Finite Group Databasebeta

← #6345335

Subgroups — #6345335

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3108C211yesmaximal, characteristic
2#60288C211yesmaximal, characteristic
3C2×#2984C211yesmaximal, characteristic, complemented
4#18058C214yesmaximal, complemented
5#18056C214yesmaximal, complemented
6#18017C218yesmaximal, complemented
7#17943C214yesmaximal, complemented
8#17911C214yesmaximal, complemented
9#17818C211yesmaximal, characteristic, complemented
10#17502C211yesmaximal, characteristic
11C2×#1491C211yesmaximal, characteristic, complemented
12#17314C211yesmaximal, characteristic, complemented
13C2×#529C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15#3108C412yes—
16#3108C414yes—
17#3108C412yes—
18#3108C2212yes—
19#3108C2214yes—
20#3108C2212yes—
21C2×#483C411yescharacteristic
22C2×#480C411yescharacteristic
23C2×#479C412yes—
24C2×#464C412yes—
25C2×#451C411yescharacteristic
26C2×#448C411yescharacteristic
27#2994C2214yes—
28#2992—22no—
29#2984—22no—
30#2984—22no—
31#2984C2214yes—
32#2984C2212yes—
33#2984C2214yescomplemented
34#2974C2212yes—
35#2973—21no—
36#2972—21no—
37#2968—22no—
38#2966C2214yes—
39C2×#446—22no—
40C2×#446C2212yescomplemented
41C2×C8×Q8C2212yescomplemented
42C2×C4×Q16C2212yes—
43C2×C4×SD16C2212yes—
44#2924—22no—
45C2×#432—21no—
46C2×#432C2211yescharacteristic
47C2×#432C2211yescharacteristic
48C2×#431—21no—
49C2×#431C2211yescharacteristic
50C2×#431C2211yescharacteristic
51C2×C4×M16C2211yescharacteristic
52#1591C2218yescomplemented
53#1590C2218yescomplemented
54#1588C2218yescomplemented
55#1587C2218yescomplemented
56#1587C2218yescomplemented
57#1585C2218yescomplemented
58C8×Q16C2218yescomplemented
59C8×SD16C2218yescomplemented
60#1573C2214yescomplemented
61#1573C2214yescomplemented
62#1568C2214yes—
63#1494C2214yes—
64#1491C2214yes—
65#1491C2212yes—
66#1491C2212yescomplemented
67#1491C2212yescomplemented
68#1490C2212yes—
69#1483C2214yes—
70C2×Q8⋊C8C2214yescomplemented
71#1450C2214yescomplemented
72C8×M16C2214yes—
73C8×M16C2214yescomplemented
74C23×Q8C2×C411yescharacteristic
75C22×Q16C2×C412yes—
76C22×SD16C2×C412yes—
77C22×M16—21no—
78#529—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82#529C2×C412yes—
83#529C2×C412yes—
84#529C2312yes—
85#529C2312yes—
86C2×#126—21no—
87C2×#125C2×C411yescharacteristic
88C2×#124C2×C411yescharacteristic
89C2×#123—21no—
90C2×C22⋊Q8C2×C411yescharacteristic
91C2×#121C2×C411yescharacteristic
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8C2×C412yes—
99C2×C4×Q8C2×C411yescharacteristic
100C2×C4×Q8C2×C412yes—
101C2×C4×Q8C2×C411yescharacteristic
102C2×C4×Q8C2311yescharacteristic
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×D4—21no—
108C2×C4×D4C2311yescharacteristic
109C2×#117—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2×C411yescharacteristic
112C22×C42—21no—
113C22×C42C2311yescharacteristic
114#483C2×C412yes—
115#483C2×C414yes—
116#483C2×C412yes—
117#480C2×C412yes—
118#480C2×C414yes—
119#480C2×C412yes—
