Finite Group Databasebeta

← #362620

Subgroups — #362620

Order 512.

1728 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#436C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1809C212yesmaximal, complemented
5#13906C211yesmaximal, characteristic
6#13906C211yesmaximal, characteristic
7#11096C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10841C211yesmaximal, characteristic, complemented
10#10722C212yesmaximal, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C23×M16C2212yes—
13C22×C4×D4C2211yescharacteristic
14C2×#436—24no—
15C2×#436—24no—
16C2×#436C418yescomplemented
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#390—21no—
23C2×#390—21no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1948—24no—
33#1947—24no—
34#1809—28no—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43#1439—22no—
44#1439—22no—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C23×D4D411yescharacteristic
48C24×C4C2312yes—
49C22×M16—24no—
50C22×M16—24no—
51C22×M16—24no—
52C22×M16—24no—
53C22×M16—42no—
54C22×M16—42no—
55C22×M16C2×C418yes—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4C2×C414yes—
64C2×C4×D4Q814yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#436—232no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—42no—
91C2×C4⋊C8C2×C414yes—
92C2×C4⋊C8D414yes—
93C2×C4⋊C8D414yes—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8C2×C418yes—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99C2×C4×C8C2×C414yes—
100#390—44no—
101#390—44no—
102#384—22no—
103#384—22no—
104#384—22no—
105#384—22no—
106#380—24no—
107#380—24no—
108#380—24no—
109#380—24no—
110#380—42no—
111#380—42no—
112#380—42no—
113#380—42no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—42no—
131#340D414yescomplemented
132C25C2×D412yes—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4C4⋊C414yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4C22×C414yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×Q814yes—
163C23×C4C2411yescharacteristic
164C2×M16—216no—
165C2×M16—216no—
166C2×M16—48no—
167C2×M16—48no—
168C2×M16—48no—
169C2×M16—48no—
170C2×M16—48no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C22×C412yes—
186C22×C8C22×C414yes—
187C22×C8C22×C412yes—
188C22×C8C2×D414yes—
189C22×C8C2×D414yes—
190C22×C8C2×D412yes—
191C22×C8#4112yes—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—81no—
239C2×C4⋊C4—81no—
240C2×C4⋊C4—81no—
241C2×C4⋊C4—81no—
242C2×C4⋊C4C4⋊C414yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4#4112yes—
246C2×C4⋊C4#4112yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4—82no—
260C2×C22⋊C4—82no—
261C2×C22⋊C4C4⋊C418yes—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42C22×C412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×Q812yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274#107—48no—
275#106—48no—
276C4⋊C8C22⋊C4116yescomplemented
277C22⋊C8—216no—
278C4×C8—28no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—42no—
284#95—42no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24C2×C4⋊C414yes—
300C24C22×D411yescharacteristic
301C2×D4—28no—
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
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—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4C2×C4⋊C414yes—
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4#12112yes—
392C22×C4#12112yes—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4AGL(1,ℤ/8)12yes—
396C22×C4AGL(1,ℤ/8)12yes—
397C22×C4#13712yes—
398C22×C4#13712yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413M16—432no—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8C2×C22⋊C418yes—
434C2×C8C4×D414yes—
435C4⋊C4—28no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C22⋊C4—216no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C42—84no—
470C42—84no—
471C42C2×C22⋊C414yes—
472C42#11714yes—
473C42C22⋊Q814yes—
474C42C22⋊Q814yes—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23C22×C4⋊C411yescharacteristic
511C23C2×#12111yescharacteristic
512C23C2×#12111yescharacteristic
513C23D4211yescharacteristic
