Finite Group Databasebeta

← #363882

Subgroups — #363882

Order 512.

2846 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#1899C212yesmaximal
3C2×#1820C211yesmaximal, characteristic, complemented
4C2×#1740C211yesmaximal, characteristic, complemented
5#13870C212yesmaximal, complemented
6#11381C212yesmaximal, complemented
7#10966C212yesmaximal
8#10761C212yesmaximal
9#10726C212yesmaximal, complemented
10C2×#552—28no—
11C2×#552—24no—
12C2×#552—24no—
13C2×#552—24no—
14C2×#552—24no—
15C22×#126C411yescharacteristic, complemented
16C22×#126C411yescharacteristic, complemented
17C22×#125C411yescharacteristic, complemented
18C22×#125C411yescharacteristic, complemented
19C22×#125C411yescharacteristic, complemented
20C22×#123C2211yescharacteristic
21C22×#123C2211yescharacteristic
22C22×C22⋊Q8C411yescharacteristic, complemented
23C22×C22⋊Q8C2211yescharacteristic
24C22×C4×Q8C411yescharacteristic, complemented
25C22×C4×D4C2211yescharacteristic
26C22×#117C411yescharacteristic, complemented
27C22×#117C2212yes—
28C22×#117C2211yescharacteristic
29C22×#117C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31C22×#99C2214yes—
32C2×#381C2212yes—
33C2×#381C2214yes—
34#1899—28no—
35#1820—24no—
36#1740—24no—
37C2×#342C2212yes—
38C2×#342C2212yes—
39C2×#342C2212yes—
40C2×#342C2212yes—
41C2×#342C2214yes—
42#1441—24no—
43#1436—24no—
44C22⋊#95C2214yes—
45C23×Q8C2×C411yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51#552—232no—
52#552—232no—
53#552—232no—
54#552—232no—
55C2×#126—24no—
56C2×#126—24no—
57C2×#126—22no—
58C2×#126—22no—
59C2×#126—24no—
60C2×#126—24no—
61C2×#126—24no—
62C2×#125—24no—
63C2×#125—24no—
64C2×#125—24no—
65C2×#125—24no—
66C2×#125—22no—
67C2×#125—22no—
68C2×#125—24no—
69C2×#125—24no—
70C2×#125—24no—
71C2×#125—24no—
72C2×#123—24no—
73C2×#123—24no—
74C2×#123—22no—
75C2×#123—22no—
76C2×#123—22no—
77C2×#123—22no—
78C2×#123—22no—
79C2×#123—22no—
80C2×#123D418yes—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C4×Q8—24no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×#117—24no—
97C2×#117—22no—
98C2×#117—24no—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117—24no—
107C2×#117—24no—
108C2×#117—24no—
109C2×#117—28no—
110C2×#117—44no—
111C2×#117D414yes—
112C2×#117D414yes—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4C2×C411yescharacteristic
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⋊C4C2×C411yescharacteristic
131C22×C4⋊C4C2×C411yescharacteristic
132C22×C4⋊C4C2×C411yescharacteristic
133C22×C4⋊C4D414yes—
134C22×C4⋊C4D414yes—
135C22×C22⋊C4C2×C411yescharacteristic
136C22×C22⋊C4C2×C411yescharacteristic
137C22×C22⋊C4C2×C411yescharacteristic
138C22×C22⋊C4C2×C411yescharacteristic
139C22×C22⋊C4C2×C411yescharacteristic
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2311yescharacteristic
143C22×C22⋊C4C2311yescharacteristic
144C22×C22⋊C4C2311yescharacteristic
145C22×C42—22no—
146C22×C42C2×C411yescharacteristic
147C22×C42C2×C411yescharacteristic
148C22×C42C2×C411yescharacteristic
149C22×C42C2×C411yescharacteristic
150C22×C42C2×C411yescharacteristic
151C2×#99—28no—
152C2×#99—28no—
153C2×#99—44no—
154#381—24no—
155#381—24no—
156#381—28no—
157#381—44no—
158C2×#95—28no—
159C2×#95—22no—
160C2×#95—28no—
161C2×#95—22no—
162#342—24no—
163#342—24no—
164#342—28no—
165#342—24no—
166#342—24no—
167C25C22×C411yescharacteristic
168C25C2411yescharacteristic
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—44no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—22no—
176C22×D4—24no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4C2×D414yes—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C22×C411yescharacteristic
202C23×C4C22×C411yescharacteristic
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4C2×D414yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219#126—232no—
220#126—216no—
221#126—216no—
222#125—232no—
223#125—232no—
224#125—216no—
225#125—216no—
226#123—216no—
227#123—216no—
228#123—28no—
229#123—28no—
230#123—28no—
231#123—28no—
232C22⋊Q8—232no—
233C22⋊Q8—216no—
234C22⋊Q8—216no—
235C4×Q8—216no—
236C4×Q8—216no—
237C4×D4—216no—
238C4×D4—216no—
239#117—216no—
240#117—216no—
241#117—216no—
242#117—216no—
243#117—28no—
244#117—28no—
245#117—28no—
246#117—28no—
247#117—216no—
248#117—416no—
249#117—48no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
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—21no—
279C2×C4⋊C4—21no—
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—48no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4C22⋊C412yes—