120#479C2×C414yes—
121#479C2×C414yes—
122#479C2×C414yes—
123#479C2×C414yes—
124#464C2×C414yes—
125#464C2×C414yes—
126#464C2×C414yes—
127#464C2×C414yes—
128#451C2×C412yes—
129#451C2×C414yes—
130#451C2×C412yes—
131#448C2×C412yes—
132#448C2×C414yes—
133#448C2×C412yes—
134#446—28no—
135#446—28no—
136#446—24no—
137#446—24no—
138#446C2318yes—
139#446C2314yes—
140C8×Q8—24no—
141C8×Q8C2318yes—
142C8×Q8C2314yes—
143C4×Q16C2×C414yes—
144C4×Q16C2×C414yes—
145C4×Q16C2314yes—
146C4×Q16C2314yes—
147C4×SD16C2×C414yes—
148C4×SD16C2×C414yes—
149C4×SD16C2314yes—
150C4×SD16C2314yes—
151#436C2314yes—
152#435—24no—
153#435—24no—
154#435—24no—
155C8×D4—24no—
156C8×D4—24no—
157C8×D4C2314yes—
158#433—24no—
159#433—24no—
160#432—21no—
161#432—21no—
162#432—21no—
163#432—21no—
164#432—24no—
165#432—24no—
166#432—22no—
167#432C2312yes—
168#432C2312yes—
169#432C2312yes—
170#432C2312yes—
171#431—24no—
172#431—24no—
173#431—22no—
174#431C2×C412yes—
175#431C2×C412yes—
176#431D412yes—
177#431D412yes—
178#431C2312yes—
179#431C2312yes—
180#431C2312yes—
181#431C2312yes—
182C2×#107C2×C412yes—
183C2×#106C2×C412yes—
184#424—24no—
185C2×C4⋊C8—22no—
186C2×C4⋊C8—22no—
187C2×C4⋊C8C2×C412yes—
188C2×C4⋊C8D412yes—
189C2×C4⋊C8C2312yes—
190C2×C4⋊C8C2312yes—
191C2×#103C2×C412yes—
192C2×#103C2×C412yes—
193C2×#103C2×C412yes—
194C2×#102C2×C412yes—
195#407—22no—
196C2×C22⋊C8—21no—
197C2×C22⋊C8—21no—
198C2×C22⋊C8—22no—
199C2×C22⋊C8C2×C411yescharacteristic
200C2×C22⋊C8C2×C411yescharacteristic
201C2×C22⋊C8D411yescharacteristic
202C2×C22⋊C8D411yescharacteristic
203#405—24no—
204C4×M16—21no—
205C4×M16—21no—
206C4×M16C2312yes—
207C4×M16C2312yes—
208C2×#97—22no—
209C2×#97—21no—
210C2×#97—21no—
211C2×#97—21no—
212C2×C4×C8—22no—
213C2×C4×C8C2312yes—
214C2×C4×C8C2312yes—
215C2×C4×C8C2312yes—
216#335C2318yes—
217#334C2318yes—
218Q8⋊C8C2318yes—
219Q8⋊C8C2318yes—
220Q8⋊C8C2318yes—
221D4⋊C8C2318yes—
222#322C2318yes—
223C82C2318yes—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8—44no—
228C22×Q8C22×C411yescharacteristic
229C22×Q8C22×C412yes—
230C22×Q8C22×C412yes—
231C22×Q8C22×C411yescharacteristic
232C22×Q8C22×C411yescharacteristic
233C22×D4—21no—
234C22×D4C22×C411yescharacteristic
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C22×C411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4#4111yescharacteristic
241C2×Q16—44no—
242C2×Q16C22×C414yes—
243C2×Q16C22×C414yes—
244C2×SD16—44no—
245C2×SD16C22×C414yes—
246C2×SD16C22×C414yes—
247C2×M16—28no—
248C2×M16—22no—
249C2×M16—22no—
250C2×M16—24no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—24no—
255C22×C8C22×C412yes—
256C22×C8C22×C412yes—
257C22×C8C2×D412yes—
258C22×C8C2×D412yes—
259#126—22no—
260#126—22no—
261#126—22no—
262#126—22no—
263#125—21no—
264#125—21no—
265#125C22×C412yes—
266#125C22×C412yes—
267#124—21no—
268#124—21no—
269#124C22×C412yes—
270#124C22×C412yes—
271#123—22no—
272#123—22no—
273#123—22no—
274#123—22no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8C22×C412yes—
278C22⋊Q8C22×C412yes—
