Finite Group Databasebeta

← #6392726

Subgroups — #6392726

Order 512.

1899 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60232C211yesmaximal, characteristic
2C2×#2981C211yesmaximal, characteristic
3#60144C211yesmaximal, characteristic
4#22719C212yesmaximal, complemented
5#22718C214yesmaximal, complemented
6#22717C212yesmaximal, complemented
7#22669C212yesmaximal, complemented
8#22668C214yesmaximal, complemented
9D4×#108C212yesmaximal, complemented
10#21994C211yesmaximal, characteristic, complemented
11#21993C211yesmaximal, characteristic, complemented
12#21950C214yesmaximal, complemented
13#21371C211yesmaximal, characteristic, complemented
14#21370C211yesmaximal, characteristic, complemented
15#20852C214yesmaximal, complemented
16C2×#568C2212yes—
17C2×#532C2211yescharacteristic
18#2981—22no—
19#2981—22no—
20#2981C412yes—
21#2981C412yes—
22#2981C414yes—
23#2981C414yes—
24#2981C414yes—
25#2980—22no—
26#2979—21no—
27#2979—21no—
28#2935—21no—
29#2935—21no—
30#2935—22no—
31D4×M16C2212yes—
32D4×M16C2212yes—
33D4×M16C2212yes—
34D4×M16C2212yes—
35D4×M16C2212yes—
36D4×M16C2214yes—
37D4×M16C2212yes—
38#2932—22no—
39#2931—22no—
40C2×#436C2212yes—
41C2×#435C2214yes—
42C2×C8×D4C2212yes—
43#2920C2211yescharacteristic
44#2920C2211yescharacteristic
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47C2×#430C2214yes—
48C22×#108C2214yes—
49C2×#423C2211yescharacteristic
50#2879C2214yes—
51#2879C2212yes—
52C2×#408C2214yes—
53C2×C4×M16C2211yescharacteristic
54#1950C2212yes—
55#1950C2212yes—
56#1950C2212yes—
57#1950C2212yes—
58#1950C2212yes—
59#1950C2214yes—
60#1950C2212yes—
61#1855C2212yes—
62#1855C2212yes—
63#1855C2214yes—
64#1855C2214yes—
65#1855C2214yes—
66#1855C2214yes—
67#1855C2214yes—
68#1855C2214yes—
69#1855C2212yes—
70#1855C2212yes—
71#1855C2214yes—
72#1855C2214yes—
73#1855C2214yes—
74#1855C2214yes—
75#1849C2212yes—
76#1848C2211yescharacteristic
77#1848C2211yescharacteristic
78#1838C2212yes—
79#1837C2211yescharacteristic
80#1837C2211yescharacteristic
81#1811C2214yes—
82#1811C2214yes—
83#1810C2214yes—
84#1810C2214yes—
85#1778C2214yes—
86C4×#108C2212yes—
87C4×#108C2212yes—
88#1738C2214yes—
89C2×#341C2214yes—
90C22×#41C2312yes—
91#568—48no—
92C2×#131—24no—
93C2×#131—22no—
94C2×#131—22no—
95C22×M16C2311yescharacteristic
96C22×M16C2314yes—
97C22×M16C2314yes—
98C22×M16C2311yescharacteristic
99C22×M16C2312yes—
100C22×M16C2314yes—
101C22×M16C2314yes—
102C22×M16C2314yes—
103C23×C8C2314yes—
104#532C2×C411yescharacteristic
105#532C2×C411yescharacteristic
106#532C2×C412yes—
107#532C2×C412yes—
108#532C2×C412yes—
109#532D412yes—
110#532D412yes—
111#532D411yescharacteristic
112#532D411yescharacteristic
113#532D412yes—
114C2×C4⋊Q8Q811yescharacteristic
115C2×#127Q811yescharacteristic
