Finite Group Databasebeta

← #185630

Subgroups — #185630

Order 512.

2469 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2#21759C212yesmaximal, complemented
3C2×#1799C211yesmaximal, characteristic
4C2×#1520C212yesmaximal, complemented
5C2×#1459C211yesmaximal, characteristic
6#8380C214yesmaximal, complemented
7#8379C214yesmaximal, complemented
8#3501—24no—
9#3501—24no—
10#3501—24no—
11#3501—24no—
12C2×#554—22no—
13C2×#554—22no—
14C2×#554C412yescomplemented
15C2×#554C2212yes—
16C2×D4×Q8C412yescomplemented
17C2×D4×Q8C2212yes—
18C2×#544—21no—
19C2×#543—21no—
20C2×#543—21no—
21C2×#533C412yescomplemented
22C22×C22⋊Q8C412yescomplemented
23C22×C22⋊Q8C2211yescharacteristic
24C22×C22⋊Q8C2211yescharacteristic
25C2×#518—21no—
26C22×C4×D4C2211yescharacteristic
27C22×#99—21no—
28C22×C22⋊C8C2212yes—
29C2×#380C2212yes—
30#1892—24no—
31#1872—24no—
32#1824C2214yescomplemented
33#1799—22no—
34#1799C2214yes—
35#1787C2214yescomplemented
36C2×#342—21no—
37#1520—24no—
38#1520C2218yescomplemented
39C2×#328—24no—
40#1459—22no—
41#1459C2214yes—
42C2×#324—21no—
43C2×#323—21no—
44C23×Q8C2×C411yescharacteristic
45C23×Q8C2×C412yescomplemented
46C23×Q8C2311yescharacteristic
47C23×D4D411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51#554—216no—
52#554—216no—
53#554—216no—
54#554—216no—
55#554—28no—
56#554—28no—
57#554—28no—
58#554—28no—
59D4×Q8—28no—
60D4×Q8—28no—
61D4×Q8—28no—
62D4×Q8—28no—
63#544—28no—
64#544—28no—
65#543—28no—
66#543—28no—
67#543—28no—
68#543—28no—
69#533—24no—
70#533—24no—
71#533—24no—
72#533—24no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C2×C412yescomplemented
76C2×C4⋊Q8C2×C412yescomplemented
77C2×#125—22no—
78C2×#125—22no—
79C2×#125—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C2×C412yes—
104C2×C22⋊Q8C2×C412yes—
105C2×C22⋊Q8C2×C412yes—
106C2×C22⋊Q8C2×C412yescomplemented
107C2×C22⋊Q8C2×C412yescomplemented
108C2×C22⋊Q8D414yes—
109C2×C22⋊Q8D414yes—
110C2×C22⋊Q8D414yes—
111C2×C22⋊Q8D414yes—
112C2×C22⋊Q8C2312yes—
113#518—28no—
114#518—28no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2×C412yescomplemented
117C2×C4×Q8C2×C412yescomplemented
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4C2×C412yes—
127C2×C4×D4D412yes—
128C2×C4×D4D412yes—
129C2×C4×D4C2312yes—
130C2×#117—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4C2×C411yescharacteristic
136C22×C4⋊C4C2×C412yescomplemented
137C22×C4⋊C4D411yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C22⋊C4—22no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C22⋊C4—22no—
143C22×C22⋊C4D411yescharacteristic
144C22×C22⋊C4D411yescharacteristic
145C22×C42C2×C411yescharacteristic
146C2×#103—28no—
147C2×#99—24no—
148C2×#99—44no—
149C2×C22⋊C8—22no—
150C2×C22⋊C8—22no—
151C2×C22⋊C8—22no—
152C2×C22⋊C8—22no—
153C2×C22⋊C8—22no—
154C2×C22⋊C8C2314yes—
155#380—22no—
156#380—22no—
157#380—22no—
158#380C2314yes—
159C2×#95—22no—
160#342—24no—
161#340—28no—
162#328—216no—
163#324—24no—
164#323—24no—
165C25C22⋊C411yescharacteristic
166C25C2×D411yescharacteristic
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—22no—
170C22×Q8—21no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—21no—
174C22×Q8—22no—
175C22×Q8—82no—
176C22×Q8—82no—
177C22×Q8—84no—
