Finite Group Databasebeta

← #6489552

Subgroups — #6489552

Order 512.

2964 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#3236C211yesmaximal, characteristic, complemented
3C2×#3229C211yesmaximal, characteristic, complemented
4#35325C211yesmaximal, characteristic
5#32984C212yesmaximal
6#32982C212yesmaximal
7#29643C212yesmaximal, complemented
8#29641C212yesmaximal, complemented
9#25914C214yesmaximal, complemented
10#25820C214yesmaximal, complemented
11#24310C218yesmaximal, complemented
12#21840C211yesmaximal, characteristic, complemented
13#21836C211yesmaximal, characteristic, complemented
14#20717C211yesmaximal, characteristic
15C2×#544—28no—
16C2×#544—28no—
17C2×#544—22no—
18C2×#544—22no—
19C2×#544C2214yes—
20C2×#544C2214yes—
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8C2211yescharacteristic
24C22×C4⋊Q8C2211yescharacteristic
25C22×#125—21no—
26C22×#125C2211yescharacteristic
27C22×#125C2211yescharacteristic
28C22×C22⋊Q8—21no—
29C22×C22⋊Q8—21no—
30C22×#117C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3236—28no—
34#3229—28no—
35C2×#501C2212yescomplemented
36C2×#498C2212yescomplemented
37C2×#491C2212yescomplemented
38C2×#487C2212yescomplemented
39C2×#484C2212yes—
40C2×#483C2212yes—
41C2×#483C2212yes—
42C2×#483C2212yes—
43C2×#479C2214yescomplemented
44C2×#475C2214yescomplemented
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2855C2212yes—
48#2848C2212yes—
49#2628C2212yes—
50#2621C2212yes—
51#2611—22no—
52#2599—22no—
53#2583—21no—
54#2561—21no—
55#2484—21no—
56#2482—21no—
57#2323C2214yes—
58#2283C2211yescharacteristic
59#2059C2218yescomplemented
60#2058C2218yescomplemented
61#2026C2218yescomplemented
62#2024C2218yescomplemented
63#1993C2214yescomplemented
64#1991C2214yescomplemented
65#1989C2214yescomplemented
66#1987C2214yescomplemented
67#1873C2218yes—
68#1872C2218yes—
69#1854C2218yes—
70#1853C2218yes—
71#1824C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1823C2212yes—
75#1817C2212yes—
76#1798C2214yes—
77#1779C2212yes—
78#1750C2214yes—
79C23×Q8—21no—
80C23×Q8—21no—
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83#544—232no—
84#544—232no—
85#544—216no—
86#544—216no—
87#544—216no—
88#544—216no—
89#533—24no—
90#533—24no—
91C2×C4⋊Q8—216no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8D418yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2314yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—216no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—22no—
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—24no—
115C2×#125—21no—
116C2×#125D418yes—
117C2×#125C2314yes—
118C2×#125C2312yes—
119C2×#125C2312yes—
120C2×C22⋊Q8—216no—
121C2×C22⋊Q8—216no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—24no—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8C2312yes—
128C2×C22⋊Q8C2314yes—
129C2×C4×Q8—21no—
130C2×C4×Q8C2312yes—
131C2×#117—22no—
132C2×#117—21no—
133C2×#117—24no—
134C2×#117—24no—
135C2×#117—24no—
136C2×#117—24no—
137C2×#117D412yes—
138C2×#117D412yes—
139C2×#117C2314yes—
140C2×#117C2312yes—
141C2×#117C2312yes—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—22no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—24no—
151C22×C4⋊C4—24no—
152C22×C4⋊C4D411yescharacteristic
153C22×C4⋊C4D411yescharacteristic
154C22×C4⋊C4C2314yes—
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C22⋊C4—21no—
160C22×C22⋊C4D411yescharacteristic
161C22×C22⋊C4D411yescharacteristic
162C22×C42—21no—
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165#501—28no—
166#498—28no—
167#491—28no—
168#487—28no—
169#484—28no—
170#483—28no—
171#483—28no—
172#483—28no—
173#479—216no—
174#475—216no—
175#432—24no—
176#431—24no—
177C2×#107C2314yes—
178C2×#106C2314yes—
179C2×#106C2314yes—
180C2×#106C2314yes—
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×C22⋊C8C2318yes—
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2312yes—
190C2×#97C2312yes—
191C2×C4×C8C2312yes—
192#400—22no—
193#399—22no—
194#398—22no—
195#398—22no—
196#397—22no—
197#397—22no—
198#395—22no—
199#393—22no—
200#391C2314yes—
201#389—24no—
202#389C2314yes—
203#388—24no—
204#388—24no—
205#387—24no—
206#387—24no—
207#385—22no—
208#385—22no—
209#385—24no—
210#383—21no—
211#382C2312yes—
212#381—22no—
213#381—21no—
214#380C2314yes—
215#380C2312yes—
216#379—22no—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219#376—21no—
220#340C2318yes—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25C2×D411yescharacteristic
226C22×Q8—28no—
227C22×Q8—28no—
228C22×Q8—24no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—21no—
232C22×Q8—84no—
233C22×Q8—84no—
234C22×Q8C2412yes—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—28no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—28no—
243C23×C4—24no—
