Finite Group Databasebeta

← #6385973

Subgroups — #6385973

Order 512.

4448 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2#34178C212yesmaximal, complemented
3#22406C2116yesmaximal, complemented
4#21793C212yesmaximal, complemented
5#21747C214yesmaximal, complemented
6#20722C212yesmaximal, complemented
7#20584C214yesmaximal, complemented
8C23×#41C2212yes—
9C2×#547—24no—
10C2×#547—22no—
11C2×#547—28no—
12C2×#547—22no—
13C2×#547C418yescomplemented
14C2×#547C2218yes—
15C22×C4⋊Q8C411yescharacteristic, complemented
16C22×#123C412yescomplemented
17C22×#123C2212yes—
18C22×C22⋊Q8C412yescomplemented
19C22×#121C411yescharacteristic, complemented
20C22×#121C2211yescharacteristic
21C22×C4×D4C411yescharacteristic, complemented
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C411yescharacteristic, complemented
24C23×C4⋊C4C2211yescharacteristic
25C2×#410C22116yescomplemented
26#2345—28no—
27#2344C22116yescomplemented
28#2316—22no—
29#2313—24no—
30#2280C2214yes—
31#2280C2218yes—
32#2274—22no—
33#1964C22116yescomplemented
34#1899C22132yescomplemented
35#1819C2214yes—
36#1799C2218yescomplemented
37#1782C22116yescomplemented
38#1781C2218yes—
39#1781C2214yes—
40C4×#99C2218yescomplemented
41C22×#41—24no—
42C22×#41—28no—
43C22×#41—24no—
44C22×#41—416no—
45C22×#41C2×C418yes—
46C22×#41C2318yes—
47C23×Q8C2×C412yes—
48C23×D4C2×C412yes—
49C23×D4C2×C411yescharacteristic
50C23×D4C2311yescharacteristic
51C23×D4C2312yes—
52C24×C4C2×C412yes—
53C24×C4C2×C412yes—
54C24×C4C2×C411yescharacteristic
55C24×C4C2312yes—
56C24×C4C2311yescharacteristic
57#547—216no—
58#547—216no—
59#547—232no—
60#547—216no—
61#547—232no—
62#547—216no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—24no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8C2×C418yes—
67C2×#123—24no—
68C2×#123—28no—
69C2×#123—24no—
70C2×#123—24no—
71C2×#123—24no—
72C2×#123—28no—
73C2×#123C2×C4116yes—
74C2×#123C2×C418yes—
75C2×#123C2318yes—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8C2×C4116yes—
80C2×#121—22no—
81C2×#121—24no—
82C2×#121—22no—
83C2×#121—22no—
84C2×#121—24no—
85C2×#121—22no—
86C2×#121C2×C418yes—
87C2×#121C2×C414yes—
88C2×#121C2314yes—
89C2×C4×D4—22no—
90C2×C4×D4—24no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4—24no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4C2×C418yes—
97C2×C4×D4C2×C414yes—
98C2×C4×D4C2314yes—
99C2×#117—24no—
100C2×#117C23116yes—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C4⋊C4C2×C412yes—
107C22×C4⋊C4C2×C418yes—
108C22×C4⋊C4C2×C411yescharacteristic
109C22×C4⋊C4C2×C412yes—
110C22×C4⋊C4C2×C411yescharacteristic
111C22×C4⋊C4C2×C411yescharacteristic
112C22×C4⋊C4C2×C412yes—
113C22×C4⋊C4C2×C414yes—
114C22×C4⋊C4D414yes—
115C22×C4⋊C4D412yes—
116C22×C4⋊C4D418yes—
117C22×C4⋊C4D412yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2×C412yes—
120C22×C22⋊C4C2×C411yescharacteristic
121C22×C22⋊C4C2×C412yes—
122C22×C22⋊C4C2×C412yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2×C411yescharacteristic
125#410—432no—
126C2×#99C23116yes—
127C2×#99C23132yes—
128#389—216no—
129#387—232no—
130#381—24no—
131#381C23116yes—
132#380—28no—
133#380—28no—
134#380—24no—
135#380C2318yes—
136#379C2318yes—
137C4×C4⋊C4—216no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4C2318yes—
142#342C23116yes—
143C25C22×C411yescharacteristic
144C25C22×C412yes—
145C25C2411yescharacteristic
146C2×#41—232no—
147C2×#41—216no—
148C2×#41—216no—
149C2×#41—216no—
150C2×#41—216no—
151C2×#41—416no—
