Finite Group Databasebeta

← #366950

Subgroups — #366950

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic, complemented
3C2×#1949C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5C2×#1768C211yesmaximal, characteristic, complemented
6#14282C211yesmaximal, characteristic, complemented
7#13785C211yesmaximal, characteristic
8#11379C211yesmaximal, characteristic, complemented
9#11377C211yesmaximal, characteristic, complemented
10#11123C211yesmaximal, characteristic
11#11121C211yesmaximal, characteristic
12#10927C211yesmaximal, characteristic
13#10925C211yesmaximal, characteristic
14#10863C211yesmaximal, characteristic, complemented
15#10862C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#474—24no—
19C2×#474—24no—
20C2×#474C418yescomplemented
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#97C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×#376C2211yescharacteristic
38#1987—24no—
39#1949—24no—
40#1804—24no—
41#1768—24no—
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52#1446—22no—
53#1445—22no—
54#1437—22no—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C22⋊#95—22no—
62C23×Q8—21no—
63C23×D4C2311yescharacteristic
64C24×C4—21no—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8C2×C414yes—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4C2×C414yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#474—232no—
97C2×#107—22no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#107—22no—
102C2×#107—22no—
103C2×#107—41no—
104C2×#107C2×C414yes—
105C2×#107C2×C414yes—
106C2×#106—22no—
107C2×#106—22no—
108C2×#106—41no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8C2×C414yes—
112C2×C4⋊C8D414yes—
113C2×C4⋊C8D414yes—
114C2×C4⋊C8D414yes—
115C2×#102—22no—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—22no—
119C2×#102—22no—
120C2×#102—22no—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—41no—
124C2×#102—41no—
125C2×#102C2×C414yes—
126C2×#102C2×C414yes—
127C2×#97—22no—
128C2×#97—22no—
129C2×#97—41no—
130C2×C4×C8—22no—
131C2×C4×C8—22no—
132C2×C4×C8—41no—
133#390—22no—
134#390—22no—
135#390—22no—
136#390Q814yescomplemented
137#382—22no—
138#382—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—22no—
141#376—22no—
142#376—22no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95D412yes—
153C2×#95D412yes—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#340—22no—
160#340—22no—
161#340—22no—
162#336—22no—
163#336—22no—
164C25—21no—
165C22×Q8—24no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—82no—
169C22×Q8—82no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4C22×C412yes—
182C22×D4C2×Q812yes—
183C22×D4#4112yes—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—24no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—82no—
195C23×C4—82no—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2411yescharacteristic
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8C22×C412yes—
222C22×C8C22×C412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C4⋊Q8—28no—
230C4⋊Q8—28no—
231C4⋊Q8—44no—
232C4⋊Q8—44no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—44no—
236C4×D4—44no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C22×C412yes—
279C2×C4⋊C4C22×C412yes—
280C2×C4⋊C4C2×Q812yes—
281C2×C4⋊C4#4112yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—82no—
288C2×C22⋊C4—82no—
289C2×C22⋊C4—82no—
290C2×C22⋊C4—82no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—42no—
302C2×C42—82no—
303C2×C42C22×C412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×Q812yes—
308C2×C42#4112yes—
309C2×C42#4112yes—
310C2×C42#4112yes—
311#107—28no—
312#107—28no—
313#107—44no—
314#106—44no—
315C4⋊C8—28no—
316C4⋊C8C22⋊C4116yescomplemented
317#102—28no—
318#102—28no—
319#102—44no—
320#102—44no—
321#97—44no—
322C4×C8—44no—
323#95—24no—
324#95—24no—
325#95—42no—
326#95—42no—
327#95—42no—
328#95—82no—
329#95—82no—
330#95—82no—
331#95—82no—
332#95—82no—
333#95—82no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—41no—
338C24—41no—
339C24—41no—
340C24—41no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24C22×D411yescharacteristic
346C2×Q8—48no—
347C2×Q8—88no—
348C2×Q8—84no—
349C2×Q8—84no—
350C2×Q8—88no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4C4×Q814yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
