Finite Group Databasebeta

← #188219

Subgroups — #188219

Order 512.

2400 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×#41C211yesmaximal, characteristic
2#21373C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic, complemented
5C2×#1481C211yesmaximal, characteristic, complemented
6C2×#1487C211yesmaximal, characteristic, complemented
7C22×Q8⋊C8C211yesmaximal, characteristic, complemented
8#8314C214yesmaximal, complemented
9#8311C214yesmaximal, complemented
10Q8×#41—24no—
11Q8×#41—24no—
12Q8×#41—24no—
13Q8×#41—24no—
14C2×Q82C411yescharacteristic
15C2×#557—21no—
16C2×#557C411yescharacteristic
17C2×#554—22no—
18C2×#554—21no—
19C2×#554C411yescharacteristic
20C2×#554C412yes—
21C2×D4×Q8C412yes—
22C2×D4×Q8C411yescharacteristic
23C2×#533—22no—
24C2×#533—21no—
25C2×#533C2212yes—
26C2×#533C2211yescharacteristic
27C2×C4×#41—21no—
28C2×C4×#41C2211yescharacteristic
29C22×C4×Q8C2212yes—
30C22×C4×Q8C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#423C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C2×#408—21no—
35C22×C4×C8C2211yescharacteristic
36#1984—21no—
37#1954—21no—
38#1869—22no—
39#1798—22no—
40#1797C2214yes—
41C4×C22⋊C8C2214yes—
42#1729—21no—
43#1493—24no—
44#1487—24no—
45#1481—24no—
46C2×Q8⋊C8—24no—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2214yes—
52C2×Q8⋊C8C2214yes—
53C22×#41D411yescharacteristic
54C23×Q8C2×C412yes—
55C23×Q8C2×C411yescharacteristic
56C22×M16—21no—
57C23×C8—21no—
58C23×C8—21no—
59Q82—24no—
60Q82—24no—
61#557—28no—
62#557—28no—
63#557—24no—
64#557—24no—
65#554—28no—
66#554—28no—
67#554—216no—
68#554—216no—
69#554—24no—
70#554—28no—
71#554—24no—
72#554—28no—
73D4×Q8—28no—
74D4×Q8—24no—
75D4×Q8—28no—
76D4×Q8—24no—
77#533—216no—
78#533—216no—
79#533—28no—
80#533—28no—
81#533—24no—
82#533—24no—
83#533—24no—
84#533—24no—
85#533—22no—
86#533—22no—
87#533D414yes—
88#533D414yes—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8C2×C412yes—
94C2×C4⋊Q8C2×C411yescharacteristic
95C2×C4⋊Q8C2×C412yes—
96C2×C4⋊Q8C2×C411yescharacteristic
97C2×#125—22no—
98C2×#125—21no—
99C2×#125—22no—
100C2×#125—21no—
101C2×#125—22no—
102C2×#125—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2×C412yes—
112C2×C22⋊Q8C2×C412yes—
113C2×C22⋊Q8C2×C412yes—
114C4×#41—28no—
115C4×#41—28no—
116C4×#41—22no—
117C4×#41—22no—
118C4×#41—22no—
119C4×#41—22no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—24no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8—22no—
127C2×C4×Q8—21no—
128C2×C4×Q8—22no—
129C2×C4×Q8—21no—
130C2×C4×Q8—22no—
131C2×C4×Q8—22no—
132C2×C4×Q8—22no—
133C2×C4×Q8—21no—
134C2×C4×Q8C2×C412yes—
135C2×C4×Q8C2×C412yes—
136C2×C4×Q8C2×C411yescharacteristic
137C2×C4×Q8C2×C411yescharacteristic
138C2×C4×Q8C2×C411yescharacteristic
139C2×C4×Q8D414yes—
140C2×C4×Q8D414yes—
141C2×C4×Q8D411yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2311yescharacteristic
145C2×C4×D4—22no—
146C2×C4×D4—21no—
147C2×C4×D4—22no—
148C2×C4×D4—21no—
149C2×C4×D4C2×C412yes—
150C2×C4×D4C2×C411yescharacteristic
151C2×C4×D4D412yes—
152C2×C4×D4D411yescharacteristic
153C2×#117—22no—
154C2×#117—21no—
155C2×#117D412yes—
156C2×#117D411yescharacteristic
157C22×C4⋊C4—22no—
158C22×C4⋊C4—21no—
159C22×C4⋊C4C2×C412yes—
160C22×C4⋊C4C2×C411yescharacteristic
161C22×C42—22no—
162C22×C42—21no—
163C22×C42C2312yes—
164C22×C42C2311yescharacteristic
165#431—22no—
166#431—22no—
167#423—22no—
168#423—22no—
169C2×C4⋊C8—22no—
170C2×C4⋊C8—22no—
171C2×C4⋊C8—21no—
172C2×C4⋊C8—21no—
173C2×C4⋊C8—21no—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176#408—44no—
177C2×C22⋊C8—24no—
178C2×C22⋊C8—21no—
179C2×C22⋊C8—24no—
180C2×C4×C8—22no—
181C2×C4×C8—22no—
182C2×C4×C8—21no—
183C2×C4×C8C2312yes—
184C2×C4×C8C2312yes—
185#336—24no—
186#336—24no—
187Q8⋊C8—24no—
