Finite Group Databasebeta

← #6484508

Subgroups — #6484508

Order 512.

1376 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#35283C211yesmaximal, characteristic
3#33212C211yesmaximal, characteristic
4#33210C211yesmaximal, characteristic
5#33209C211yesmaximal, characteristic
6#33207C211yesmaximal, characteristic
7#30299C214yesmaximal
8#29325C214yesmaximal, complemented
9#29324C214yesmaximal, complemented
10#29208C211yesmaximal, characteristic, complemented
11#29207C211yesmaximal, characteristic, complemented
12#29207C211yesmaximal, characteristic, complemented
13#29206C211yesmaximal, characteristic, complemented
14#28689C211yesmaximal, characteristic, complemented
15#28688C211yesmaximal, characteristic, complemented
16#28688C211yesmaximal, characteristic, complemented
17#28687C211yesmaximal, characteristic, complemented
18#21966C214yesmaximal
19C4×#424C211yesmaximal, characteristic
20C2×#500C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#498C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#424C2212yes—
25C2×C4×M16C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2647C2212yes—
36#2647C2212yes—
37#2629C2212yes—
38#2628C2212yes—
39#2511—21no—
40#2511—21no—
41#2507—21no—
42#2507—21no—
43#2497—21no—
44#2497—21no—
45#2492—21no—
46#2492—21no—
47#2451—21no—
48#2451—21no—
49#2446—21no—
50#2446—21no—
51#2429—21no—
52#2429—21no—
53#2423—21no—
54#2423—21no—
55#2331C2212yes—
56#2330C2212yes—
57#2314C2212yes—
58#2314C2212yes—
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×#117C2211yescharacteristic
64#2085C2214yes—
65#2085C2214yes—
66#2084C2214yes—
67#2083C2214yes—
68#2059C2214yes—
69#2059C2214yes—
70#2058C2214yes—
71#2057C2214yes—
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1949C2214yes—
81#1949C2214yes—
82#1949C2214yes—
83#1949C2214yes—
84#1948C2214yes—
85#1948C2214yes—
86#1947C2214yes—
87#1947C2214yes—
88#1944C2214yes—
89#1943C2214yes—
90#1942C2214yes—
91#1941C2214yes—
92#1845C2211yescharacteristic
93#1845C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1809C2214yes—
97#1809C2214yes—
98#1808C2214yes—
99#1807C2214yes—
100#1768C2212yes—
101#1768C2212yes—
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106#1729C2212yes—
107C22×M16C2312yes—
108C23×C8C2312yes—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×#125C2312yes—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#500—42no—
131#500—42no—
132#498—42no—
133#498—42no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142#424Q8116yescomplemented
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C4×M16—42no—
148C2×#97C2312yes—
149C2×C4×C8—42no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2312yes—
155#400—22no—
156#400—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#399—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#396—22no—
168#396—22no—
169#395—22no—
170#393—22no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—22no—
184#391—22no—
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#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#384—21no—
220#384—21no—
221#384—21no—
222#384—21no—
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2312yes—
228#384C2312yes—
229#384C2312yes—
230#382C2312yes—
231#382C2312yes—
232#382C2312yes—
233#380—21no—
234#380—21no—
235#380—21no—
236#380—21no—
237#380C2312yes—
238#380C2312yes—
239#380C2312yes—
240#380C2312yes—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257#376C2311yescharacteristic
258#376C2311yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—22no—
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×M16—42no—
282C2×M16—42no—
283C2×M16C2×Q818yes—
284C22×C8—42no—
285C22×C8—42no—
286C22×C8C2×Q818yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—44no—
300C4⋊Q8—82no—
301C4⋊Q8—82no—
302C4⋊Q8—82no—
303C4⋊Q8—82no—
304#125—44no—
305C22⋊Q8—82no—
306C22⋊Q8—82no—
307C22⋊Q8—82no—
308C22⋊Q8—82no—
309C4×Q8—82no—
310C4×Q8—82no—
311C4×Q8—82no—
312C4×Q8—82no—
313#117—42no—
314#117C2×Q818yes—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
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—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
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⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4C2×D411yescharacteristic
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C22⋊C4C2×D411yescharacteristic
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—44no—
387#107—44no—
388#107—44no—
389#107—44no—
390#106—44no—
391#106—44no—
392#106—44no—
393#106—44no—
394C4⋊C8C2×Q818yes—
395C4⋊C8C2×Q818yes—
396C4⋊C8C2×Q818yes—
397C4⋊C8C2×Q818yes—
398#97—42no—
399C4×C8—22no—
400C4×C8—22no—
401C4×C8—22no—
402C4×C8—22no—
403C4×C8—42no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
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#95—22no—
425#95—22no—
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434C24C22×D411yescharacteristic
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—82no—
449C2×Q8—82no—
450C22×C4—21no—
451C22×C4—21no—
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—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
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×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×Q814yes—
481C22×C4C2511yescharacteristic
482M16—48no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—48no—
492C2×C8C22×Q814yes—
493C2×C8C22×Q814yes—
494C2×C8C22×Q814yes—
495C2×C8C22×Q814yes—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—41no—
509C4⋊C4—41no—
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—82no—
526C4⋊C4—82no—
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⋊C4C4⋊Q814yes—
540C4⋊C4C4⋊Q814yes—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4C4⋊Q814yes—
548C22⋊C4C4⋊Q814yes—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42—22no—
562C42—22no—
563C42—82no—
564C42—82no—
565C42—82no—
566C42—82no—
567C42C22×Q814yes—
568C42C22×Q814yes—
569C42H(5,𝔽2)12yes—
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—84no—
582Q8—84no—
583Q8—164no—
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—24no—
595C2×C4—24no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
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—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—44no—
615C2×C4—44no—
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—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4C2×C4⋊Q812yes—
641C2×C4C2×C4⋊Q812yes—
642C2×C4C2×C4⋊Q812yes—
643C2×C4C2×C4⋊Q812yes—
644C2×C4C2×C4⋊Q812yes—
645C2×C4#53712yes—
646C2×C4#53712yes—
647C2×C4#57612yes—
648C2×C4#57712yes—
649C2×C4#57712yes—
650C2×C4#57812yes—
651C2×C4C23×Q812yes—
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×2−1+411yescharacteristic
654C2×C4#58512yes—
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×#53711yescharacteristic
668C22#344911yescharacteristic
669C22C2×#57611yescharacteristic
670C22C2×#57711yescharacteristic
671C22C2×#57711yescharacteristic
672C22C2×#57811yescharacteristic
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—41no—
679C4—41no—
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#325714yes—
693C4#325814yes—
694C4#350112yes—
695C4#350212yes—
696C2—42no—
697C2#3517311yescharacteristic
698C2#6066511yescharacteristic
699C2#6066711yescharacteristic
700C2C2×#325711yescharacteristic
701C2C2×#325811yescharacteristic
702C2#6100611yescharacteristic
703C2#6100811yescharacteristic
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
8. #29325C2semidirect
9. #29324C2semidirect
10. #29208C2semidirect
11. #29207C2semidirect
12. #29207C2semidirect
13. #29206C2semidirect
14. #28689C2semidirect
15. #28688C2semidirect
16. #28688C2semidirect
17. #28687C2semidirect
142. #424Q8semidirect

Maximal subgroups

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

subgroupindex
#352842
#352832
#332122
#332102
#332092
#332072
#302992
#293252
#293242
#292082
#292072
#292062
#286892
#286882
#286872
#219662
C4×#4242