Finite Group Databasebeta

← #226000

Subgroups — #226000

Order 512.

659 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22735C211yesmaximal, characteristic, complemented
2#22651C211yesmaximal, characteristic
3#21279C211yesmaximal, characteristic
4C2×#1588C211yesmaximal, characteristic
5C2×#1582C211yesmaximal, characteristic
6#9599C211yesmaximal, characteristic, complemented
7#9598C211yesmaximal, characteristic, complemented
8#8850C211yesmaximal, characteristic, complemented
9#8849C211yesmaximal, characteristic
10#8844C211yesmaximal, characteristic, complemented
11#8843C211yesmaximal, characteristic
12#8808C211yesmaximal, characteristic, complemented
13#8569C211yesmaximal, characteristic
14#8568C211yesmaximal, characteristic, complemented
15#8263C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#424—21no—
22#2295C2211yescharacteristic
23#2293C2211yescharacteristic
24#1973C411yescharacteristic
25#1973C411yescharacteristic
26#1944C411yescharacteristic
27#1944C411yescharacteristic
28#1940—21no—
29#1939—21no—
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1936C411yescharacteristic
35#1936C411yescharacteristic
36#1935C411yescharacteristic
37#1935C411yescharacteristic
38#1851—21no—
39#1850—21no—
40#1842C2211yescharacteristic
41#1832—21no—
42#1831C2211yescharacteristic
43#1798—21no—
44C4×#107C2211yescharacteristic
45#1768—21no—
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#102C2211yescharacteristic
50C4×#97C2211yescharacteristic
51#1588—24no—
52#1582—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×M16—21no—
71C23×C8—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75#446—22no—
76#446—41no—
77C8×Q8—22no—
78C8×Q8—41no—
79C4×SD16D414yes—
80C4×SD16Q814yes—
81#436—22no—
82#436—41no—
83C8×D4—22no—
84C8×D4—41no—
85C2×#107C2×C411yescharacteristic
86C2×#107C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88#424—44no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8D411yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×#97C2×C411yescharacteristic
114C2×#97C2×C411yescharacteristic
115C2×#97C2×C411yescharacteristic
116C2×#97C2×C411yescharacteristic
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#388—21no—
133#388—21no—
134#387—21no—
135#387—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—21no—
140#376—21no—
141C43C2311yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336D411yescharacteristic
156#336D411yescharacteristic
157#335—22no—
158#335—41no—
159#334—22no—
160#334—41no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165#322—22no—
166#322—22no—
167C22×Q8C22×C411yescharacteristic
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C2×SD16—41no—
171C2×SD16—41no—
172C2×SD16C4⋊C414yes—
173C2×M16—42no—
174C2×M16—42no—
175C2×M16—81no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8—81no—
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8C2×D412yes—
196C4×Q8C2×Q812yes—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4C2×D412yes—
200C4×D4C2×Q812yes—
201#117—42no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42C2411yescharacteristic
229#107—41no—
230#107—41no—
231#106C4⋊C414yes—
232C4⋊C8—22no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—41no—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—81no—
251#103—22no—
252#103—41no—
253#103C4⋊C414yes—
254#102—22no—
255#102—41no—
256#102C4⋊C414yes—
257C22⋊C8—42no—
258C22⋊C8—42no—
259#97—22no—
260#97—41no—
261#97—41no—
262#97—41no—
263#97—42no—
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—21no—
273C4×C8—21no—
274C4×C8—42no—
275C4×C8C2×D412yes—
276C4×C8C2×Q812yes—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287C24—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8C2×C4⋊C412yes—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4C2×C4⋊C412yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320SD16—44no—
321M16—84no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—84no—
361C2×C8C2×C4⋊C412yes—
362C2×C8C2×C4⋊C412yes—
363C2×C8#13112yes—
364C2×C8AGL(1,ℤ/8)12yes—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—81no—
381C4⋊C4C2×C4⋊C412yes—
382C4⋊C4C2×C4⋊C412yes—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—81no—
389C42—22no—
390C42—22no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42C22×D411yescharacteristic
406C42C22×Q811yescharacteristic
407C42H(5,𝔽2)11yescharacteristic
408C422−1+411yescharacteristic
409C23—22no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23C2×C4×D411yescharacteristic
417Q8—22no—
418Q8—22no—
419Q8—41no—
420Q8—41no—
421D4—22no—
422D4—22no—
423D4—41no—
424D4—41no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4C22×C4⋊C411yescharacteristic
465C2×C4#52011yescharacteristic
466C2×C4D4211yescharacteristic
467C2×C4#54711yescharacteristic
468C2×C4D4×Q811yescharacteristic
469C2×C4#55411yescharacteristic
470C2×C4C2×#13111yescharacteristic
471C2×C4#56811yescharacteristic
472C2×C4#56811yescharacteristic
473C2×C4#56811yescharacteristic
474C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
475C2×C4#57511yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57811yescharacteristic
478C8—22no—
479C8—22no—
480C8—22no—
481C8—22no—
482C8—42no—
483C8—41no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—81no—
488C8—81no—
489C8—81no—
490C8—81no—
491C8—81no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22D4×C4⋊C411yescharacteristic
500C22#293211yescharacteristic
501C22#293511yescharacteristic
502C22C4×AGL(1,ℤ/8)11yescharacteristic
503C22#294711yescharacteristic
504C22#295511yescharacteristic
505C22#295611yescharacteristic
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—42no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—41no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4—81no—
519C4#290511yescharacteristic
520C4#290611yescharacteristic
521C4#299211yescharacteristic
522C4#299511yescharacteristic
523C4#326511yescharacteristic
524C4#326911yescharacteristic
525C4#338611yescharacteristic
526C4#338811yescharacteristic
527C2—82no—
528C2#1798311yescharacteristic
529C2#1798811yescharacteristic
530C2#1807611yescharacteristic
531C2#1808311yescharacteristic
532C2#2265211yescharacteristic
533C2#2267711yescharacteristic
534C2#2268711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×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. #22735C2semidirect
6. #9599C2semidirect
7. #9598C2semidirect
8. #8850C2semidirect
10. #8844C2semidirect
12. #8808C2semidirect
14. #8568C2semidirect
15. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#227352
#226512
#212792
C2×#15882
C2×#15822
#95992
#95982
#88502
#88492
#88442
#88432
#88082
#85692
#85682
#82632