Finite Group Databasebeta

← #228134

Subgroups — #228134

Order 512.

801 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23569C211yesmaximal, characteristic
2#22990C211yesmaximal, characteristic, complemented
3#22474C211yesmaximal, characteristic, complemented
4C2×#1589C211yesmaximal, characteristic, complemented
5C2×#1495C211yesmaximal, characteristic, complemented
6#9660C211yesmaximal, characteristic, complemented
7#9553C211yesmaximal, characteristic, complemented
8#9544C211yesmaximal, characteristic, complemented
9#9074C211yesmaximal, characteristic, complemented
10#8949C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8847C211yesmaximal, characteristic, complemented
13#8832C211yesmaximal, characteristic, complemented
14#8200C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436—21no—
20C2×#436C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#2059C411yescharacteristic
27#2042C411yescharacteristic
28#2036C411yescharacteristic
29#2034C411yescharacteristic
30#2025C411yescharacteristic
31#2012C411yescharacteristic
32#2000C411yescharacteristic
33#1999C411yescharacteristic
34#1984C2211yescharacteristic
35#1984C2211yescharacteristic
36#1954C2211yescharacteristic
37#1939C2211yescharacteristic
38#1923—21no—
39#1870—21no—
40#1870—21no—
41#1869—21no—
42#1851—21no—
43#1818—21no—
44#1797C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1729—21no—
51#1589—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54#1495—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C22×M16—21no—
73C22×M16—21no—
74C23×C8—21no—
75#529—22no—
76#529—22no—
77#529—41no—
78#529—41no—
79C2×#126—21no—
80C2×#125C2×C411yescharacteristic
81C2×#124C2×C411yescharacteristic
82C2×#123—21no—
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×C4×Q8C2311yescharacteristic
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4C2311yescharacteristic
90C2×C4×D4C2311yescharacteristic
91C2×#117—21no—
92C22×C42C2311yescharacteristic
93C4×SD16—41no—
94C4×SD16—41no—
95C4×D8—22no—
96C4×D8—22no—
97#436—22no—
98#436—22no—
99#436—41no—
100#436—41no—
101#436—44no—
102C8×D4—22no—
103C8×D4—41no—
104#431—22no—
105#431—41no—
106C2×#107C2×C411yescharacteristic
107C2×#106C2×C411yescharacteristic
108#423—22no—
109#423—41no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#103C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×#102C2×C411yescharacteristic
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8C2×C411yescharacteristic
133C2×C22⋊C8C2×C411yescharacteristic
134C2×C22⋊C8D411yescharacteristic
135C2×C22⋊C8D411yescharacteristic
136C2×C4×C8—21no—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143#390C2×C411yescharacteristic
144#389C2×C411yescharacteristic
145#388—21no—
146#386—21no—
147#380C2×C411yescharacteristic
148C4×C4⋊C4C2311yescharacteristic
149C4×C22⋊C4—21no—
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336C2×C411yescharacteristic
162#336C2×C411yescharacteristic
163#336D411yescharacteristic
164#336D411yescharacteristic
165#335—22no—
166#335—41no—
167Q8⋊C8—22no—
168Q8⋊C8—22no—
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—41no—
172D4⋊C8D414yescomplemented
173#322—41no—
174C82—22no—
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C22×D4C22×C411yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4#4111yescharacteristic
185C2×SD16—22no—
186C2×SD16—41no—
187C2×SD16—41no—
188C2×D8—22no—
189C2×D8—22no—
190C2×D8—41no—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16—42no—
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—81no—
198C2×M16—81no—
199C2×M16—81no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—81no—
209C22×C8C22×C411yescharacteristic
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217#126—42no—
218#126—42no—
219#125—22no—
220#125—41no—
221#124—22no—
222#124—41no—
223#123—42no—
224#123—42no—
225C22⋊Q8—22no—
226C22⋊Q8—41no—
227#121—22no—
228#121—22no—
229#121—41no—
230#121—41no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4C2×D412yes—
