Finite Group Databasebeta

← #6477326

Subgroups — #6477326

Order 512.

2920 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2C2×#3195C211yesmaximal, characteristic
3C2×#3189C211yesmaximal, characteristic
4#35311C211yesmaximal, characteristic
5#33010C212yesmaximal, complemented
6#33008C212yesmaximal, complemented
7#29643C212yesmaximal, complemented
8#29627C212yesmaximal, complemented
9#25512C218yesmaximal, complemented
10#24071C214yesmaximal, complemented
11#23943C214yesmaximal, complemented
12#21768C211yesmaximal, characteristic
13#21762C211yesmaximal, characteristic
14#20685C211yesmaximal, characteristic
15C2×#541—28no—
16C2×#541—28no—
17C2×#541—22no—
18C2×#541—22no—
19C2×#541C2214yes—
20C2×#541C2214yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#126—21no—
23C22×#126—21no—
24C22×#125—21no—
25C22×#125—21no—
26C22×#125C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C22⋊Q8C2211yescharacteristic
29C22×C22⋊Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×#117C2211yescharacteristic
33#3195—28no—
34#3189—28no—
35C2×#484C2212yes—
36C2×#484C2212yes—
37C2×#483C2212yes—
38C2×#483C2212yes—
39C2×#470C2214yes—
40C2×#467C2214yes—
41C2×#451C2212yes—
42C2×#451C2212yes—
43C2×#451C2212yes—
44C2×#451C2212yes—
45C2×#407C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47#2855C2212yes—
48#2855C2212yes—
49#2625C2214yes—
50#2503—21no—
51#2502—21no—
52#2499—22no—
53#2484—22no—
54#2483—21no—
55#2483—21no—
56#2317C2212yes—
57#2310C2212yes—
58#2280C2211yescharacteristic
59#2059C2218yescomplemented
60#2043C2218yescomplemented
61#2026C2218yescomplemented
62#2016C2218yescomplemented
63#1894C2214yes—
64#1892C2214yes—
65#1878C2218yescomplemented
66#1878C2218yescomplemented
67#1873C2214yes—
68#1873C2214yes—
69#1865C2218yescomplemented
70#1863C2218yescomplemented
71#1824C2212yes—
72#1823C2212yes—
73#1818C2214yes—
74#1817C2212yes—
75#1817C2212yes—
76#1787C2212yes—
77#1787C2212yes—
78#1729C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×Q16C2314yes—
84C22×Q16C2314yes—
85C22×M16C2312yes—
86C23×C8C2312yes—
87#541—232no—
88#541—232no—
89#541—216no—
90#541—216no—
91#541—216no—
92#541—216no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2314yes—
97C2×#126—216no—
98C2×#126—216no—
99C2×#126—24no—
100C2×#126—24no—
101C2×#126—24no—
102C2×#126—24no—
103C2×#125—216no—
104C2×#125—216no—
105C2×#125—24no—
106C2×#125—24no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—22no—
112C2×#125C2314yes—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8D418yes—
124C2×C22⋊Q8D418yes—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2312yes—
128C2×C22⋊Q8C2312yes—
129C2×C22⋊Q8C2312yes—
130C2×C22⋊Q8C2312yes—
131C2×C4×Q8—24no—
132C2×C4×Q8—24no—
133C2×C4×Q8—24no—
134C2×C4×Q8—24no—
135C2×C4×Q8—22no—
136C2×C4×Q8—22no—
137C2×C4×Q8C2314yes—
138C2×C4×Q8C2314yes—
139C2×#117—24no—
140C2×#117—24no—
141C2×#117D412yes—
142C2×#117D412yes—
143C2×#117C2312yes—
144C2×#117C2312yes—
145C2×#117C2312yes—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C22⋊C4—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C22⋊C4D411yescharacteristic
164C22×C22⋊C4D411yescharacteristic
165C22×C42—21no—
166C22×C42—21no—
167C22×C42C2311yescharacteristic
168C22×C42C2311yescharacteristic
169#484—28no—
170#484—28no—
171#483—28no—
172#483—28no—
173#470—216no—
174#467—216no—
175#451—28no—
176#451—28no—
177#451—28no—
178#451—28no—
179C2×#107C2314yes—
180C2×#106C2314yes—
181C2×#103C2314yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2318yes—
186C2×#103C2318yes—
187#407—24no—
188C2×C22⋊C8—24no—
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193#400—24no—
194#398—24no—
195#397—24no—
196#397—24no—
197#391—22no—
198#391—22no—
199#391C2314yes—
200#391C2314yes—
201#389C2314yes—
202#389C2314yes—
203#387—24no—
204#387—24no—
205#385—22no—
206#385—22no—
207#385—22no—
208#385—22no—
209#384—22no—
210#382—22no—
211#381—22no—
212#381—22no—
213#381—21no—
214#381—22no—
215#381—22no—
216#380C2312yes—
217#380C2312yes—
218#379—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221#340C2318yes—
222#340C2318yes—
223#336C2318yes—
224#336C2318yes—
225C25C2×D411yescharacteristic
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—21no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—44no—
234C22×Q8—44no—
235C22×Q8—44no—
236C22×Q8C2×D414yes—
237C22×Q8C2×D414yes—
