Finite Group Databasebeta

← #6448028

Subgroups — #6448028

Order 512.

2074 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3135C211yesmaximal, characteristic
2C2×#3078C211yesmaximal, characteristic, complemented
3#35178C211yesmaximal, characteristic
4#34213C211yesmaximal, characteristic, complemented
5#25291C214yesmaximal, complemented
6#25290C214yesmaximal, complemented
7#25287C214yesmaximal, complemented
8#25286C214yesmaximal, complemented
9#23616C212yesmaximal, complemented
10#23510C212yesmaximal, complemented
11#21262C212yesmaximal, complemented
12#21258C211yesmaximal, characteristic
13#21144C211yesmaximal, characteristic, complemented
14#21062C212yesmaximal, complemented
15C2×#1763C211yesmaximal, characteristic, complemented
16C2×#536C2212yes—
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20#3135—28no—
21C2×#491C2212yes—
22C2×#490C2212yes—
23C2×#489C2212yes—
24C2×#489C2212yes—
25#3078—28no—
26C2×#479C2214yes—
27C2×#477C2212yes—
28C2×#475C2212yes—
29C2×#428C2212yes—
30C2×#424C2212yes—
31C2×#423C2211yescharacteristic
32C2×#417C2212yes—
33C2×#417C2212yes—
34C2×#419C2212yes—
35C2×#416C2212yes—
36C2×#416C2212yes—
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#97C2211yescharacteristic
40#2854—22no—
41#2618—22no—
42#2548C2212yes—
43#2546C2212yes—
44#2545C2214yes—
45#2538—21no—
46#2538—21no—
47#2536—22no—
48#2370C2214yes—
49#2369C2212yes—
50#2365C2212yes—
51#2355—21no—
52#2355—22no—
53Q8×C4⋊C4—21no—
54#2329—22no—
55#2317C2212yes—
56#2311C2212yes—
57C4×C22⋊Q8—22no—
58#2285C2211yescharacteristic
59#2274C2212yes—
60C2×C4×C4⋊C4C2211yescharacteristic
61#2068C2214yes—
62#2067C2214yes—
63#2066C2214yes—
64#2065C2214yes—
65#2043C2214yes—
66#2043C2214yes—
67#2042C2214yes—
68#2042C2214yes—
69#1983C2214yes—
70#1983C2214yes—
71#1983C2214yes—
72#1982C2212yes—
73#1981C2212yes—
74#1981C2214yes—
75#1981C2214yes—
76#1940C2214yes—
77#1940C2214yes—
78#1928C2214yes—
79#1927C2214yes—
80#1926C2214yes—
81#1925C2214yes—
82#1763C2214yes—
83#1763C2214yes—
84C2×#340C2211yescharacteristic
85C2×#340C2211yescharacteristic
86#1729C2212yes—
87C22×#41C2312yes—
88C23×Q8C2311yescharacteristic
89C22×M16C2312yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92#536—28no—
93#536—44no—
94#533—24no—
95#533D418yes—
96#532—24no—
97#532—24no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#125C2312yes—
104C2×#124C2312yes—
105C2×#124C2314yes—
106C2×#123—24no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×#121—21no—
112C2×#121—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133#491—28no—
134#490—28no—
135#489—28no—
136#489—28no—
137#479—216no—
138#477—28no—
139#475—28no—
140#428—28no—
141C2×#107C2314yes—
142C2×#106C2314yes—
143#424—44no—
144#423—24no—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149#419—28no—
150#417—28no—
151#417—44no—
152#416—28no—
153#416—28no—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#97—24no—
173C2×#97C2312yes—
174C2×#97C2312yes—
175#398—24no—
176#396—24no—
177#396—24no—
178#394—24no—
179#391—22no—
180#391—24no—
181#391C2314yes—
182#390—22no—
183#390—22no—
184#390C2314yes—
185#389—24no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2314yes—
190#389C2314yes—
191#388—24no—
192#388—24no—
193#387—24no—
194#387—24no—
195#387—24no—
196#386—24no—
197#385—24no—
198#384—22no—
199#384D411yescharacteristic
200#384D411yescharacteristic
201#382—22no—
202#382D412yes—
203#380—22no—
204#380—22no—
205#380D412yes—
206#380D412yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—24no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—24no—
220C2×#95—21no—
221#340C2314yes—
222#340C2314yes—
223#336C2314yes—
224#336C2314yes—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—24no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8#4114yes—
240C22×Q8C2412yes—
241C22×Q8C2412yes—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2412yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×M16—24no—
255C2×M16—24no—
256C2×M16—24no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C4⋊Q8—28no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—44no—