246C4×D4#4112yes—
247#117—42no—
248#117—42no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42C2411yescharacteristic
279#107—22no—
280#106—41no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—42no—
289C4⋊C8—42no—
290C4⋊C8—42no—
291C4⋊C8C2×D412yes—
292#103—22no—
293#103—22no—
294#103—41no—
295#102—22no—
296#102—22no—
297#102—22no—
298#102—22no—
299#102—41no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—41no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309C22⋊C8—81no—
310C22⋊C8—81no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—42no—
323C4×C8C2×D412yes—
324#95—21no—
325#95—21no—
326#95—21no—
327C24—21no—
328C24—21no—
329C24C4×D411yescharacteristic
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—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—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—81no—
352C2×D4—81no—
353C2×D4C4×D412yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4C4×D411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4111yescharacteristic
394SD16—44no—
395D8—44no—
396M16—84no—
397M16—84no—
398M16—84no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—84no—
439C2×C8#13112yes—
440C2×C8#13112yes—
441C2×C8AGL(1,ℤ/8)12yes—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4C4×D412yes—
457C4⋊C4#13112yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42C22×D411yescharacteristic
481C42C2×#4111yescharacteristic
482C42H(5,𝔽2)11yescharacteristic
483C42H(5,𝔽2)11yescharacteristic
484C23—22no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C2×C4×D411yescharacteristic
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510D4—22no—
511D4—22no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—84no—
519D4—84no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4C2×C4×D411yescharacteristic
568C2×C4#51811yescharacteristic
569C2×C4D4211yescharacteristic
570C2×C4#54611yescharacteristic
571C2×C4#54611yescharacteristic
572C2×C4#55111yescharacteristic
573C2×C4C2×#13111yescharacteristic
574C2×C4C2×#13111yescharacteristic
575C2×C4C2×#13111yescharacteristic
576C2×C4#56811yescharacteristic
577C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
578C2×C4#57511yescharacteristic
579C2×C4#57611yescharacteristic
580C2×C4#57811yescharacteristic
581C8—22no—
582C8—22no—
583C8—22no—
584C8—42no—
585C8—41no—
586C8—41no—
587C8—41no—
588C8—42no—
589C8—41no—
590C8—41no—
591C8—81no—
592C8—81no—
593C8—81no—
594C8—81no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—42no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—41no—
603C22—42no—
604C22—41no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22#235811yescharacteristic
617C22#293111yescharacteristic
618C22#293311yescharacteristic
619C22C4×AGL(1,ℤ/8)11yescharacteristic
620C22#295111yescharacteristic
621C22#295511yescharacteristic
622C22#295711yescharacteristic
623C4—21no—
624C4—21no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4#293111yescharacteristic
635C4#293311yescharacteristic
636C4#297611yescharacteristic
637C4#297911yescharacteristic
638C4#326411yescharacteristic
639C4#326911yescharacteristic
640C4#332211yescharacteristic
641C4#333011yescharacteristic
642C2—81no—
643C2—82no—
644C2—161no—
645C2#1796611yescharacteristic
646C2#1797311yescharacteristic
647C2#1800811yescharacteristic
648C2#1803311yescharacteristic
649C2#2285311yescharacteristic
650C2#2287811yescharacteristic
651C2#2288911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#1999C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22990C2semidirect
3. #22474C2semidirect
4. C2×#1589C2semidirect
5. C2×#1495C2semidirect
6. #9660C2semidirect
7. #9553C2semidirect
8. #9544C2semidirect
9. #9074C2semidirect
10. #8949C2semidirect
11. #8947C2semidirect
12. #8847C2semidirect
13. #8832C2semidirect
14. #8200C2semidirect
15. #8148C2semidirect
172. D4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#235692
#229902
#224742
C2×#15892
C2×#14952
#96602
#95532
#95442
#90742
#89492
#89472
#88472
#88322
#82002
#81482