Finite Group Databasebeta

← #368637

Subgroups — #368637

Order 512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1970C211yesmaximal, characteristic
2C2×#1970C211yesmaximal, characteristic
3C2×#1970C211yesmaximal, characteristic
4C2×C4×#107C211yesmaximal, characteristic, complemented
5C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
6#14022C212yesmaximal, complemented
7#11189C212yesmaximal, complemented
8#11076C212yesmaximal, complemented
9#10931C212yesmaximal, complemented
10#10862C211yesmaximal, characteristic, complemented
11#10862C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C4×C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#388—22no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1970—24no—
30#1970—24no—
31#1970C418yescomplemented
32C4×#107—24no—
33C8×C4⋊C4—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1440—22no—
40#1439—22no—
41#1437—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×D4C2312yes—
45C22×D8—22no—
46C22×D8—22no—
47C22×D8—22no—
48C22×D8—22no—
49C22×D8—22no—
50C22×D8—22no—
51C22×D8C2×C414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C42—22no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107C2×C414yes—
79C2×#107D414yes—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102C2×C418yes—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99C2×C4×C8C2×C414yes—
100C2×C4×C8D414yes—
101#390—24no—
102#390—24no—
103#390—24no—
104#390—24no—
105#390—42no—
106#390C2×C418yes—
107#388—48no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—42no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—42no—
121#340—42no—
122#336—22no—
123#336—22no—
124C25—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4C22×C414yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C2×D8—28no—
150C2×D8—28no—
151C2×D8—28no—
152C2×D8—44no—
153C2×D8—44no—
154C2×D8—44no—
155C2×D8—44no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C22×C412yes—
175C22×C8C22×C412yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8#4112yes—
180C22×C8#4112yes—
181C22×C8#4112yes—
182C22×C8#4112yes—
183C22×C8#4112yes—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4D814yescomplemented
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C2×D414yes—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4—82no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—82no—
243C2×C42C22×C412yes—
244C2×C42C2×D412yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247#107—28no—
248#107—28no—
249#107—44no—
250C4⋊C8—28no—
251#102—48no—
252#102—48no—
253#102—48no—
254C4×C8—28no—
255C4×C8—28no—
256C4×C8—44no—
257#95—22no—
258#95—22no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—82no—
263#95—82no—
264C24—24no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24C22×D411yescharacteristic
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4#12112yes—
364C22×C4#12112yes—
365C22×C4C2×D812yes—
366C22×C4C2×D812yes—
367C22×C4C2×D812yes—
368C22×C4#13512yes—
369C22×C4#13512yes—
370C22×C4#13512yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C42−1+411yescharacteristic
385D8—416no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8#11714yes—
408C2×C8C4×D414yes—
409C2×C8C4×D414yes—
410C2×C8#12114yes—
411C2×C8#12314yes—
412C2×C8#12414yes—
413C2×C8#12614yes—
414C4⋊C4—28no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C22⋊C4—416no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—88no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23C2×C4×D411yescharacteristic
483C23C2×#12111yescharacteristic
484C23#53211yescharacteristic
485C23D4211yescharacteristic
486C23#54711yescharacteristic
487C23#55011yescharacteristic
488C23#55011yescharacteristic
489C23C22×D811yescharacteristic
490C23C2×#13511yescharacteristic
491C23C2×#13511yescharacteristic
492C23C2×#13511yescharacteristic