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—44no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D414yes—
256C23×C4C2×D414yes—
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×Q16—28no—
261C2×Q16—28no—
262C2×Q16—48no—
263C2×Q16—48no—
264C2×M16—24no—
265C2×M16—44no—
266C22×C8—24no—
267C22×C8—44no—
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—216no—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273#126—232no—
274#126—232no—
275#126—232no—
276#126—232no—
277#125—232no—
278#125—232no—
279#125—232no—
280#125—232no—
281#125—216no—
282#125—28no—
283#125—28no—
284C22⋊Q8—28no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C22⋊Q8—28no—
288C22⋊Q8—24no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C22⋊Q8C2×D418yes—
299C22⋊Q8C2×D418yes—
300C22⋊Q8C2×D418yes—
301C22⋊Q8C2×D418yes—
302C4×Q8—216no—
303C4×Q8—216no—
304C4×Q8—28no—
305C4×Q8—28no—
306C4×Q8—28no—
307C4×Q8—28no—
308#117—28no—
309#117—28no—
310#117—24no—
311#117—24no—
312#117—24no—
313#117—24no—
314#117—44no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2414yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—28no—
358C2×C22⋊C4—28no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C42—28no—
376C2×C42—28no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—24no—
382C2×C42—24no—
383C2×C42—22no—
384C2×C42C2×D412yes—
385C2×C42C2×D412yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388#107—28no—
389#106—28no—
390#103—28no—
391#103—28no—
392#103—28no—
393#103—28no—
394#103—48no—
395#103—48no—
396C22⋊C8—24no—
397C22⋊C8—24no—
398C22⋊C8—24no—
399C22⋊C8—24no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408C24—24no—
409C24—42no—
410C24—42no—
411C24#12014yes—
412C24C22×D411yescharacteristic
413C24C22×D411yescharacteristic
414C24C22×D411yescharacteristic
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—28no—
422C2×Q8—28no—
423C2×Q8—28no—
424C2×Q8—48no—
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—48no—
429C2×Q8—48no—
430C2×Q8—44no—
431C2×Q8—42no—
432C2×Q8—84no—
433C2×Q8—84no—
434C2×Q8C22×D414yes—
435C2×Q8C22×D414yes—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—82no—
471C22×C4#12014yes—
472C22×C4#12014yes—
473C22×C4#12014yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C2511yescharacteristic
488Q16—88no—
489Q16—88no—
490M16—84no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—84no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—28no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4C22×D414yes—
523C4⋊C4C22×D414yes—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—82no—
537C42—216no—
538C42—216no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—44no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—28no—
555C23—44no—
556C23—44no—
557C23—88no—
558C23C2×#12012yes—
559C23C2×#12012yes—
560C23C2×#12012yes—
561C23#57814yes—
562C23C23×D411yescharacteristic
563Q8—28no—
564Q8—28no—
565Q8—48no—
566Q8—48no—
567Q8—44no—
568Q8—44no—
569Q8—44no—
570Q8—44no—
571Q8—84no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
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—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4C2×#12012yes—
618C2×C4C2×#12012yes—
619C2×C4C2×#12012yes—
620C2×C4C23×D411yescharacteristic
621C2×C4C23×D411yescharacteristic
622C2×C4#58514yes—
623C8—44no—
624C8—44no—
625C22—22no—
626C22—28no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—44no—
635C22—44no—
636C22—88no—
637C22#301114yes—
638C22C22×#12011yescharacteristic
639C22#344912yes—
640C22C2×#57812yes—
641C22#358612yes—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—82no—
654C4—84no—
655C4—82no—
656C4#348512yes—
657C4#348512yes—
658C2—24no—
659C2—42no—
660C2—42no—
661C2#3508811yescharacteristic
662C2C2×#301111yescharacteristic
663C2#6036811yescharacteristic
664C2#6098314yes—
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
5. #33010C2semidirect
6. #33008C2semidirect
7. #29643C2semidirect
8. #29627C2semidirect
9. #25512C2semidirect
10. #24071C2semidirect
11. #23943C2semidirect
59. #2059C22semidirect
60. #2043C22semidirect
61. #2026C22semidirect
62. #2016C22semidirect
65. #1878C22semidirect
66. #1878C22semidirect
69. #1865C22semidirect
70. #1863C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#31952
C2×#31892
#353112
#330102
#330082
#296432
#296272
#255122
#240712
#239432
#217682
#217622
#206852