188Q8⋊C8—24no—
189Q8⋊C8—28no—
190Q8⋊C8—24no—
191Q8⋊C8—24no—
192Q8⋊C8—28no—
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—22no—
196C2×#41—22no—
197C2×#41SD1614yes—
198C2×#41Q1614yes—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—21no—
206C22×Q8—24no—
207C22×Q8—28no—
208C22×Q8—42no—
209C22×Q8—44no—
210C22×Q8C22⋊C414yes—
211C22×Q8C22⋊C411yescharacteristic
212C22×Q8C22×C412yes—
213C22×Q8C22×C411yescharacteristic
214C22×Q8C22×C411yescharacteristic
215C22×D4C22⋊C412yes—
216C22×D4C22⋊C411yescharacteristic
217C23×C4—22no—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—21no—
221C23×C4C22×C412yes—
222C23×C4C22×C411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C2×M16—42no—
228C2×M16—42no—
229C2×M16—42no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—44no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8—42no—
238C22×C8—42no—
239C4⋊Q8—216no—
240C4⋊Q8—28no—
241C4⋊Q8—216no—
242C4⋊Q8—28no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8C22⋊C414yes—
250C4⋊Q8C22⋊C414yes—
251#125—216no—
252#125—216no—
253#125—28no—
254#125—28no—
255#125—28no—
256#125—28no—
257#125—24no—
258#125—24no—
259C22⋊Q8—216no—
260C22⋊Q8—216no—
261C22⋊Q8—216no—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264C22⋊Q8—216no—
265C22⋊Q8—216no—
266C22⋊Q8—216no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8C22⋊C418yes—
275C4×Q8—28no—
276C4×Q8—28no—
277C4×Q8—28no—
278C4×Q8—28no—
279C4×Q8—28no—
280C4×Q8—28no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8C22⋊C414yes—
306C4×Q8C22⋊C414yes—
307C4×Q8C2×D412yes—
308C4×Q8C2×D412yes—
309C4×Q8C2×D414yes—
310C4×Q8C2×D414yes—
311C4×Q8C2×D412yes—
312C4×Q8C2×D412yes—
313C4×D4—216no—
314C4×D4—28no—
315C4×D4—216no—
316C4×D4—28no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325#117—28no—
326#117—28no—
327#117—24no—
328#117—24no—
329#117—22no—
330#117—22no—
331#117—24no—
332#117—24no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4C22⋊C411yescharacteristic
359C2×C4⋊C4C22⋊C412yes—
360C2×C4⋊C4C22⋊C411yescharacteristic
361C2×C4⋊C4C22⋊C412yes—
362C2×C4⋊C4C22⋊C414yes—
363C2×C4⋊C4C22×C411yescharacteristic
364C2×C4⋊C4C22×C412yes—
365C2×C4⋊C4C22×C411yescharacteristic
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4C22⋊C412yes—
370C2×C22⋊C4C22⋊C412yes—
371C2×C22⋊C4C22⋊C412yes—
372C2×C42—28no—
373C2×C42—28no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—24no—
379C2×C42—24no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42C22×C412yes—
388C2×C42C22×C411yescharacteristic
389C2×C42C2×D412yes—
390C2×C42C2×D412yes—
391C2×C42C2×D412yes—
392C2×C42C2×D412yes—
393C2×C42C2×D412yes—
394C2×C42C2×D412yes—
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2411yescharacteristic
398C4⋊C8—22no—
399C4⋊C8—22no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402C4⋊C8—42no—
403C4⋊C8—42no—
404C4⋊C8—42no—
405C22⋊C8—48no—
406C22⋊C8—42no—
407C22⋊C8—48no—
408C4×C8—22no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—22no—
412C4×C8—42no—
413C24C2×C22⋊C412yes—
414C24C2×C22⋊C411yescharacteristic
415#41—28no—
416#41—28no—
417#41—28no—
418#41—28no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—28no—
430C2×Q8—22no—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—44no—
435C2×Q8—48no—
436C2×Q8—44no—
437C2×Q8—48no—
438C2×Q8—416no—
439C2×Q8—48no—
440C2×Q8—48no—
441C2×Q8—82no—
442C2×Q8—84no—
443C2×Q8#10314yes—
444C2×Q8C2×C22⋊C414yes—
445C2×Q8C2×C22⋊C412yes—
446C2×Q8C2×C22⋊C412yes—
447C2×Q8#13114yes—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4#10318yes—
453C2×D4#10314yes—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—28no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—28no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—48no—
480C22×C4—82no—
481C22×C4—84no—