514C23#54611yescharacteristic
515C23#54611yescharacteristic
516C23#54711yescharacteristic
517C23C2×AGL(1,ℤ/8)11yescharacteristic
518C23C2×AGL(1,ℤ/8)11yescharacteristic
519C23C2×#13711yescharacteristic
520C23C2×#13711yescharacteristic
521C23#57611yescharacteristic
522C23#57711yescharacteristic
523C23#57711yescharacteristic
524C23#57811yescharacteristic
525D4—416no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4#38014yes—
590C2×C4#38014yes—
591C2×C4#42814yes—
592C2×C4#42814yes—
593C2×C4#46814yes—
594C2×C4#46814yes—
595C2×C4#46914yes—
596C2×C4#47014yes—
597C2×C4C22×C22⋊C411yescharacteristic
598C2×C4C2×#11711yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C4×#4111yescharacteristic
601C2×C4#51811yescharacteristic
602C2×C4#52011yescharacteristic
603C2×C4C2×C22⋊Q811yescharacteristic
604C2×C4C2×C22⋊Q811yescharacteristic
605C2×C4#52911yescharacteristic
606C2×C4#52911yescharacteristic
607C2×C4#53211yescharacteristic
608C2×C4#53211yescharacteristic
609C2×C4#54011yescharacteristic
610C2×C4#54011yescharacteristic
611C2×C4#54211yescharacteristic
612C2×C4#54211yescharacteristic
613C2×C4D4211yescharacteristic
614C2×C4D4211yescharacteristic
615C2×C4#54611yescharacteristic
616C2×C4#54611yescharacteristic
617C2×C4#54811yescharacteristic
618C2×C4#54811yescharacteristic
619C2×C4D4×Q811yescharacteristic
620C2×C4#55011yescharacteristic
621C2×C4#55011yescharacteristic
622C2×C4#55411yescharacteristic
623C2×C4#55411yescharacteristic
624C2×C4#55411yescharacteristic
625C8—28no—
626C8—24no—
627C8—42no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C2×#38011yescharacteristic
644C22D4×C4⋊C411yescharacteristic
645C22#234911yescharacteristic
646C22#242711yescharacteristic
647C22#242711yescharacteristic
648C22#242811yescharacteristic
649C22#242811yescharacteristic
650C22C2×#42811yescharacteristic
651C22C2×#42811yescharacteristic
652C22#291211yescharacteristic
653C22#291311yescharacteristic
654C22C2×#46811yescharacteristic
655C22C2×#46811yescharacteristic
656C22C2×#46911yescharacteristic
657C22C2×#47011yescharacteristic
658C22#306211yescharacteristic
659C22#306211yescharacteristic
660C22#306311yescharacteristic
661C22#306311yescharacteristic
662C22#327211yescharacteristic
663C22#327411yescharacteristic
664C22#327611yescharacteristic
665C22#327811yescharacteristic
666C22#332411yescharacteristic
667C22#332511yescharacteristic
668C22#332811yescharacteristic
669C22#332911yescharacteristic
670C22#333011yescharacteristic
671C22#333111yescharacteristic
672C22#333211yescharacteristic
673C22#333311yescharacteristic
674C22#335311yescharacteristic
675C22#335411yescharacteristic
676C22#335611yescharacteristic
677C22#335711yescharacteristic
678C4—24no—
679C4—44no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—162no—
685C4—162no—
686C4#180918yescomplemented
687C4#229311yescharacteristic
688C4#231911yescharacteristic
689C4D4×C22⋊C411yescharacteristic
690C4#234111yescharacteristic
691C4#245211yescharacteristic
692C4#245211yescharacteristic
693C4#245411yescharacteristic
694C4#245411yescharacteristic
695C2—84no—
696C2#1392011yescharacteristic
697C2C2×#180911yescharacteristic
698C2#2066411yescharacteristic
699C2#2266311yescharacteristic
700C2#2266511yescharacteristic
701C2#2271311yescharacteristic
702C2#2271511yescharacteristic
703C2#2534611yescharacteristic
704C2#2534711yescharacteristic
705C2#2534811yescharacteristic
706C2#2534911yescharacteristic
707C2#2539211yescharacteristic
708C2#2539311yescharacteristic
709C2#2539411yescharacteristic
710C2#2539511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1948C2semidirect
3. C2×#1947C2semidirect
4. C2×#1809C2semidirect
7. #11096C2semidirect
8. #11095C2semidirect
9. #10841C2semidirect
10. #10722C2semidirect
11. C4×#340C2semidirect
16. C2×#436C4semidirect
131. #340D4semidirect
276. C4⋊C8C22⋊C4semidirect
686. C4#1809semidirect

Maximal subgroups

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

subgroupindex
C22×#4362
C2×#19482
C2×#19472
C2×#18092
#139062
#110962
#110952
#108412
#107222
C4×#3402