Finite Group Databasebeta

← #364829

Subgroups — #364829

Order 512.

1680 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
1C22×#477C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C4×#391C211yesmaximal, characteristic
6#11219C212yesmaximal
7#11067C212yesmaximal
8#10932C212yesmaximal
9#10907C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10832C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×#477—24no—
16C2×#477—24no—
17C2×#477C418yescomplemented
18C22×#107C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C4×C4⋊Q8—24no—
24C4×C4⋊Q8—24no—
25C2×#391—21no—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389—21no—
29C2×#389C2212yes—
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C43C2211yescharacteristic
35#1987—24no—
36#1803—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1436—21no—
45#1436—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2211yescharacteristic
48C4×#95—21no—
49C4×#95—21no—
50C23×Q8—21no—
51C23×Q8C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yes—
58C2×C4⋊Q8D414yes—
59C2×C4⋊Q8D414yes—
60C2×C4×Q8—28no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—44no—
67C2×C4×Q8—44no—
68C2×C4×Q8C2×C414yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C42—22no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#477—232no—
90C2×#107—24no—
91C2×#107—24no—
92C2×#107C2×C418yes—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8C2×C414yes—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—42no—
104C2×#103C2×C418yes—
105#391—28no—
106#391—22no—
107#391—22no—
108#391—22no—
109#391Q814yescomplemented
110#389—28no—
111#389—24no—
112#389—24no—
113#384—22no—
114#384D414yes—
115C4×C4⋊C4—28no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C43—22no—
132C43—22no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—42no—
138#336—24no—
139#336—42no—
140C22×Q8—24no—
141C22×Q8—24no—
142C22×Q8—24no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8C22×C412yes—
159C22×Q8C2×Q812yes—
160C22×Q8#4112yes—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C22×C414yes—
188C4⋊Q8—28no—
189C4⋊Q8—28no—
190C4⋊Q8—28no—
191C4⋊Q8C22⋊C4116yes—
192C4×Q8—28no—
193C4×Q8—28no—
194C4×Q8—432no—
195C4×Q8—44no—
196C4×Q8—44no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
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—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×Q812yes—
241C2×C4⋊C4#4112yes—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—28no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—44no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—44no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42Q1614yes—
273C2×C42C22×C412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×Q812yes—
278C2×C42#4112yes—
279C2×C42#4112yes—
280C2×C42#4112yes—
281#107—216no—
282C4⋊C8—28no—
283C4⋊C8—44no—
284#103—216no—
285#103—48no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—42no—
297C24C22×D411yescharacteristic
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—216no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×Q8—48no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8C4×Q814yescomplemented
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4C4×D412yes—
402C22×C4C4×D412yes—
403C22×C4#12012yes—
404C22×C4#12012yes—
405C22×C4#12112yes—
406C22×C4#12112yes—
407C22×C4C22⋊Q812yes—
408C22×C4C22⋊Q812yes—
409C22×C4C2×Q1612yes—
410C22×C4C2×Q1612yes—
411C22×C4C2×Q1612yes—
412C22×C4#13512yes—
413C22×C4AGL(1,ℤ/8)12yes—
414C22×C4AGL(1,ℤ/8)12yes—
415C22×C4#13712yes—
416C22×C4#13712yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×Q811yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4H(5,𝔽2)11yescharacteristic
428C22×C42−1+411yescharacteristic
429C22×C42−1+411yescharacteristic
430C22×C42−1+411yescharacteristic
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—216no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—42no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4C2×C22⋊C418yes—
470C42—28no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—28no—
476C42—44no—
477C42—416no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—48no—
483C42—44no—
484C42—84no—