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—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—48no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—88no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—88no—
436C22×C4C4×D412yes—
437C22×C4C4×D412yes—
438C22×C4#12112yes—
439C22×C4#12112yes—
440C22×C4C22⋊Q814yes—
441C22×C4#12712yes—
442C22×C4#12712yes—
443C22×C4C23×C411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×Q811yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C42−1+411yescharacteristic
455C22×C42−1+411yescharacteristic
456C22×C42−1+411yescharacteristic
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8C2×C22⋊C414yes—
484C2×C8C2×C22⋊C414yes—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—48no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C22⋊C4—48no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—84no—
526C42C2×C22⋊C414yes—
527C42C4×D414yes—
528C42#12114yes—
529C42#12414yes—
530C23—24no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—81no—
544C23—81no—
545C23—81no—
546C23—81no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23—82no—
553C23C2×C4×D411yescharacteristic
554C23C2×#12111yescharacteristic
555C23C2×C22⋊Q811yescharacteristic
556C23#52511yescharacteristic
557C23C2×#12711yescharacteristic
558C23#53611yescharacteristic
559C23#53711yescharacteristic
560C23#57612yes—
561C23#57612yes—
562C23#57712yes—
563C23#57812yes—
564Q8—816no—
565D4—24no—
566D4—24no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—816no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4#39014yes—
632C2×C4#39014yes—
633C2×C4C22×C22⋊C411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×Q811yescharacteristic
636C2×C4C4×#4111yescharacteristic
637C2×C4#51911yescharacteristic
638C2×C4#52011yescharacteristic
639C2×C4C2×#12111yescharacteristic
640C2×C4C2×#12411yescharacteristic
641C2×C4#52911yescharacteristic
642C2×C4#53311yescharacteristic
643C2×C4#54111yescharacteristic
644C2×C4#54211yescharacteristic
645C2×C4#54611yescharacteristic
646C2×C4#54611yescharacteristic
647C2×C4#54611yescharacteristic
648C2×C4#54711yescharacteristic
649C2×C4#54711yescharacteristic
650C2×C4#54711yescharacteristic
651C2×C4#54811yescharacteristic
652C2×C4#54811yescharacteristic
653C2×C4#54811yescharacteristic
654C2×C4D4×Q811yescharacteristic
655C2×C4D4×Q811yescharacteristic
656C2×C4D4×Q811yescharacteristic
657C2×C4#55211yescharacteristic
658C2×C4#55211yescharacteristic
659C2×C4#55411yescharacteristic
660C2×C4#55611yescharacteristic
661C8—24no—
662C8—24no—
663C8—42no—
664C8—42no—
665C22—48no—
666C22—44no—
667C22—84no—
668C22—82no—
669C22—82no—
670C22—84no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22C2×#39011yescharacteristic
677C22C4×C22⋊Q811yescharacteristic
678C22#232311yescharacteristic
679C22#241511yescharacteristic
680C22#241711yescharacteristic
681C22#254011yescharacteristic
682C22#254211yescharacteristic
683C22#294712yes—
684C22#294812yes—
685C22#305812yes—
686C22#306312yes—
687C22#308112yes—
688C22#308312yes—
689C22#311812yes—
690C22#312412yes—
691C22#315411yescharacteristic
692C22#315511yescharacteristic
693C22#315511yescharacteristic
694C22#315611yescharacteristic
695C22#324112yes—
696C22#324312yes—
697C22#325712yes—
698C22#326212yes—
699C4—24no—
700C4—24no—
701C4—44no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—161no—
709C4—161no—
710C4#233911yescharacteristic
711C4Q8×C22⋊C411yescharacteristic
712C4#236011yescharacteristic
713C4#236211yescharacteristic
714C4#242711yescharacteristic
715C4#242911yescharacteristic
716C4#248911yescharacteristic
717C4#249211yescharacteristic
718C2—84no—
719C2#1399711yescharacteristic
720C2#2123311yescharacteristic
721C2#2123411yescharacteristic
722C2#2162412yes—
723C2#2212712yes—
724C2#2471212yes—
725C2#2484712yes—
726C2#2839311yescharacteristic
727C2#2839411yescharacteristic
728C2#2852311yescharacteristic
729C2#2852511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1987C2semidirect
3. C2×#1949C2semidirect
5. C2×#1768C2semidirect
6. #14282C2semidirect
8. #11379C2semidirect
9. #11377C2semidirect
14. #10863C2semidirect
15. #10862C2semidirect
20. C2×#474C4semidirect
136. #390Q8semidirect
316. C4⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#19872
C2×#19492
C2×#18042
C2×#17682
#142822
#137852
#113792
#113772
#111232
#111212
#109272
#109252
#108632
#108622