Finite Group Databasebeta

← #6667370

Subgroups — #6667370

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35318C211yesmaximal, characteristic
2#35294C211yesmaximal, characteristic, complemented
3#32826C212yesmaximal, complemented
4#32821C212yesmaximal, complemented
5#29928C212yesmaximal
6#29923C212yesmaximal, complemented
7#29856C212yesmaximal, complemented
8#29849C212yesmaximal
9#29699C212yesmaximal, complemented
10#29696C212yesmaximal
11#29216C212yesmaximal, complemented
12#28760C212yesmaximal, complemented
13#28723C211yesmaximal, characteristic, complemented
14#28720C211yesmaximal, characteristic, complemented
15#28520C211yesmaximal, characteristic, complemented
16#28511C211yesmaximal, characteristic, complemented
17#28291C212yesmaximal, complemented
18#28288C212yesmaximal, complemented
19#20813C211yesmaximal, characteristic, complemented
20C2×#542C2211yescharacteristic
21C2×#541C2211yescharacteristic
22C2×#491C2212yes—
23C2×#488C2212yes—
24C2×#477C2212yes—
25C2×#477C2212yes—
26C2×#475C2212yes—
27C2×#474C2212yes—
28C2×#433C2212yes—
29#2868C2211yescharacteristic
30#2862C2211yescharacteristic
31#2861C2211yescharacteristic
32#2859C2212yes—
33#2859C2211yescharacteristic
34#2857C2211yescharacteristic
35#2850C2211yescharacteristic
36#2660C2211yescharacteristic
37#2658C2211yescharacteristic
38#2629C2212yes—
39#2628C2212yes—
40#2621C2211yescharacteristic
41#2620C2211yescharacteristic
42#2570—22no—
43#2556—22no—
44#2553—22no—
45#2550—22no—
46#2521—22no—
47#2519—22no—
48#2441—22no—
49#2439—22no—
50#2331C2212yes—
51#2330C2212yes—
52#2327C2211yescharacteristic
53#2326C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×#124C2211yescharacteristic
56#2285C2212yes—
57C4×#117C2211yescharacteristic
58#2095C2212yes—
59#2095C2212yes—
60#2095C2212yes—
61#2094C2212yes—
62#2072C2212yes—
63#2072C2212yes—
64#2072C2212yes—
65#2071C2212yes—
66#2068C2214yes—
67#2068C2214yes—
68#2067C2214yes—
69#2066C2214yes—
70#2059C2214yes—
71#2057C2214yes—
72#2043C2214yes—
73#2042C2214yes—
74#2036C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2033C2214yes—
78#1992C2212yes—
79#1991C2212yes—
80#1983C2214yes—
81#1981C2214yes—
82#1959C2212yes—
83#1956C2212yes—
84#1949C2214yes—
85#1947C2214yes—
86#1847C2212yes—
87#1845C2212yes—
88#1843C2212yes—
89#1831C2212yes—
90#1805C2212yes—
91#1805C2212yes—
92#1805C2212yes—
93#1804C2212yes—
94#1798C2212yes—
95#1797C2212yes—
96#1768C2212yes—
97#1767C2211yescharacteristic
98C4×C4⋊C8C2211yescharacteristic
99#542—44no—
100#541—44no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—21no—
108C2×#126—21no—
109C2×#126—21no—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125—22no—
113C2×#125C2312yes—
114C2×#125C2311yescharacteristic
115C2×#124—21no—
116C2×#124C2311yescharacteristic
117C2×#124C2311yescharacteristic
118C2×#124C2311yescharacteristic
119C2×#123—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8C2311yescharacteristic
122C2×C22⋊Q8C2311yescharacteristic
123C2×C22⋊Q8C2311yescharacteristic
124C2×#121C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117—22no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#491—44no—
134#488—44no—
135#477—44no—
136#477—44no—
137#475—44no—
138#474—44no—
139#433—44no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2314yes—
157C2×#97C2312yes—
158C2×C4×C8C2312yes—
159#400—22no—
160#399—22no—
161#399—22no—
162#399—22no—
163#398—24no—
164#398—22no—
165#398—22no—
166#397—22no—
167#396—22no—
168#396—24no—
169#395—22no—
170#395—22no—
171#394—22no—
172#393—22no—
173#391—22no—
174#391—24no—
175#391C2311yescharacteristic
176#391C2311yescharacteristic
177#391C2311yescharacteristic
178#391C2311yescharacteristic
179#391C2311yescharacteristic
180#391C2311yescharacteristic
181#390C2311yescharacteristic
182#390C2311yescharacteristic
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—24no—
190#389C2314yes—
191#389C2314yes—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#387—24no—
198#387—24no—
199#387—22no—
200#387—22no—
201#386—22no—
202#385—22no—
203#384—22no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#384C2312yes—
