Finite Group Databasebeta

← #219472

Subgroups — #219472

Order 512.

712 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22262C211yesmaximal, characteristic, complemented
2#22210C211yesmaximal, characteristic, complemented
3C4×#496C211yesmaximal, characteristic
4#9645C211yesmaximal, characteristic, complemented
5#9644C211yesmaximal, 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#8786C211yesmaximal, characteristic, complemented
15#8134C211yesmaximal, characteristic, complemented
16C2×#496C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C2×#436C2211yescharacteristic
24C2×C8×D4C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C4×#124C2211yescharacteristic
27#1984C2211yescharacteristic
28#1973C411yescharacteristic
29#1973C411yescharacteristic
30#1972C411yescharacteristic
31#1972C411yescharacteristic
32#1961C411yescharacteristic
33#1959C411yescharacteristic
34#1954C2211yescharacteristic
35#1954C2211yescharacteristic
36#1954C2211yescharacteristic
37#1923—21no—
38#1923—21no—
39#1870—21no—
40#1870—21no—
41#1869—21no—
42#1869—21no—
43#1851—21no—
44#1851—21no—
45#1847C411yescharacteristic
46#1842C2211yescharacteristic
47#1831C2211yescharacteristic
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
82#496—24no—
83C8×Q8—41no—
84C8×Q8—41no—
85C4×SD16—41no—
86C4×SD16—41no—
87C4×SD16—41no—
88C4×SD16—41no—
89C4×D8—41no—
90C4×D8—41no—
91C4×D8—41no—
92C4×D8—41no—
93#436—41no—
94#436—41no—
95#436—41no—
96#436—41no—
97C8×D4—41no—
98C8×D4—41no—
99C2×#107C2×C411yescharacteristic
100C2×#107C2×C411yescharacteristic
101C2×#106C2×C411yescharacteristic
102C2×#106C2×C411yescharacteristic
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×C22⋊C8—21no—
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×#97C2×C411yescharacteristic
128C2×#97C2×C411yescharacteristic
129C2×#97C2×C411yescharacteristic
130C2×#97C2×C411yescharacteristic
131C2×C4×C8—21no—
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138#391C2×C411yescharacteristic
139#390—21no—
140#390—21no—
141#390C2×C411yescharacteristic
142#390C2×C411yescharacteristic
143#390C2×C411yescharacteristic
144#388—21no—
145#388—21no—
146#384C2×C411yescharacteristic
147#382C2×C411yescharacteristic
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C43C2311yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336D411yescharacteristic
158#336D411yescharacteristic
159#336D411yescharacteristic
160#336D411yescharacteristic
161Q8⋊C8—41no—
162Q8⋊C8—41no—
163D4⋊C8—41no—
164D4⋊C8—41no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168D4⋊C8—41no—
169C22×Q8C22×C411yescharacteristic
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C2×SD16—22no—
179C2×SD16—22no—
180C2×SD16—41no—
181C2×SD16—41no—
182C2×SD16—41no—
183C2×SD16M1614yescomplemented
184C2×D8—22no—
185C2×D8—22no—
186C2×D8—22no—
187C2×D8—41no—
188C2×D8—41no—
189C2×D8—41no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—81no—
193C2×M16—81no—
194C2×M16—81no—
195C2×M16—81no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—42no—
203C22×C8—81no—
204C22×C8—81no—
205C22×C8C22×C411yescharacteristic
206C22×C8C22×C411yescharacteristic
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213#127—22no—
214#127—41no—
215#124—22no—
216#124—41no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×D4—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—81no—
234C4×D4—81no—
235C4×D4—81no—
236C4×D4—81no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C22⋊C4—21no—
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×C42—21no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42C2411yescharacteristic
263#107—41no—
264#107—41no—
265#106—41no—
266#106—41no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—42no—
276C4⋊C8—81no—
277C4⋊C8—81no—
278#103—41no—
279#103—41no—
280#102—41no—
281#102—41no—
282#102—41no—
283#102—41no—
284#102—41no—
285#102—41no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289C22⋊C8—81no—
290C22⋊C8—81no—
291C22⋊C8—81no—
292C22⋊C8—81no—
293C22⋊C8—81no—
294C22⋊C8—81no—
295#97—22no—
296#97—22no—
297#97—41no—
298#97—41no—
299C4×C8—21no—
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—42no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8C2×M1612yes—
325C2×D4—21no—
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—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4C2×M1612yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—41no—
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×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379SD16—44no—
380SD16—44no—
381D8—44no—
382D8—44no—
383M16—84no—
384M16—84no—
385C2×C8—21no—
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—41no—
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—84no—
426C2×C8C2×M1612yes—
427C2×C8#13112yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—81no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C22⋊C4—81no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42H(5,𝔽2)11yescharacteristic
467C42H(5,𝔽2)11yescharacteristic
468C42H(5,𝔽2)11yescharacteristic
469C42H(5,𝔽2)11yescharacteristic
470C23—22no—
471C23—41no—
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—
491C23C2×C4×D411yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—84no—
509D4—84no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
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—81no—
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#51811yescharacteristic
552C2×C4#51811yescharacteristic
553C2×C4#53511yescharacteristic
554C2×C4#53511yescharacteristic
555C2×C4#54011yescharacteristic
556C2×C4#54011yescharacteristic
557C2×C4C22×M1611yescharacteristic
558C2×C4C2×#13111yescharacteristic
559C2×C4#56811yescharacteristic
560C2×C4#56811yescharacteristic
561C2×C4#57611yescharacteristic
562C2×C4#57611yescharacteristic
563C2×C4#57711yescharacteristic
564C2×C4#57711yescharacteristic
565C8—22no—
566C8—22no—
567C8—41no—
568C8—41no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—41no—
573C8—81no—
574C8—81no—
575C8—81no—
576C8—81no—
577C8—81no—
578C8—81no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
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#232811yescharacteristic
601C22D4×M1611yescharacteristic
602C22#293511yescharacteristic
603C22#294611yescharacteristic
604C22#294611yescharacteristic
605C22#295511yescharacteristic
606C22#295511yescharacteristic
607C4—21no—
608C4—21no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4#297311yescharacteristic
619C4#297411yescharacteristic
620C4#298011yescharacteristic
621C4#298111yescharacteristic
622C4#322011yescharacteristic
623C4#322111yescharacteristic
624C4#322411yescharacteristic
625C4#322411yescharacteristic
626C2—82no—
627C2—161no—
628C2—161no—
629C2#1805211yescharacteristic
630C2#1805211yescharacteristic
631C2#1808011yescharacteristic
632C2#1808211yescharacteristic
633C2#2226611yescharacteristic
634C2#2227911yescharacteristic
635C2#2228111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#496C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22262C2semidirect
2. #22210C2semidirect
4. #9645C2semidirect
5. #9644C2semidirect
6. #8951C2semidirect
7. #8951C2semidirect
8. #8951C2semidirect
9. #8949C2semidirect
10. #8948C2semidirect
11. #8948C2semidirect
12. #8947C2semidirect
13. #8945C2semidirect
14. #8786C2semidirect
15. #8134C2semidirect
183. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
#222622
#222102
C4×#4962
#96452
#96442
#89512
#89492
#89482
#89472
#89452
#87862
#81342