298C2×C4⋊C4C22⋊C412yes—
299C2×C4⋊C4C22⋊C414yes—
300C2×C4⋊C4C22⋊C414yes—
301C2×C4⋊C4C22⋊C412yes—
302C2×C4⋊C4C22⋊C412yes—
303C2×C4⋊C4C22⋊C414yes—
304C2×C4⋊C4C22⋊C414yes—
305C2×C4⋊C4C22⋊C414yes—
306C2×C4⋊C4C22⋊C414yes—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4C22⋊C414yes—
340C2×C22⋊C4C22⋊C414yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D414yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4#4118yes—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—48no—
363C2×C42—44no—
364C2×C42C22⋊C414yes—
365C2×C42C22⋊C414yes—
366#99—416no—
367#95—416no—
368#95—416no—
369#95—44no—
370#95—44no—
371C24—21no—
372C24—21no—
373C24—22no—
374C24—22no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24C2×C22⋊C412yes—
380C24C23×C411yescharacteristic
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383C24C22×D411yescharacteristic
384C24C22×D412yes—
385C24C2×#4114yes—
386C24H(5,𝔽2)12yes—
387C242−1+412yes—
388C2×Q8—28no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—44no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4C2×C22⋊C412yes—
489C22×C4C2×C22⋊C412yes—
490C22×C4C2×C22⋊C412yes—
491C22×C4C2×C22⋊C412yes—
492C22×C4C2×C22⋊C412yes—
493C22×C4C2×C22⋊C412yes—
494C22×C4C2×C22⋊C412yes—
495C22×C4C2×C22⋊C412yes—
496C22×C4C2×C22⋊C411yescharacteristic
497C22×C4C2×C22⋊C411yescharacteristic
498C22×C4C2×C22⋊C412yes—
499C22×C4C2×C22⋊C412yes—
500C22×C4C2×C22⋊C412yes—
501C22×C4C2×C22⋊C412yes—
502C22×C4C2×C22⋊C412yes—
503C22×C4C2×C22⋊C412yes—
504C22×C4C2×C22⋊C412yes—
505C22×C4C2×C22⋊C412yes—
506C22×C4#12014yes—
507C22×C4#12014yes—
508C22×C4#12014yes—
509C22×C4#12014yes—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—28no—
517C4⋊C4—28no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—28no—
529C4⋊C4—416no—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—28no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—28no—
542C22⋊C4—28no—
543C22⋊C4—28no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C42—28no—
551C42—28no—
552C42—28no—
553C42—24no—
554C42—24no—
555C42—24no—
556C42—24no—
557C42—416no—
558C23—24no—
559C23—22no—
560C23—24no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—48no—
571C23—44no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—41no—
582C23—84no—
583C23—84no—
584C23C22×C22⋊C411yescharacteristic
585C23C22×C22⋊C411yescharacteristic
586C23C22×C22⋊C411yescharacteristic
587C23C4×#4112yes—
588C23#52012yes—
589C23C2×#12012yes—
590C23C2×#12012yes—
591C23#52512yes—
592C23#53212yes—
593C23#53511yescharacteristic
594C23#53611yescharacteristic
595C23#53712yes—
596C23#54614yes—
597C23#54614yes—
598C23#54714yes—
599C23#54714yes—
600Q8—416no—
601D4—44no—
602D4—44no—
603D4—48no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—48no—
638C2×C4—48no—
639C2×C4—416no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—48no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—416no—
673C2×C4—48no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—88no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4—88no—
680C2×C4—88no—
681C2×C4—84no—
682C2×C4#37912yes—
683C2×C4#37912yes—
684C2×C4#37914yes—
685C2×C4#41014yes—
686C22—21no—
687C22—21no—
688C22—22no—
689C22—22no—
690C22—21no—
691C22—21no—
692C22—22no—
693C22—22no—
694C22—28no—
695C22—28no—
696C22—24no—
697C22—24no—
698C22—42no—
699C22—42no—
700C22—41no—
701C22—41no—
702C22—42no—
703C22—44no—
704C22—44no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—84no—
709C22—82no—
710C22—82no—
711C22—84no—
712C22C2×#37911yescharacteristic
713C22#229711yescharacteristic
714C22#231111yescharacteristic
715C22#233912yes—
716C22#233912yes—
717C22#240412yes—
718C22#240612yes—
719C22#243014yes—
720C22C2×#41012yes—
721C22#288412yes—
722C4—24no—
723C4—22no—
724C4—24no—
725C4—22no—
726C4—42no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—42no—
736C4—88no—
737C4—164no—
738C2—22no—
739C2—24no—
740C2—21no—
741C2—22no—
742C2—21no—
743C2—22no—
744C2—82no—
745C2#1395611yescharacteristic
746C2#2058411yescharacteristic
747C2#2058711yescharacteristic
748C2#2247814yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1820C2semidirect
4. C2×#1740C2semidirect
5. #13870C2semidirect
6. #11381C2semidirect
9. #10726C2semidirect
15. C22×#126C4semidirect
16. C22×#126C4semidirect
17. C22×#125C4semidirect
18. C22×#125C4semidirect
19. C22×#125C4semidirect
22. C22×C22⋊Q8C4semidirect
24. C22×C4×Q8C4semidirect
26. C22×#117C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#18992
C2×#18202
C2×#17402
#138702
#113812
#109662
#107612
#107262