Finite Group Databasebeta

← #231449

Subgroups — #231449

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23209C211yesmaximal, characteristic, complemented
2#23205C211yesmaximal, characteristic, complemented
3C4×#495C211yesmaximal, characteristic
4C2×#1584C211yesmaximal, characteristic, complemented
5C2×#1581C211yesmaximal, characteristic
6#9676C211yesmaximal, characteristic, complemented
7#9676C211yesmaximal, characteristic, complemented
8#9663C211yesmaximal, characteristic
9#9662C211yesmaximal, characteristic, complemented
10#8952C211yesmaximal, characteristic, complemented
11#8949C211yesmaximal, characteristic, complemented
12#8948C211yesmaximal, characteristic, complemented
13#8945C211yesmaximal, characteristic, complemented
14#8274C211yesmaximal, characteristic, complemented
15#8261C211yesmaximal, 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#1923—21no—
41#1851—21no—
42#1850—21no—
43#1846C411yescharacteristic
44#1767C2211yescharacteristic
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48#1753—21no—
49#1752—21no—
50C42×C8C2211yescharacteristic
51#1584—24no—
52#1581—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×D8C2×C411yescharacteristic
73C22×M16—21no—
74C23×C8—21no—
75C2×#124C2×C411yescharacteristic
76C2×#124C2×C411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81#495—24no—
82#446—22no—
83#446—41no—
84C8×Q8—22no—
85C8×Q8—41no—
86C4×Q16—22no—
87C4×Q16—22no—
88C4×SD16—41no—
89C4×SD16—41no—
90C4×SD16—41no—
91C4×SD16—41no—
92C4×D8—22no—
93C4×D8—22no—
94#436—22no—
95#436—41no—
96C8×D4—22no—
97C8×D4—41no—
98C2×#107C2×C411yescharacteristic
99C2×#107C2×C411yescharacteristic
100C2×#106C2×C411yescharacteristic
101C2×#106C2×C411yescharacteristic
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×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×#97—21no—
126C2×#97—21no—
127C2×#97—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2×C411yescharacteristic
130C2×C4×C8C2×C411yescharacteristic
131C2×C4×C8C2×C411yescharacteristic
132C2×C4×C8C2×C411yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139#391C2×C411yescharacteristic
140#391C2×C411yescharacteristic
141#390C2×C411yescharacteristic
142#390C2×C411yescharacteristic
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#382C2×C411yescharacteristic
148#382C2×C411yescharacteristic
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C43C2311yescharacteristic
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336D411yescharacteristic
162#336D411yescharacteristic
163#335—22no—
164#335D414yes—
165Q8⋊C8—22no—
166Q8⋊C8—22no—
167Q8⋊C8—41no—
168Q8⋊C8—41no—
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—41no—
172D4⋊C8—41no—
173#322—22no—
174#322—22no—
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×D4C22×C411yescharacteristic
178C22×D4C22×C411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C2×Q16—22no—
182C2×Q16—22no—
183C2×Q16—41no—
184C2×SD16—22no—
185C2×SD16—41no—
186C2×SD16—41no—
187C2×SD16—41no—
188C2×SD16—41no—
189C2×SD16M1614yescomplemented
190C2×D8—22no—
191C2×D8—22no—
192C2×D8—41no—
193C2×M16—42no—
194C2×M16—42no—
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—42no—
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#124—22no—
214#124—22no—
215#124—41no—
216#124—41no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—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—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42#4111yescharacteristic
257C2×C42C2411yescharacteristic
258#107—22no—
259#107—22no—
260#106—41no—
261#106—41no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—42no—
271C4⋊C8—42no—
272C4⋊C8—42no—
273C4⋊C8C2×D412yes—
274C4⋊C8C2×D412yes—
275#103—22no—
276#103—22no—
277#103—41no—
278#103—41no—
279#102—22no—
280#102—22no—
281#102—41no—
282#102—41no—
283C22⋊C8—42no—
284C22⋊C8—42no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287C22⋊C8—81no—
288C22⋊C8—81no—
289#97—42no—
290#97—81no—
291#97—81no—
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—41no—
304C4×C8—41no—
305C4×C8—41no—
306C4×C8—42no—
307C4×C8C2×D412yes—
308C4×C8#4112yes—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317C24—21no—
318C24—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8C2×M1612yes—
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×D4C2×M1612yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—41no—
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×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368Q16—44no—
369SD16—44no—
370SD16—44no—
371D8—44no—
372M16—84no—
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—22no—
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
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—41no—
410C2×C8—84no—
411C2×C8C4×D412yes—
412C2×C8C4×D412yes—
413C2×C8C2×M1612yes—
414C2×C8#13512yes—
415C2×C8#13512yes—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—81no—
439C22⋊C4—81no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—22no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42C22×D411yescharacteristic
455C42C2×#4111yescharacteristic
456C42H(5,𝔽2)11yescharacteristic
457C42H(5,𝔽2)11yescharacteristic
458C23—22no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23C2×C4×D411yescharacteristic
473Q8—22no—
474Q8—22no—
475Q8—41no—
476Q8—41no—
477Q8—41no—
478Q8—41no—
479Q8—41no—
480Q8—41no—
481D4—22no—
482D4—22no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—41no—
488D4—41no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
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—41no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4#44412yes—
529C2×C4C2×C4×D411yescharacteristic
530C2×C4#51811yescharacteristic
531C2×C4D4211yescharacteristic
532C2×C4#55011yescharacteristic
533C2×C4#55011yescharacteristic
534C2×C4#55911yescharacteristic
535C2×C4C22×M1611yescharacteristic
536C2×C4#56811yescharacteristic
537C2×C4#56811yescharacteristic
538C2×C4#56811yescharacteristic
539C2×C4C2×#13511yescharacteristic
540C2×C4C2×#13511yescharacteristic
541C2×C4#57611yescharacteristic
542C2×C4#57811yescharacteristic
543C8—22no—
544C8—22no—
545C8—22no—
546C8—22no—
547C8—22no—
548C8—41no—
549C8—41no—
550C8—42no—
551C8—42no—
552C8—41no—
553C8—81no—
554C8—81no—
555C8—81no—
556C8—81no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#236611yescharacteristic
572C22#293211yescharacteristic
573C22D4×M1611yescharacteristic
574C22C4×#13511yescharacteristic
575C22#295111yescharacteristic
576C22C2×#44411yescharacteristic
577C22#295711yescharacteristic
578C4—22no—
579C4—21no—
580C4—21no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4#293211yescharacteristic
591C4D4×M1611yescharacteristic
592C4#297311yescharacteristic
593C4#298011yescharacteristic
594C4#328011yescharacteristic
595C4#328511yescharacteristic
596C4#339311yescharacteristic
597C4#339611yescharacteristic
598C2—82no—
599C2—161no—
600C2#1797811yescharacteristic
601C2#1799211yescharacteristic
602C2#1803711yescharacteristic
603C2#1806111yescharacteristic
604C2#2317511yescharacteristic
605C2#2318011yescharacteristic
606C2#2318611yescharacteristic
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×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23209C2semidirect
2. #23205C2semidirect
4. C2×#1584C2semidirect
6. #9676C2semidirect
7. #9676C2semidirect
9. #9662C2semidirect
10. #8952C2semidirect
11. #8949C2semidirect
12. #8948C2semidirect
13. #8945C2semidirect
14. #8274C2semidirect
15. #8261C2semidirect
189. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
#232092
#232052
C4×#4952
C2×#15842
C2×#15812
#96762
#96632
#96622
#89522
#89492
#89482
#89452
#82742
#82612