212yes—
29#2567C2212yes—
30#2567C2214yes—
31C2×#395—22no—
32#2355C2211yescharacteristic
33#2355C2211yescharacteristic
34#2355C2212yes—
35#2355C2212yes—
36Q8×C4⋊C4C2211yescharacteristic
37Q8×C4⋊C4C2211yescharacteristic
38#2345—22no—
39#2345—21no—
40Q8×C22⋊C4—21no—
41#2295—22no—
42#2295—21no—
43C2×#391—21no—
44C2×#389C2212yes—
45C2×#389C2212yes—
46#2285C2211yescharacteristic
47#2285C2212yes—
48C2×#384—21no—
49C42×Q8—21no—
50C4×#117C2212yes—
51C4×#117C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2085C2214yes—
58#2085C2214yes—
59#2084C2214yes—
60#2083C2214yes—
61#1993C2214yes—
62#1990C2212yes—
63#1990C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1985C2214yes—
69#1984C2212yes—
70#1984C2212yes—
71#1949C2214yes—
72#1948C2212yes—
73#1947C2212yes—
74#1921C2214yes—
75#1921C2214yes—
76#1920C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1918C2214yes—
82#1776C2212yes—
83#1776C2212yes—
84C4×#107C2212yes—
85C4×#106C2212yes—
86#1736C2212yes—
87#1736C2211yescharacteristic
88#1736C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1736C2211yescharacteristic
93#1728C2212yes—
94#1728C2212yes—
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97C22×M16C2311yescharacteristic
98C22×M16C2312yes—
99C22×M16C2311yescharacteristic
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D412yes—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D412yes—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C42—21no—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#428—24no—
131#428—28no—
132#428Q818yescomplemented
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139#423—24no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C4×M16—24no—
143C2×#97C2312yes—
144C2×C4×C8C2312yes—
145#395—24no—
146#395—24no—
147#395—24no—
148#395—24no—
149#391—22no—
150#391—24no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391—22no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#384—22no—
175#384—24no—
176#384—22no—
177#384C2312yes—
178#384C2312yes—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#382C2312yes—
182#382C2312yes—
183#382C2312yes—
184#380—22no—
185#380—21no—
186#380—22no—
187#380—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4—22no—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212#376—22no—
213#376—21no—
214C2×#95—21no—
215C2×#95—22no—
216C2×#95—22no—
217C43—22no—
218C43—21no—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D412yes—
247C22×Q8C2×D411yescharacteristic
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—24no—
260C2×M16—24no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—24no—
264C2×M16C2×Q814yes—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4×Q8—24no—
275C4×Q8—28no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—24no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×Q8—44no—
285#117—22no—
286#117—24no—
287#117—22no—
288#117—24no—
289#117—22no—
290#117C2×Q814yes—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4—84no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×Q814yes—
325C2×C4⋊C4#4114yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—44no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42#4114yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—24no—
365#107—28no—
366#107C2×Q818yescomplemented
367#106—24no—
368#106—28no—
369#106C2×Q818yescomplemented
370C4⋊C8—24no—
371C4⋊C8—24no—
372#97—24no—
373C4×C8—24no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394C24C22×D411yescharacteristic
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—42no—
403C2×Q8—416no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—816no—
411C2×Q8#13714yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4#12314yes—
441C22×C4#12314yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×Q812yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C42−1+412yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458M16—28no—
459M16—44no—
460M16—48no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8C22×Q814yes—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4C22×Q812yes—
509C4⋊C4C22×Q814yes—
510C4⋊C4C22×Q812yes—
511C4⋊C4C2×#4114yes—
512C22⋊C4—24no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C42—24no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—44no—
523C42—44no—
524C42—44no—
525C42—42no—
526C42—44no—
527C42—42no—
528C42—44no—
529C42—42no—
530C42—81no—
531C42—81no—
532C42—82no—
533C42—82no—
534C42#12314yes—
535C42#12314yes—
536C42C2×#4112yes—
537C42C2×#4112yes—
538C42C2×#4112yes—
539C42C2×#4112yes—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23C2×#12312yes—
545C23#54812yes—
546C23D4×Q812yes—
547C23C23×D411yescharacteristic
548Q8—28no—
549Q8—48no—
550Q8—44no—
551Q8—88no—
552Q8—88no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—84no—
603C2×C4—88no—
604C2×C4C2×#12312yes—
605C2×C4C2×#12312yes—
606C2×C4C2×#12312yes—
607C2×C4C2×#12312yes—
608C2×C4C2×#12312yes—
609C2×C4#53314yes—
610C2×C4#54114yes—
611C2×C4#54812yes—
612C2×C4D4×Q812yes—
613C2×C4#55614yes—
614C2×C4#55714yes—
615C2×C4C2×#13714yes—
616C2×C4C2×#13712yes—
617C2×C4#57512yes—
618C2×C4C23×Q811yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C8—24no—
625C8—42no—
626C8—44no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22#244614yes—
634C22C22×#12311yescharacteristic
635C22C2×#54811yescharacteristic
636C22C2×D4×Q811yescharacteristic
637C22C22×#13711yescharacteristic
638C22C2×#57511yescharacteristic
639C22#358511yescharacteristic
640C22#358511yescharacteristic
641C4—22no—
642C4—24no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—44no—
649C4—42no—
650C4—44no—
651C4—42no—
652C4—84no—
653C4—82no—
654C4—81no—
655C4—81no—
656C4—82no—
657C4#244614yes—
658C4#309514yes—
659C4C2×#53311yescharacteristic
660C4C2×#54111yescharacteristic
661C4C2×#55611yescharacteristic
662C4C2×#55711yescharacteristic
663C2—24no—
664C2C2×#244611yescharacteristic
665C2C2×#309511yescharacteristic
666C2#6051811yescharacteristic
667C2#6131811yescharacteristic
668C2#6131911yescharacteristic
669C2#6136711yescharacteristic
670C2Q8×#13711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26295C2semidirect
4. #26293C2semidirect
5. #26282C2semidirect
6. #26280C2semidirect
7. #26279C2semidirect
8. #26277C2semidirect
132. #428Q8semidirect
366. #107C2×Q8semidirect
369. #106C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
Q8×#1172
#262952
#262932
#262822
#262802
#262792
#262772
#237472
#236572
#213242
#213132
#213032
#211702
#210342
C4×#4282