Finite Group Databasebeta

← #6382891

Subgroups — #6382891

Order 512.

2112 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3139C211yesmaximal, characteristic
2#34145C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic
4#23233C214yesmaximal, complemented
5#23227C214yesmaximal, complemented
6#23222C212yesmaximal, complemented
7#23221C212yesmaximal, complemented
8#23217C212yesmaximal, complemented
9#23214C212yesmaximal, complemented
10#21312C211yesmaximal, characteristic
11#21259C211yesmaximal, characteristic
12#21094C212yesmaximal
13#21061C212yesmaximal
14#20874C212yesmaximal
15C4×#428C211yesmaximal, characteristic
16C4⋊C4×M16C211yesmaximal, characteristic
17C4×#419C212yesmaximal
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#3139C4116yescomplemented
22C2×#491C2214yes—
23C2×#487C2212yes—
24C2×#487C2212yes—
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#419C2212yes—
29C2×#419C2212yes—
30C2×#419C2212yes—
31C2×#419C2212yes—
32C2×C4×M16C2212yes—
33C2×C4×M16C2211yescharacteristic
34#2370C2212yes—
35#2370C2212yes—
36#2369C2211yescharacteristic
37#2369C2211yescharacteristic
38#2368—22no—
39#2365—21no—
40#2365—22no—
41#2365—21no—
42#2365C2211yescharacteristic
43#2365C2211yescharacteristic
44#2365C2212yes—
45#2365C2212yes—
46#2365C2214yes—
47#2360—21no—
48#2360—21no—
49#2296C2212yes—
50#2295—21no—
51#2295—22no—
52C2×#391C2212yes—
53C2×#389—21no—
54C2×#387—22no—
55C42×Q8—21no—
56C4×#117C2211yescharacteristic
57C4×#117C2212yes—
58C2×C4×C4⋊C4—21no—
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61#1989C2214yes—
62#1988C2212yes—
63#1987C2212yes—
64#1981C2212yes—
65#1981C2212yes—
66#1981C2212yes—
67#1981C2212yes—
68#1935C2214yes—
69#1935C2214yes—
70#1933C2214yes—
71#1933C2214yes—
72#1928C2214yes—
73#1928C2214yes—
74#1927C2214yes—
75#1927C2214yes—
76#1865C2214yes—
77#1865C2214yes—
78#1863C2214yes—
79#1863C2214yes—
80#1776C2212yes—
81#1776C2212yes—
82C4×#107C2212yes—
83C4×#106C2212yes—
84#1771C2214yes—
85#1770C2212yes—
86C8×C4⋊C4C2212yes—
87#1763C2214yes—
88#1763C2214yes—
89C4×#103C2214yes—
90C4×#103C2214yes—
91#1736C2212yes—
92#1736C2212yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1728C2212yes—
96C23×Q8C2312yes—
97C22×M16C2312yes—
98C22×M16C2311yescharacteristic
99C22×M16C2311yescharacteristic
100#533—24no—
101#533C2×C418yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8C2×C418yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#125C2312yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—28no—
120C2×C4×Q8—44no—
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2314yes—
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42—22no—
134C22×C42—21no—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#491C2×C4132yescomplemented
138#487C2×C4116yescomplemented
139#487C2×C4116yescomplemented
140#428—24no—
141#428—24no—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#423—24no—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149#419—28no—
150#419—44no—
151#419C2×C4116yes—
152#419C2×C4116yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C4×M16—44no—
162C4×M16C2×C418yes—
163C2×#97C2314yes—
164C2×#97C2312yes—
165C2×C4×C8C2314yes—
166C2×C4×C8C2312yes—
167#391—21no—
168#391—21no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391C2314yes—
174#391C2314yes—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—24no—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#387—22no—
187#387—22no—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#385—22no—
197#385—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—24no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229#376—21no—
230#376—22no—
231C2×#95—22no—
232C2×#95—21no—
233C2×#95—22no—
234C2×#95D412yes—
235C43—22no—
236C43—21no—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#340C2312yes—
243#340C2312yes—
244#340C2312yes—
245#336C2314yes—
246#336C2314yes—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×Q8—21no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8—24no—
254C22×Q8—24no—
255C22×Q8C22×C418yes—
256C22×Q8C2414yes—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16C22×C414yes—
273C2×M16C22×C414yes—
274C22×C8C2414yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C4⋊Q8—28no—
278C4⋊Q8—28no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8C22×C418yes—
284C4⋊Q8C22×C418yes—
285C4⋊Q8C22×C414yes—
286C4⋊Q8C22×C414yes—
287#125—24no—
