Finite Group Databasebeta

← #218958

Subgroups — #218958

Order 512.

745 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23697C211yesmaximal, characteristic
2#21487C211yesmaximal, characteristic, complemented
3C8×#121C211yesmaximal, characteristic, complemented
4#9546C211yesmaximal, characteristic, complemented
5#9542C211yesmaximal, characteristic, complemented
6#9538C211yesmaximal, characteristic, complemented
7#9534C211yesmaximal, characteristic, complemented
8#9078C211yesmaximal, characteristic, complemented
9#9078C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8847C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic, complemented
14#8600C211yesmaximal, characteristic, complemented
15#7848C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×#435—21no—
19C2×C8×D4—21no—
20C2×C8×D4C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#2087C411yescharacteristic
27#2076C411yescharacteristic
28#2065C411yescharacteristic
29#2065C411yescharacteristic
30#2054C411yescharacteristic
31#2034C411yescharacteristic
32#2034C411yescharacteristic
33#1999C411yescharacteristic
34#1954C2211yescharacteristic
35#1954C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1923—21no—
39#1923—21no—
40#1869—21no—
41#1869—21no—
42#1797C2211yescharacteristic
43#1797C2211yescharacteristic
44C4×#106C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1753—21no—
50C8×C22⋊C4—21no—
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C22×M16—21no—
69C22×M16—21no—
70C23×C8—21no—
71C23×C8D411yescharacteristic
72C23×C8D411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2311yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C4×D8—41no—
81C4×D8—41no—
82#436—41no—
83#436—41no—
84#435—44no—
85C8×D4—41no—
86C8×D4—41no—
87C8×D4—44no—
88#431—41no—
89#431—41no—
90C2×#107C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8C2×C411yescharacteristic
116C2×C22⋊C8C2×C411yescharacteristic
117C4×M16—41no—
118C4×M16—41no—
119C2×#97—21no—
120C2×#97—21no—
121C2×C4×C8—21no—
122C2×C4×C8—41no—
123C2×C4×C8—41no—
124C2×C4×C8D411yescharacteristic
125C2×C4×C8D411yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132#390C2×C411yescharacteristic
133#390C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#388—21no—
137#388—21no—
138#386—21no—
139#386—21no—
140#380C2×C411yescharacteristic
141#380C2×C411yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#335—41no—
159#334—41no—
160D4⋊C8—41no—
161D4⋊C8—41no—
162D4⋊C8—41no—
163D4⋊C8—41no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C22×C411yescharacteristic
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4111yescharacteristic
173C2×D8—22no—
174C2×D8—41no—
175C2×D8—41no—
176C2×M16—24no—
177C2×M16—42no—
178C2×M16—42no—
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—42no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—41no—
191C22×C8—41no—
192C22×C8—41no—
193C22×C8—41no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8C22×C411yescharacteristic
198C22×C8C22×C411yescharacteristic
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205#121—22no—
206#121—22no—
207#121—41no—
208#121—41no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42C2411yescharacteristic
249#107—41no—
250#106—22no—
251#106—41no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264#102—22no—
265#102—22no—
266#102—41no—
267#102—41no—
268#102—41no—
269#102—41no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280#97—42no—
281#97—42no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—41no—
295C4×C8—41no—
296C4×C8—42no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303C24—21no—
304C24—21no—
305C24C4×D411yescharacteristic
306C2×D4—24no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4#13112yes—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4111yescharacteristic
364D8—44no—
365M16—48no—
366M16—84no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—44no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—44no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—84no—
410C2×C8AGL(1,ℤ/8)12yes—
411C2×C8AGL(1,ℤ/8)12yes—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4#13112yes—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42H(5,𝔽2)11yescharacteristic
449C42H(5,𝔽2)11yescharacteristic
450C42H(5,𝔽2)11yescharacteristic
451C422−1+411yescharacteristic
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23#44412yes—
473C23C2×C4×D411yescharacteristic
474D4—22no—
475D4—22no—
476D4—48no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—84no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4#44312yes—
531C2×C4#51811yescharacteristic
532C2×C4#52011yescharacteristic
533C2×C4#53411yescharacteristic
534C2×C4#53711yescharacteristic
535C2×C4#53811yescharacteristic
536C2×C4#53911yescharacteristic
537C2×C4C2×#13111yescharacteristic
538C2×C4C2×#13111yescharacteristic
539C2×C4#56811yescharacteristic
540C2×C4#56811yescharacteristic
541C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
542C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
543C2×C4#57511yescharacteristic
544C2×C4#57511yescharacteristic
545C8—22no—
546C8—22no—
547C8—42no—
548C8—42no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—81no—
556C8—81no—
557C8—81no—
558C8—81no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22#232511yescharacteristic
581C22#292911yescharacteristic
582C22#293011yescharacteristic
583C22C4×AGL(1,ℤ/8)11yescharacteristic
584C22C4×AGL(1,ℤ/8)11yescharacteristic
585C22C2×#44311yescharacteristic
586C22C2×#44411yescharacteristic
587C4—21no—
588C4—21no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4#297611yescharacteristic
599C4#298111yescharacteristic
600C4#299211yescharacteristic
601C4#299311yescharacteristic
602C4#319211yescharacteristic
603C4#319711yescharacteristic
604C4#325111yescharacteristic
605C4#325111yescharacteristic
606C2—42no—
607C2—82no—
608C2—161no—
609C2#1791211yescharacteristic
610C2#1791411yescharacteristic
611C2#1793811yescharacteristic
612C2#1794511yescharacteristic
613C2#2217311yescharacteristic
614C2#2218711yescharacteristic
615C2#2220411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#1999C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21487C2semidirect
3. C8×#121C2semidirect
4. #9546C2semidirect
5. #9542C2semidirect
6. #9538C2semidirect
7. #9534C2semidirect
8. #9078C2semidirect
9. #9078C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8847C2semidirect
13. #8843C2semidirect
14. #8600C2semidirect
15. #7848C2semidirect

Maximal subgroups

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

subgroupindex
#236972
#214872
C8×#1212
#95462
#95422
#95382
#95342
#90782
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