Finite Group Databasebeta

← #6386157

Subgroups — #6386157

Order 512.

3364 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#2974C211yesmaximal, characteristic
3#34194C211yesmaximal, characteristic
4#22502C214yesmaximal, complemented
5#22497C218yesmaximal, complemented
6#22487C214yesmaximal, complemented
7#21840C211yesmaximal, characteristic, complemented
8#21833C211yesmaximal, characteristic, complemented
9#21765C212yesmaximal, complemented
10#21762C212yesmaximal, complemented
11#20740C211yesmaximal, characteristic, complemented
12#20735C211yesmaximal, characteristic, complemented
13#20700C212yesmaximal, complemented
14C2×#1823C212yesmaximal, complemented
15C2×#554—28no—
16C2×#554—22no—
17C2×#554—21no—
18C2×#554—21no—
19C2×#554C4116yescomplemented
20C2×#554C2212yes—
21C2×#554C2214yes—
22C2×#554C2212yes—
23C22×C4⋊Q8C2211yescharacteristic
24C22×#125—21no—
25C22×C22⋊Q8—22no—
26C22×C22⋊Q8C2212yes—
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4—21no—
29C22×C4×D4C2211yescharacteristic
30C23×C4⋊C4C2212yes—
31#2974—28no—
32C2×#436C2212yes—
33C2×#435C2214yes—
34C2×C8×D4C2212yes—
35C2×#432C2211yescharacteristic
36C2×#431C2211yescharacteristic
37C2×#407C2212yes—
38C22×C22⋊C8C2212yes—
39#2354C2212yes—
40#2352C2214yes—
41D4×C4⋊C4C2212yes—
42#2345—21no—
43#2342—22no—
44D4×C22⋊C4—21no—
45#2319—21no—
46#2318—22no—
47#2285C2211yescharacteristic
48#2283C2211yescharacteristic
49#2280C2212yes—
50C2×#380C2212yes—
51#2275—21no—
52#1993C2214yescomplemented
53#1990C2214yescomplemented
54#1988C2214yescomplemented
55#1987C2214yescomplemented
56#1894C2218yescomplemented
57#1892C2218yescomplemented
58#1873C2218yescomplemented
59#1872C2218yescomplemented
60#1824C2218yescomplemented
61#1823C2212yes—
62#1823C2212yes—
63#1823C2212yes—
64#1823C2212yes—
65#1823C2212yes—
66#1823C2211yescharacteristic
67#1823C2211yescharacteristic
68#1823C2218yescomplemented
69#1805C2214yescomplemented
70#1803C2214yescomplemented
71#1787C2211yescharacteristic
72#1787C2212yes—
73#1787C2211yescharacteristic
74#1763C2214yescomplemented
75C4×#103C2214yescomplemented
76#1736C2218yescomplemented
77C2×#340C2218yescomplemented
78C23×Q8—21no—
79C23×Q8C2311yescharacteristic
80C23×D4D411yescharacteristic
81C24×C4—22no—
82C24×C4C2312yes—
83C22×M16C2314yes—
84C23×C8C2314yes—
85#554—216no—
86#554—232no—
87#554—216no—
88#554—28no—
89#554—216no—
90#554—28no—
91#554C2×C4116yes—
92#554C2×C4132yescomplemented
93#554C2×C4116yescomplemented
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8C2×C418yescomplemented
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2312yes—
100C2×#125—216no—
101C2×#125—22no—
102C2×#125—24no—
103C2×#125—22no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—232no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2×C4116yescomplemented
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2314yes—
115C2×C4×Q8—24no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2×C418yescomplemented
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—216no—
122C2×C4×D4—22no—
123C2×C4×D4—24no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4C2×C418yes—
127C2×C4×D4D418yes—
128C2×C4×D4D412yes—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2312yes—
133C2×C4×D4C2311yescharacteristic
134C2×#117—21no—
135C2×#117C2314yes—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—28no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4—22no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—22no—
143C22×C4⋊C4C2×C4116yescomplemented
144C22×C4⋊C4D411yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2314yes—
149C22×C4⋊C4C2314yes—
150C22×C22⋊C4—22no—
151C22×C22⋊C4—22no—
152C22×C22⋊C4D412yes—
153C22×C42—21no—
154C22×C42C2311yescharacteristic
155#436—28no—
156#435—216no—
157C8×D4—28no—