116C2×#124Q812yes—
117C2×#121Q812yes—
118C2×#121Q812yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C22×C42C2311yescharacteristic
122#436—22no—
123#436—22no—
124#436—24no—
125#436—22no—
126#436—22no—
127#436—22no—
128#436—22no—
129#436C2×C414yes—
130#436C2×C414yes—
131#436C2×C418yes—
132#435—24no—
133#435—24no—
134#435—24no—
135#435—24no—
136#435—24no—
137#435—24no—
138#435—24no—
139#435—24no—
140#435C2×C418yes—
141#435C2×C418yes—
142#435C2×C418yes—
143#435C2×C418yes—
144C8×D4—22no—
145C8×D4—22no—
146C8×D4—24no—
147C8×D4—22no—
148C8×D4—22no—
149C8×D4—22no—
150C8×D4—22no—
151C8×D4C2×C414yes—
152C8×D4C2×C414yes—
153C8×D4C2×C418yes—
154#432—21no—
155#432—21no—
156#432—21no—
157#432—21no—
158#432C2×C412yes—
159#432C2×C412yes—
160#432C2×C412yes—
161#432C2×C412yes—
162#431—21no—
163#431—21no—
164#431—21no—
165#431—21no—
166#431C2×C412yes—
167#431C2×C412yes—
168#431C2×C412yes—
169#431C2×C412yes—
170#430—48no—
171C2×#108—48no—
172C2×#108C2312yes—
173C2×#108C2314yes—
174C2×#108C2312yes—
175C2×#108C2312yes—
176C2×#108C2314yes—
177C2×#108C2314yes—
178C2×#108C2314yes—
179C2×#108C2312yes—
180C2×#108C2314yes—
181C2×#108C2314yes—
182#423—21no—
183#423—21no—
184#423C2311yescharacteristic
185#423C2311yescharacteristic
186#423C2311yescharacteristic
187#423C2311yescharacteristic
188#423C2312yes—
189#423C2311yescharacteristic
190#423C2311yescharacteristic
191C2×C4⋊C8—21no—
192C2×C4⋊C8—21no—
193C2×C4⋊C8C2312yes—
194C2×C4⋊C8C2312yes—
195#408—24no—
196#408—22no—
197#408—22no—
198#408—24no—
199#408—24no—
200#408—24no—
201#408—24no—
202#408C2×C418yes—
203#408C2×C418yes—
204#408C2×C418yes—
205#408C2×C418yes—
206#407C2314yes—
207#407C2314yes—
208#407C2314yes—
209#407C2314yes—
210#407C2312yes—
211#407C2312yes—
212C2×C22⋊C8C2312yes—
213C2×C22⋊C8C2312yes—
214C2×C22⋊C8C2314yes—
215C2×C22⋊C8C2314yes—
216C4×M16—21no—
217C4×M16—21no—
218C4×M16C2312yes—
219C4×M16C2312yes—
220C4×M16C2312yes—
221C4×M16C2312yes—
222C2×#97—21no—
223C2×#97C2312yes—
224C2×C4×C8—21no—
225C2×C4×C8C2311yescharacteristic
226C2×C4×C8C2311yescharacteristic
227#341C2312yes—
228#341C2314yes—
229#341C2314yes—
230#341C2314yes—
231#341C2312yes—
232#341C2314yes—
233#341C2314yes—
234#341C2312yes—
235#341C2314yes—
236#341C2312yes—
237C2×#41—44no—
238C2×#41—44no—
239C2×#41C22×C412yes—
240C2×#41C22×C412yes—
241C2×#41C22×C414yes—
242C2×#41C2×D414yes—
243C2×#41C2×D412yes—
244C2×#41C2×D412yes—
245C22×Q8C2×Q812yes—
246C22×D4C2×Q812yes—
247C22×D4C2×Q814yes—
248C22×D4C2×Q812yes—
249C22×D4C2×Q812yes—
250C23×C4C2×Q811yescharacteristic
251C23×C4C2×Q811yescharacteristic
252C23×C4C2414yes—
253C23×C4C2411yescharacteristic
254#131—416no—
255#131—48no—
256#131—48no—
257C2×M16—21no—
258C2×M16—24no—
259C2×M16—22no—
