Finite Group Databasebeta

← #353371

Subgroups — #353371

Order 512.

4198 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#519C211yesmaximal, characteristic
2C22×#419C211yesmaximal, characteristic
3C23×#103C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5C2×#1735C211yesmaximal, characteristic
6C2×#1729C211yesmaximal, characteristic
7#13795C211yesmaximal, characteristic
8#10789C218yesmaximal, complemented
9C24×Q8C2211yescharacteristic
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C2×#519—24no—
13C2×#519—24no—
14C2×#519—24no—
15C2×#519—24no—
16C2×#519—24no—
17C2×#519C418yescomplemented
18C22×C4×Q8—22no—
19C22×C4×Q8C2214yes—
20C22×#117—22no—
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C2×#419—24no—
26C2×#419—24no—
27C2×#419C418yescomplemented
28C22×#103—24no—
29C22×#103—24no—
30C22×#103C418yescomplemented
31C22×#103C2214yes—
32C22×#103C2214yes—
33#1736—24no—
34#1735—24no—
35C2×#340C2214yes—
36C2×#340C2214yes—
37#1729—24no—
38C2×#336C2214yes—
39C4⋊#95—22no—
40C4⋊#95C2214yes—
41C22⋊#95—22no—
42C23×Q8—21no—
43C23×Q8—22no—
44C23×Q8—22no—
45C23×Q8—22no—
46C23×Q8—22no—
47C23×Q8—22no—
48C23×Q8—22no—
49C23×Q8—22no—
50C23×Q8C2×C414yes—
51C23×Q8C2×C414yes—
52C23×Q8C2×C414yes—
53C23×Q8D414yes—
54C23×Q8D414yes—
55C23×Q8D414yes—
56C23×Q8Q814yes—
57C23×Q8C2312yes—
58C24×C4—21no—
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C22×M16—22no—
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—22no—
67C22×M16C2×C414yes—
68C23×C8—22no—
69C23×C8—22no—
70C23×C8—22no—
71C23×C8—22no—
72C23×C8—22no—
73C23×C8—22no—
74C23×C8C2×C414yes—
75C23×C8C2312yes—
76C23×C8C2312yes—
77#519—232no—
78#519—232no—
79#519—232no—
80#519—232no—
81C2×C4×Q8—216no—
82C2×C4×Q8—216no—
83C2×C4×Q8—28no—
84C2×C4×Q8—28no—
85C2×C4×Q8—216no—
86C2×C4×Q8—28no—
87C2×C4×Q8—28no—
88C2×C4×Q8—28no—
89C2×C4×Q8C2×C4116yescomplemented
90C2×#117—216no—
91C2×#117—216no—
92C2×#117—22no—
93C2×#117—22no—
94C2×#117—22no—
95C2×#117—22no—
96C2×#117—22no—
97C2×#117—22no—
98C2×#117—22no—
99C2×#117—22no—
100C2×#117—216no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117C2×C414yes—
106C2×#117C2×C414yes—
107C2×#117C2×C414yes—
108C2×#117D414yes—
109C2×#117D414yes—
110C2×#117D414yes—
111C22×C4⋊C4—24no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4C2×C414yes—
119C22×C4⋊C4D412yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4D412yes—
126C22×C42—24no—
127C22×C42—21no—
128C22×C42—21no—
129C22×C42C2312yes—
130#419—232no—
131C2×#103—232no—
132C2×#103—28no—
133C2×#103—28no—
134C2×#103—28no—
135C2×#103—28no—
136C2×#103C2×C4116yes—
137C2×#103C2×C4116yescomplemented
138C2×#95—28no—
139#340—28no—
140#340—28no—
141#336—28no—
142C25C2×D411yescharacteristic
143C22×Q8—24no—
144C22×Q8—28no—
145C22×Q8—28no—
146C22×Q8—28no—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—28no—
150C22×Q8—28no—
151C22×Q8—28no—
152C22×Q8—28no—
153C22×Q8—28no—
154C22×Q8—28no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—28no—
158C22×Q8—28no—
159C22×Q8—24no—
160C22×Q8—28no—
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—24no—
164C22×Q8—28no—
165C22×Q8—28no—
166C22×Q8—28no—
167C22×Q8—28no—
168C22×Q8—28no—
169C22×Q8—28no—
170C22×Q8—28no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8C42116yes—
179C22×Q8C22⋊C4116yes—
180C22×Q8C22⋊C4116yes—
181C22×Q8C22⋊C4116yes—
182C22×Q8C4⋊C4116yes—
183C22×Q8C4⋊C4116yes—
184C22×Q8C4⋊C4116yes—
185C22×Q8C22×C414yes—
186C22×Q8C22×C414yes—
187C22×Q8C22×C414yes—
188C22×Q8C2×D414yes—
189C22×Q8C2×D414yes—
190C22×Q8C2×D414yes—