158#432—24no—
159#431—24no—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2312yes—
162C2×#103C2318yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165#407—28no—
166C2×C22⋊C8—28no—
167C2×C22⋊C8C2311yescharacteristic
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2311yescharacteristic
170C2×C22⋊C8C2314yes—
171C2×C22⋊C8C2314yes—
172C2×#97C2312yes—
173C2×C4×C8C2312yes—
174#390—22no—
175#389—22no—
176#388—24no—
177#388—24no—
178#387—24no—
179#386—24no—
180#385—24no—
181#384C2312yes—
182#383—21no—
183#382—21no—
184#382C2312yes—
185#381—22no—
186#381—21no—
187#380—22no—
188#380—21no—
189#380—22no—
190#380C2311yescharacteristic
191#380C2312yes—
192#380C2311yescharacteristic
193#380C2314yes—
194#380C2314yes—
195#379—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—24no—
204#376—21no—
205C2×#95—22no—
206#340C2318yes—
207#340C2314yes—
208#340C2314yes—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213C25C2×D412yes—
214C22×Q8—28no—
215C22×Q8—22no—
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—44no—
220C22×Q8—82no—
221C22×Q8—82no—
222C22×Q8C22×C414yes—
223C22×Q8C2412yes—
224C22×D4—28no—
225C22×D4C22⋊C418yes—
226C22×D4C2×D412yes—
227C22×D4C2×D411yescharacteristic
228C22×D4C2×D411yescharacteristic
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—21no—
232C23×C4—216no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—48no—
237C23×C4—84no—
238C23×C4—84no—
239C23×C4C22×C418yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D418yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C2×M16—28no—
250C2×M16—48no—
251C22×C8—28no—
252C22×C8—48no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8C22×C418yes—
261C4⋊Q8C22×C418yes—
262#125—216no—
263#125—232no—
264#125—216no—
265C22⋊Q8—216no—
266C22⋊Q8—216no—
267C22⋊Q8—28no—
268C22⋊Q8—28no—
269C22⋊Q8—232no—
270C22⋊Q8—232no—
271C22⋊Q8—232no—
272C22⋊Q8—232no—
273C22⋊Q8C22×C4116yes—
274C22⋊Q8C22×C4116yes—
275C4×Q8—28no—
276C4×Q8—28no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8C22×C418yes—
280C4×Q8C22×C418yes—
281C4×D4—216no—
282C4×D4—232no—
283C4×D4—216no—
284C4×D4—48no—
285C4×D4C22⋊C4116yes—
286C4×D4C22⋊C418yes—
287C4×D4C22⋊C418yes—
288C4×D4C22×C414yes—
289C4×D4C22×C418yes—
290C4×D4C22×C414yes—
291C4×D4C2×D414yes—
292C4×D4C2×D418yes—
293C4×D4C2×D414yes—
294#117—48no—
295#117—82no—
296C2×C4⋊C4—216no—
297C2×C4⋊C4—216no—
298C2×C4⋊C4—216no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—216no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4C22⋊C418yes—
321C2×C4⋊C4C22×C414yes—
322C2×C4⋊C4C22×C418yes—
323C2×C4⋊C4C22×C4116yes—
324C2×C4⋊C4C22×C4116yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C22⋊C4—216no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—84no—
343C2×C22⋊C4C22⋊C4116yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—28no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42C22×C414yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D414yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C4⋊C8—24no—
361C4⋊C8—24no—
362#103—44no—
363#103—48no—
364#103—44no—
365C22⋊C8—22no—
366C22⋊C8—24no—
367C22⋊C8—28no—
368C22⋊C8—22no—
369C22⋊C8—28no—
370#97—24no—
371C4×C8—24no—
372#95—21no—
373#95—22no—
374#95—24no—
375#95—21no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381C24—28no—
382C24—24no—
383C24—24no—
384C24—44no—
385C24C2×C22⋊C418yes—
386C24C22×D412yes—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—48no—
396C2×Q8—44no—
397C2×Q8—48no—
398C2×Q8—816no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8C23×C414yes—
404C2×D4—28no—
405C2×D4—232no—
406C2×D4—24no—
407C2×D4C2×C22⋊C418yes—
408C2×D4C2×C22⋊C414yes—
409C2×D4C2×C22⋊C414yes—
410C2×D4#13718yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—216no—