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—21no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—28no—
266C2×M16—28no—
267C2×M16—44no—
268C2×M16—44no—
269C2×M16—42no—
270C2×M16—44no—
271C2×M16—44no—
272C2×M16—44no—
273C2×M16—44no—
274C2×M16—44no—
275C2×M16—44no—
276C2×M16—44no—
277C2×M16C22×C418yes—
278C2×M16C22×C418yes—
279C2×M16C2412yes—
280C2×M16C2412yes—
281C2×M16C2412yes—
282C2×M16C2412yes—
283C2×M16C2412yes—
284C2×M16C2412yes—
285C2×M16C2412yes—
286C2×M16C2412yes—
287C22×C8—21no—
288C22×C8—22no—
289C22×C8—24no—
290C22×C8—24no—
291C22×C8—21no—
292C22×C8—24no—
293C22×C8—24no—
294C22×C8—44no—
295C22×C8—44no—
296C22×C8C22×C418yes—
297C22×C8C22×C418yes—
298C22×C8C2412yes—
299C22×C8C2414yes—
300C22×C8C2412yes—
301C4⋊Q8C4⋊C412yes—
302C4⋊Q8C4⋊C412yes—
303C4⋊Q8C4⋊C411yescharacteristic
304C4⋊Q8C4⋊C411yescharacteristic
305C4⋊Q8C4⋊C412yes—
306#127C4⋊C412yes—
307#127C4⋊C411yescharacteristic
308#127C4⋊C411yescharacteristic
309#127C4⋊C412yes—
310#127C4⋊C412yes—
311#124C4⋊C414yes—
312#124C4⋊C414yes—
313#124C4⋊C412yes—
314#124C4⋊C412yes—
315#124C4⋊C412yes—
316#124C4⋊C412yes—
317#121C4⋊C412yes—
318#121C4⋊C412yes—
319#121C4⋊C412yes—
320#121C4⋊C412yes—
321#121C4⋊C414yes—
322#121C4⋊C412yes—
323#121C4⋊C412yes—
324#121C4⋊C414yes—
325#121C4⋊C412yes—
326#121C4⋊C412yes—
327#121C4⋊C414yes—
328#121C4⋊C414yes—
329C4×D4C22×C412yes—
330C4×D4C22×C412yes—
331C4×D4C22×C414yes—
332C4×D4C22×C412yes—
333C4×D4C22×C412yes—
334C4×D4C22×C414yes—
335C4×D4C2×D412yes—
336C4×D4C2×D412yes—
337C4×D4C2×D414yes—
338C4×D4C2×D412yes—
339C4×D4C2×D412yes—
340C4×D4C2×D414yes—
341C2×C4⋊C4C2×Q812yes—
342C2×C4⋊C4C2×Q812yes—
343C2×C22⋊C4C2×Q814yes—
344C2×C22⋊C4C2×Q814yes—
345C2×C42C22×C411yescharacteristic
346C2×C42C22×C411yescharacteristic
347C2×C42C22×C411yescharacteristic
348C2×C42C22×C411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×Q811yescharacteristic
354C2×C42C2×Q811yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#108—24no—
358#108—28no—
359#108—24no—
360#108—44no—
361#108—44no—
362#108—42no—
363#108—44no—
364#108—42no—
365#108—44no—
366#108—44no—
367C4⋊C8—21no—
368C4⋊C8—21no—
369C4⋊C8—22no—
370C4⋊C8—22no—
371C4⋊C8—21no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—21no—
375C4⋊C8—21no—
376C4⋊C8—21no—
377C4⋊C8—21no—
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—21no—
381C4⋊C8C22×C414yes—
382C4⋊C8C22×C414yes—
383C4⋊C8C22×C414yes—
384C4⋊C8C22×C414yes—
385C22⋊C8—22no—
386C22⋊C8—22no—
387C22⋊C8—22no—
388C22⋊C8—22no—
389C22⋊C8—24no—
390C22⋊C8—24no—
391C22⋊C8—24no—
392C22⋊C8—24no—
393C22⋊C8—24no—
394C22⋊C8—24no—
395C22⋊C8—24no—
396C22⋊C8—24no—
397C22⋊C8C22×C414yes—
398C22⋊C8C22×C414yes—
399C22⋊C8C22×C414yes—
400C22⋊C8C22×C414yes—
401C22⋊C8C22×C418yes—
402C22⋊C8C22×C418yes—
403C22⋊C8C22×C418yes—