482C22×C4C2×C22⋊C412yes—
483C22×C4C2×C22⋊C412yes—
484C22×C4C2×C22⋊C412yes—
485C22×C4C2×C22⋊C412yes—
486C22×C4C2×C22⋊C411yescharacteristic
487C22×C4C2×C22⋊C411yescharacteristic
488C22×C4C2×C22⋊C412yes—
489C22×C4C2×C22⋊C411yescharacteristic
490C22×C4#12012yes—
491C22×C4#12012yes—
492C22×C4#12012yes—
493C22×C4#12012yes—
494C22×C4C2×SD1614yes—
495C22×C4C2×SD1612yes—
496C22×C4C2×Q1614yes—
497C22×C4C2×Q1612yes—
498C22×C4#13712yes—
499C22×C4#13712yes—
500C22×C4C23×C411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503M16—84no—
504C2×C8—24no—
505C2×C8—42no—
506C2×C8—48no—
507C2×C8—42no—
508C2×C8—44no—
509C2×C8—42no—
510C2×C8—42no—
511C2×C8—42no—
512C2×C8—42no—
513C2×C8—42no—
514C2×C8—42no—
515C2×C8—42no—
516C2×C8—84no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—28no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4C2×C22⋊C412yes—
546C4⋊C4C2×C22⋊C414yes—
547C4⋊C4C2×C22⋊C412yes—
548C22⋊C4—28no—
549C22⋊C4—28no—
550C22⋊C4—28no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—24no—
554C42—28no—
555C42—28no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—24no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—24no—
569C42—24no—
570C42C2×C22⋊C412yes—
571C42C2×C22⋊C412yes—
572C42#12012yes—
573C42#12012yes—
574C42#12012yes—
575C42#12012yes—
576C42C22×D411yescharacteristic
577C42C22×D411yescharacteristic
578C42C22×D411yescharacteristic
579C42C22×D411yescharacteristic
580C23—22no—
581C23—22no—
582C23—24no—
583C23—22no—
584C23—24no—
585C23—22no—
586C23#37912yes—
587C23C2×#10314yes—
588C23C2×#10312yes—
589C23#41912yes—
590C23C22×C22⋊C411yescharacteristic
591Q8—22no—
592Q8—24no—
593Q8—28no—
594Q8—22no—
595Q8—22no—
596Q8—28no—
597Q8—48no—
598Q8—416no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—84no—
603Q8—88no—
604Q8#40814yescomplemented
605D4—216no—
606D4—28no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—48no—
633C2×C4—416no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—84no—
645C2×C4—88no—
646C2×C4#37912yes—
647C2×C4#37912yes—
648C2×C4#37912yes—
649C2×C4C2×#10314yes—
650C2×C4C2×#10312yes—
651C2×C4#41912yes—
652C2×C4#44814yes—
653C2×C4#44814yes—
654C2×C4#45114yes—
655C2×C4#45114yes—
656C2×C4C22×C22⋊C411yescharacteristic
657C2×C4C22×C22⋊C411yescharacteristic
658C2×C4C2×#12011yescharacteristic
659C2×C4C2×#12011yescharacteristic
660C2×C4C2×#12011yescharacteristic
661C2×C4C2×#12011yescharacteristic
662C2×C4C2×#13112yes—
663C2×C4#56812yes—
664C2×C4C22×SD1611yescharacteristic
665C2×C4C22×Q1611yescharacteristic
666C2×C4C2×#13711yescharacteristic
667C2×C4C2×#13711yescharacteristic
668C8—44no—
669C8—82no—
670C8—82no—
671C8—82no—
672C22—24no—
673C22—22no—
674C22—24no—
675C22—22no—
676C22—28no—
677C22—24no—
678C22—42no—
679C22—44no—
680C22—42no—
681C22#178714yes—
682C22C2×#37911yescharacteristic
683C22#288012yes—
684C22C22×#10311yescharacteristic
685C22C2×#41911yescharacteristic
686C22#290112yes—
687C4—22no—
688C4—24no—
689C4—24no—
690C4—21no—
691C4—21no—
692C4—22no—
693C4—42no—
694C4—42no—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—44no—
700C4#178714yes—
701C4C2×#40811yescharacteristic
702C4#287911yescharacteristic
703C4#288012yes—
704C4C2×#44811yescharacteristic
705C4C2×#44811yescharacteristic
706C4C2×#45111yescharacteristic
707C4C2×#45111yescharacteristic
708C2—24no—
709C2—44no—
710C2#1750912yes—
711C2#1751412yes—
712C2#2058111yescharacteristic
713C2C2×#178711yescharacteristic
714C2#2060611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1493C2semidirect
5. C2×#1481C2semidirect
6. C2×#1487C2semidirect
7. C22×Q8⋊C8C2semidirect
8. #8314C2semidirect
9. #8311C2semidirect
604. Q8#408semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#412
#213732
#206292
C2×#14932
C2×#14812
C2×#14872
C22×Q8⋊C82
#83142
#83112