279#121—21no—
280#121—21no—
281#121C22×C412yes—
282#121C22×C412yes—
283C4×Q8—21no—
284C4×Q8—22no—
285C4×Q8—21no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—21no—
289C4×Q8—22no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8C22×C412yes—
302C4×Q8C22×C412yes—
303C4×Q8C22×C412yes—
304C4×Q8C22×C412yes—
305C4×Q8C22×C412yes—
306C4×Q8C22×C412yes—
307C4×Q8C22×C412yes—
308C4×Q8C2412yes—
309C4×Q8C2412yes—
310C4×Q8C2412yes—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4C22×C412yes—
318C4×D4C2412yes—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C22×C412yes—
339C2×C4⋊C4C22×C411yescharacteristic
340C2×C4⋊C4C22×C411yescharacteristic
341C2×C4⋊C4C22×C412yes—
342C2×C4⋊C4C22×C411yescharacteristic
343C2×C4⋊C4C22×C411yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42C22×C411yescharacteristic
359C2×C42C22×C411yescharacteristic
360C2×C42C22×C411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42#4111yescharacteristic
365C2×C42#4111yescharacteristic
366C2×C42#4111yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107C22×C414yes—
371#107C22×C414yes—
372#106C22×C414yes—
373#106C22×C414yes—
374C4⋊C8—28no—
375C4⋊C8—28no—
376C4⋊C8—24no—
377C4⋊C8—24no—
378C4⋊C8—22no—
379C4⋊C8—22no—
380C4⋊C8C22×C414yes—
381C4⋊C8C22×C414yes—
382C4⋊C8C2×D414yes—
383C4⋊C8C2×D414yes—
384C4⋊C8C2414yes—
385C4⋊C8C2414yes—
386#103C22×C414yes—
387#103C22×C414yes—
388#103C22×C414yes—
389#103C22×C414yes—
390#103C22×C414yes—
391#103C22×C414yes—
392#102C22×C414yes—
393#102C22×C414yes—
394C22⋊C8—22no—
395C22⋊C8—22no—
396C22⋊C8—22no—
397C22⋊C8—22no—
398C22⋊C8—24no—
399C22⋊C8—24no—
400C22⋊C8C22×C412yes—
401C22⋊C8C22×C412yes—
402C22⋊C8C22×C412yes—
403C22⋊C8C22×C412yes—
404C22⋊C8C2×D412yes—
405C22⋊C8C2×D412yes—
406C22⋊C8C2×D412yes—
407C22⋊C8C2×D412yes—
408#97—28no—
409#97—24no—
410#97—22no—
411#97—22no—
412#97—22no—
413#97—22no—
414#97—22no—
415#97—22no—
416C4×C8—28no—
417C4×C8—24no—
418C4×C8—22no—
419C4×C8C22×C414yes—
420C4×C8C2×D414yes—
421C4×C8C2414yes—
422C4×C8C2414yes—
423C4×C8C2414yes—
424C24—21no—
425C24C4×D411yescharacteristic
426C2×Q8—24no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8C23×C412yes—
442C2×Q8C23×C412yes—
443C2×Q8C23×C412yes—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4C23×C412yes—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—82no—
469C22×C4C4×D411yescharacteristic
470C22×C4C4×D411yescharacteristic
471C22×C4C4×D411yescharacteristic
472C22×C4C4×D411yescharacteristic
473C22×C4C4×D411yescharacteristic
474C22×C4C4×D411yescharacteristic
475C22×C4C4×D411yescharacteristic
476C22×C4C4×D411yescharacteristic
477C22×C4C4×D411yescharacteristic
478C22×C4C23×C411yescharacteristic
479C22×C4C23×C411yescharacteristic
480C22×C4C23×C411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4111yescharacteristic
485C22×C4C2×#4111yescharacteristic
486C22×C4C2×#4111yescharacteristic
487Q16—28no—
488Q16—44no—
489SD16—28no—
490SD16—44no—
491M16—28no—
492M16—28no—
493M16—84no—