152C2×#41—416no—
153C2×#41—416no—
154C2×#41—432no—
155C2×#41—464no—
156C2×#41—432no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—48no—
160C22×Q8C22×C418yes—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—21no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—44no—
172C22×D4—48no—
173C22×D4—48no—
174C22×D4C22×C412yes—
175C22×D4C22×C418yes—
176C22×D4C22×C414yes—
177C22×D4C22×C414yes—
178C22×D4C2414yes—
179C22×D4C2412yes—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—48no—
190C23×C4—44no—
191C23×C4—48no—
192C23×C4—48no—
193C23×C4—44no—
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C418yes—
196C23×C4C22×C418yes—
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C412yes—
201C23×C4C22×C414yes—
202C23×C4C22×C411yescharacteristic
203C23×C4C22×C412yes—
204C23×C4C22×C412yes—
205C23×C4C22×C412yes—
206C23×C4C22×C414yes—
207C23×C4C2×D412yes—
208C23×C4C2×D414yes—
209C23×C4C2×D412yes—
210C23×C4C2×D414yes—
211C23×C4C2×D418yes—
212C23×C4C2×D414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C4⋊Q8—216no—
216C4⋊Q8—216no—
217#123—216no—
218#123—216no—
219#123—216no—
220#123—216no—
221#123—232no—
222#123—232no—
223C22⋊Q8—232no—
224C22⋊Q8—232no—
225#121—28no—
226#121—28no—
227#121—28no—
228#121—28no—
229#121—216no—
230#121—216no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—216no—
236C4×D4—28no—
237C4×D4—28no—
238#117—416no—
239#117—432no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4C22⋊C412yes—
256C2×C4⋊C4C22⋊C414yes—
257C2×C4⋊C4C22⋊C411yescharacteristic
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C412yes—
260C2×C4⋊C4C22⋊C414yes—
261C2×C4⋊C4C22⋊C412yes—
262C2×C4⋊C4C22⋊C414yes—
263C2×C4⋊C4C22⋊C414yes—
264C2×C4⋊C4C22⋊C418yes—
265C2×C4⋊C4C22⋊C418yes—
266C2×C4⋊C4C22⋊C418yes—
267C2×C4⋊C4C22⋊C4116yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C418yes—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4C22×C414yes—
274C2×C4⋊C4C22×C418yes—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D4116yes—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C22×C414yes—
293C2×C22⋊C4C22×C418yes—
294C2×C22⋊C4C22×C418yes—
295C2×C22⋊C4C22×C412yes—
296C2×C22⋊C4C24116yes—
297C2×C22⋊C4C2412yes—
298C2×C42—21no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42—28no—
302C2×C42—216no—
303C2×C42C22×C414yes—
304#99—232no—
305#99—432no—
306#95—216no—
307#95—216no—
308#95—28no—
309#95—28no—
310C24—21no—
311C24—21no—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—42no—
317C24—42no—
318C24—44no—
319C24—42no—
320C24—42no—
321C24—44no—
322C24—44no—
323C24C23×C411yescharacteristic
324C24C23×C414yes—
325C24C23×C411yescharacteristic
326C24C23×C412yes—
327C24C22×D412yes—
328C24C22×D412yes—
329C24C2511yescharacteristic
330#41—464no—
331#41—464no—
332#41—464no—
333#41—464no—
334C2×Q8—216no—
335C2×Q8—216no—
336C2×Q8—48no—
337C2×Q8—416no—
338C2×Q8—416no—
339C2×D4—216no—
340C2×D4—216no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—28no—
346C2×D4—216no—
347C2×D4—28no—
348C2×D4—216no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—216no—
352C2×D4—24no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—44no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—44no—
363C2×D4H(5,𝔽2)18yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—216no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—216no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—44no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—48no—
397C22×C4—416no—