288#125C22×C418yes—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×Q8—48no—
298C4×Q8—48no—
299C4×Q8—44no—
300C4×Q8—44no—
301#117—24no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117C22×C418yes—
306#117#4114yes—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—48no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4#4114yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—21no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—48no—
367C2×C42—44no—
368C2×C42C22×C414yes—
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42#4114yes—
374C2×C42C2412yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—24no—
379#107—24no—
380#106—24no—
381#106—24no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384#103—28no—
385#103—28no—
386#103—44no—
387#103—44no—
388#103C22×C4116yes—
389#103C22×C4116yes—
390#103C22×C4116yes—
391#103C22×C4116yes—
392#97—44no—
393#97C22×C418yes—
394C4×C8—44no—
395C4×C8C22×C418yes—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95—21no—
403#95—21no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—21no—
409#95—21no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417#95C2×D412yes—
418C24C22×D411yescharacteristic
419C2×Q8—24no—
420C2×Q8—28no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—216no—
424C2×Q8—216no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—48no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—81no—
436C2×Q8—81no—
437C2×Q8C23×C418yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4C4×D414yes—
467C22×C4C4×D414yes—
468C22×C4C23×C412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C22×C4C2511yescharacteristic
484M16—28no—
485M16—28no—
486M16—48no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8C23×C414yes—
495C2×C8C23×C414yes—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—42no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4#13714yes—
529C4⋊C4C23×C414yes—
530C4⋊C4C23×C414yes—
531C4⋊C4C2×#4112yes—
532C4⋊C4C2×#4112yes—
533C4⋊C4C2×#4114yes—
534C22⋊C4—28no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C42—28no—
538C42—24no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—48no—
546C42—48no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42—44no—
551C42—42no—
552C42—44no—
553C42—42no—
554C42—82no—
555C42—81no—
556C42—81no—
557C42—82no—
558C42C4×D414yes—
559C42C4×D414yes—
560C42C23×C412yes—
561C42C23×C412yes—
562C42C2×#4112yes—
563C42C2×#4112yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23C2×C4×D412yes—
569C23#54612yes—
570C23#54812yes—
571C23C23×D411yescharacteristic
572Q8—28no—
573Q8—28no—
574Q8—416no—
575Q8—48no—
576Q8—44no—
577Q8—44no—
578Q8—84no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—416no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4C2×C4×D412yes—
631C2×C4C2×C4×D412yes—
632C2×C4C2×C4×D412yes—
633C2×C4C2×C4×D412yes—
634C2×C4C2×C4×D412yes—
635C2×C4C4×#4114yes—
636C2×C4#52014yes—
637C2×C4#54612yes—
638C2×C4#54812yes—
639C2×C4#55614yes—
640C2×C4#55914yes—
641C2×C4C2×#13714yes—
642C2×C4C2×#13712yes—
643C2×C4#57512yes—
644C2×C4C24×C411yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C8—24no—
651C8—24no—
652C8—44no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22#236714yes—
660C22C22×C4×D411yescharacteristic
661C22C2×#54611yescharacteristic
662C22C2×#54811yescharacteristic
663C22C22×#13711yescharacteristic
664C22C2×#57511yescharacteristic
665C22#358611yescharacteristic
666C22#358611yescharacteristic
667C4—22no—
668C4—22no—
669C4—24no—
670C4—22no—
671C4—44no—
672C4—48no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—82no—
681C4—81no—
682C4—81no—
683C4#236714yes—
684C4C4×#13714yes—
685C4C2×C4×#4111yescharacteristic
686C4C2×#52011yescharacteristic
687C4C2×#55611yescharacteristic
688C4C2×#55911yescharacteristic
689C2—24no—
690C2C2×#236711yescharacteristic
691C2C2×C4×#13711yescharacteristic
692C2#6016911yescharacteristic
693C2#6115711yescharacteristic
694C2#6115911yescharacteristic
695C2#6132211yescharacteristic
696C2#6132311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #23233C2semidirect
5. #23227C2semidirect
6. #23222C2semidirect
7. #23221C2semidirect
8. #23217C2semidirect
9. #23214C2semidirect
21. #3139C4semidirect
137. #491C2×C4semidirect
138. #487C2×C4semidirect
139. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
#341452
C2×#23652
#232332
#232272
#232222
#232212
#232172
#232142
#213122
#212592
#210942
#210612
#208742
C4×#4282
C4⋊C4×M162
C4×#4192