Finite Group Databasebeta

← #6478029

Subgroups — #6478029

Order 512.

1594 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2C2×#2370C211yesmaximal, characteristic, complemented
3#33147C211yesmaximal, characteristic, complemented
4#33107C211yesmaximal, characteristic
5#33096C212yesmaximal, complemented
6#29966C211yesmaximal, characteristic, complemented
7#29754C211yesmaximal, characteristic
8#29640C212yesmaximal, complemented
9#27150C214yesmaximal, complemented
10#27022C214yesmaximal, complemented
11#26880C214yesmaximal, complemented
12#26712C214yesmaximal, complemented
13#22082C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21578C212yesmaximal, complemented
16C2×#539—21no—
17C22×#125C2211yescharacteristic
18C2×#481C2212yes—
19C2×#480C2212yes—
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22#2854C2212yes—
23#2639—21no—
24#2618—21no—
25#2560—22no—
26#2558—22no—
27#2512C2212yes—
28#2506C2212yes—
29#2452C2212yes—
30#2445C2212yes—
31#2370—22no—
32#2370—22no—
33#2370C2214yes—
34#2370C2214yes—
35#2325—21no—
36C4×#121C2212yes—
37C2×#389—21no—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#387—21no—
41C2×#387—21no—
42C2×C4×C4⋊C4—21no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47#2099C2212yes—
48#2086C2214yes—
49#2083C2212yes—
50#2067C2214yes—
51#2059C2214yes—
52#2059C2214yes—
53#2057C2212yes—
54#2041C2212yes—
55#2025C2214yes—
56#2025C2214yes—
57#1988C2212yes—
58#1982C2212yes—
59#1948C2212yes—
60#1948C2212yes—
61#1944C2214yes—
62#1936C2214yes—
63#1926C2214yes—
64#1926C2214yes—
65#1926C2214yes—
66#1925C2214yes—
67#1921C2214yes—
68#1921C2214yes—
69#1920C2214yes—
70#1919C2214yes—
71#1919C2214yes—
72#1917C2214yes—
73#1917C2214yes—
74#1874C2212yes—
75#1874C2212yes—
76#1862C2214yes—
77#1853C2212yes—
78#1852C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1776C2214yes—
85#1762C2214yes—
86#1749C2212yes—
87#539—48no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×#127C2311yescharacteristic
90C2×#126—22no—
91C2×#125—22no—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—22no—
95C2×#125C2314yes—
96C2×#123—22no—
97C2×#123—22no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—22no—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C42—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#481—28no—
122#480—28no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×#97C2314yes—
136#400—22no—
137#398—22no—
138#397—22no—
139#396—22no—
140#396—22no—
141#394—22no—
142#394—22no—
143#393—22no—
144#391—21no—
145#390—21no—
146#390—21no—
147#390C2312yes—
148#390C2312yes—
149#390C2312yes—
150#390C2312yes—
151#389—24no—
152#389—24no—
153#389—21no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2314yes—
163#389C2312yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#387—24no—
169#387—24no—
170#387—22no—
171#387—24no—
172#387—24no—
173#386—22no—
174#386—22no—
175#386—22no—
176#385—22no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#380—21no—
180#380C2312yes—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—24no—
186C4×C4⋊C4—24no—
187C4×C4⋊C4D412yes—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2314yes—
191C4×C4⋊C4C2314yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217C22×Q8—21no—
218C22×D4—22no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4C2412yes—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—22no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—42no—
237#127—42no—
238#126—48no—
239#125—216no—
240#125—28no—
241#125—28no—
242#125—48no—
243#123—48no—
244#123—48no—
245C22⋊Q8—48no—
246C22⋊Q8—44no—
247#121—24no—
248#121—24no—
249#121—44no—
250#121—44no—
251C4×D4—44no—
252C4×D4—48no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4#4114yes—
286C2×C4⋊C4#4114yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—44no—
315C2×C42—82no—
316C2×C42—82no—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2412yes—
322C2×C42C2412yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325#107—28no—
326#107—44no—
327#106—28no—
328#106—28no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331#102—28no—
332#102—28no—
333#102—28no—
334#102—44no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337#97—44no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95C2×D412yes—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355C24—22no—
356C24C22×D411yescharacteristic
357C2×Q8—42no—
358C2×Q8—81no—
359C2×Q8—81no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—42no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—42no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—81no—
372C2×D4—81no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4AGL(1,ℤ/8)14yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2511yescharacteristic
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C4⋊C4—24no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—84no—
462C4⋊C4—81no—
463C4⋊C4C2×#4114yes—
464C4⋊C4C2×#4114yes—
465C4⋊C42−1+414yes—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—24no—
481C42—22no—
482C42—22no—
483C42—41no—
484C42—41no—
485C42—42no—
486C42—48no—
487C42—44no—
488C42—82no—
489C42—82no—
490C42—84no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23#52512yes—
503C23C2×AGL(1,ℤ/8)12yes—
504C23#57512yes—
505C23C23×D411yescharacteristic
506Q8—162no—
507D4—44no—
508D4—44no—
509D4—88no—
510D4—162no—
511D4—162no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—84no—
569C2×C4—164no—
570C2×C4#52512yes—
571C2×C4C2×AGL(1,ℤ/8)12yes—
572C2×C4C2×AGL(1,ℤ/8)12yes—
573C2×C4C22×#4111yescharacteristic
574C2×C4C22×#4111yescharacteristic
575C2×C4C2×2−1+412yes—
576C2×C4#58512yes—
577C8—44no—
578C8—44no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22#311514yes—
593C22C2×#52511yescharacteristic
594C22#348511yescharacteristic
595C22#348611yescharacteristic
596C22C22×AGL(1,ℤ/8)11yescharacteristic
597C22C2×#57511yescharacteristic
598C22#358511yescharacteristic
599C22#358611yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—44no—
609C4—44no—
610C4—42no—
611C4—44no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4#346112yes—
617C4#350712yes—
618C2—24no—
619C2—162no—
620C2#3509211yescharacteristic
621C2C2×#311511yescharacteristic
622C2#6056511yescharacteristic
623C2#6170711yescharacteristic
624C2#6171611yescharacteristic
625C2#6182811yescharacteristic
626C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2370C2semidirect
3. #33147C2semidirect
5. #33096C2semidirect
6. #29966C2semidirect
8. #29640C2semidirect
9. #27150C2semidirect
10. #27022C2semidirect
11. #26880C2semidirect
12. #26712C2semidirect
13. #22082C2semidirect
14. #21916C2semidirect
15. #21578C2semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#23702
#331472
#331072
#330962
#299662
#297542
#296402
#271502
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