Finite Group Databasebeta

← #188206

Subgroups — #188206

Order 512.

2598 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×#554C211yesmaximal, characteristic
2#21740C212yesmaximal
3C2×#1472C212yesmaximal, complemented
4C2×#1463C211yesmaximal, characteristic, complemented
5C22×#324C211yesmaximal, characteristic, complemented
6#8308C212yesmaximal, complemented
7#8306C212yesmaximal, complemented
8#8303C212yesmaximal, complemented
9#8299C212yesmaximal, complemented
10C2×#554—28no—
11C2×#554—24no—
12C2×#554—24no—
13C2×#554—24no—
14C2×#554—24no—
15C22×C4⋊Q8C411yescharacteristic
16C22×#125C411yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C411yescharacteristic
19C22×C22⋊Q8C411yescharacteristic
20C22×C22⋊Q8C411yescharacteristic
21C22×C22⋊Q8C411yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C22⋊Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C411yescharacteristic
26C22×C4×D4C411yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C2×#407C2212yes—
31C22×C22⋊C8C2212yes—
32#1870—24no—
33#1869—24no—
34#1817C2214yes—
35#1779C2214yes—
36#1472—28no—
37#1463—24no—
38C2×#324—24no—
39C2×#324C2212yes—
40C2×#324C2212yes—
41C2×#324C2214yes—
42C2×#324C2214yes—
43C2×#324C2212yes—
44C2×#324C2212yes—
45C23×Q8C2×C411yescharacteristic
46C23×Q8C2×C411yescharacteristic
47C23×Q8C2×C411yescharacteristic
48C23×D4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C22×M16—22no—
56C23×C8—22no—
57#554—232no—
58#554—232no—
59#554—232no—
60#554—232no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8—24no—
64C2×#125—24no—
65C2×#125—24no—
66C2×#125—24no—
67C2×#125—24no—
68C2×#125—22no—
69C2×#125—22no—
70C2×#125—24no—
71C2×C22⋊Q8—24no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—24no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8D418yes—
94C2×C4×Q8—24no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—24no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4D414yes—
110C2×C4×D4D414yes—
111C22×C4⋊C4—24no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4C2×C411yescharacteristic
119C22×C4⋊C4C2×C411yescharacteristic
120C22×C4⋊C4C2×C411yescharacteristic
121C22×C4⋊C4C2×C411yescharacteristic
122C22×C4⋊C4C2×C411yescharacteristic
123C22×C4⋊C4C2×C411yescharacteristic
124C22×C4⋊C4C2×C411yescharacteristic
125C22×C4⋊C4C2×C411yescharacteristic
126C22×C4⋊C4C2×C411yescharacteristic
127C22×C4⋊C4C2×C411yescharacteristic
128C22×C4⋊C4C2×C411yescharacteristic
129C22×C4⋊C4C2×C411yescharacteristic
130C22×C4⋊C4D414yes—
131C22×C4⋊C4D414yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4C2×C411yescharacteristic
137C22×C22⋊C4C2×C411yescharacteristic
138C22×C22⋊C4C2×C411yescharacteristic
139C22×C22⋊C4C2×C411yescharacteristic
140C22×C22⋊C4C2×C411yescharacteristic
141C22×C22⋊C4C2×C411yescharacteristic
142C22×C42C2×C411yescharacteristic
143C22×C42C2×C411yescharacteristic
144C22×C42C2311yescharacteristic
145#407—24no—
146#407—24no—
147C2×C22⋊C8—24no—
148C2×C22⋊C8—24no—
149C2×C22⋊C8—22no—
150C2×C22⋊C8—22no—
151C2×C22⋊C8—28no—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154#336—28no—
155#324—24no—
156#324—24no—
157#324—24no—
158#324—24no—
159#324—28no—
160#324—28no—
161C25C22×C411yescharacteristic
162C25C22×C411yescharacteristic
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—42no—
173C22×Q8—82no—
174C22×Q8—82no—
175C22×Q8C22⋊C414yes—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—28no—
179C22×D4C22⋊C412yes—
180C22×D4C22⋊C412yes—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—82no—
201C23×C4—82no—
202C23×C4—82no—
203C23×C4—82no—
204C23×C4C22⋊C414yes—
205C23×C4C22⋊C414yes—
206C23×C4C22×C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C411yescharacteristic
215C23×C4C22×C411yescharacteristic
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C2×D414yes—
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2×D414yes—
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2411yescharacteristic
225C2×M16—44no—
226C2×M16—84no—
227C22×C8—24no—
228C22×C8—24no—
229C22×C8—44no—
230C22×C8—84no—
231C4⋊Q8—232no—
232#125—232no—
233#125—216no—
234#125—216no—
235C22⋊Q8—216no—
236C22⋊Q8—216no—