191C22×Q8C2×Q814yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—84no—
207C23×C4C22×C412yes—
208C23×C4C22×C412yes—
209C23×C4C22×C412yes—
210C23×C4C2×D414yes—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×Q812yes—
219C23×C4#4112yes—
220C23×C4#4112yes—
221C23×C4#4112yes—
222C23×C4#4112yes—
223C23×C4C2411yescharacteristic
224C2×M16—28no—
225C2×M16—28no—
226C2×M16—28no—
227C2×M16—44no—
228C2×M16—44no—
229C2×M16—44no—
230C2×M16—44no—
231C22×C8—28no—
232C22×C8—28no—
233C22×C8—28no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—44no—
247C22×C8—44no—
248C22×C8—44no—
249C22×C8—44no—
250C22×C8C22×C414yes—
251C22×C8C22×C414yes—
252C4×Q8—232no—
253C4×Q8—232no—
254C4×Q8—2128no—
255C4×Q8—232no—
256C4×Q8—232no—
257#117—2128no—
258#117—28no—
259#117—28no—
260#117—28no—
261#117—28no—
262#117—28no—
263#117—28no—
264#117—28no—
265#117—28no—
266#117—28no—
267#117—44no—
268#117—44no—
269#117C22⋊C4116yes—
270C2×C4⋊C4—216no—
271C2×C4⋊C4—216no—
272C2×C4⋊C4—216no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C22⋊C418yes—
296C2×C4⋊C4C22×C414yes—
297C2×C4⋊C4C22×C414yes—
298C2×C4⋊C4C22×C414yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D414yes—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4C22⋊C418yes—
310C2×C42—216no—
311C2×C42—216no—
312C2×C42—216no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42C22×C414yes—
324C2×C42C2×D414yes—
325#103—232no—
326#103—232no—
327#95—88no—
328C24—24no—
329C24—24no—
330C24C2×C22⋊C412yes—
331C24C4×D412yes—
332C24C4×D412yes—
333C24#12012yes—
334C24#12012yes—
335C24#12112yes—
336C24C22⋊Q812yes—
337C24C22×D411yescharacteristic
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—232no—
360C2×Q8—216no—
361C2×Q8—216no—
362C2×Q8—232no—
363C2×Q8—216no—
364C2×Q8—232no—
365C2×Q8—232no—
366C2×Q8—232no—
367C2×Q8—216no—
368C2×Q8—216no—
369C2×Q8—232no—
370C2×Q8—232no—
371C2×Q8—216no—
372C2×Q8—216no—
373C2×Q8#95164yescomplemented
374C2×Q8C2×C4218yes—
375C2×Q8C2×C22⋊C418yes—
376C2×Q8C2×C22⋊C418yes—
377C2×Q8C2×C22⋊C418yes—
378C2×Q8C2×C4⋊C418yes—
379C2×Q8C2×C4⋊C418yes—
380C2×Q8C2×C4⋊C418yes—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4C2×C4214yes—
418C22×C4C2×C22⋊C418yes—
419C22×C4C2×C22⋊C414yes—
420C22×C4C2×C22⋊C414yes—
421C22×C4C2×C22⋊C414yes—
422C22×C4C2×C22⋊C414yes—
423C22×C4C2×C22⋊C412yes—
424C22×C4C2×C4⋊C414yes—
425C22×C4C2×C4⋊C414yes—
426C22×C4C2×C4⋊C414yes—
427C22×C4#11714yes—
428C22×C4#11714yes—
429C22×C4C4×D414yes—
430C22×C4C4×D414yes—
431C22×C4C4×D414yes—
432C22×C4C4×D414yes—
433C22×C4C4×D414yes—
434C22×C4C4×D414yes—
435C22×C4C4×D412yes—
436C22×C4C4×D412yes—
437C22×C4#12014yes—
438C22×C4#12014yes—
439C22×C4#12012yes—
440C22×C4#12012yes—
441C22×C4#12114yes—
442C22×C4#12114yes—
443C22×C4#12114yes—
444C22×C4#12112yes—
445C22×C4C22⋊Q814yes—
446C22×C4C22⋊Q814yes—
447C22×C4C22⋊Q814yes—
448C22×C4C22⋊Q812yes—
449C22×C4#12314yes—
450C22×C4#12314yes—
451C22×C4#12314yes—
452C22×C4#12314yes—
453C22×C4C23×C411yescharacteristic
454C22×C4C23×C411yescharacteristic
455C22×C4C23×C411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×Q811yescharacteristic
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4111yescharacteristic
465C22×C4C2×#4111yescharacteristic
466C22×C4C2×#4111yescharacteristic
467M16—416no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—416no—
483C4⋊C4—264no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—216no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4C2×C22⋊C418yes—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—48no—