244C23×C4—84no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2412yes—
253C22×C8C2414yes—
254C22×C8C2414yes—
255C4⋊Q8—232no—
256C4⋊Q8—232no—
257C4⋊Q8—216no—
258C4⋊Q8—28no—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8C2×D418yes—
269C4⋊Q8C2×D418yes—
270#125—232no—
271#125—232no—
272#125—216no—
273#125—28no—
274#125—28no—
275#125—28no—
276#125—28no—
277#125—24no—
278#125—24no—
279#125—44no—
280#125C2×D418yes—
281#125C2×D418yes—
282C22⋊Q8—232no—
283C22⋊Q8—232no—
284C22⋊Q8—232no—
285C22⋊Q8—232no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—216no—
289C22⋊Q8—44no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—44no—
293#117—216no—
294#117—28no—
295#117—28no—
296#117—28no—
297#117—28no—
298#117—24no—
299#117—24no—
300#117—24no—
301#117—24no—
302#117—44no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—216no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2×D414yes—
346C2×C4⋊C4C2×D414yes—
347C2×C4⋊C4C2×D414yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2414yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—28no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C42—28no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—24no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42C2×D412yes—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D414yes—
376C2×C42C2×D414yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381#106—28no—
382#106—28no—
383#106—28no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386#103—28no—
387#103—28no—
388#103—28no—
389#103—28no—
390C22⋊C8—24no—
391C22⋊C8—24no—
392C22⋊C8—216no—
393#97—24no—
394C4×C8—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403C24—22no—
404C24—24no—
405C24—24no—
406C24—42no—
407C24C22×D412yes—
408C24C22×D411yescharacteristic
409C2×Q8—216no—
410C2×Q8—28no—
411C2×Q8—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—28no—
415C2×Q8—42no—
416C2×Q8—82no—
417C2×Q8—816no—
418C2×Q8—84no—
419C2×Q8—84no—
420C2×Q8—88no—
421C2×Q8—816no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—216no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—88no—
456C22×C4—816no—
457C22×C4#12014yes—
458C22×C4#12014yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C42−1+418yes—
470C22×C4C2511yescharacteristic
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—28no—
474C2×C8—28no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—28no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4C22×D414yes—
508C4⋊C4C22×D414yes—
509C4⋊C4C22×D414yes—
510C4⋊C4C22×D414yes—
511C22⋊C4—216no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—84no—
520C42—216no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—44no—
525C42—82no—
526C42#12014yes—
527C42#12014yes—
528C42C22×D412yes—
529C42C22×D412yes—
530C42C22×D412yes—
531C42C22×D412yes—
532C23—22no—
533C23—22no—
534C23—28no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—416no—
540C23—44no—
541C23—44no—
542C23C2×#12012yes—
543C23#53614yes—
544C23#57714yes—
545C23C23×D411yescharacteristic
546Q8—44no—
547Q8—44no—
548Q8—88no—
549Q8—88no—
550Q8—84no—
551Q8—816no—
552C2×C4—216no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—816no—
593C2×C4C2×#12012yes—
594C2×C4C2×#12012yes—
595C2×C4C2×#12012yes—
596C2×C4C2×#12012yes—
597C2×C4C2×#12012yes—
598C2×C4#53614yes—
599C2×C4C23×D411yescharacteristic
600C2×C4C23×D411yescharacteristic
601C2×C4C2×2−1+414yes—
602C8—44no—
603C8—44no—
604C22—22no—
605C22—28no—
606C22—22no—
607C22—24no—
608C22—28no—
609C22—416no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22#300914yes—
614C22C22×#12011yescharacteristic
615C22C2×#53612yes—
616C22C2×#57712yes—
617C22#358612yes—
618C4—22no—
619C4—24no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—42no—
625C4—44no—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—44no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4#348612yes—
634C4#348612yes—
635C2—24no—
636C2—44no—
637C2#3524211yescharacteristic
638C2C2×#300911yescharacteristic
639C2#6036811yescharacteristic
640C2#6087214yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3236C2semidirect
3. C2×#3229C2semidirect
7. #29643C2semidirect
8. #29641C2semidirect
9. #25914C2semidirect
10. #25820C2semidirect
11. #24310C2semidirect
12. #21840C2semidirect
13. #21836C2semidirect
35. C2×#501C22semidirect
36. C2×#498C22semidirect
37. C2×#491C22semidirect
38. C2×#487C22semidirect
43. C2×#479C22semidirect
44. C2×#475C22semidirect
59. #2059C22semidirect
60. #2058C22semidirect
61. #2026C22semidirect
62. #2024C22semidirect
63. #1993C22semidirect
64. #1991C22semidirect
65. #1989C22semidirect
66. #1987C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#32362
C2×#32292
#353252
#329842
#329822
#296432
#296412
#259142
#258202
#243102
#218402
#218362
#207172