Finite Group Databasebeta

← #219449

Subgroups — #219449

Order 512.

677 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#495C211yesmaximal, characteristic
2#21634C211yesmaximal, characteristic, complemented
3#21633C211yesmaximal, characteristic, complemented
4#9645C211yesmaximal, characteristic, complemented
5#9645C211yesmaximal, characteristic, complemented
6#8952C211yesmaximal, characteristic, complemented
7#8951C211yesmaximal, characteristic
8#8950C211yesmaximal, characteristic, complemented
9#8949C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic
12#8946C211yesmaximal, characteristic, complemented
13#8945C211yesmaximal, characteristic, complemented
14#8785C211yesmaximal, characteristic, complemented
15#8132C211yesmaximal, characteristic, complemented
16C2×#495C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×#436C2211yescharacteristic
24C2×C8×D4C2211yescharacteristic
25C4×#124C2211yescharacteristic
26C4×#124C2211yescharacteristic
27#1984C2211yescharacteristic
28#1984C2211yescharacteristic
29#1970C411yescharacteristic
30#1969C411yescharacteristic
31#1968C411yescharacteristic
32#1968C411yescharacteristic
33#1958C411yescharacteristic
34#1958C411yescharacteristic
35#1954C2211yescharacteristic
36#1954C2211yescharacteristic
37#1924—21no—
38#1924—21no—
39#1924—21no—
40#1924—21no—
41#1846C411yescharacteristic
42#1831C2211yescharacteristic
43#1831C2211yescharacteristic
44#1753—21no—
45#1753—21no—
46#1752—21no—
47#1752—21no—
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊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×Q16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C2×#124C2×C411yescharacteristic
74C2×#124C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×C4×D4C2311yescharacteristic
79#495—24no—
80#446—41no—
81#446—41no—
82C8×Q8—41no—
83C8×Q8—41no—
84C4×Q16—41no—
85C4×Q16—41no—
86C4×SD16—41no—
87C4×SD16—41no—
88C4×SD16—41no—
89C4×SD16—41no—
90C4×D8—41no—
91C4×D8—41no—
92#436—41no—
93#436—41no—
94C8×D4—41no—
95C8×D4—41no—
96C2×#107C2×C411yescharacteristic
97C2×#107C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99C2×#106C2×C411yescharacteristic
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×#97—21no—
121C2×#97D411yescharacteristic
122C2×#97D411yescharacteristic
123C2×#97D411yescharacteristic
124C2×#97D411yescharacteristic
125C2×C4×C8—21no—
126C2×C4×C8C2×C411yescharacteristic
127C2×C4×C8C2×C411yescharacteristic
128C2×C4×C8C2×C411yescharacteristic
129C2×C4×C8C2×C411yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136#391C2×C411yescharacteristic
137#391C2×C411yescharacteristic
138#390C2×C411yescharacteristic
139#390C2×C411yescharacteristic
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#382C2×C411yescharacteristic
145#382C2×C411yescharacteristic
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
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#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161Q8⋊C8—41no—
162Q8⋊C8—41no—
163Q8⋊C8—41no—
164Q8⋊C8—41no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168D4⋊C8—41no—
169C22×Q8C22×C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C22×D4C22×C411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C2×Q16—22no—
176C2×Q16—22no—
177C2×Q16—41no—
178C2×SD16—22no—
179C2×SD16—41no—
180C2×SD16—41no—
181C2×SD16—41no—
182C2×SD16—41no—
183C2×SD16M1614yescomplemented
184C2×D8—22no—
185C2×D8—22no—
186C2×D8—41no—
187C2×M16—42no—
188C2×M16—42no—
189C2×M16—42no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207#124—22no—
208#124—22no—
209#124—41no—
210#124—41no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42#4111yescharacteristic
251C2×C42C2411yescharacteristic
252#107—41no—
253#107—41no—
254#106—41no—
255#106—41no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266#103—41no—
267#103—41no—
268#103—41no—
269#103—41no—
270#102—41no—
271#102—41no—
272#102—41no—
273#102—41no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280#97—22no—
281#97—22no—
282#97—41no—
283#97—41no—
284#97—42no—
285C4×C8—22no—
286C4×C8—22no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—41no—
300C4×C8—41no—
301C4×C8—42no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310C24—21no—
311C24—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8C2×M1612yes—
324C2×D4—21no—
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×D4C2×M1612yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4C4×D411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361Q16—44no—
362SD16—44no—
363SD16—44no—
364D8—44no—
365M16—84no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—84no—
405C2×C8C2×M1612yes—
406C2×C8#13112yes—
407C2×C8#13512yes—
408C2×C8#13512yes—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C22⋊C4—81no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42H(5,𝔽2)11yescharacteristic
448C42H(5,𝔽2)11yescharacteristic
449C42H(5,𝔽2)11yescharacteristic
450C42H(5,𝔽2)11yescharacteristic
451C23—22no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23C2×C4×D411yescharacteristic
466Q8—22no—
467Q8—22no—
468Q8—41no—
469Q8—41no—
470Q8—41no—
471Q8—41no—
472Q8—41no—
473Q8—41no—
474D4—22no—
475D4—22no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—41no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4#51811yescharacteristic
523C2×C4#51811yescharacteristic
524C2×C4#53511yescharacteristic
525C2×C4#53511yescharacteristic
526C2×C4#54011yescharacteristic
527C2×C4#54011yescharacteristic
528C2×C4C22×M1611yescharacteristic
529C2×C4C2×#13111yescharacteristic
530C2×C4#56811yescharacteristic
531C2×C4#56811yescharacteristic
532C2×C4C2×#13511yescharacteristic
533C2×C4C2×#13511yescharacteristic
534C2×C4#57511yescharacteristic
535C2×C4#57511yescharacteristic
536C8—22no—
537C8—22no—
538C8—22no—
539C8—22no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—41no—
544C8—81no—
545C8—81no—
546C8—81no—
547C8—81no—
548C8—81no—
549C8—81no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22#232811yescharacteristic
565C22D4×M1611yescharacteristic
566C22#293511yescharacteristic
567C22C4×#13511yescharacteristic
568C22#294311yescharacteristic
569C22#295711yescharacteristic
570C22#295711yescharacteristic
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4#297311yescharacteristic
584C4#297411yescharacteristic
585C4#298011yescharacteristic
586C4#298111yescharacteristic
587C4#321111yescharacteristic
588C4#321111yescharacteristic
589C4#321211yescharacteristic
590C4#321211yescharacteristic
591C2—82no—
592C2—161no—
593C2#1803711yescharacteristic
594C2#1804111yescharacteristic
595C2#1806911yescharacteristic
596C2#1807511yescharacteristic
597C2#2226611yescharacteristic
598C2#2227111yescharacteristic
599C2#2228211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
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. #21634C2semidirect
3. #21633C2semidirect
4. #9645C2semidirect
5. #9645C2semidirect
6. #8952C2semidirect
8. #8950C2semidirect
9. #8949C2semidirect
10. #8948C2semidirect
12. #8946C2semidirect
13. #8945C2semidirect
14. #8785C2semidirect
15. #8132C2semidirect
183. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4952
#216342
#216332
#96452
#89522
#89512
#89502
#89492
#89482
#89472
#89462
#89452
#87852
#81322