Finite Group Databasebeta

← #228313

Subgroups — #228313

Order 512.

701 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23083C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic, complemented
3C4×#481C211yesmaximal, characteristic
4#9596C211yesmaximal, characteristic, complemented
5#9554C211yesmaximal, characteristic, complemented
6#9541C211yesmaximal, characteristic, complemented
7#9534C211yesmaximal, characteristic, complemented
8#9534C211yesmaximal, characteristic, complemented
9#9074C211yesmaximal, characteristic
10#8845C211yesmaximal, characteristic, complemented
11#8831C211yesmaximal, characteristic, complemented
12C8×#106C211yesmaximal, characteristic, complemented
13#8200C211yesmaximal, characteristic, complemented
14#8148C211yesmaximal, characteristic, complemented
15#7892C211yesmaximal, characteristic, complemented
16C2×#481C411yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×#431—21no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C4×#121C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1954—21no—
24#1954C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1926C411yescharacteristic
29#1926C411yescharacteristic
30#1923—21no—
31#1919C411yescharacteristic
32#1919C411yescharacteristic
33#1916—21no—
34#1916—21no—
35#1874C411yescharacteristic
36#1869—21no—
37#1853C411yescharacteristic
38#1824C411yescharacteristic
39#1817—21no—
40#1797C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C4×#106C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C8×C22⋊C4—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
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—
68C2×#121C2×C411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4C2311yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#481C818yescomplemented
76C8×D4—22no—
77C8×D4—41no—
78#431—22no—
79#431—22no—
80#431—41no—
81#431—44no—
82#431—41no—
83C2×#106C2×C411yescharacteristic
84C2×#106C2×C411yescharacteristic
85C2×#106C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8C2×C411yescharacteristic
106C2×C22⋊C8C2×C411yescharacteristic
107C2×C22⋊C8D411yescharacteristic
108C2×C22⋊C8D411yescharacteristic
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#390C2×C411yescharacteristic
120#389C2×C411yescharacteristic
121#388—21no—
122#386—21no—
123#380C2×C411yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—21no—
132C2×#95—21no—
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
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#336D411yescharacteristic
151#336D411yescharacteristic
152#335—41no—
153#334—22no—
154D4⋊C8—22no—
155D4⋊C8—41no—
156#322—41no—
157C82—22no—
158C22×D4—21no—
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4#4111yescharacteristic
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—81no—
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183#121—22no—
184#121C2×C814yes—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—81no—
190C4×D4—81no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4C2×C814yescomplemented
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42C2411yescharacteristic
225#106—22no—
226#106—41no—
227#106C2×C814yescomplemented
228#106C2×C814yescomplemented
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—81no—
240C4⋊C8#4112yes—
241#102—22no—
242#102—41no—
243#102C2×C814yes—
244#102C2×C814yes—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—41no—
248C22⋊C8—42no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—81no—
254C22⋊C8C2×C814yescomplemented
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—42no—
267C4×C8—42no—
268C4×C8—42no—
269C4×C8—81no—
270C4×C8#4112yes—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278C24—21no—
279C24C4×D411yescharacteristic
280C2×D4—24no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—81no—
287C2×D4—81no—
288C2×D4C22×C812yes—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—81no—
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C22×C812yes—
326C22×C4#13112yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C2×C8—21no—
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—22no—
341C2×C8—21no—
342C2×C8—22no—
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—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
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—84no—
370C2×C8C22×C812yes—
371C2×C8C22×C812yes—
372C2×C8AGL(1,ℤ/8)12yes—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—24no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—42no—
385C4⋊C4—81no—
386C4⋊C4C22×C812yes—
387C4⋊C4C22×C812yes—
388C4⋊C4C22×C812yes—
389C4⋊C4#13112yes—
390C22⋊C4—24no—
391C22⋊C4—42no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—81no—
395C22⋊C4—81no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—42no—
403C42—42no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42C2×#4111yescharacteristic
411C42C2×#4111yescharacteristic
412C42H(5,𝔽2)11yescharacteristic
413C422−1+411yescharacteristic
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23C8×D412yes—
428C23C2×C4×D411yescharacteristic
429D4—22no—
430D4—22no—
431D4—41no—
432D4—41no—
433D4—84no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—48no—
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×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—84no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4C8×D412yes—
482C2×C4C4×#4111yescharacteristic
483C2×C4#52011yescharacteristic
484C2×C4#54611yescharacteristic
485C2×C4#54711yescharacteristic
486C2×C4#55111yescharacteristic
487C2×C4#55211yescharacteristic
488C2×C4C23×C811yescharacteristic
489C2×C4C2×#13111yescharacteristic
490C2×C4C2×#13111yescharacteristic
491C2×C4#56811yescharacteristic
492C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
493C2×C4#57511yescharacteristic
494C2×C4#57711yescharacteristic
495C2×C4#57711yescharacteristic
496C8—22no—
497C8—22no—
498C8—22no—
499C8—42no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—42no—
506C8—81no—
507C8—81no—
508C8—81no—
509C8—81no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22#235911yescharacteristic
525C22C2×C8×D411yescharacteristic
526C22#293311yescharacteristic
527C22C4×AGL(1,ℤ/8)11yescharacteristic
528C22#294911yescharacteristic
529C22#295511yescharacteristic
530C22#295711yescharacteristic
531C4—22no—
532C4—21no—
533C4—21no—
534C4—42no—
535C4—41no—
536C4—41no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4C8×#4111yescharacteristic
544C4#296811yescharacteristic
545C4#298811yescharacteristic
546C4#299411yescharacteristic
547C4#332311yescharacteristic
548C4#333211yescharacteristic
549C4#334511yescharacteristic
550C4#334811yescharacteristic
551C2—42no—
552C2—82no—
553C2C8×AGL(1,ℤ/8)11yescharacteristic
554C2#1785511yescharacteristic
555C2#1800811yescharacteristic
556C2#1803011yescharacteristic
557C2#2290011yescharacteristic
558C2#2294311yescharacteristic
559C2#2295111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C2×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. #22599C2semidirect
4. #9596C2semidirect
5. #9554C2semidirect
6. #9541C2semidirect
7. #9534C2semidirect
8. #9534C2semidirect
10. #8845C2semidirect
11. #8831C2semidirect
12. C8×#106C2semidirect
13. #8200C2semidirect
14. #8148C2semidirect
15. #7892C2semidirect
75. #481C8semidirect
197. C2×C4⋊C4C2×C8semidirect
227. #106C2×C8semidirect
228. #106C2×C8semidirect
254. C22⋊C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#230832
#225992
C4×#4812
#95962
#95542
#95412
#95342
#90742
#88452
#88312
C8×#1062
#82002
#81482
#78922