237C22⋊Q8—232no—
238C22⋊Q8—232no—
239C22⋊Q8—216no—
240C22⋊Q8—216no—
241C22⋊Q8—216no—
242C22⋊Q8—216no—
243C22⋊Q8—28no—
244C22⋊Q8—28no—
245C22⋊Q8—28no—
246C22⋊Q8—28no—
247C4×Q8—216no—
248C4×Q8—216no—
249C4×D4—216no—
250C4×D4—216no—
251C4×D4—216no—
252C4×D4—216no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—28no—
256C4×D4—28no—
257C2×C4⋊C4—216no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
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—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4C22⋊C414yes—
306C2×C4⋊C4C22⋊C414yes—
307C2×C4⋊C4C22⋊C414yes—
308C2×C4⋊C4C22⋊C414yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D414yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4C22⋊C414yes—
332C2×C22⋊C4C22⋊C414yes—
333C2×C22⋊C4C22⋊C412yes—
334C2×C22⋊C4C22⋊C412yes—
335C2×C22⋊C4C22⋊C414yes—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42C2×D414yes—
346C22⋊C8—24no—
347C22⋊C8—24no—
348C22⋊C8—24no—
349C22⋊C8—24no—
350C22⋊C8—44no—
351C22⋊C8—44no—
352C22⋊C8—88no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—42no—
362C24—42no—
363C24C2×C22⋊C411yescharacteristic
364C24C2×C22⋊C412yes—
365C24C2×C22⋊C412yes—
366C24C2×C22⋊C412yes—
367C24C2×C22⋊C411yescharacteristic
368C24C23×C411yescharacteristic
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—84no—
377C2×Q8—84no—
378C2×Q8—84no—
379C2×Q8—84no—
380C2×Q8—84no—
381C2×Q8—84no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4#10118yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—82no—
450C22×C4—84no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—84no—
455C22×C4—82no—
456C22×C4—84no—
457C22×C4—82no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4C2×C22⋊C412yes—
463C22×C4C2×C22⋊C412yes—
464C22×C4C2×C22⋊C412yes—
465C22×C4C2×C22⋊C412yes—
466C22×C4C2×C22⋊C412yes—
467C22×C4C2×C22⋊C412yes—
468C22×C4C2×C22⋊C412yes—
469C22×C4C2×C22⋊C412yes—
470C22×C4C2×C22⋊C412yes—
471C22×C4C2×C22⋊C412yes—
472C22×C4C2×C22⋊C412yes—
473C22×C4C2×C22⋊C412yes—
474C22×C4C2×C22⋊C412yes—
475C22×C4C2×C22⋊C412yes—
476C22×C4#12014yes—
477C22×C4#12014yes—
478C22×C4#12014yes—
479C22×C4#12014yes—
480C22×C4#13114yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485M16—88no—
486C2×C8—48no—
487C2×C8—48no—
488C2×C8—44no—
489C2×C8—44no—
490C2×C8—88no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—28no—
520C42—28no—
521C42—28no—
522C42—24no—
523C42—24no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—48no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23#37912yes—
554C23#37912yes—
555C23C2×#10114yes—
556C23C2×#10114yes—
557C23C22×C22⋊C411yescharacteristic
558C23C22×C22⋊C411yescharacteristic
559C23C22×C22⋊C411yescharacteristic
560C23C2×#13112yes—
561C23#56812yes—
562Q8—416no—
563Q8—88no—
564Q8—88no—
565D4—216no—
566D4—216no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—416no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4#37914yes—
635C2×C4#40814yes—
636C2×C4C2×#10114yes—
637C2×C4C2×#12012yes—
638C2×C4C2×#12012yes—
639C8—84no—
640C8—84no—
641C22—24no—
642C22—24no—
643C22—28no—
644C22—22no—
645C22—21no—
646C22—21no—
647C22—22no—
648C22—21no—
649C22—21no—
650C22—28no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—24no—
655C22—44no—
656C22—44no—
657C22—48no—
658C22—44no—
659C22—44no—
660C22—48no—
661C22—44no—
662C22—44no—
663C22—42no—
664C22—42no—
665C22#178414yes—
666C22C2×#37911yescharacteristic
667C22C2×#40811yescharacteristic
668C22#287911yescharacteristic
669C22#288012yes—
670C22#288012yes—
671C22#288312yes—
672C22C22×#10112yes—
673C4—24no—
674C4—22no—
675C4—22no—
676C4—24no—
677C4—24no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C2—21no—
688C2—21no—
689C2—22no—
690C2—24no—
691C2—44no—
692C2—42no—
693C2#1742614yes—
694C2#2058111yescharacteristic
695C2#2058511yescharacteristic
696C2C2×#178411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1472C2semidirect
4. C2×#1463C2semidirect
5. C22×#324C2semidirect
6. #8308C2semidirect
7. #8306C2semidirect
8. #8303C2semidirect
9. #8299C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
#217402
C2×#14722
C2×#14632
C22×#3242
#83082
#83062
#83032
#82992