Finite Group Databasebeta

← #230053

Subgroups — #230053

Order 512.

608 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#472C211yesmaximal, characteristic
2#17893C211yesmaximal, characteristic, complemented
3#17877C211yesmaximal, characteristic, complemented
4#17873C211yesmaximal, characteristic, complemented
5#17799C211yesmaximal, characteristic, complemented
6#9663C211yesmaximal, characteristic, complemented
7#9610C211yesmaximal, characteristic, complemented
8#9552C211yesmaximal, characteristic, complemented
9#9550C211yesmaximal, characteristic, complemented
10#8956C211yesmaximal, characteristic, complemented
11#8956C211yesmaximal, characteristic, complemented
12#8837C211yesmaximal, characteristic, complemented
13#8835C211yesmaximal, characteristic, complemented
14#8296C211yesmaximal, characteristic, complemented
15#8186C211yesmaximal, characteristic, complemented
16C2×#472C411yescharacteristic
17#2989C2211yescharacteristic
18#2989C2211yescharacteristic
19#2975C2211yescharacteristic
20#2972C2211yescharacteristic
21C4×AGL(1,ℤ/8)C2211yescharacteristic
22C4×AGL(1,ℤ/8)C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1978C411yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1952C411yescharacteristic
29#1931C411yescharacteristic
30#1931C411yescharacteristic
31#1907C411yescharacteristic
32#1907C411yescharacteristic
33#1816C411yescharacteristic
34C4×#108C2211yescharacteristic
35C4×#100C2211yescharacteristic
36C4×#100C2211yescharacteristic
37#1594—21no—
38#1592—21no—
39#1590—21no—
40#1589—21no—
41#1586—21no—
42#1586—21no—
43#1586—21no—
44#1585—21no—
45#1582—21no—
46#1581—21no—
47#1581—21no—
48#1581—21no—
49#1569—21no—
50#1569—21no—
51#1568—21no—
52#1567—21no—
53#1565C2211yescharacteristic
54#1565C2211yescharacteristic
55#1565C2211yescharacteristic
56#1565C2211yescharacteristic
57#1564C2211yescharacteristic
58C2×#335C2211yescharacteristic
59#1478C2211yescharacteristic
60#1478C2211yescharacteristic
61#1476C2211yescharacteristic
62#1476C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65#1452C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1278C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×AGL(1,ℤ/8)C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C411yescharacteristic
77C22×D8C2×C411yescharacteristic
78C2×#131C2×C411yescharacteristic
79C2×#131C2×C411yescharacteristic
80C4×#41C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83#472—24no—
84#446—21no—
85#446—21no—
86#442—21no—
87#442—21no—
88#440—21no—
89#440—21no—
90C4×SD16—21no—
91C4×SD16—21no—
92C4×D8—21no—
93C4×D8—21no—
94C4×D8—21no—
95C4×D8—21no—
96C4×D8—21no—
97C4×D8—21no—
98#436—21no—
99#436—21no—
100#436—21no—
101#436—21no—
102#436—21no—
103#436—21no—
104#435—41no—
105#435—41no—
106C8×D4—21no—
107C8×D4—21no—
108C8×D4—21no—
109C8×D4—21no—
110#432—21no—
111#432—21no—
112#431C2311yescharacteristic
113#431C2311yescharacteristic
114#431C2311yescharacteristic
115#431C2311yescharacteristic
116C2×#108C2×C411yescharacteristic
117C2×#108C2×C411yescharacteristic
118#428C2×C411yescharacteristic
119#428C2×C411yescharacteristic
120C2×#107C2×C411yescharacteristic
121#423—21no—
122#423—21no—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125#418C2×C411yescharacteristic
126#418C2×C411yescharacteristic
127#417C2×C411yescharacteristic
128#417C2×C411yescharacteristic
129C2×#102C2×C411yescharacteristic
130C2×#102C2×C411yescharacteristic
131C2×#100C2×C411yescharacteristic
132C2×#100C2×C411yescharacteristic
133C2×#100C2×C411yescharacteristic
134C2×#100C2×C411yescharacteristic
135#407—41no—
136C2×C22⋊C8—41no—
137C2×C22⋊C8—41no—
138C2×C22⋊C8—41no—
139#405—21no—
140#405—21no—
141#405C2×C411yescharacteristic
142#405C2×C411yescharacteristic
143C4×M16—21no—
144C4×M16C2311yescharacteristic
145C4×M16C2311yescharacteristic
146C4×M16C2311yescharacteristic
147C2×C4×C8—21no—
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151#343C2×C411yescharacteristic
152#343C2×C411yescharacteristic
153#343C2×C411yescharacteristic
154#343C2×C411yescharacteristic
155#341C2×C411yescharacteristic
156#341C2×C411yescharacteristic
157#335—21no—
158#335—21no—
159#335—21no—
160#335—21no—
161#335—21no—
162#335—21no—
163#335D411yescharacteristic
164#335D411yescharacteristic
165#334—21no—
166#334—21no—
167#334—21no—
168#334—21no—
169#334D411yescharacteristic
170#334D411yescharacteristic
171Q8⋊C8—21no—
172Q8⋊C8D411yescharacteristic
173Q8⋊C8D411yescharacteristic
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8D411yescharacteristic
184D4⋊C8D411yescharacteristic
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191#322—21no—
192C82—21no—
193C2×#41C22×C411yescharacteristic
194C22×D4C22×C411yescharacteristic
195C22×D4C22×C411yescharacteristic
196C23×C4—21no—
197C23×C4—21no—
198AGL(1,ℤ/8)—22no—
199AGL(1,ℤ/8)—22no—
200AGL(1,ℤ/8)—41no—
201AGL(1,ℤ/8)—41no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×D8—21no—
205C2×D8—21no—
206C2×D8—22no—