501C42—264no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—216no—
507C42—48no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23C4×C22⋊C414yes—
519C23#37914yes—
520C23#38014yes—
521C23#38514yes—
522C23#38514yes—
523C23#38614yes—
524C23#38614yes—
525C23C22×C22⋊C411yescharacteristic
526C23C2×C4×D411yescharacteristic
527C23C2×C4×D411yescharacteristic
528C23C2×#12011yescharacteristic
529C23C2×#12011yescharacteristic
530C23C2×#12111yescharacteristic
531C23C2×C22⋊Q811yescharacteristic
532C23#57714yes—
533C23#57814yes—
534Q8—264no—
535Q8—28no—
536Q8—28no—
537Q8—28no—
538Q8—28no—
539Q8—28no—
540Q8—28no—
541Q8—28no—
542Q8C2×#95116yes—
543C2×C4—264no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—216no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—816no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—82no—
569C2×C4—84no—
570C2×C4—82no—
571C2×C4C2×#9518yes—
572C2×C4C4×C22⋊C414yes—
573C2×C4C4×C22⋊C418yes—
574C2×C4#37914yes—
575C2×C4#38014yes—
576C2×C4#38018yes—
577C2×C4#38118yes—
578C2×C4#38514yes—
579C2×C4#38514yes—
580C2×C4#38518yes—
581C2×C4#38518yes—
582C2×C4#38614yes—
583C2×C4#38614yes—
584C2×C4#38618yes—
585C2×C4#38618yes—
586C2×C4C22×C4211yescharacteristic
587C2×C4C22×C22⋊C411yescharacteristic
588C2×C4C22×C22⋊C411yescharacteristic
589C2×C4C22×C22⋊C411yescharacteristic
590C2×C4C22×C22⋊C411yescharacteristic
591C2×C4C22×C4⋊C411yescharacteristic
592C2×C4C22×C4⋊C411yescharacteristic
593C2×C4C22×C4⋊C411yescharacteristic
594C2×C4C2×#11711yescharacteristic
595C2×C4C2×#11711yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×#12011yescharacteristic
603C2×C4C2×#12011yescharacteristic
604C2×C4C2×#12111yescharacteristic
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×C22⋊Q811yescharacteristic
608C2×C4C2×C22⋊Q811yescharacteristic
609C2×C4C2×C22⋊Q811yescharacteristic
610C2×C4C2×#12311yescharacteristic
611C2×C4C2×#12311yescharacteristic
612C2×C4C2×#12311yescharacteristic
613C2×C4C2×#12311yescharacteristic
614C8—44no—
615C8—44no—
616C22—28no—
617C22—28no—
618C22—28no—
619C22—28no—
620C22—28no—
621C22—28no—
622C22—24no—
623C22—24no—
624C22C22⋊#9518yes—
625C22C2×C4×C22⋊C411yescharacteristic
626C22C2×#37911yescharacteristic
627C22C2×#38011yescharacteristic
628C22C2×#38511yescharacteristic
629C22C2×#38511yescharacteristic
630C22C2×#38611yescharacteristic
631C22C2×#38611yescharacteristic
632C22#289814yes—
633C22#294714yes—
634C22#294912yes—
635C22#295012yes—
636C22#301012yes—
637C22#301112yes—
638C22#301314yes—
639C22#304312yes—
640C22#304412yes—
641C22#308214yes—
642C4—216no—
643C4—48no—
644C4—48no—
645C4—84no—
646C4—84no—
647C4C22⋊#9518yes—
648C4C22×#9511yescharacteristic
649C4C2×C4×C22⋊C411yescharacteristic
650C4C2×#38011yescharacteristic
651C4C2×#38111yescharacteristic
652C4C2×#38511yescharacteristic
653C4C2×#38511yescharacteristic
654C4C2×#38611yescharacteristic
655C4C2×#38611yescharacteristic
656C2—28no—
657C2C2×C22⋊#9511yescharacteristic
658C2#2041712yes—
659C2#2060312yes—
660C2#2065112yes—
661C2#2087712yes—
662C2#2088012yes—
663C2#2093012yes—
664C2#2093511yescharacteristic
665C2#2093611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #10789C2semidirect
17. C2×#519C4semidirect
27. C2×#419C4semidirect
30. C22×#103C4semidirect
89. C2×C4×Q8C2×C4semidirect
137. C2×#103C2×C4semidirect
373. C2×Q8#95semidirect

Maximal subgroups

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

subgroupindex
C22×#5192
C22×#4192
C23×#1032
C2×#17362
C2×#17352
C2×#17292
#137952
#107892