178C22×Q8C22⋊C414yes—
179C22×Q8C22×C412yes—
180C22×Q8C22×C412yes—
181C22×Q8C2×D414yes—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—24no—
185C22×D4C22⋊C412yes—
186C22×D4SD1614yes—
187C22×D4Q1614yes—
188C22×D4C2×D412yes—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—21no—
192C23×C4—22no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—42no—
200C23×C4—82no—
201C23×C4—82no—
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22⋊C411yescharacteristic
205C23×C4C22⋊C412yes—
206C23×C4C22×C412yes—
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C22×C8—24no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8—84no—
221C4⋊Q8—216no—
222C4⋊Q8—216no—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—24no—
226C4⋊Q8—24no—
227#125—28no—
228#125—28no—
229#125—216no—
230#125—216no—
231C22⋊Q8—28no—
232C22⋊Q8—28no—
233C22⋊Q8—28no—
234C22⋊Q8—28no—
235C22⋊Q8—28no—
236C22⋊Q8—28no—
237C22⋊Q8—216no—
238C22⋊Q8—216no—
239C22⋊Q8—216no—
240C22⋊Q8—216no—
241C22⋊Q8—216no—
242C22⋊Q8—216no—
243C22⋊Q8—216no—
244C22⋊Q8—216no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8C22⋊C414yes—
264C22⋊Q8C22⋊C414yes—
265C22⋊Q8C22⋊C414yes—
266C22⋊Q8C22⋊C414yes—
267C22⋊Q8C22⋊C418yes—
268C22⋊Q8C2×D418yes—
269C4×Q8—28no—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×D4—216no—
276C4×D4—216no—
277C4×D4—216no—
278C4×D4—216no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287#117—28no—
288#117—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—84no—
318C2×C4⋊C4C22⋊C412yes—
319C2×C4⋊C4C22⋊C414yes—
320C2×C4⋊C4C22×C412yes—
321C2×C4⋊C4C22×C412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D414yes—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—42no—
344C2×C22⋊C4C22⋊C412yes—
345C2×C22⋊C4C22⋊C412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42C22⋊C412yes—
355C2×C42C22×C412yes—
356#103—88no—
357#99—48no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C22⋊C8—42no—
361C22⋊C8—42no—
362#95—24no—
363#95—42no—
364C24—21no—
365C24—21no—
366C24—22no—
367C24—81no—
368C24—81no—
369C24—81no—
370C24—81no—
371C24#10314yescomplemented
372C24C2×C22⋊C412yes—
373C24C2×C22⋊C411yescharacteristic
374C24C2×C22⋊C411yescharacteristic
375C24#12012yes—
376C24#12012yes—
377C24C2×SD1612yes—
378C24C2×Q1612yes—
379C24#13714yes—
380C24C22×D411yescharacteristic
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—84no—
391C2×Q8—84no—
392C2×Q8—82no—
393C2×Q8—82no—
394C2×Q8—84no—
395C2×Q8—88no—
396C2×Q8—84no—
397C2×Q8—88no—
398C2×Q8—84no—
399C2×Q8C2×C22⋊C414yes—
400C2×Q8C2×C22⋊C414yes—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4#10318yescomplemented
410C2×D4C2×SD1614yes—
411C2×D4C2×Q1614yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—81no—
443C22×C4—82no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—84no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—84no—
455C22×C4—82no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4#10314yescomplemented
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C22⋊C412yes—
465C22×C4C2×C22⋊C412yes—
466C22×C4C2×C22⋊C412yes—
467C22×C4C2×C22⋊C412yes—
468C22×C4C2×C22⋊C411yescharacteristic
469C22×C4C2×C22⋊C411yescharacteristic
470C22×C4C2×C22⋊C412yes—
471C22×C4C2×C22⋊C411yescharacteristic