272C4⋊Q8C2×D414yes—
273C4⋊Q8C2×D414yes—
274#127—22no—
275#127—22no—
276#125—24no—
277#125C2×D418yes—
278#124—28no—
279#124—24no—
280#124—24no—
281#124—44no—
282#123—416no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—48no—
286C22⋊Q8—48no—
287#121—44no—
288#121—44no—
289C4×Q8—28no—
290C4×Q8—28no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×Q8—84no—
298C4×D4—44no—
299C4×D4—44no—
300#117—24no—
301#117—24no—
302#117—24no—
303#117C2×D418yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4#4114yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4C2×D414yes—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42—82no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D414yes—
357C2×C42#4114yes—
358C2×C42C2414yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—28no—
362#106—28no—
363C4⋊C8—24no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366C4⋊C8—44no—
367#103—28no—
368#103—24no—
369#103—24no—
370#103—24no—
371#103—24no—
372#103—28no—
373#103—28no—
374#103—28no—
375#103—28no—
376#103—44no—
377#102—24no—
378#102—24no—
379#102—24no—
380#102—24no—
381#102—28no—
382#102—44no—
383#97—24no—
384#97—24no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95C2×D414yes—
394#95C2×D414yes—
395C24—22no—
396C24C22×D411yescharacteristic
397#41—48no—
398#41—48no—
399#41—48no—
400#41—48no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—28no—
406C2×Q8—28no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—81no—
424C2×Q8—81no—
425C2×Q8C2×#4114yes—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—81no—
440C2×D4—81no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4#12114yes—
468C22×C4#12114yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483M16—48no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—44no—
494C2×C8—44no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4C22×D414yes—
524C4⋊C4C22×D414yes—
525C4⋊C4C2×#4114yes—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—28no—
535C42—24no—
536C42—24no—
537C42—44no—
538C42—44no—
539C42—44no—
540C42—84no—
541C42—82no—
542C42—84no—
543C42#12114yes—
544C42#12114yes—
545C42C22×D412yes—
546C42C22×D412yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23C2×#12112yes—
561C23#54612yes—
562C23#54812yes—
563C23C23×D411yescharacteristic
564Q8—24no—
565Q8—24no—
566Q8—48no—
567Q8—48no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—84no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—84no—
580D4—84no—
581D4—88no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—88no—
624C2×C4C2×#12112yes—
625C2×C4C2×#12112yes—
626C2×C4C2×#12112yes—
627C2×C4C2×#12112yes—
628C2×C4C2×#12112yes—
629C2×C4#52914yes—
630C2×C4#54214yes—
631C2×C4#54612yes—
632C2×C4#54812yes—
633C2×C4#54814yes—
634C2×C4#55014yes—
635C2×C4C23×D411yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+411yescharacteristic
641C8—42no—
642C8—42no—
643C8—44no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22#242614yes—
658C22C22×#12111yescharacteristic
659C22C2×#54611yescharacteristic
660C22C2×#54811yescharacteristic
661C22#358511yescharacteristic
662C22#358611yescharacteristic
663C22#358611yescharacteristic
664C22#358611yescharacteristic
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#242614yes—
679C4C2×#52911yescharacteristic
680C4C2×#54211yescharacteristic
681C4C2×#54811yescharacteristic
682C4C2×#55011yescharacteristic
683C2—24no—
684C2—162no—
685C2C2×#242611yescharacteristic
686C2#6042511yescharacteristic
687C2#6042711yescharacteristic
688C2#6122011yescharacteristic
689C2#6122111yescharacteristic
690C2#6133311yescharacteristic
691C2#6133511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3078C2semidirect
4. #34213C2semidirect
5. #25291C2semidirect
6. #25290C2semidirect
7. #25287C2semidirect
8. #25286C2semidirect
9. #23616C2semidirect
10. #23510C2semidirect
11. #21262C2semidirect
13. #21144C2semidirect
14. #21062C2semidirect
15. C2×#1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31352
C2×#30782
#351782
#342132
#252912
#252902
#252872
#252862
#236162
#235102
#212622
#212582
#211442
#210622
C2×#17632