Finite Group Databasebeta

← #219460

Subgroups — #219460

Order 512.

734 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23796C211yesmaximal, characteristic
2#21682C211yesmaximal, characteristic, complemented
3#21634C211yesmaximal, characteristic, complemented
4#9642C211yesmaximal, characteristic, complemented
5#9640C211yesmaximal, characteristic, complemented
6#8951C211yesmaximal, characteristic, complemented
7#8951C211yesmaximal, characteristic, complemented
8#8951C211yesmaximal, characteristic, complemented
9#8949C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8948C211yesmaximal, characteristic, complemented
12#8947C211yesmaximal, characteristic, complemented
13#8945C211yesmaximal, characteristic, complemented
14#8789C211yesmaximal, characteristic, complemented
15#8134C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C2×#435—21no—
24C2×#435—21no—
25C2×C8×D4C2211yescharacteristic
26C4×#127C2211yescharacteristic
27C4×#124C2211yescharacteristic
28#2095C411yescharacteristic
29#2094C411yescharacteristic
30#2092C411yescharacteristic
31#2092C411yescharacteristic
32#2014C411yescharacteristic
33#2012C411yescharacteristic
34#2012C411yescharacteristic
35#2011C411yescharacteristic
36#1984C2211yescharacteristic
37#1955—21no—
38#1955—21no—
39#1954C2211yescharacteristic
40#1954C2211yescharacteristic
41#1954C2211yescharacteristic
42#1924—21no—
43#1924—21no—
44#1842C2211yescharacteristic
45#1831C2211yescharacteristic
46#1753—21no—
47#1753—21no—
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
50C4×#97C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C22×M16—21no—
73C23×C8—21no—
74C2×#127C2×C411yescharacteristic
75C2×#124C2×C411yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C8×Q8—41no—
83C8×Q8—41no—
84C4×SD16—41no—
85C4×SD16—41no—
86C4×SD16—41no—
87C4×SD16—41no—
88C4×D8—41no—
89C4×D8—41no—
90C4×D8—41no—
91C4×D8—41no—
92#436—41no—
93#436—41no—
94#436—41no—
95#436—41no—
96#435—44no—
97#435—44no—
98C8×D4—41no—
99C8×D4—41no—
100C2×#107C2×C411yescharacteristic
101C2×#107C2×C411yescharacteristic
102C2×#106C2×C411yescharacteristic
103C2×#106C2×C411yescharacteristic
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
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×#97D411yescharacteristic
129C2×#97D411yescharacteristic
130C2×#97D411yescharacteristic
131C2×#97D411yescharacteristic
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#336—21no—
158#336C2×C411yescharacteristic
159#336C2×C411yescharacteristic
160#336C2×C411yescharacteristic
161#336C2×C411yescharacteristic
162Q8⋊C8—41no—
163Q8⋊C8—41no—
164D4⋊C8—41no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168D4⋊C8—41no—
169D4⋊C8—41no—
170C22×Q8C22×C411yescharacteristic
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C22×C411yescharacteristic
174C22×D4C22×C411yescharacteristic
175C22×D4C22×C411yescharacteristic
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C2×SD16—22no—
180C2×SD16—22no—
181C2×SD16—22no—
182C2×SD16—41no—
183C2×SD16—41no—
184C2×SD16—41no—
185C2×D8—22no—
186C2×D8—41no—
187C2×D8—41no—
188C2×D8—41no—
189C2×D8—41no—
190C2×D8—41no—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16—42no—
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—42no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8C22×C411yescharacteristic
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214#127—22no—
215#127—41no—
216#124—22no—
217#124—41no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—42no—
235C4×D4—42no—
236C4×D4—42no—
237C4×D4—42no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
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×C42—21no—
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42C2411yescharacteristic
264#107—41no—
265#107—41no—
266#106—41no—
267#106—41no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—42no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279#103—41no—
280#103—41no—
281#102—41no—
282#102—41no—
283#102—41no—
284#102—41no—
285#102—41no—
286#102—41no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293C22⋊C8—42no—
294C22⋊C8—42no—
295C22⋊C8—42no—
296#97—22no—
297#97—22no—
298#97—41no—
299#97—41no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
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—42no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8#13112yes—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
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—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—81no—
348C2×D4—81no—
349C2×D4#13112yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
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×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4111yescharacteristic
380SD16—44no—
381SD16—44no—
382D8—44no—
383D8—44no—
384M16—84no—
385M16—84no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
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—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
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—84no—
427C2×C8AGL(1,ℤ/8)12yes—
428C2×C8AGL(1,ℤ/8)12yes—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—81no—
454C22⋊C4—81no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42H(5,𝔽2)11yescharacteristic
468C42H(5,𝔽2)11yescharacteristic
469C42H(5,𝔽2)11yescharacteristic
470C42H(5,𝔽2)11yescharacteristic
471C23—22no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
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—
492C23C2×C4×D411yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497D4—22no—
498D4—22no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—84no—
510D4—84no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
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—81no—
541C2×C4—81no—
542C2×C4—81no—
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#51811yescharacteristic
553C2×C4#51811yescharacteristic
554C2×C4#53511yescharacteristic
555C2×C4#53511yescharacteristic
556C2×C4#54011yescharacteristic
557C2×C4#54011yescharacteristic
558C2×C4C2×#13111yescharacteristic
559C2×C4C2×#13111yescharacteristic
560C2×C4#56811yescharacteristic
561C2×C4#56811yescharacteristic
562C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
563C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
564C2×C4#57511yescharacteristic
565C2×C4#57511yescharacteristic
566C8—22no—
567C8—22no—
568C8—41no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—81no—
575C8—81no—
576C8—81no—
577C8—81no—
578C8—81no—
579C8—81no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22#232811yescharacteristic
602C22#293311yescharacteristic
603C22#293311yescharacteristic
604C22C4×AGL(1,ℤ/8)11yescharacteristic
605C22C4×AGL(1,ℤ/8)11yescharacteristic
606C22#295711yescharacteristic
607C22#295711yescharacteristic
608C4—21no—
609C4—21no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4#297611yescharacteristic
620C4#297611yescharacteristic
621C4#297711yescharacteristic
622C4#297811yescharacteristic
623C4#321311yescharacteristic
624C4#321411yescharacteristic
625C4#321711yescharacteristic
626C4#321911yescharacteristic
627C2—82no—
628C2—161no—
629C2—161no—
630C2#1800611yescharacteristic
631C2#1800811yescharacteristic
632C2#1801111yescharacteristic
633C2#1801111yescharacteristic
634C2#2226711yescharacteristic
635C2#2227211yescharacteristic
636C2#2228211yescharacteristic
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×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21682C2semidirect
3. #21634C2semidirect
4. #9642C2semidirect
5. #9640C2semidirect
6. #8951C2semidirect
7. #8951C2semidirect
8. #8951C2semidirect
9. #8949C2semidirect
10. #8948C2semidirect
11. #8948C2semidirect
12. #8947C2semidirect
13. #8945C2semidirect
14. #8789C2semidirect
15. #8134C2semidirect

Maximal subgroups

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

subgroupindex
#237962
#216822
#216342
#96422
#96402
#89512
#89492
#89482
#89472
#89452
#87892
#81342