472C22×C4C2×C22⋊C411yescharacteristic
473C22×C4#12012yes—
474C22×C4#12012yes—
475C22×C4#12012yes—
476C22×C4#12012yes—
477C22×C4#12012yes—
478C22×C4#12012yes—
479C22×C4C23×C411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C2×C8—48no—
484C2×C8—44no—
485C2×C8—84no—
486C2×C8—84no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—48no—
508C4⋊C4—84no—
509C4⋊C4C2×C22⋊C414yes—
510C4⋊C4C2×C22⋊C414yes—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—28no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—28no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—84no—
526C42—28no—
527C42—28no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—81no—
549C23—81no—
550C23—81no—
551C23—81no—
552C23—82no—
553C23—82no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23#37911yescharacteristic
558C23#37911yescharacteristic
559C23#37912yes—
560C23C2×#10314yes—
561C23C2×#10312yes—
562C23C2×#10312yes—
563C23#41912yes—
564C23#41912yes—
565C23#44814yes—
566C23#44814yes—
567C23#45114yes—
568C23#45114yes—
569C23C22×C22⋊C411yescharacteristic
570C23C2×#12012yes—
571C23C22×SD1611yescharacteristic
572C23C22×Q1611yescharacteristic
573C23C2×#13712yes—
574Q8—28no—
575Q8—44no—
576Q8—44no—
577Q8—44no—
578Q8—84no—
579Q8—84no—
580Q8—88no—
581Q8—88no—
582D4—28no—
583D4—28no—
584D4—216no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—88no—
632C2×C4—88no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—168no—
637C2×C4#37911yescharacteristic
638C2×C4#37911yescharacteristic
639C2×C4#37912yes—
640C2×C4C2×#10314yes—
641C2×C4C22×C22⋊C411yescharacteristic
642C2×C4C22×C22⋊C411yescharacteristic
643C2×C4C2×#12012yes—
644C8—84no—
645C22—21no—
646C22—22no—
647C22—22no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—22no—
652C22—22no—
653C22—21no—
654C22—21no—
655C22—24no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—41no—
660C22—41no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—42no—
670C22—42no—
671C22—84no—
672C22—84no—
673C22#178712yes—
674C22#178712yes—
675C22#178714yes—
676C22C2×#37911yescharacteristic
677C22C22×#10311yescharacteristic
678C22C2×#41911yescharacteristic
679C22C2×#44812yes—
680C22C2×#45112yes—
681C4—22no—
682C4—21no—
683C4—21no—
684C4—21no—
685C4—22no—
686C4—24no—
687C4—42no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—42no—
693C4—164no—
694C4#288312yes—
695C2—22no—
696C2—21no—
697C2—21no—
698C2—22no—
699C2—22no—
700C2—22no—
701C2—24no—
702C2—44no—
703C2#2058511yescharacteristic
704C2C2×#178711yescharacteristic
705C2#2062311yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21759C2semidirect
4. C2×#1520C2semidirect
6. #8380C2semidirect
7. #8379C2semidirect
14. C2×#554C4semidirect
16. C2×D4×Q8C4semidirect
21. C2×#533C4semidirect
22. C22×C22⋊Q8C4semidirect
32. #1824C22semidirect
35. #1787C22semidirect
38. #1520C22semidirect
45. C23×Q8C2×C4semidirect
75. C2×C4⋊Q8C2×C4semidirect
76. C2×C4⋊Q8C2×C4semidirect
106. C2×C22⋊Q8C2×C4semidirect
107. C2×C22⋊Q8C2×C4semidirect
116. C2×C4×Q8C2×C4semidirect
117. C2×C4×Q8C2×C4semidirect
136. C22×C4⋊C4C2×C4semidirect
371. C24#103semidirect
409. C2×D4#103semidirect
461. C22×C4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
#217592
C2×#17992
C2×#15202
C2×#14592
#83802
#83792