Finite Group Databasebeta

← #6805694

Subgroups — #6805694

Order 512.

1516 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic, complemented
2#36562C211yesmaximal, characteristic, complemented
3#33298C211yesmaximal, characteristic, complemented
4#32440C214yesmaximal, complemented
5#32436C214yesmaximal, complemented
6#30212C212yesmaximal, complemented
7#29212C212yesmaximal, complemented
8#28727C211yesmaximal, characteristic, complemented
9#28621C212yesmaximal
10#28521C211yesmaximal, characteristic, complemented
11#24696C212yesmaximal, complemented
12#24692C212yesmaximal, complemented
13#24340C212yesmaximal, complemented
14#24339C212yesmaximal, complemented
15#22223C212yesmaximal, complemented
16#21701C211yesmaximal, characteristic
17#21346C211yesmaximal, characteristic, complemented
18C2×#557C2211yescharacteristic
19C2×#554C2211yescharacteristic
20C2×#496C2211yescharacteristic
21C2×#489C2212yes—
22C2×#476C2212yes—
23C2×#474C2212yes—
24C2×#442C2212yes—
25C2×#440C2212yes—
26#2862C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859C2212yes—
29#2858C2212yes—
30#2592—22no—
31#2580C2212yes—
32#2567C2211yescharacteristic
33#2566C2212yes—
34#2565C2212yes—
35#2556—22no—
36#2556—22no—
37#2542C2211yescharacteristic
38#2513C2214yes—
39#2448—22no—
40#2447—22no—
41#2429—22no—
42#2367—22no—
43#2360—22no—
44#2357C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#127C2211yescharacteristic
47C4×#124C2211yescharacteristic
48C4×#124C2212yes—
49C42×Q8C2211yescharacteristic
50#2092C2214yes—
51#2074C2212yes—
52#2073C2212yes—
53#2071C2214yes—
54#2071C2214yes—
55#2059C2214yes—
56#2059C2214yes—
57#2055C2214yes—
58#2054C2214yes—
59#2025C2214yes—
60#2025C2214yes—
61#1993C2212yes—
62#1993C2212yes—
63#1993C2212yes—
64#1993C2212yes—
65#1993C2212yes—
66#1985C2212yes—
67#1970C2212yes—
68#1968C2212yes—
69#1961C2211yescharacteristic
70#1961C2212yes—
71#1959C2211yescharacteristic
72#1948C2214yes—
73#1947C2214yes—
74#1940C2212yes—
75#1928C2214yes—
76#1926C2214yes—
77#1916C2214yes—
78#1880C2214yes—
79#1879C2214yes—
80#1865C2214yes—
81#1862C2214yes—
82#1847C2211yescharacteristic
83#1847C2212yes—
84#1835C2214yes—
85#1805C2212yes—
86#1804C2212yes—
87#1767C2212yes—
88#1767C2211yescharacteristic
89#1745C2212yes—
90C4×#97C2211yescharacteristic
91C22×SD16C2312yes—
92C22×D8C2312yes—
93#557—44no—
94#554—44no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#125—22no—
101C2×#125C2312yes—
102C2×#125C2312yes—
103C2×#124—22no—
104C2×#124—22no—
105C2×#124C2311yescharacteristic
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8D412yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C22×C4⋊C4—21no—
120#496—24no—
121#489—28no—
122#476—44no—
123#474—44no—
124#442—44no—
125#440—44no—
126C2×#107C2312yes—
127C2×#107C2314yes—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#97C2312yes—
141C2×#97C2311yescharacteristic
142C2×#97C2312yes—
143C2×#97C2311yescharacteristic
144C2×C4×C8C2312yes—
145#399—22no—
146#399—21no—
147#399—21no—
148#398—22no—
149#398—22no—
150#397—24no—
151#397—24no—
152#396—22no—
153#396—22no—
154#395—21no—
155#395—21no—
156#393—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2311yescharacteristic
163#391C2312yes—
164#390—22no—
165#390—22no—
166#390—22no—
167#390C2312yes—
168#390C2312yes—
169#390C2314yes—
170#390C2311yescharacteristic
171#390C2314yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2314yes—
179#388—24no—
180#388—24no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—24no—
185#387—22no—
186#387—22no—
187#385—24no—
188#385—22no—
189#385—22no—
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#382C2312yes—
194#382C2312yes—
195#382C2312yes—
196#382C2311yescharacteristic
197#382C2312yes—
198#381—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C43C2311yescharacteristic
211C43C2312yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220#336C2312yes—
221#336C2314yes—
222C22×Q8—21no—
223C22×Q8—22no—
224C22×Q8—21no—
225C22×Q8C2×D411yescharacteristic