494C2×C8—28no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8—42no—
501C2×C8—42no—
502C2×C8—44no—
503C2×C8—84no—
504C2×C8—82no—
505C2×C8C23×C414yes—
506C2×C8C23×C414yes—
507C2×C8C22×D414yes—
508C2×C8C22×D414yes—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4C23×C412yes—
532C4⋊C4C23×C412yes—
533C4⋊C4C23×C412yes—
534C4⋊C4C23×C412yes—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42C4×D411yescharacteristic
556C42C4×D411yescharacteristic
557C42C4×D411yescharacteristic
558C42C4×D411yescharacteristic
559C42C4×D411yescharacteristic
560C42C4×D411yescharacteristic
561C42C23×C411yescharacteristic
562C42C23×C411yescharacteristic
563C42C23×C411yescharacteristic
564C42C22×D411yescharacteristic
565C42C22×D411yescharacteristic
566C42C22×D411yescharacteristic
567C42C2×#4111yescharacteristic
568C42C2×#4111yescharacteristic
569C42C2×#4111yescharacteristic
570C42C2511yescharacteristic
571C23—22no—
572C23—41no—
573C23—41no—
574C23—42no—
575C23—81no—
576C23—81no—
577C23#44314yes—
578C23C2×C4×D411yescharacteristic
579C23C2×C4×D411yescharacteristic
580C23C2×C4×D411yescharacteristic
581Q8—24no—
582Q8—22no—
583Q8—22no—
584Q8—24no—
585Q8—22no—
586Q8—44no—
587Q8—44no—
588Q8—42no—
589Q8—81no—
590Q8#56814yes—
591D4—24no—
592D4—44no—
593D4—44no—
594D4—81no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—82no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C2×C4×D411yescharacteristic
637C2×C4C2×C4×D411yescharacteristic
638C2×C4C2×C4×D411yescharacteristic
639C2×C4C2×C4×D411yescharacteristic
640C2×C4C2×C4×D411yescharacteristic
641C2×C4C2×C4×D411yescharacteristic
642C2×C4C24×C411yescharacteristic
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C8—28no—
646C8—24no—
647C8—24no—
648C8—24no—
649C8—24no—
650C8—44no—
651C8—44no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—162no—
663C22C2×#44312yes—
664C22C2×#44312yes—
665C22C2×#44312yes—
666C22C22×C4×D411yescharacteristic
667C4—21no—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—24no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4C2×#56812yes—
685C4#358612yes—
686C2—22no—
687C2—22no—
688C2—22no—
689C2—82no—
690C2#6014111yescharacteristic
691C2#6016911yescharacteristic
692C2C22×#44311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2984C2semidirect
4. #18058C2semidirect
5. #18056C2semidirect
6. #18017C2semidirect
7. #17943C2semidirect
8. #17911C2semidirect
9. #17818C2semidirect
11. C2×#1491C2semidirect
12. #17314C2semidirect
33. #2984C22semidirect
40. C2×#446C22semidirect
41. C2×C8×Q8C22semidirect
52. #1591C22semidirect
53. #1590C22semidirect
54. #1588C22semidirect
55. #1587C22semidirect
56. #1587C22semidirect
57. #1585C22semidirect
58. C8×Q16C22semidirect
59. C8×SD16C22semidirect
60. #1573C22semidirect
61. #1573C22semidirect
66. #1491C22semidirect
67. #1491C22semidirect
70. C2×Q8⋊C8C22semidirect
71. #1450C22semidirect
73. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31082
#602882
C2×#29842
#180582
#180562
#180172
#179432
#179112
#178182
#175022
C2×#14912
#173142