209#382—22no—
210#382—22no—
211#382—22no—
212#382C2312yes—
213#382C2311yescharacteristic
214#382C2311yescharacteristic
215#380—22no—
216#380—22no—
217#380C2314yes—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D412yes—
222C4×C4⋊C4C2314yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226#376C2311yescharacteristic
227#376C2311yescharacteristic
228C43C2311yescharacteristic
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2411yescharacteristic
240C22×Q8C2411yescharacteristic
241C22×Q8C2411yescharacteristic
242C22×D4—21no—
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2411yescharacteristic
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—84no—
257#126—44no—
258#126—44no—
259#126—44no—
260#125—24no—
261#125—24no—
262#125—42no—
263#125—44no—
264#125—44no—
265#125—84no—
266#124—22no—
267#124—24no—
268#124—22no—
269#124—42no—
270#124—44no—
271#123—44no—
272C22⋊Q8—42no—
273C22⋊Q8—42no—
274C22⋊Q8—42no—
275C22⋊Q8—44no—
276#121—42no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×D4—24no—
281#117—24no—
282#117—22no—
283#117—22no—
284#117—84no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2414yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C2×D411yescharacteristic
331C2×C22⋊C4C2×D411yescharacteristic
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—44no—
350#107—44no—
351#107—44no—
352#106—44no—
353C4⋊C8—24no—
354C4⋊C8—24no—
355C4⋊C8—28no—
356C4⋊C8—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#102—44no—
364#102—44no—
365C22⋊C8—44no—
366#97—24no—
367C4×C8—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—24no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C24C22×D411yescharacteristic
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—84no—
403C2×Q8—82no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—82no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—82no—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C2511yescharacteristic
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—84no—
494C42—24no—
495C42—24no—
496C42—24no—
497C42—21no—
498C42—22no—
499C42—22no—
500C42—21no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—84no—
507C42H(5,𝔽2)14yes—
508C42H(5,𝔽2)12yes—
509C422−1+414yes—
510C422−1+412yes—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23C23×D411yescharacteristic
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—164no—
529Q8—162no—
530D4—42no—
531D4—42no—
532D4—162no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—162no—
578C2×C4#53512yes—
579C2×C4#53512yes—
580C2×C4#53614yes—
581C2×C4#53714yes—
582C2×C4#57814yes—
583C2×C4C2×H(5,𝔽2)11yescharacteristic
584C2×C4C2×H(5,𝔽2)12yes—
585C2×C4C2×2−1+412yes—
586C2×C4C2×2−1+411yescharacteristic
587C8—44no—
588C8—44no—
589C22—42no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—42no—
594C22—82no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—82no—
599C22C2×#53511yescharacteristic
600C22C2×#53611yescharacteristic
601C22C2×#53711yescharacteristic
602C22C2×#57812yes—
603C22#358512yes—
604C4—24no—
605C4—22no—
606C4—22no—
607C4—24no—
608C4—24no—
609C4—42no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—84no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—84no—
622C4#354812yes—
623C4#355012yes—
624C2—81no—
625C2—161no—
626C2#3764011yescharacteristic
627C2#6082111yescharacteristic
628C2#6082511yescharacteristic
629C2#6087412yes—
630C2#6093012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35294C2semidirect
3. #32826C2semidirect
4. #32821C2semidirect
6. #29923C2semidirect
7. #29856C2semidirect
9. #29699C2semidirect
11. #29216C2semidirect
12. #28760C2semidirect
13. #28723C2semidirect
14. #28720C2semidirect
15. #28520C2semidirect
16. #28511C2semidirect
17. #28291C2semidirect
18. #28288C2semidirect
19. #20813C2semidirect

Maximal subgroups

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

subgroupindex
#353182
#352942
#328262
#328212
#299282
#299232
#298562
#298492
#296992
#296962
#292162
#287602
#287232
#287202
#285202
#285112
#282912
#282882
#208132