226C22×Q8C2411yescharacteristic
227C22×D4—22no—
228C22×D4C2412yes—
229C22×D4C2411yescharacteristic
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—22no—
233C2×SD16—24no—
234C2×SD16—24no—
235C2×SD16—42no—
236C2×D8—24no—
237C2×D8—24no—
238C2×D8—42no—
239C22×C8C2412yes—
240C22×C8C2414yes—
241C22×C8C2412yes—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—44no—
247#127—22no—
248#127—22no—
249#125—44no—
250#125—44no—
251#125—48no—
252#124—22no—
253#124—48no—
254#124—84no—
255#124#4114yes—
256C22⋊Q8—48no—
257C22⋊Q8—48no—
258C22⋊Q8—44no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—84no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—44no—
269C4×D4—84no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42—24no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2412yes—
319C2×C42C2412yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—42no—
324#107—44no—
325#106—42no—
326#106—44no—
327C4⋊C8—28no—
328C4⋊C8—24no—
329#103—28no—
330#103—42no—
331#102—28no—
332#102—42no—
333#102—42no—
334#102—42no—
335#102—44no—
336#102—44no—
337#97—24no—
338#97—22no—
339#97—24no—
340#97—22no—
341C4×C8—24no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95C2×D411yescharacteristic
356#95C2×D411yescharacteristic
357#95C2×D412yes—
358#95C2×D412yes—
359C24—21no—
360C24—22no—
361C2×Q8—22no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—42no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—44no—
372C2×Q8—82no—
373C2×Q8C2×#4112yes—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4C2×#4112yes—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—84no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C2511yescharacteristic
410SD16—48no—
411D8—48no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C82−1+414yes—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—84no—
449C22⋊C4—28no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—82no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C42—24no—
462C42—24no—
463C42—22no—
464C42—24no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—44no—
471C42—42no—
472C42—84no—
473C42C2×#4112yes—
474C42H(5,𝔽2)14yes—
475C422−1+412yes—
476C23—22no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23C23×D411yescharacteristic
487Q8—24no—
488Q8—44no—
489Q8—42no—
490Q8—84no—
491Q8—88no—
492Q8—162no—
493D4—24no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—88no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—84no—
537C2×C4—164no—
538C2×C4—164no—
539C2×C4#53512yes—
540C2×C4#53512yes—
541C2×C4#54814yes—
542C2×C4#57514yes—
543C2×C4C22×#4111yescharacteristic
544C2×C4C2×H(5,𝔽2)12yes—
545C2×C4C2×2−1+412yes—
546C2×C4C2×2−1+411yescharacteristic
547C8—24no—
548C8—44no—
549C8—42no—
550C22—44no—
551C22—42no—
552C22—82no—
553C22—81no—
554C22—81no—
555C22—84no—
556C22—82no—
557C22—84no—
558C22—82no—
559C22—82no—
560C22C2×#53511yescharacteristic
561C22C2×#54811yescharacteristic
562C22#349911yescharacteristic
563C22C2×#57512yes—
564C22#358512yes—
565C4—24no—
566C4—22no—
567C4—24no—
568C4—22no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—81no—
575C4—82no—
576C4—84no—
577C4—81no—
578C4—82no—
579C4—82no—
580C4#352012yes—
581C4#355012yes—
582C2—82no—
583C2—162no—
584C2#3916711yescharacteristic
585C2#6080211yescharacteristic
586C2#6082511yescharacteristic
587C2#6132412yes—
588C2#6250612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#496C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36609C2semidirect
2. #36562C2semidirect
3. #33298C2semidirect
4. #32440C2semidirect
5. #32436C2semidirect
6. #30212C2semidirect
7. #29212C2semidirect
8. #28727C2semidirect
10. #28521C2semidirect
11. #24696C2semidirect
12. #24692C2semidirect
13. #24340C2semidirect
14. #24339C2semidirect
15. #22223C2semidirect
17. #21346C2semidirect

Maximal subgroups

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

subgroupindex
#366092
#365622
#332982
#324402
#324362
#302122
#292122
#287272
#286212
#285212
#246962
#246922
#243402
#243392
#222232
#217012
#213462