Finite Group Databasebeta

← #6495056

Subgroups — #6495056

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33311C211yesmaximal, characteristic, complemented
5#33294C211yesmaximal, characteristic, complemented
6#33289C211yesmaximal, characteristic, complemented
7#30315C214yesmaximal, complemented
8#29208C211yesmaximal, characteristic, complemented
9#29207C211yesmaximal, characteristic, complemented
10#29207C211yesmaximal, characteristic, complemented
11#29206C211yesmaximal, characteristic, complemented
12#28698C214yesmaximal, complemented
13#27950C211yesmaximal, characteristic
14#27949C211yesmaximal, characteristic
15#27949C211yesmaximal, characteristic
16#27948C211yesmaximal, characteristic
17#25822C214yesmaximal
18#21992C214yesmaximal
19C4×#433C211yesmaximal, characteristic
20C2×#544C2212yes—
21C2×#501C2214yes—
22C2×#499C2214yes—
23C2×#433C2212yes—
24#2868C2211yescharacteristic
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2660C2212yes—
33#2660C2212yes—
34#2647C2212yes—
35#2629C2212yes—
36#2552—21no—
37#2552—21no—
38#2551—21no—
39#2551—21no—
40#2507—21no—
41#2507—21no—
42#2497—21no—
43#2497—21no—
44#2446—21no—
45#2446—21no—
46#2440—21no—
47#2440—21no—
48#2440—21no—
49#2440—21no—
50#2429—21no—
51#2429—21no—
52#2330C2212yes—
53#2317C2212yes—
54#2314C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C4×#117C2211yescharacteristic
60#2085C2214yes—
61#2085C2214yes—
62#2084C2214yes—
63#2083C2214yes—
64#2059C2214yes—
65#2059C2214yes—
66#2058C2214yes—
67#2057C2214yes—
68#1992C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2214yes—
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1949C2214yes—
79#1949C2214yes—
80#1948C2214yes—
81#1947C2214yes—
82#1854C2214yes—
83#1854C2214yes—
84#1853C2214yes—
85#1852C2214yes—
86#1847C2214yes—
87#1846C2214yes—
88#1845C2211yescharacteristic
89#1845C2211yescharacteristic
90#1845C2211yescharacteristic
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1844C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1836C2214yes—
97#1834C2214yes—
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107#544—48no—
108C2×C4⋊Q8—22no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2312yes—
119C2×#125—22no—
120C2×#125C2312yes—
121C2×#125C2312yes—
122C2×#125C2312yes—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8—22no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×#117C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C42C2311yescharacteristic
136#501—48no—
137#499—48no—
138#433Q8116yescomplemented
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×#97C2312yes—
152C2×#97C2312yes—
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2312yes—
158C2×C4×C8C2312yes—
159#400—22no—
160#400—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#396—22no—
170#396—22no—
171#395—22no—
172#395—22no—
173#395—22no—
174#395—22no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—22no—
198#389—22no—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#388—22no—
204#388—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#385—22no—
210#385—22no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384—21no—
215#384—21no—
216#384—21no—
217#384—21no—
218#384—21no—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2312yes—
222#384C2312yes—
223#384C2312yes—
224#384C2312yes—
225#384C2312yes—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#382C2312yes—
231#382C2312yes—
232#382C2312yes—
233#382C2312yes—
234#380—21no—
235#380—21no—
236#380—21no—
237#380—21no—
238#380C2312yes—
239#380C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254#376C2311yescharacteristic
255#376C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×Q8—21no—
272C22×Q8—22no—
273C22×Q8—22no—
274C23×C4—22no—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C4⋊Q8—28no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—44no—
