Finite Group Databasebeta

← #225916

Subgroups — #225916

Order 512.

755 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22701C211yesmaximal, characteristic, complemented
2D4×C4⋊C8C211yesmaximal, characteristic, complemented
3#21118C211yesmaximal, characteristic
4C2×#1581C211yesmaximal, characteristic
5C2×#1580C211yesmaximal, characteristic
6#9597C211yesmaximal, characteristic, complemented
7#9596C211yesmaximal, characteristic, complemented
8#8847C211yesmaximal, characteristic, complemented
9#8845C211yesmaximal, characteristic, complemented
10#8845C211yesmaximal, characteristic, complemented
11#8843C211yesmaximal, characteristic, complemented
12#8812C211yesmaximal, characteristic, complemented
13#8567C211yesmaximal, characteristic, complemented
14#8565C211yesmaximal, characteristic, complemented
15#8265C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C22×C4⋊C8—21no—
22#2293C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1926C411yescharacteristic
29#1926C411yescharacteristic
30#1921C411yescharacteristic
31#1921C411yescharacteristic
32#1879C411yescharacteristic
33#1879C411yescharacteristic
34#1864C411yescharacteristic
35#1864C411yescharacteristic
36#1851—21no—
37#1850—21no—
38#1850—21no—
39#1850—21no—
40#1842C2211yescharacteristic
41#1831C2211yescharacteristic
42#1798—21no—
43#1797—21no—
44C4×#106C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1744C2211yescharacteristic
50#1729—21no—
51#1581—24no—
52#1580—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×M16—21no—
71C23×C8—21no—
72C23×C8—21no—
73C23×C8—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C22×C42—21no—
80C4×D8D414yes—
81C4×D8Q814yes—
82#436—22no—
83#436—41no—
84C8×D4—22no—
85C8×D4—22no—
86C8×D4—22no—
87C8×D4—41no—
88C8×D4—41no—
89C8×D4—41no—
90C2×#107C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—44no—
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#390C2×C411yescharacteristic
124#390C2×C411yescharacteristic
125#389C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#388—21no—
128#388—21no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C43C2311yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336C2×C411yescharacteristic
152#336C2×C411yescharacteristic
153#336C2×C411yescharacteristic
154#336C2×C411yescharacteristic
155#336D411yescharacteristic
156#336D411yescharacteristic
157#335—22no—
158#335—41no—
159#334—22no—
160#334—41no—
161D4⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165#322—22no—
166#322—22no—
167C22×D4C22×C411yescharacteristic
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C2×D8—22no—
174C2×D8—22no—
175C2×D8C4⋊C414yes—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—81no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—81no—
192C22×C8—81no—
193C22×C8—81no—
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—41no—
210C4×D4C2×D412yes—
211C4×D4C2×D412yes—
212C4×D4C2×Q812yes—
213C4×D4C2×Q812yes—
214#117—81no—
215#117—81no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245#107C4⋊C414yes—
246#106—22no—
247#106—41no—
248C4⋊C8—22no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—41no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C4⋊C8—81no—
263C4⋊C8—81no—
264C4⋊C8—81no—
265#102—22no—
266#102—22no—
267#102—22no—
268#102—41no—
269#102C4⋊C414yes—
270#102C4⋊C414yes—
271C22⋊C8—42no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C4×C8—21no—
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×C8C2×D412yes—
286C4×C8C2×Q812yes—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293C24—21no—
294C24—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4C2×C4⋊C412yes—
304C2×D4C2×C4⋊C412yes—
305C2×D4#13112yes—
306C2×D4#13112yes—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—81no—
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338D8—28no—
339M16—84no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
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—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—84no—
380C2×C8—84no—
381C2×C8—84no—
382C2×C8C2×C4⋊C412yes—
383C2×C8C2×C4⋊C412yes—
384C2×C8AGL(1,ℤ/8)12yes—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—81no—
395C4⋊C4—81no—
396C4⋊C4C2×C4⋊C412yes—
397C4⋊C4C2×C4⋊C412yes—
398C4⋊C4#13112yes—
399C22⋊C4—24no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—81no—
407C22⋊C4—81no—
408C22⋊C4—81no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—22no—
418C42—41no—
419C42—42no—
420C42—42no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42C22×D411yescharacteristic
426C42C22×Q811yescharacteristic
427C42H(5,𝔽2)11yescharacteristic
428C422−1+411yescharacteristic
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23C2×C4×D411yescharacteristic
444D4—22no—
445D4—22no—
446D4—22no—
447D4—22no—
448D4—22no—
449D4—41no—
450D4—41no—
451D4#42414yes—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—48no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
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—84no—
493C2×C4#44412yes—
494C2×C4C22×C4⋊C411yescharacteristic
495C2×C4#52011yescharacteristic
496C2×C4D4211yescharacteristic
497C2×C4#54711yescharacteristic
498C2×C4D4×Q811yescharacteristic
499C2×C4#55411yescharacteristic
500C2×C4C2×#13111yescharacteristic
501C2×C4C2×#13111yescharacteristic
502C2×C4C2×#13111yescharacteristic
503C2×C4#56811yescharacteristic
504C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
505C2×C4#57511yescharacteristic
506C2×C4#57611yescharacteristic
507C2×C4#57711yescharacteristic
508C8—22no—
509C8—22no—
510C8—22no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—81no—
517C8—81no—
518C8—81no—
519C8—81no—
520C8—81no—
521C8—81no—
522C22—24no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22D4×C4⋊C411yescharacteristic
537C22#292911yescharacteristic
538C22#293311yescharacteristic
539C22C4×AGL(1,ℤ/8)11yescharacteristic
540C22#294611yescharacteristic
541C22C2×#44411yescharacteristic
542C22#295711yescharacteristic
543C4—21no—
544C4—21no—
545C4—22no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4C2×#42411yescharacteristic
556C4#290611yescharacteristic
557C4#299211yescharacteristic
558C4#299411yescharacteristic
559C4#326411yescharacteristic
560C4#326711yescharacteristic
561C4#338411yescharacteristic
562C4#338911yescharacteristic
563C2—44no—
564C2—82no—
565C2#1791411yescharacteristic
566C2#1792011yescharacteristic
567C2#1800811yescharacteristic
568C2#1802211yescharacteristic
569C2#2264711yescharacteristic
570C2#2267611yescharacteristic
571C2#2268611yescharacteristic
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. #22701C2semidirect
2. D4×C4⋊C8C2semidirect
6. #9597C2semidirect
7. #9596C2semidirect
8. #8847C2semidirect
9. #8845C2semidirect
10. #8845C2semidirect
11. #8843C2semidirect
12. #8812C2semidirect
13. #8567C2semidirect
14. #8565C2semidirect
15. #8265C2semidirect

Maximal subgroups

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

subgroupindex
#227012
D4×C4⋊C82
#211182
C2×#15812
C2×#15802
#95972
#95962
#88472
#88452
#88432
#88122
#85672
#85652
#82652