Finite Group Databasebeta

← #231014

Subgroups — #231014

Order 512.

711 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22989C211yesmaximal, characteristic
2#22956C211yesmaximal, characteristic, complemented
3C4×#465C211yesmaximal, characteristic
4#9556C211yesmaximal, characteristic, complemented
5#9555C211yesmaximal, characteristic, complemented
6#9539C211yesmaximal, characteristic, complemented
7#9538C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic, complemented
9#8949C211yesmaximal, characteristic, complemented
10#8939C211yesmaximal, characteristic
11#8937C211yesmaximal, characteristic, complemented
12#8209C211yesmaximal, characteristic, complemented
13#8208C211yesmaximal, characteristic, complemented
14#8144C211yesmaximal, characteristic, complemented
15C2×#1277C211yesmaximal, characteristic, complemented
16C2×#465C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#435—21no—
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#1984C2211yescharacteristic
27#1968C411yescharacteristic
28#1954C2211yescharacteristic
29#1941C411yescharacteristic
30#1940—21no—
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1934C411yescharacteristic
34#1933C411yescharacteristic
35#1924—21no—
36#1917—21no—
37#1893C411yescharacteristic
38#1892C411yescharacteristic
39#1869—21no—
40#1851—21no—
41#1850—21no—
42#1807C411yescharacteristic
43C4×#106C2211yescharacteristic
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1752—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×M16—21no—
69C23×C8—21no—
70C23×C8—21no—
71C23×C8C2×C411yescharacteristic
72C23×C8C2×C411yescharacteristic
73C2×C22⋊Q8C2×C411yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#465—24no—
80#446—22no—
81#446—41no—
82C4×SD16—41no—
83C4×SD16—41no—
84#436—22no—
85#436—41no—
86#435—44no—
87C2×#106C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—41no—
98C2×C4⋊C8—41no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8D411yescharacteristic
114C2×C22⋊C8D411yescharacteristic
115C2×#97—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—41no—
119C2×C4×C8—41no—
120C2×C4×C8C2×C411yescharacteristic
121C2×C4×C8C2×C411yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#391C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#388—21no—
133#387—21no—
134#386—21no—
135#385—21no—
136#380C2×C411yescharacteristic
137#380C2×C411yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336D411yescharacteristic
157#336D411yescharacteristic
158Q8⋊C8—22no—
159Q8⋊C8—41no—
160D4⋊C8—22no—
161D4⋊C8—41no—
162#322—41no—
163#322—41no—
164C22×Q8C22×C411yescharacteristic
165C22×D4—21no—
166C22×D4C22×C411yescharacteristic
167C23×C4—21no—
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4111yescharacteristic
171C2×SD16—22no—
172C2×SD16—41no—
173C2×SD16—41no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—81no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—41no—
186C22×C8—41no—
187C22×C8—41no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—81no—
193C22×C8—81no—
194C22×C8M1614yescomplemented
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198C22×C8C22×C411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205#121—22no—
206#121—22no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—42no—
216C4×D4—42no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245#106—22no—
246#106—41no—
247#106—41no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8—81no—
259C4⋊C8#4112yes—
260C4⋊C8#4112yes—
261#103—22no—
262#103—22no—
263#103—41no—
264#102—22no—
265#102—22no—
266#102—41no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271C22⋊C8—42no—
272C22⋊C8—81no—
273#97—42no—
274#97—42no—
275#97—81no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—41no—
289C4×C8—41no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298C24—21no—
299C24C4×D411yescharacteristic
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C2×M1612yes—
344C22×C4#13112yes—
345C22×C4AGL(1,ℤ/8)12yes—
346C22×C4#13712yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350SD16—44no—
351M16—84no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—21no—
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—44no—
376C2×C8—44no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—84no—
394C2×C8—84no—
395C2×C8C2×M1612yes—
396C2×C8C2×M1612yes—
397C2×C8#13112yes—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—81no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—81no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42C2×#4111yescharacteristic
436C42C2×#4111yescharacteristic
437C42H(5,𝔽2)11yescharacteristic
438C42H(5,𝔽2)11yescharacteristic
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23#43612yes—
451C23#44012yes—
452C23#44412yes—
453C23C2×C4×D411yescharacteristic
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—84no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4#43612yes—
508C2×C4#44012yes—
509C2×C4C4×#4111yescharacteristic
510C2×C4#51811yescharacteristic
511C2×C4#54611yescharacteristic
512C2×C4#54611yescharacteristic
513C2×C4#55311yescharacteristic
514C2×C4#55311yescharacteristic
515C2×C4C22×M1611yescharacteristic
516C2×C4C2×#13111yescharacteristic
517C2×C4C2×#13111yescharacteristic
518C2×C4#56811yescharacteristic
519C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
520C2×C4C2×#13711yescharacteristic
521C2×C4#57711yescharacteristic
522C2×C4#57711yescharacteristic
523C8—22no—
524C8—22no—
525C8—22no—
526C8—41no—
527C8—41no—
528C8—42no—
529C8—41no—
530C8—42no—
531C8—41no—
532C8—41no—
533C8—81no—
534C8—81no—
535C8—81no—
536C8—81no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22#159214yes—
551C22#236411yescharacteristic
552C22C2×#43611yescharacteristic
553C22#293511yescharacteristic
554C22C2×#44011yescharacteristic
555C22#295211yescharacteristic
556C22C2×#44411yescharacteristic
557C22#295711yescharacteristic
558C4—22no—
559C4—22no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4#296511yescharacteristic
571C4#296811yescharacteristic
572C4#297311yescharacteristic
573C4#297711yescharacteristic
574C4#330611yescharacteristic
575C4#331011yescharacteristic
576C4#331611yescharacteristic
577C4#332011yescharacteristic
578C2—24no—
579C2—161no—
580C2C2×#159211yescharacteristic
581C2#1789911yescharacteristic
582C2#1807111yescharacteristic
583C2#1809311yescharacteristic
584C2#2312011yescharacteristic
585C2#2313411yescharacteristic
586C2#2314711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22956C2semidirect
4. #9556C2semidirect
5. #9555C2semidirect
6. #9539C2semidirect
7. #9538C2semidirect
8. #8952C2semidirect
9. #8949C2semidirect
11. #8937C2semidirect
12. #8209C2semidirect
13. #8208C2semidirect
14. #8144C2semidirect
15. C2×#1277C2semidirect
194. C22×C8M16semidirect

Maximal subgroups

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

subgroupindex
#229892
#229562
C4×#4652
#95562
#95552
#95392
#95382
#89522
#89492
#89392
#89372
#82092
#82082
#81442
C2×#12772