Finite Group Databasebeta

← #219477

Subgroups — #219477

Order 512.

665 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23799C211yesmaximal, characteristic
2#22262C211yesmaximal, characteristic
3#22240C211yesmaximal, characteristic, complemented
4#9642C211yesmaximal, characteristic
5#9641C211yesmaximal, 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#8788C211yesmaximal, characteristic, complemented
15#8135C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#127C2211yescharacteristic
26#2096C411yescharacteristic
27#2096C411yescharacteristic
28#2094C411yescharacteristic
29#2094C411yescharacteristic
30#2014C411yescharacteristic
31#2014C411yescharacteristic
32#2013C411yescharacteristic
33#2013C411yescharacteristic
34#1984C2211yescharacteristic
35#1984C2211yescharacteristic
36#1954C2211yescharacteristic
37#1954C2211yescharacteristic
38#1917—21no—
39#1917—21no—
40#1916—21no—
41#1916—21no—
42#1870—21no—
43#1870—21no—
44#1869—21no—
45#1869—21no—
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
50#1744C2211yescharacteristic
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#446—41no—
82#446—41no—
83C8×Q8—41no—
84C8×Q8—41no—
85C4×SD16—41no—
86C4×SD16—41no—
87C4×SD16—41no—
88C4×SD16—41no—
89C4×SD16—41no—
90C4×SD16—41no—
91C4×SD16—41no—
92C4×SD16—41no—
93#436—41no—
94#436—41no—
95C8×D4—41no—
96C8×D4—41no—
97C2×#106C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99C2×#106C2×C411yescharacteristic
100C2×#106C2×C411yescharacteristic
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#391C2×C411yescharacteristic
130#391C2×C411yescharacteristic
131#390—21no—
132#390—21no—
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#389—21no—
136#389—21no—
137#384C2×C411yescharacteristic
138#384C2×C411yescharacteristic
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C43C2311yescharacteristic
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336C2×C411yescharacteristic
153#336C2×C411yescharacteristic
154#336C2×C411yescharacteristic
155#336C2×C411yescharacteristic
156#336D411yescharacteristic
157#336D411yescharacteristic
158#336D411yescharacteristic
159#336D411yescharacteristic
160Q8⋊C8—41no—
161Q8⋊C8—41no—
162Q8⋊C8—41no—
163Q8⋊C8—41no—
164D4⋊C8—41no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168C22×Q8C22×C411yescharacteristic
169C22×Q8C22×C411yescharacteristic
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C2×SD16—22no—
177C2×SD16—22no—
178C2×SD16—22no—
179C2×SD16—22no—
180C2×SD16—41no—
181C2×SD16—41no—
182C2×SD16—41no—
183C2×SD16—41no—
184C2×SD16—41no—
185C2×SD16—41no—
186C2×SD16—41no—
187C2×SD16—41no—
188C2×M16—42no—
189C2×M16—81no—
190C2×M16—81no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—81no—
199C22×C8—81no—
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C4⋊Q8—22no—
209C4⋊Q8—41no—
210#127—22no—
211#127—41no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—81no—
229C4×D4—81no—
230C4×D4—81no—
231C4×D4—81no—
232C2×C4⋊C4—21no—
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⋊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×C42—21no—
251C2×C42—21no—
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42#4111yescharacteristic
257C2×C42#4111yescharacteristic
258C2×C42C2411yescharacteristic
259#106—41no—
260#106—41no—
261#106—41no—
262#106—41no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—42no—
272C4⋊C8—42no—
273#103—41no—
274#103—41no—
275#103—41no—
276#103—41no—
277#102—41no—
278#102—41no—
279#102—41no—
280#102—41no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8—81no—
284C22⋊C8—81no—
285C22⋊C8—81no—
286C22⋊C8—81no—
287#97—42no—
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—21no—
300C4×C8—42no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305C24—21no—
306C24—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8#13112yes—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—81no—
335C2×D4—81no—
336C2×D4#13112yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
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×C4—41no—
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4111yescharacteristic
362SD16—44no—
363SD16—44no—
364SD16—44no—
365SD16—44no—
366M16—84no—
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—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
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—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—84no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—81no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C22⋊C4—81no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42H(5,𝔽2)11yescharacteristic
451C42H(5,𝔽2)11yescharacteristic
452C42H(5,𝔽2)11yescharacteristic
453C42H(5,𝔽2)11yescharacteristic
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23C2×C4×D411yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—41no—
472Q8—41no—
473Q8—41no—
474Q8—41no—
475Q8—41no—
476Q8—41no—
477D4—22no—
478D4—22no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—84no—
486D4—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
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—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4#51811yescharacteristic
528C2×C4#51811yescharacteristic
529C2×C4#53511yescharacteristic
530C2×C4#53511yescharacteristic
531C2×C4#54011yescharacteristic
532C2×C4#54011yescharacteristic
533C2×C4C2×#13111yescharacteristic
534C2×C4C2×#13111yescharacteristic
535C2×C4#56811yescharacteristic
536C2×C4#56811yescharacteristic
537C2×C4#57711yescharacteristic
538C2×C4#57711yescharacteristic
539C2×C4#57711yescharacteristic
540C2×C4#57711yescharacteristic
541C8—41no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—81no—
550C8—81no—
551C8—81no—
552C8—81no—
553C8—81no—
554C8—81no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22#232811yescharacteristic
570C22#293311yescharacteristic
571C22#293311yescharacteristic
572C22#294911yescharacteristic
573C22#294911yescharacteristic
574C22#295511yescharacteristic
575C22#295511yescharacteristic
576C4—21no—
577C4—21no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4#297611yescharacteristic
589C4#297611yescharacteristic
590C4#297711yescharacteristic
591C4#297811yescharacteristic
592C4#322111yescharacteristic
593C4#322111yescharacteristic
594C4#322211yescharacteristic
595C4#322211yescharacteristic
596C2—82no—
597C2—161no—
598C2#1802811yescharacteristic
599C2#1802811yescharacteristic
600C2#1803011yescharacteristic
601C2#1803011yescharacteristic
602C2#2226711yescharacteristic
603C2#2227511yescharacteristic
604C2#2228111yescharacteristic
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
3. #22240C2semidirect
5. #9641C2semidirect
6. #8952C2semidirect
7. #8952C2semidirect
8. #8949C2semidirect
9. #8949C2semidirect
10. #8948C2semidirect
11. #8948C2semidirect
12. #8945C2semidirect
13. #8945C2semidirect
14. #8788C2semidirect
15. #8135C2semidirect

Maximal subgroups

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

subgroupindex
#237992
#222622
#222402
#96422
#96412
#89522
#89492
#89482
#89452
#87882
#81352