Finite Group Databasebeta

← #219467

Subgroups — #219467

Order 512.

699 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22262C211yesmaximal, characteristic
2#22240C211yesmaximal, characteristic, complemented
3C4×#492C211yesmaximal, characteristic
4#9648C211yesmaximal, characteristic
5#9647C211yesmaximal, characteristic, complemented
6#8952C211yesmaximal, characteristic, complemented
7#8952C211yesmaximal, characteristic, complemented
8#8949C211yesmaximal, characteristic, complemented
9#8949C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8948C211yesmaximal, characteristic, complemented
12#8945C211yesmaximal, characteristic, complemented
13#8945C211yesmaximal, characteristic, complemented
14#8785C211yesmaximal, characteristic, complemented
15#8132C211yesmaximal, characteristic, complemented
16C2×#492C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×#436C2211yescharacteristic
24C2×C8×D4C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#127C2211yescharacteristic
27#1984C2211yescharacteristic
28#1984C2211yescharacteristic
29#1968C411yescharacteristic
30#1968C411yescharacteristic
31#1968C411yescharacteristic
32#1968C411yescharacteristic
33#1957C411yescharacteristic
34#1956C411yescharacteristic
35#1954C2211yescharacteristic
36#1954C2211yescharacteristic
37#1917—21no—
38#1917—21no—
39#1916—21no—
40#1916—21no—
41#1870—21no—
42#1870—21no—
43#1869—21no—
44#1869—21no—
45#1844C411yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
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×SD16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×#127C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81#492—24no—
82#446—41no—
83#446—41no—
84C8×Q8—41no—
85C8×Q8—41no—
86C4×SD16—41no—
87C4×SD16—41no—
88C4×SD16—41no—
89C4×SD16—41no—
90C4×SD16—41no—
91C4×SD16—41no—
92C4×SD16—41no—
93C4×SD16—41no—
94#436—41no—
95#436—41no—
96C8×D4—41no—
97C8×D4—41no—
98C2×#106C2×C411yescharacteristic
99C2×#106C2×C411yescharacteristic
100C2×#106C2×C411yescharacteristic
101C2×#106C2×C411yescharacteristic
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×#97—21no—
123C2×#97—21no—
124C2×#97—21no—
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#390—21no—
139#390—21no—
140#390C2×C411yescharacteristic
141#390C2×C411yescharacteristic
142#389—21no—
143#389—21no—
144#384C2×C411yescharacteristic
145#384C2×C411yescharacteristic
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C43C2311yescharacteristic
149#336—21no—
150#336—21no—
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—
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×D4—21no—
172C22×D4—21no—
173C22×D4C22×C411yescharacteristic
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C2×SD16—22no—
178C2×SD16—22no—
179C2×SD16—22no—
180C2×SD16—22no—
181C2×SD16—22no—
182C2×SD16—41no—
183C2×SD16—41no—
184C2×SD16—41no—
185C2×SD16—41no—
186C2×SD16—41no—
187C2×SD16—41no—
188C2×SD16M1614yescomplemented
189C2×M16—42no—
190C2×M16—81no—
191C2×M16—81no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—42no—
199C22×C8—81no—
200C22×C8—81no—
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C22×C411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C4⋊Q8—22no—
210C4⋊Q8—41no—
211#127—22no—
212#127—41no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
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—81no—
230C4×D4—81no—
231C4×D4—81no—
232C4×D4—81no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
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×C42—21no—
252C2×C42—21no—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42#4111yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42C2411yescharacteristic
260#106—41no—
261#106—41no—
262#106—41no—
263#106—41no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—42no—
273C4⋊C8—42no—
274#103—41no—
275#103—41no—
276#103—41no—
277#103—41no—
278#102—41no—
279#102—41no—
280#102—41no—
281#102—41no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284C22⋊C8—81no—
285C22⋊C8—81no—
286C22⋊C8—81no—
287C22⋊C8—81no—
288#97—24no—
289#97—42no—
290#97—42no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—41no—
306C4×C8—41no—
307C4×C8—42no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312C24—21no—
313C24—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×Q8—21no—
324C2×Q8—21no—
325C2×Q8C2×M1612yes—
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—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—81no—
342C2×D4—81no—
343C2×D4C2×M1612yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
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×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369SD16—44no—
370SD16—44no—
371SD16—44no—
372SD16—44no—
373M16—84no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
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—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—84no—
415C2×C8C2×M1612yes—
416C2×C8#13112yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—22no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42H(5,𝔽2)11yescharacteristic
458C42H(5,𝔽2)11yescharacteristic
459C42H(5,𝔽2)11yescharacteristic
460C42H(5,𝔽2)11yescharacteristic
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23C2×C4×D411yescharacteristic
476Q8—22no—
477Q8—22no—
478Q8—41no—
479Q8—41no—
480Q8—41no—
481Q8—41no—
482Q8—41no—
483Q8—41no—
484D4—22no—
485D4—22no—
486D4—41no—
487D4—41no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—84no—
493D4—84no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4#44312yes—
533C2×C4#44412yes—
534C2×C4#51811yescharacteristic
535C2×C4#51811yescharacteristic
536C2×C4#53511yescharacteristic
537C2×C4#53511yescharacteristic
538C2×C4#54011yescharacteristic
539C2×C4#54011yescharacteristic
540C2×C4C22×M1611yescharacteristic
541C2×C4C2×#13111yescharacteristic
542C2×C4#56811yescharacteristic
543C2×C4#56811yescharacteristic
544C2×C4#57711yescharacteristic
545C2×C4#57711yescharacteristic
546C2×C4#57711yescharacteristic
547C2×C4#57711yescharacteristic
548C8—22no—
549C8—22no—
550C8—42no—
551C8—42no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—41no—
558C8—81no—
559C8—81no—
560C8—81no—
561C8—81no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22#232811yescharacteristic
577C22D4×M1611yescharacteristic
578C22#293511yescharacteristic
579C22#294911yescharacteristic
580C22#294911yescharacteristic
581C22C2×#44311yescharacteristic
582C22C2×#44411yescharacteristic
583C4—21no—
584C4—21no—
585C4—22no—
586C4—22no—
587C4—41no—
588C4—41no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4#297311yescharacteristic
596C4#297411yescharacteristic
597C4#298011yescharacteristic
598C4#298111yescharacteristic
599C4#322111yescharacteristic
600C4#322111yescharacteristic
601C4#322211yescharacteristic
602C4#322211yescharacteristic
603C2—82no—
604C2—161no—
605C2#1805611yescharacteristic
606C2#1806011yescharacteristic
607C2#1808511yescharacteristic
608C2#1808911yescharacteristic
609C2#2226611yescharacteristic
610C2#2227511yescharacteristic
611C2#2228011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#127C2×C4
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
2. #22240C2semidirect
5. #9647C2semidirect
6. #8952C2semidirect
7. #8952C2semidirect
8. #8949C2semidirect
9. #8949C2semidirect
10. #8948C2semidirect
11. #8948C2semidirect
12. #8945C2semidirect
13. #8945C2semidirect
14. #8785C2semidirect
15. #8132C2semidirect
188. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
#222622
#222402
C4×#4922
#96482
#96472
#89522
#89492
#89482
#89452
#87852
#81322