Finite Group Databasebeta

← #231789

Subgroups — #231789

Order 512.

792 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
1#23796C211yesmaximal, characteristic
2#23205C211yesmaximal, characteristic, complemented
3#23173C211yesmaximal, characteristic, complemented
4C2×#1581C211yesmaximal, characteristic, complemented
5C2×SD16⋊C8C211yesmaximal, characteristic, complemented
6#9678C211yesmaximal, characteristic, complemented
7#9675C211yesmaximal, characteristic, complemented
8#9661C211yesmaximal, characteristic, complemented
9#9659C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8948C211yesmaximal, characteristic, complemented
12#8947C211yesmaximal, characteristic, complemented
13#8945C211yesmaximal, characteristic, complemented
14#8279C211yesmaximal, characteristic, complemented
15#8265C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×C8×D4—21no—
23C2×C8×D4C2211yescharacteristic
24C2×C8×D4C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C4×#124C2211yescharacteristic
27#2095C411yescharacteristic
28#2094C411yescharacteristic
29#2092C411yescharacteristic
30#2092C411yescharacteristic
31#2014C411yescharacteristic
32#2012C411yescharacteristic
33#2012C411yescharacteristic
34#2011C411yescharacteristic
35#1984C2211yescharacteristic
36#1954—21no—
37#1954C2211yescharacteristic
38#1954C2211yescharacteristic
39#1954C2211yescharacteristic
40#1924—21no—
41#1923—21no—
42#1869—21no—
43#1869—21no—
44#1851—21no—
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49#1753—21no—
50#1744C2211yescharacteristic
51#1581—24no—
52SD16⋊C8—24no—
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×D8C2×C411yescharacteristic
72C22×D8C2×C411yescharacteristic
73C22×M16—21no—
74C23×C8—21no—
75C23×C8—21no—
76C2×#127C2×C411yescharacteristic
77C2×#124C2×C411yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×C4×D4C2311yescharacteristic
84C8×Q8—22no—
85C8×Q8—41no—
86C4×SD16—22no—
87C4×SD16—22no—
88C4×SD16—41no—
89C4×SD16—41no—
90C4×D8—22no—
91C4×D8—22no—
92C4×D8—41no—
93C4×D8—41no—
94#436—22no—
95#436—41no—
96C8×D4—22no—
97C8×D4—22no—
98C8×D4—41no—
99C8×D4—41no—
100C8×D4—44no—
101C2×#107C2×C411yescharacteristic
102C2×#107C2×C411yescharacteristic
103C2×#106C2×C411yescharacteristic
104C2×#106C2×C411yescharacteristic
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8D411yescharacteristic
108C2×C4⋊C8D411yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×#97—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139#391C2×C411yescharacteristic
140#390—21no—
141#390—21no—
142#390C2×C411yescharacteristic
143#390C2×C411yescharacteristic
144#390C2×C411yescharacteristic
145#388—21no—
146#388—21no—
147#384C2×C411yescharacteristic
148#382C2×C411yescharacteristic
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C43C2311yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336C2×C411yescharacteristic
158#336C2×C411yescharacteristic
159#336C2×C411yescharacteristic
160#336C2×C411yescharacteristic
161#336D411yescharacteristic
162#336D411yescharacteristic
163#335D414yescomplemented
164#334—22no—
165Q8⋊C8—22no—
166Q8⋊C8—41no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169D4⋊C8—22no—
170D4⋊C8—41no—
171D4⋊C8—41no—
172D4⋊C8—41no—
173#322—22no—
174#322—22no—
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C22×D4C22×C411yescharacteristic
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C2×SD16—22no—
185C2×SD16—22no—
186C2×SD16—22no—
187C2×SD16—22no—
188C2×SD16—22no—
189C2×SD16—41no—
190C2×D8—22no—
191C2×D8—22no—
192C2×D8—22no—
193C2×D8—41no—
194C2×D8—41no—
195C2×D8—41no—
196C2×M16—42no—
197C2×M16—42no—
198C2×M16—81no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—81no—
210C22×C8—81no—
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219#127—22no—
220#127—22no—
221#124—22no—
222#124—41no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4—81no—
242C4×D4—81no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C42—21no—
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42C2411yescharacteristic
269#107—22no—
270#107—41no—
271#106—22no—
272#106—41no—
