Finite Group Databasebeta

← #218953

Subgroups — #218953

Order 512.

729 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23696C211yesmaximal, characteristic
2#21487C211yesmaximal, characteristic, complemented
3C8×#121C211yesmaximal, characteristic, complemented
4#9546C211yesmaximal, characteristic, complemented
5#9542C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9536C211yesmaximal, characteristic, complemented
8#9076C211yesmaximal, characteristic, complemented
9#9076C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8849C211yesmaximal, characteristic, complemented
13#8845C211yesmaximal, characteristic, complemented
14#8598C211yesmaximal, characteristic, complemented
15#7849C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×#435—21no—
19C2×C8×D4—21no—
20C2×C8×D4C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#2086C411yescharacteristic
27#2075C411yescharacteristic
28#2067C411yescharacteristic
29#2067C411yescharacteristic
30#2054C411yescharacteristic
31#2033C411yescharacteristic
32#2033C411yescharacteristic
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×#107C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1753—21no—
50C8×C22⋊C4—21no—
51C2×#334C2211yescharacteristic
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—
69C23×C8—21no—
70C23×C8D411yescharacteristic
71C23×C8D411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C4×D8—41no—
80C4×D8—41no—
81#436—41no—
82#436—41no—
83#435—44no—
84C8×D4—41no—
85C8×D4—41no—
86C8×D4—44no—
87#431—41no—
88#431—41no—
89#431—41no—
90#431—41no—
91C2×#107C2×C411yescharacteristic
92C2×#107C2×C411yescharacteristic
93C2×#107C2×C411yescharacteristic
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2×C411yescharacteristic
98C2×C4⋊C8C2×C411yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
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⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8C2×C411yescharacteristic
119C2×C22⋊C8C2×C411yescharacteristic
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#334—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—42no—
177C2×M16—42no—
178C2×M16—42no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—41no—
190C22×C8—41no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—42no—
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202#121—22no—
203#121—22no—
204#121—41no—
205#121—41no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
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×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42C2411yescharacteristic
246#107—22no—
247#107—41no—
248#107—41no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262#102—22no—
263#102—22no—
264#102—41no—
265#102—41no—
266#102—41no—
267#102—41no—
268C22⋊C8—41no—
269C22⋊C8—41no—
270C22⋊C8—42no—
271C22⋊C8—42no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279#97—42no—
280C4×C8—21no—
281C4×C8—21no—
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—41no—
293C4×C8—41no—
294C4×C8—42no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301C24—21no—
302C24—21no—
303C24C4×D411yescharacteristic
304C2×D4—24no—
305C2×D4—21no—
306C2×D4—21no—
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—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4#13112yes—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
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—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
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×C4C4×D411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4AGL(1,ℤ/8)12yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4111yescharacteristic
362D8—44no—
363M16—84no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
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—44no—
387C2×C8—44no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
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—84no—
407C2×C8#13512yes—
408C2×C8#13512yes—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
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—42no—
421C4⋊C4—42no—
422C4⋊C4#13112yes—
423C22⋊C4—24no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42H(5,𝔽2)11yescharacteristic
446C42H(5,𝔽2)11yescharacteristic
447C42H(5,𝔽2)11yescharacteristic
448C422−1+411yescharacteristic
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23#44212yes—
470C23C2×C4×D411yescharacteristic
471D4—22no—
472D4—22no—
473D4—48no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—84no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—42no—
497C2×C4—42no—
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—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4#44212yes—
528C2×C4#51811yescharacteristic
529C2×C4#52011yescharacteristic
530C2×C4#53411yescharacteristic
531C2×C4#53711yescharacteristic
532C2×C4#53811yescharacteristic
533C2×C4#53911yescharacteristic
534C2×C4C2×#13111yescharacteristic
535C2×C4C2×#13111yescharacteristic
536C2×C4#56811yescharacteristic
537C2×C4#56811yescharacteristic
538C2×C4C2×#13511yescharacteristic
539C2×C4C2×#13511yescharacteristic
540C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
541C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
542C8—22no—
543C8—22no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C8—81no—
551C8—81no—
552C8—81no—
553C8—81no—
554C8—81no—
555C8—81no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22#232511yescharacteristic
578C22#292911yescharacteristic
579C22#293011yescharacteristic
580C22C4×#13511yescharacteristic
581C22C2×#44211yescharacteristic
582C22#295511yescharacteristic
583C22#295511yescharacteristic
584C4—22no—
585C4—21no—
586C4—21no—
587C4—41no—
588C4—41no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4#297611yescharacteristic
596C4#298111yescharacteristic
597C4#299211yescharacteristic
598C4#299311yescharacteristic
599C4#318811yescharacteristic
600C4#318811yescharacteristic
601C4#324711yescharacteristic
602C4#324711yescharacteristic
603C2—42no—
604C2—82no—
605C2—161no—
606C2#1790411yescharacteristic
607C2#1790711yescharacteristic
608C2#1793211yescharacteristic
609C2#1793511yescharacteristic
610C2#2217311yescharacteristic
611C2#2218411yescharacteristic
612C2#2220511yescharacteristic
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. #9540C2semidirect
7. #9536C2semidirect
8. #9076C2semidirect
9. #9076C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8849C2semidirect
13. #8845C2semidirect
14. #8598C2semidirect
15. #7849C2semidirect

Maximal subgroups

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

subgroupindex
#236962
#214872
C8×#1212
#95462
#95422
#95402
#95362
#90762
#89512
#89472
#88492
#88452
#85982
#78492