485C42C2×C22⋊C414yes—
486C42C4×D414yes—
487C42#12114yes—
488C42#12314yes—
489C23C2×C4×D411yescharacteristic
490C23C2×#12011yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×C22⋊Q811yescharacteristic
493C23#52511yescharacteristic
494C23#53611yescharacteristic
495C23#53711yescharacteristic
496C23C22×Q1611yescharacteristic
497C23C2×#13511yescharacteristic
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23C2×AGL(1,ℤ/8)11yescharacteristic
500C23C2×#13711yescharacteristic
501C23C2×#13711yescharacteristic
502C23#57511yescharacteristic
503C23#57511yescharacteristic
504Q8—24no—
505Q8—24no—
506Q8—432no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—42no—
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—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4#38614yes—
589C2×C4#38614yes—
590C2×C4C4×Q1614yes—
591C2×C4#44114yes—
592C2×C4#45114yes—
593C2×C4#45214yes—
594C2×C4#46214yes—
595C2×C4#46414yes—
596C2×C4#47714yescomplemented
597C2×C4C22×C22⋊C411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×C4×Q811yescharacteristic
600C2×C4C4×#4111yescharacteristic
601C2×C4#51911yescharacteristic
602C2×C4#52011yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×#12311yescharacteristic
605C2×C4#52911yescharacteristic
606C2×C4#53311yescharacteristic
607C2×C4#54111yescharacteristic
608C2×C4#54211yescharacteristic
609C2×C4#54611yescharacteristic
610C2×C4#54711yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4#54811yescharacteristic
613C2×C4#54811yescharacteristic
614C2×C4#54811yescharacteristic
615C2×C4#54811yescharacteristic
616C2×C4D4×Q811yescharacteristic
617C2×C4D4×Q811yescharacteristic
618C2×C4D4×Q811yescharacteristic
619C2×C4#55011yescharacteristic
620C2×C4#55011yescharacteristic
621C2×C4#55211yescharacteristic
622C2×C4#55211yescharacteristic
623C2×C4#55411yescharacteristic
624C2×C4#55611yescharacteristic
625C8—44no—
626C8—82no—
627C22C2×#38611yescharacteristic
628C22C4×C22⋊Q811yescharacteristic
629C22#232311yescharacteristic
630C22#240711yescharacteristic
631C22#240911yescharacteristic
632C22#253611yescharacteristic
633C22#253811yescharacteristic
634C22C2×C4×Q1611yescharacteristic
635C22C2×#44111yescharacteristic
636C22C4×AGL(1,ℤ/8)11yescharacteristic
637C22C4×AGL(1,ℤ/8)11yescharacteristic
638C22C2×#45111yescharacteristic
639C22C2×#45211yescharacteristic
640C22#300311yescharacteristic
641C22#300511yescharacteristic
642C22C2×#46211yescharacteristic
643C22C2×#46411yescharacteristic
644C22#303611yescharacteristic
645C22#303611yescharacteristic
646C22C2×#47711yescharacteristic
647C22#307611yescharacteristic
648C22#307811yescharacteristic
649C22#307911yescharacteristic
650C22#310411yescharacteristic
651C22#310611yescharacteristic
652C22#311211yescharacteristic
653C22#311611yescharacteristic
654C22#323011yescharacteristic
655C22#323311yescharacteristic
656C22#323711yescharacteristic
657C22#323811yescharacteristic
658C22#324711yescharacteristic
659C22#324911yescharacteristic
660C22#325211yescharacteristic
661C22#325411yescharacteristic
662C4—24no—
663C4—24no—
664C4—42no—
665C4—42no—
666C4—44no—
667C4—48no—
668C4—42no—
669C4—82no—
670C4—84no—
671C4—82no—
672C4—82no—
673C4#187818yes—
674C4Q8×C22⋊C411yescharacteristic
675C4#234111yescharacteristic
676C4#236011yescharacteristic
677C4#236211yescharacteristic
678C4#242111yescharacteristic
679C4#242311yescharacteristic
680C4#242611yescharacteristic
681C4#242911yescharacteristic
682C2#1396911yescharacteristic
683C2C2×#187811yescharacteristic
684C2#2092511yescharacteristic
685C2C4×#47711yescharacteristic
686C2#2161611yescharacteristic
687C2#2211311yescharacteristic
688C2#2211911yescharacteristic
689C2#2415511yescharacteristic
690C2#2415611yescharacteristic
691C2#2430211yescharacteristic
692C2#2430311yescharacteristic
693C2#2820411yescharacteristic
694C2#2820611yescharacteristic
695C2#2828211yescharacteristic
696C2#2828411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#477C4semidirect
109. #391Q8semidirect
332. C2×Q8C4×Q8semidirect
596. C2×C4#477semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×C4×C4⋊Q82
C2×#19872
C2×#18032
C4×#3912
#112192
#110672
#109322
#109072
#108492
#108322