398C22×C4—48no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—416no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—416no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C414yes—
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C414yes—
419C22×C4C2×C22⋊C414yes—
420C22×C4C2×C22⋊C414yes—
421C22×C4C2×C22⋊C414yes—
422C22×C4C2×C22⋊C418yes—
423C22×C4C2×C22⋊C418yes—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C23×C414yes—
426C22×C4C23×C412yes—
427C22×C4C23×C414yes—
428C22×C4C23×C414yes—
429C22×C4C23×C412yes—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C22×D418yes—
434C22×C4C22×D418yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—216no—
443C4⋊C4—416no—
444C4⋊C4—816no—
445C4⋊C4C2×C22⋊C418yes—
446C4⋊C4C2×C22⋊C414yes—
447C4⋊C4C2×C22⋊C414yes—
448C4⋊C4C2×C22⋊C418yes—
449C4⋊C4C2×C22⋊C418yes—
450C4⋊C4C2×C22⋊C4116yes—
451C4⋊C4C2×C22⋊C4116yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—28no—
457C22⋊C4—216no—
458C22⋊C4—216no—
459C22⋊C4—232no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C42—24no—
463C42—24no—
464C42—416no—
465C42—816no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—41no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—416no—
490C23—416no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—816no—
503C23C22×C22⋊C412yes—
504C23C22×C22⋊C411yescharacteristic
505C23C22×C22⋊C411yescharacteristic
506C23#51814yes—
507C23C24×C411yescharacteristic
508C23C23×D412yes—
509C23C2×H(5,𝔽2)14yes—
510Q8—432no—
511D4—216no—
512D4—216no—
513D4—432no—
514D4—48no—
515D4—416no—
516D4—416no—
517D4—48no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—216no—
527C2×C4—216no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—416no—
548C2×C4—416no—
549C2×C4—416no—
550C2×C4—416no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—416no—
564C2×C4#41014yescomplemented
565C2×C4C22×C22⋊C412yes—
566C2×C4C22×C22⋊C414yes—
567C2×C4C22×C22⋊C414yes—
568C2×C4C22×C22⋊C414yes—
569C2×C4C22×C22⋊C418yes—
570C2×C4C22×C22⋊C412yes—
571C2×C4#51814yes—
572C22—28no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—21no—
577C22—21no—
578C22—22no—
579C22—21no—
580C22—21no—
581C22—22no—
582C22—24no—
583C22—22no—
584C22—22no—
585C22—44no—
586C22—48no—
587C22—48no—
588C22—44no—
589C22—48no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—88no—
599C22—88no—
600C22—88no—
601C22C2×#41012yes—
602C22#288412yes—
603C22C23×C22⋊C411yescharacteristic
604C22C2×#51812yes—
605C22#348414yes—
606C4—24no—
607C4—24no—
608C4—24no—
609C4—28no—
610C4—22no—
611C4—24no—
612C4—44no—
613C4—42no—
614C4—42no—
615C4—816no—
616C4C2×#41014yes—
617C2—24no—
618C2—22no—
619C2—22no—
620C2—22no—
621C2—42no—
622C2—44no—
623C2—84no—
624C2#3417111yescharacteristic
625C2C22×#41011yescharacteristic
626C2C2×#288411yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC22×#547C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34178C2semidirect
3. #22406C2semidirect
4. #21793C2semidirect
5. #21747C2semidirect
6. #20722C2semidirect
7. #20584C2semidirect
13. C2×#547C4semidirect
15. C22×C4⋊Q8C4semidirect
16. C22×#123C4semidirect
18. C22×C22⋊Q8C4semidirect
19. C22×#121C4semidirect
21. C22×C4×D4C4semidirect
23. C23×C4⋊C4C4semidirect
25. C2×#410C22semidirect
27. #2344C22semidirect
33. #1964C22semidirect
34. #1899C22semidirect
36. #1799C22semidirect
37. #1782C22semidirect
40. C4×#99C22semidirect
564. C2×C4#410semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
#341782
#224062
#217932
#217472
#207222
#205842