404C22⋊C8C22×C418yes—
405#97—22no—
406#97—22no—
407#97—22no—
408#97—22no—
409#97C22×C414yes—
410#97C22×C414yes—
411C4×C8—22no—
412C4×C8—22no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8C22×C412yes—
416C4×C8C22×C412yes—
417C4×C8C22×C412yes—
418C4×C8C22×C412yes—
419C24C22×Q814yes—
420C24C22×Q811yescharacteristic
421#41—44no—
422#41—42no—
423#41—42no—
424#41—44no—
425#41—44no—
426#41—44no—
427#41—44no—
428#41—44no—
429#41—42no—
430#41—42no—
431C2×Q8—44no—
432C2×Q8C2×C4⋊C414yes—
433C2×Q8C2×C4⋊C412yes—
434C2×Q8C2×C4⋊C412yes—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—82no—
440C2×D4C2×C4⋊C412yes—
441C2×D4C2×C4⋊C412yes—
442C2×D4C2×C4⋊C414yes—
443C2×D4C2×C4⋊C414yes—
444C2×D4C2×C4⋊C412yes—
445C2×D4C2×C4⋊C412yes—
446C2×D4C2×C4⋊C414yes—
447C2×D4C2×C4⋊C414yes—
448C2×D4C2×C4⋊C414yes—
449C2×D4C2×C4⋊C414yes—
450C2×D4C2×C4⋊C414yes—
451C2×D4C2×C4⋊C412yes—
452C2×D4C2×C4⋊C412yes—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4C2×C4⋊C411yescharacteristic
462C22×C4C2×C4⋊C411yescharacteristic
463C22×C4C2×C4⋊C411yescharacteristic
464C22×C4C2×C4⋊C411yescharacteristic
465C22×C4C2×C4⋊C411yescharacteristic
466C22×C4C2×C4⋊C411yescharacteristic
467C22×C4C2×C4⋊C411yescharacteristic
468C22×C4C2×C4⋊C411yescharacteristic
469C22×C4C23×C411yescharacteristic
470C22×C4C23×C411yescharacteristic
471C22×C4C23×C414yes—
472C22×C4C23×C414yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×Q811yescharacteristic
478C22×C4C22×Q811yescharacteristic
479C22×C4C22×Q814yes—
480C22×C4C22×Q811yescharacteristic
481C22×C4C22×Q811yescharacteristic
482C22×C4C22×Q811yescharacteristic
483C22×C4C2511yescharacteristic
484M16—24no—
485M16—22no—
486M16—22no—
487M16—24no—
488M16—22no—
489M16—24no—
490M16—24no—
491M16—22no—
492M16—22no—
493M16—22no—
494M16—24no—
495M16—22no—
496M16—48no—
497M16—48no—
498M16—48no—
499M16—48no—
500M16—42no—
501M16—44no—
502M16—84no—
503M16—84no—
504M16H(5,𝔽2)14yes—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—48no—
521C2×C8—48no—
522C2×C8—48no—
523C2×C8—48no—
524C2×C8—42no—
525C2×C8—44no—
526C2×C8—84no—
527C2×C8—84no—
528C2×C8C23×C414yes—
529C2×C8C23×C414yes—
530C2×C8C23×C418yes—
531C4⋊C4C2×C4⋊C414yes—
532C4⋊C4C2×C4⋊C414yes—
533C4⋊C4C2×C4⋊C412yes—
534C4⋊C4C2×C4⋊C412yes—
535C4⋊C4C2×C4⋊C412yes—
536C4⋊C4C2×C4⋊C412yes—
537C22⋊C4C2×C4⋊C414yes—
538C22⋊C4C2×C4⋊C414yes—
539C22⋊C4C2×C4⋊C414yes—
540C22⋊C4C2×C4⋊C414yes—
541C22⋊C4C2×C4⋊C414yes—
542C22⋊C4C2×C4⋊C414yes—
543C22⋊C4C2×C4⋊C414yes—
544C22⋊C4C2×C4⋊C414yes—
545C42C2×C4⋊C411yescharacteristic
546C42C2×C4⋊C411yescharacteristic
547C42C2×C4⋊C411yescharacteristic
548C42C2×C4⋊C411yescharacteristic
549C42C2×C4⋊C411yescharacteristic
550C42C2×C4⋊C411yescharacteristic
551C42C2×C4⋊C411yescharacteristic
552C42C2×C4⋊C411yescharacteristic