493C23#57611yescharacteristic
494C23#57611yescharacteristic
495C23#57611yescharacteristic
496C23#57811yescharacteristic
497D4—44no—
498D4—44no—
499D4—44no—
500D4—44no—
501D4—82no—
502D4—82no—
503D4—82no—
504D4—82no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4C4×D814yes—
567C2×C4#46614yes—
568C2×C4C2×#11711yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C4×#4111yescharacteristic
572C2×C4#51811yescharacteristic
573C2×C4#52011yescharacteristic
574C2×C4C2×#12111yescharacteristic
575C2×C4C2×#12311yescharacteristic
576C2×C4#52511yescharacteristic
577C2×C4C2×#12411yescharacteristic
578C2×C4C2×#12611yescharacteristic
579C2×C4#52911yescharacteristic
580C2×C4#53511yescharacteristic
581C2×C4#53711yescharacteristic
582C2×C4#54011yescharacteristic
583C2×C4#54211yescharacteristic
584C2×C4#54611yescharacteristic
585C2×C4#54811yescharacteristic
586C2×C4#55011yescharacteristic
587C2×C4#55311yescharacteristic
588C2×C4#55311yescharacteristic
589C2×C4#55511yescharacteristic
590C2×C4#55611yescharacteristic
591C2×C4#55611yescharacteristic
592C2×C4#55911yescharacteristic
593C2×C4#55911yescharacteristic
594C2×C4#55911yescharacteristic
595C2×C4#55911yescharacteristic
596C8—42no—
597C8—42no—
598C8—42no—
599C8#38818yescomplemented
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—162no—
608C22—162no—
609C22—162no—
610C22—162no—
611C22—162no—
612C22—162no—
613C22—162no—
614C22—162no—
615C22C4×#12111yescharacteristic
616C22#235411yescharacteristic
617C22#236611yescharacteristic
618C22#241611yescharacteristic
619C22#244511yescharacteristic
620C22#251211yescharacteristic
621C22#254811yescharacteristic
622C22C2×C4×D811yescharacteristic
623C22C4×#13511yescharacteristic
624C22#294811yescharacteristic
625C22#295111yescharacteristic
626C22C2×#46611yescharacteristic
627C22#305011yescharacteristic
628C22#305811yescharacteristic
629C22#306111yescharacteristic
630C22#316811yescharacteristic
631C22#316911yescharacteristic
632C22#317511yescharacteristic
633C22#317611yescharacteristic
634C22D4×D811yescharacteristic
635C22#328011yescharacteristic
636C22#328511yescharacteristic
637C22#328711yescharacteristic
638C22#333411yescharacteristic
639C22#333911yescharacteristic
640C22#333911yescharacteristic
641C22#334111yescharacteristic
642C22#339111yescharacteristic
643C22#339111yescharacteristic
644C22#339311yescharacteristic
645C22#339311yescharacteristic
646C22#339611yescharacteristic
647C22#339611yescharacteristic
648C22#339611yescharacteristic
649C22#339611yescharacteristic
650C4—24no—
651C4—24no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—84no—
657C4—84no—
658C4—162no—
659C4C2×#38811yescharacteristic
660C4#232411yescharacteristic
661C4#235711yescharacteristic
662C4#236711yescharacteristic
663C4#241411yescharacteristic
664C4#244811yescharacteristic
665C4#251311yescharacteristic
666C4#259211yescharacteristic
667C2—164no—
668C2#1402211yescharacteristic
669C2C4×#46611yescharacteristic
670C2#2152411yescharacteristic
671C2#2280711yescharacteristic
672C2#2281011yescharacteristic
673C2#2318511yescharacteristic
674C2#2318611yescharacteristic
675C2#2477811yescharacteristic
676C2#2478011yescharacteristic
677C2#2625311yescharacteristic
678C2#2625411yescharacteristic
679C2#2801311yescharacteristic
680C2#2801411yescharacteristic
681C2#2873611yescharacteristic
682C2#2874011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×C4×#107C2semidirect
5. C2×C8×C4⋊C4C2semidirect
6. #14022C2semidirect
7. #11189C2semidirect
8. #11076C2semidirect
9. #10931C2semidirect
10. #10862C2semidirect
11. #10862C2semidirect
31. #1970C4semidirect
205. C2×C4⋊C4D8semidirect
599. C8#388semidirect

Maximal subgroups

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

subgroupindex
C2×#19702
C2×C4×#1072
C2×C8×C4⋊C42
#140222
#111892
#110762
#109312
#108622