207C2×D8—21no—
208C2×D8—21no—
209C2×D8—41no—
210#131—22no—
211#131—41no—
212#131—41no—
213#131—41no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—41no—
217C2×M16—41no—
218C2×M16—41no—
219C2×M16—41no—
220C2×M16C22×C411yescharacteristic
221C2×M16C22×C411yescharacteristic
222C2×M16C22×C411yescharacteristic
223C2×M16C22×C411yescharacteristic
224C2×M16C22×C411yescharacteristic
225C2×M16C22×C411yescharacteristic
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—41no—
233C22×C8—41no—
234C22×C8C22×C411yescharacteristic
235C22×C8C22×C411yescharacteristic
236C4×Q8C2×D411yescharacteristic
237C4×Q8#4111yescharacteristic
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4C2×D411yescharacteristic
249C4×D4#4111yescharacteristic
250#117—21no—
251#117C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C42—21no—
257C2×C42C2411yescharacteristic
258#108—22no—
259#108—41no—
260#107—21no—
261#107—21no—
262#107—21no—
263#107—21no—
264#106—21no—
265#106—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8C2×D411yescharacteristic
274C4⋊C8C2×D411yescharacteristic
275C4⋊C8C2×D411yescharacteristic
276C4⋊C8C2×D411yescharacteristic
277C4⋊C8C2×D411yescharacteristic
278C4⋊C8C2×D411yescharacteristic
279#103—21no—
280#103—21no—
281#102—21no—
282#102—21no—
283#102—21no—
284#102—21no—
285#102—21no—
286#102—21no—
287#102—21no—
288#102—21no—
289#102—21no—
290#102—21no—
291#100—22no—
292#100—41no—
293#100—41no—
294#100M1614yescomplemented
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C22⋊C8—41no—
304C22⋊C8—41no—
305C22⋊C8—82no—
306C22⋊C8—82no—
307#97—21no—
308#97—21no—
309#97—21no—
310#97—21no—
311#97—21no—
312#97—41no—
313#97—41no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8#4111yescharacteristic
331C4×C8#4111yescharacteristic
332C24—21no—
333C24—21no—
334#41—21no—
335#41—21no—
336#41—21no—
337#41—21no—
338C2×Q8C4×D411yescharacteristic
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4C4×D411yescharacteristic
352C2×D4C2×M1612yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4C23×C411yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C42−1+411yescharacteristic
369SD16—42no—
370SD16—42no—
371D8—42no—
372D8—42no—
373D8—42no—
374D8—42no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—42no—
386M16—42no—
387M16—161no—
388M16—161no—
389M16—161no—
390M16—161no—
391M16C2×M1612yes—
392M16C2×M1612yes—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—22no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—81no—
420C2×C8—81no—
421C2×C8—81no—
422C2×C8—161no—
423C2×C8—161no—
424C2×C8C4×D411yescharacteristic
425C2×C8C4×D411yescharacteristic
426C2×C8C4×D411yescharacteristic
427C2×C8C4×D411yescharacteristic
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4C4×D411yescharacteristic
435C4⋊C4C4×D411yescharacteristic
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42C22×D411yescharacteristic
447C42C22×D411yescharacteristic
448C42C2×#4111yescharacteristic
449C42C2×#4111yescharacteristic
450C23—21no—
451C23—22no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23#52011yescharacteristic
462Q8—21no—
463Q8—21no—
464D4—21no—
465D4—21no—
466D4—22no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—161no—
496C2×C4—161no—
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4D4211yescharacteristic
500C2×C4#54711yescharacteristic
501C2×C4#54811yescharacteristic
502C2×C4#55311yescharacteristic
503C2×C4C22×M1611yescharacteristic
504C2×C4#56811yescharacteristic
505C2×C4#56811yescharacteristic
506C2×C4#56811yescharacteristic
507C8—21no—
508C8—21no—
509C8—21no—
510C8—21no—
511C8—22no—
512C8—22no—
513C8—42no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—81no—
527C8—81no—
528C8—81no—
529C22—21no—
530C22—21no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—42no—
535C22—82no—
536C22—82no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—161no—
543C22—161no—
544C22#236111yescharacteristic
545C22#298911yescharacteristic
546C22#299511yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4#293011yescharacteristic
556C4#293211yescharacteristic
557C4D4×M1611yescharacteristic
558C4D4×M1611yescharacteristic
559C4#329011yescharacteristic
560C4#329211yescharacteristic
561C2—21no—
562C2—21no—
563C2—81no—
564C2—82no—
565C2—161no—
566C2#2298811yescharacteristic
567C2#2304311yescharacteristic
568C2#2304611yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radicalC2×#472C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17893C2semidirect
3. #17877C2semidirect
4. #17873C2semidirect
5. #17799C2semidirect
6. #9663C2semidirect
7. #9610C2semidirect
8. #9552C2semidirect
9. #9550C2semidirect
10. #8956C2semidirect
11. #8956C2semidirect
12. #8837C2semidirect
13. #8835C2semidirect
14. #8296C2semidirect
15. #8186C2semidirect
294. #100M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4722
#178932
#178772
#178732
#177992
#96632
#96102
#95522
#95502
#89562
#88372
#88352
#82962
#81862