553C42C23×C411yescharacteristic
554C42C23×C411yescharacteristic
555C42C23×C411yescharacteristic
556C42C23×C411yescharacteristic
557C42C22×D411yescharacteristic
558C42C22×D411yescharacteristic
559C42C22×D411yescharacteristic
560C42C22×D411yescharacteristic
561C23—22no—
562C23—24no—
563C23—24no—
564C23—41no—
565C23—41no—
566C23—44no—
567C23—44no—
568C23C22×C4⋊C414yes—
569C23C22×C4⋊C414yes—
570C23C22×C4⋊C411yescharacteristic
571C23C22×C4⋊C411yescharacteristic
572C23C23×Q811yescharacteristic
573Q8—44no—
574Q8—42no—
575Q8—42no—
576D4—24no—
577D4—24no—
578D4—44no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—44no—
583D4—44no—
584D4—44no—
585D4—44no—
586D4—44no—
587D4—42no—
588D4—42no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—21no—
606C2×C4—24no—
607C2×C4—21no—
608C2×C4—24no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4C22×C4⋊C411yescharacteristic
619C2×C4C22×C4⋊C411yescharacteristic
620C2×C4C22×C4⋊C411yescharacteristic
621C2×C4C22×C4⋊C411yescharacteristic
622C2×C4C22×C4⋊C411yescharacteristic
623C2×C4C22×C4⋊C411yescharacteristic
624C2×C4C22×C4⋊C411yescharacteristic
625C2×C4C22×C4⋊C411yescharacteristic
626C2×C4C22×C4⋊C411yescharacteristic
627C2×C4C22×C4⋊C414yes—
628C2×C4C22×C4⋊C414yes—
629C2×C4C22×C4⋊C411yescharacteristic
630C2×C4C24×C411yescharacteristic
631C2×C4C23×D411yescharacteristic
632C2×C4C2×H(5,𝔽2)12yes—
633C2×C4#58512yes—
634C8—24no—
635C8—28no—
636C8—24no—
637C8—24no—
638C8—42no—
639C8—42no—
640C8—42no—
641C8—42no—
642C8—42no—
643C8—42no—
644C8—42no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—24no—
650C22—24no—
651C22—21no—
652C22—21no—
653C22—24no—
654C22—24no—
655C22—21no—
656C22—21no—
657C22—24no—
658C22—24no—
659C22—82no—
660C22—84no—
661C22—84no—
662C22C23×C4⋊C411yescharacteristic
663C22#350112yes—
664C4—21no—
665C4—21no—
666C4—24no—
667C4—24no—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4#291612yes—
676C4C4×H(5,𝔽2)12yes—
677C4#348411yescharacteristic
678C4#348511yescharacteristic
679C2—22no—
680C2—41no—
681C2—41no—
682C2—44no—
683C2—44no—
684C2#3424811yescharacteristic
685C2C2×#291611yescharacteristic
686C2#6011311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C23×C4⋊C4
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22719C2semidirect
5. #22718C2semidirect
6. #22717C2semidirect
7. #22669C2semidirect
8. #22668C2semidirect
9. D4×#108C2semidirect
10. #21994C2semidirect
11. #21993C2semidirect
12. #21950C2semidirect
13. #21371C2semidirect
14. #21370C2semidirect
15. #20852C2semidirect

Maximal subgroups

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

subgroupindex
#602322
C2×#29812
#601442
#227192
#227182
#227172
#226692
#226682
D4×#1082
#219942
#219932
#219502
#213712
#213702
#208522