288C4⋊Q8—48no—
289C4⋊Q8—82no—
290C4⋊Q8—82no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293C4⋊Q8—82no—
294C4⋊Q8—82no—
295#125—28no—
296#125—28no—
297#125—44no—
298#125—48no—
299C22⋊Q8—48no—
300C22⋊Q8—48no—
301C22⋊Q8—82no—
302C22⋊Q8—82no—
303C4×Q8—82no—
304C4×Q8—82no—
305#117—24no—
306#117—44no—
307#117C2×Q818yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
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—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—82no—
347C2×C4⋊C4—82no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C22⋊C4C2×D411yescharacteristic
365C2×C22⋊C4C2×D411yescharacteristic
366C2×C22⋊C4C2×D411yescharacteristic
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388#107—44no—
389#107—44no—
390#107—44no—
391#107—44no—
392#106—44no—
393#106—44no—
394#106—44no—
395#106—44no—
396C4⋊C8C2×Q818yes—
397C4⋊C8C2×Q818yes—
398C22⋊C8C2×Q818yes—
399C22⋊C8C2×Q818yes—
400#97—24no—
401#97C2×Q818yes—
402C4×C8—24no—
403C4×C8—22no—
404C4×C8—22no—
405C4×C8—22no—
406C4×C8—22no—
407C4×C8C2×Q818yes—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432C24C22×D411yescharacteristic
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—82no—
442C2×Q8—82no—
443C2×Q8—82no—
444C2×Q8—82no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—44no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×Q814yes—
479C22×C4C2511yescharacteristic
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8C22×Q814yes—
489C2×C8C22×Q814yes—
490C2×C8C22×Q814yes—
491C2×C8C22×Q814yes—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4C4⋊Q814yes—
542C4⋊C4C4⋊Q814yes—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—44no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4C4⋊Q814yes—
549C22⋊C4C4⋊Q814yes—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—21no—
562C42—21no—
563C42—44no—
564C42—82no—
565C42—82no—
566C42C22×Q814yes—
567C42C22×Q814yes—
568C42H(5,𝔽2)14yes—
569C422−1+412yes—
570C422−1+412yes—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23C2×C4⋊Q812yes—
578C23C23×D411yescharacteristic
579Q8—84no—
580Q8—84no—
581Q8—164no—
582Q8—164no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—164no—
638C2×C4C2×C4⋊Q812yes—
639C2×C4C2×C4⋊Q812yes—
640C2×C4C2×C4⋊Q812yes—
641C2×C4C2×C4⋊Q812yes—
642C2×C4C2×C4⋊Q812yes—
643C2×C4#53512yes—
644C2×C4#53512yes—
645C2×C4#53612yes—
646C2×C4#53612yes—
647C2×C4#57612yes—
648C2×C4#57712yes—
649C2×C4#57712yes—
650C2×C4#57812yes—
651C2×C4C23×Q812yes—
652C2×C4C2×H(5,𝔽2)12yes—
653C2×C4C2×2−1+411yescharacteristic
654C2×C4C2×2−1+411yescharacteristic
655C8—24no—
656C8—24no—
657C8—24no—
658C8—24no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22C22×C4⋊Q811yescharacteristic
667C22C2×#53511yescharacteristic
668C22C2×#53611yescharacteristic
669C22C2×#57611yescharacteristic
670C22C2×#57711yescharacteristic
671C22C2×#57711yescharacteristic
672C22C2×#57811yescharacteristic
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4#324114yes—
693C4#324214yes—
694C4#350312yes—
695C4#350312yes—
696C2—42no—
697C2#3528111yescharacteristic
698C2#6066511yescharacteristic
699C2#6066711yescharacteristic
700C2#6082211yescharacteristic
701C2#6082411yescharacteristic
702C2C2×#324111yescharacteristic
703C2C2×#324211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33316C2semidirect
4. #33311C2semidirect
5. #33294C2semidirect
6. #33289C2semidirect
7. #30315C2semidirect
8. #29208C2semidirect
9. #29207C2semidirect
10. #29207C2semidirect
11. #29206C2semidirect
12. #28698C2semidirect
138. #433Q8semidirect

Maximal subgroups

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

subgroupindex
#352842
#352752
#333162
#333112
#332942
#332892
#303152
#292082
#292072
#292062
#286982
#279502
#279492
#279482
#258222
#219922
C4×#4332