Finite Group Databasebeta

← C8×#478 #212304

Subgroups — C8×#478 #212304

Order 512.

738 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#125C211yesmaximal, characteristic, complemented
2C4×#478C211yesmaximal, characteristic
3C4×C8×D4C211yesmaximal, characteristic
4#9866C211yesmaximal, characteristic, complemented
5#9603C211yesmaximal, characteristic, complemented
6#9536C211yesmaximal, characteristic, complemented
7#9534C211yesmaximal, characteristic, complemented
8#8845C211yesmaximal, characteristic
9#8843C211yesmaximal, characteristic
10#8565C211yesmaximal, characteristic
11C8×#107C211yesmaximal, characteristic, complemented
12C8×#106C211yesmaximal, characteristic, complemented
13C8×C4⋊C8C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic
15C8×#102C211yesmaximal, characteristic
16C2×#478C411yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C22×C4×C8—21no—
20C4×#125C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#1989C411yescharacteristic
23#1949C411yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1921C411yescharacteristic
27#1919C411yescharacteristic
28#1916—21no—
29#1916—21no—
30#1864C411yescharacteristic
31#1862C411yescharacteristic
32#1831C2211yescharacteristic
33#1804C411yescharacteristic
34C4×#107C2211yescharacteristic
35C4×#106C2211yescharacteristic
36C8×C4⋊C4—21no—
37C8×C4⋊C4—21no—
38C8×C4⋊C4C2211yescharacteristic
39C8×C4⋊C4C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C8×C22⋊C4—21no—
48C8×C22⋊C4—21no—
49C4×C22⋊C8—21no—
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×C82C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C2×#125C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C22×C42—21no—
74#478C818yesdirect product factor
75C8×D4—22no—
76C8×D4—22no—
77C8×D4—22no—
78C8×D4Q814yes—
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2×C411yescharacteristic
87C2×C4⋊C8C2×C411yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—44no—
104C2×C4×C8D411yescharacteristic
105C2×C4×C8D411yescharacteristic
106C2×C4×C8D411yescharacteristic
107C2×C4×C8D411yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#389C2×C411yescharacteristic
115#389C2×C411yescharacteristic
116#387—21no—
117#387—21no—
118#382C2×C411yescharacteristic
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C43C2311yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336C2×C411yescharacteristic
149#336C2×C411yescharacteristic
150#335—22no—
151#334—22no—
152D4⋊C8—22no—
153D4⋊C8—22no—
154C82—22no—
155C82—22no—
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C22×C8—24no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180#125—22no—
181#125C2×C814yescomplemented
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4C2×C814yes—
188C4×D4C2×Q812yes—
189C4×D4#4112yes—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42C2411yescharacteristic
218#107—22no—
219#107C2×C814yescomplemented
220#106—22no—
221#106C2×C814yescomplemented
222C4⋊C8—24no—
223C4⋊C8—22no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8C2×C814yescomplemented
233C4⋊C8C2×Q812yes—
234C4⋊C8#4112yes—
235#102—22no—
236#102—22no—
237#102C2×C814yes—
238#102C2×C814yes—
239C22⋊C8—24no—
240C22⋊C8—24no—
241C4×C8—22no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—41no—
255C4×C8—42no—
256C4×C8—42no—
257C4×C8—42no—
258C4×C8—42no—
259C4×C8—42no—
260C4×C8C2×Q812yes—
261C4×C8#4112yes—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271C24—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4C4×Q812yes—
276C2×D4C22×C812yes—
277C2×D4#13112yes—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—48no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—48no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8C22⋊Q812yes—
342C2×C8C22⋊Q812yes—
343C2×C8C22×C812yes—
344C2×C8C22×C812yes—
345C2×C8#13512yes—
346C2×C8#13512yes—
347C2×C8AGL(1,ℤ/8)12yes—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4C4×Q812yes—
362C4⋊C4C22×C812yes—
363C4⋊C4C22×C812yes—
364C4⋊C4C22×C812yes—
365C4⋊C4#13112yes—
366C22⋊C4—24no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—24no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—22no—
377C42—41no—
378C42—42no—
379C42—42no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42C22×C812yes—
387C42#13112yes—
388C42C22×Q811yescharacteristic
389C42C2×#4111yescharacteristic
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C23—22no—
393C23—22no—
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23C2×C4×D411yescharacteristic
400D4—22no—
401D4—22no—
402D4—22no—
403D4C8×Q814yes—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4C8×D412yes—
443C2×C4C8×D412yes—
444C2×C4#44312yes—
445C2×C4C2×C4×Q811yescharacteristic
446C2×C4C4×#4111yescharacteristic
447C2×C4C2×C22⋊Q811yescharacteristic
448C2×C4#52511yescharacteristic
449C2×C4#52911yescharacteristic
450C2×C4#53311yescharacteristic
451C2×C4C23×C811yescharacteristic
452C2×C4C2×#13111yescharacteristic
453C2×C4C2×#13111yescharacteristic
454C2×C4C2×#13111yescharacteristic
455C2×C4C2×#13511yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
458C2×C4#57511yescharacteristic
459C8—22no—
460C8—22no—
461C8—22no—
462C8—22no—
463C8—42no—
464C8—42no—
465C8—42no—
466C8—41no—
467C8—42no—
468C8—41no—
469C8—41no—
470C8—81no—
471C8—81no—
472C8#47814yesdirect product factor
473C22—24no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22C4×C22⋊Q811yescharacteristic
481C22C2×C8×D411yescharacteristic
482C22#292911yescharacteristic
483C22C4×#13511yescharacteristic
484C22C4×AGL(1,ℤ/8)11yescharacteristic
485C22C2×#44311yescharacteristic
486C22#295511yescharacteristic
487C4—21no—
488C4—21no—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—41no—
497C4—42no—
498C4—81no—
499C4—81no—
500C4C2×C8×Q811yescharacteristic
501C4#296011yescharacteristic
502C4C8×#4111yescharacteristic
503C4#296611yescharacteristic
504C4C2×#47811yescharacteristic
505C4#307611yescharacteristic
506C4#310211yescharacteristic
507C4#310511yescharacteristic
508C2—44no—
509C2C8×#13511yescharacteristic
510C2C8×AGL(1,ℤ/8)11yescharacteristic
511C2#1790411yescharacteristic
512C2#1791011yescharacteristic
513C2C8×C22⋊Q811yescharacteristic
514C2C4×#47811yescharacteristic
515C2#2162711yescharacteristic
subgroupG/H
centerC22×C8C2×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C8 ◁ C42×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C8×#125C2semidirect
4. #9866C2semidirect
5. #9603C2semidirect
6. #9536C2semidirect
7. #9534C2semidirect
11. C8×#107C2semidirect
12. C8×#106C2semidirect
13. C8×C4⋊C8C2semidirect
74. #478C8direct (N × H)
181. #125C2×C8semidirect
219. #107C2×C8semidirect
221. #106C2×C8semidirect
232. C4⋊C8C2×C8semidirect
472. C8#478direct (N × H)

Maximal subgroups

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

subgroupindex
C8×#1252
C4×#4782
C4×C8×D42
#98662
#96032
#95362
#95342
#88452
#88432
#85652
C8×#1072
C8×#1062
C8×C4⋊C82
C8×#1022