273C4⋊C8—22no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—41no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283C4⋊C8C2×D412yes—
284C4⋊C8C2×D412yes—
285#103—22no—
286#103—41no—
287#102—22no—
288#102—22no—
289#102—22no—
290#102—41no—
291#102—41no—
292#102—41no—
293C22⋊C8—42no—
294C22⋊C8—42no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299C22⋊C8—81no—
300C22⋊C8—81no—
301C22⋊C8—81no—
302#97—81no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—42no—
314C4×C8—42no—
315C4×C8C2×D412yes—
316C4×C8#4112yes—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8#13112yes—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—81no—
351C2×D4—81no—
352C2×D4#13112yes—
353C2×D4#13112yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4C4×D411yescharacteristic
378C22×C4C4×D411yescharacteristic
379C22×C4C4×D411yescharacteristic
380C22×C4C4×D411yescharacteristic
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384SD16—28no—
385SD16—44no—
386D8—44no—
387D8—44no—
388M16—84no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—84no—
429C2×C8—84no—
430C2×C8C4×D412yes—
431C2×C8C4×D412yes—
432C2×C8AGL(1,ℤ/8)12yes—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—42no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—81no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—42no—
456C22⋊C4—81no—
457C22⋊C4—81no—
458C22⋊C4—81no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42C22×D411yescharacteristic
472C42C2×#4111yescharacteristic
473C42H(5,𝔽2)11yescharacteristic
474C42H(5,𝔽2)11yescharacteristic
475C23—22no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C2×C4×D411yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—41no—
500Q8—41no—
501D4—22no—
502D4—22no—
503D4—22no—
504D4—22no—
505D4—48no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—84no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4#44412yes—
556C2×C4C2×C4×D411yescharacteristic
557C2×C4#51811yescharacteristic
558C2×C4D4211yescharacteristic
559C2×C4#55011yescharacteristic
560C2×C4#55011yescharacteristic
561C2×C4#55911yescharacteristic
562C2×C4C2×#13111yescharacteristic
563C2×C4C2×#13111yescharacteristic
564C2×C4C2×#13111yescharacteristic
565C2×C4#56811yescharacteristic
566C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
567C2×C4#57511yescharacteristic
568C2×C4#57611yescharacteristic
569C2×C4#57711yescharacteristic
570C8—22no—
571C8—22no—
572C8—22no—
573C8—41no—
574C8—41no—
575C8—41no—
576C8—42no—
577C8—42no—
578C8—41no—
579C8—41no—
580C8—81no—
581C8—81no—
582C8—81no—
583C8—81no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22#236611yescharacteristic
606C22#292911yescharacteristic
607C22#293311yescharacteristic
608C22C4×AGL(1,ℤ/8)11yescharacteristic
609C22#294611yescharacteristic
610C22C2×#44411yescharacteristic
611C22#295711yescharacteristic
612C4—21no—
613C4—21no—
614C4—22no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—41no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4#292911yescharacteristic
624C4#293311yescharacteristic
625C4#297611yescharacteristic
626C4#297711yescharacteristic
627C4#326411yescharacteristic
628C4#327111yescharacteristic
629C4#339711yescharacteristic
630C4#340311yescharacteristic
631C2—82no—
632C2—82no—
633C2—161no—
634C2#1791411yescharacteristic
635C2#1791811yescharacteristic
636C2#1801111yescharacteristic
637C2#1802511yescharacteristic
638C2#2317711yescharacteristic
639C2#2318011yescharacteristic
640C2#2319511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#2011C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23205C2semidirect
3. #23173C2semidirect
4. C2×#1581C2semidirect
5. C2×SD16⋊C8C2semidirect
6. #9678C2semidirect
7. #9675C2semidirect
8. #9661C2semidirect
9. #9659C2semidirect
10. #8951C2semidirect
11. #8948C2semidirect
12. #8947C2semidirect
13. #8945C2semidirect
14. #8279C2semidirect
15. #8265C2semidirect
163. #335D4semidirect

Maximal subgroups

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

subgroupindex
#237962
#232052
#231732
C2×#15812
C2×SD16⋊C82
#96782
#96752
#96612
#96592
#89512
#89482
#89472
#89452
#82792
#82652