424C22×C4—216no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—832no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C2×C22⋊C414yes—
447C22×C4C2×C22⋊C414yes—
448C22×C4C2×C22⋊C418yes—
449C22×C4C2×C22⋊C418yes—
450C22×C4C2×C22⋊C418yes—
451C22×C4C23×C412yes—
452C22×C4C23×C414yes—
453C22×C4C23×C414yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C2511yescharacteristic
465M16—88no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—28no—
471C2×C8—24no—
472C2×C8—88no—
473C4⋊C4—24no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—28no—
487C4⋊C4—24no—
488C4⋊C4—44no—
489C4⋊C4—84no—
490C4⋊C4—82no—
491C4⋊C4C2×C22⋊C418yes—
492C4⋊C4C2×C22⋊C414yes—
493C4⋊C4C2×C22⋊C414yes—
494C4⋊C4C23×C414yes—
495C4⋊C4C23×C418yes—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—48no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4C2×C22⋊C418yes—
505C22⋊C4C2×C22⋊C418yes—
506C22⋊C4C2×C22⋊C418yes—
507C22⋊C4C2×C22⋊C418yes—
508C42—28no—
509C42—28no—
510C42—44no—
511C42—84no—
512C42—82no—
513C42C2×C22⋊C414yes—
514C42C2×C22⋊C414yes—
515C42C23×C412yes—
516C42C23×C412yes—
517C42C22×D412yes—
518C42C22×D412yes—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—216no—
525C23—216no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—48no—
531C23—416no—
532C23—416no—
533C23C22×C22⋊C414yes—
534C23C22×C22⋊C414yes—
535C23C22×C22⋊C412yes—
536C23C2×#13718yes—
537C23#57612yes—
538C23#57712yes—
539C23C23×D411yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—48no—
543Q8—44no—
544Q8—88no—
545Q8—88no—
546D4—216no—
547D4—28no—
548D4#419116yes—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—416no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—816no—
589C2×C4—816no—
590C2×C4—168no—
591C2×C4C22×C22⋊C412yes—
592C2×C4C22×C22⋊C412yes—
593C2×C4C22×C22⋊C412yes—
594C2×C4C22×C22⋊C412yes—
595C2×C4C22×C22⋊C414yes—
596C2×C4C22×C22⋊C414yes—
597C2×C4C22×C22⋊C412yes—
598C2×C4C2×#13714yes—
599C2×C4C24×C411yescharacteristic
600C2×C4C23×D411yescharacteristic
601C2×C4#58514yes—
602C8—42no—
603C8—44no—
604C8—42no—
605C22—28no—
606C22—24no—
607C22—24no—
608C22—28no—
609C22—22no—
610C22—24no—
611C22—22no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—432no—
617C22—44no—
618C22—44no—
619C22C2×#41918yes—
620C22#289714yes—
621C22C23×C22⋊C411yescharacteristic
622C22#348512yes—
623C22C22×#13712yes—
624C22C2×#57611yescharacteristic
625C22C2×#57711yescharacteristic
626C4—22no—
627C4—24no—
628C4—24no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—44no—
633C4—44no—
634C4—48no—
635C4—84no—
636C4—82no—
637C4—82no—
638C4C2×#41914yes—
639C4#342812yes—
640C4#348512yes—
641C2—24no—
642C2—28no—
643C2—44no—
644C2#3417211yescharacteristic
645C2C22×#41911yescharacteristic
646C2C2×#289711yescharacteristic
647C2#6172612yes—
648C2#6173512yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22502C2semidirect
5. #22497C2semidirect
6. #22487C2semidirect
7. #21840C2semidirect
8. #21833C2semidirect
9. #21765C2semidirect
10. #21762C2semidirect
11. #20740C2semidirect
12. #20735C2semidirect
13. #20700C2semidirect
14. C2×#1823C2semidirect
19. C2×#554C4semidirect
52. #1993C22semidirect
53. #1990C22semidirect
54. #1988C22semidirect
55. #1987C22semidirect
56. #1894C22semidirect
57. #1892C22semidirect
58. #1873C22semidirect
59. #1872C22semidirect
60. #1824C22semidirect
68. #1823C22semidirect
69. #1805C22semidirect
70. #1803C22semidirect
74. #1763C22semidirect
75. C4×#103C22semidirect
76. #1736C22semidirect
77. C2×#340C22semidirect
92. #554C2×C4semidirect
93. #554C2×C4semidirect
97. C2×C4⋊Q8C2×C4semidirect
112. C2×C22⋊Q8C2×C4semidirect
118. C2×C4×Q8C2×C4semidirect
143. C22×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#29742
#341942
#225022
#224972
#224872
#218402
#218332
#217652
#217622
#207402
#207352
#207002
C2×#18232