Finite Group Databasebeta

← #6486127

Subgroups — #6486127

Order 512.

1361 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35287C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic, complemented
3#33284C212yesmaximal, complemented
4#33261C212yesmaximal, complemented
5#30365C212yesmaximal, complemented
6#30356C212yesmaximal, complemented
7#29518C212yesmaximal, complemented
8#29514C212yesmaximal, complemented
9#29226C211yesmaximal, characteristic, complemented
10#29209C211yesmaximal, characteristic, complemented
11#29145C211yesmaximal, characteristic, complemented
12#28508C211yesmaximal, characteristic, complemented
13#27901C212yesmaximal
14#27885C212yesmaximal, complemented
15#24712C214yesmaximal, complemented
16#22127C214yesmaximal, complemented
17#20468C211yesmaximal, characteristic, complemented
18C2×#474C2214yes—
19C2×#433C2212yes—
20#2868C2211yescharacteristic
21#2862C2211yescharacteristic
22#2860C2212yes—
23#2857C2212yes—
24#2854C2212yes—
25#2647C2212yes—
26#2647C2211yescharacteristic
27#2641C2211yescharacteristic
28#2629C2211yescharacteristic
29#2628C2212yes—
30#2619C2211yescharacteristic
31#2593—22no—
32#2554—22no—
33#2511—22no—
34#2492—22no—
35#2468—22no—
36#2457—22no—
37#2451—22no—
38#2423—22no—
39#2331C2212yes—
40#2324C2212yes—
41#2319C2212yes—
42#2314C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×#127C2211yescharacteristic
45C4×#125C2211yescharacteristic
46C4×#117C2211yescharacteristic
47#2099C2212yes—
48#2099C2212yes—
49#2093C2212yes—
50#2093C2212yes—
51#2076C2214yes—
52#2075C2214yes—
53#2058C2212yes—
54#2057C2212yes—
55#2034C2214yes—
56#2033C2214yes—
57#2023C2212yes—
58#2023C2212yes—
59#1992C2212yes—
60#1991C2212yes—
61#1987C2214yes—
62#1961C2212yes—
63#1957C2212yes—
64#1949C2214yes—
65#1944C2214yes—
66#1943C2214yes—
67#1942C2214yes—
68#1942C2214yes—
69#1936C2214yes—
70#1936C2214yes—
71#1934C2214yes—
72#1934C2214yes—
73#1926C2214yes—
74#1925C2214yes—
75#1921C2214yes—
76#1918C2214yes—
77#1846C2212yes—
78#1845C2212yes—
79#1836C2212yes—
80#1836C2212yes—
81#1804C2214yes—
82#1768C2212yes—
83#1767C2211yescharacteristic
84C4×C4⋊C8C2211yescharacteristic
85#1750C2212yes—
86#1749C2211yescharacteristic
87C4×C22⋊C8C2211yescharacteristic
88#1745C2211yescharacteristic
89#1744C2211yescharacteristic
90#1726C2212yes—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×#127C2311yescharacteristic
95C2×#126—22no—
96C2×#125C2311yescharacteristic
97C2×#123—22no—
98C2×C22⋊Q8—22no—
99C2×#121C2312yes—
100C2×C4×Q8—22no—
101C2×C4×D4—22no—
102C2×C4×D4C2312yes—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#474—216no—
107#433—28no—
108C2×#107C2314yes—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2312yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×C22⋊C8C2312yes—
119C2×C22⋊C8C2312yes—
120C2×#97C2312yes—
121C2×#97C2312yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124C2×C4×C8C2312yes—
125C2×C4×C8C2312yes—
126#400—22no—
127#400—22no—
128#400—24no—
129#399—22no—
130#398—22no—
131#398—24no—
132#398—22no—
133#396—22no—
134#395—22no—
135#394—22no—
136#394—22no—
137#394—22no—
138#393—22no—
139#392—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—24no—
145#390—22no—
146#390—22no—
147#390C2312yes—
148#390C2312yes—
149#390C2312yes—
150#390C2312yes—
151#389—22no—
152#389—22no—
153#389C2312yes—
154#389C2314yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2314yes—
159#388—22no—
160#388—24no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#387—24no—
169#387—22no—
170#387—24no—
171#386—22no—
172#386—22no—
173#385—22no—
174#384—22no—
175#384C2311yescharacteristic
176#384C2312yes—
177#384C2312yes—
178#382C2311yescharacteristic
179#382C2312yes—
180#382C2312yes—
181#381—22no—
182#380—22no—
183#380—22no—
184#380C2314yes—
185#380C2312yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4D412yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2314yes—
192C4×C22⋊C4D412yes—
193C4×C22⋊C4D412yes—
194#376C2311yescharacteristic
195#376C2311yescharacteristic
196C43C2311yescharacteristic
197C43C2311yescharacteristic
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209#336C2311yescharacteristic
210#336C2312yes—
211#336C2312yes—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4C2412yes—
217C23×C4—22no—
218C23×C4C2×D412yes—
219C23×C4C2411yescharacteristic
220C22×C8C2412yes—
221C22×C8C2412yes—
222C22×C8C2412yes—
223C22×C8C2412yes—
224C4⋊Q8—28no—
225C4⋊Q8—44no—
226C4⋊Q8—84no—
227#127—22no—
228#127—22no—
229#126—84no—
230#125—22no—
231#125—22no—
232#123—84no—
233C22⋊Q8—84no—
234#121—44no—
235C4×Q8—84no—
236C4×D4—28no—
237C4×D4—84no—
238#117—24no—
239#117—22no—
240#117—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2414yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4C2×D411yescharacteristic
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—42no—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291#107—28no—
292#107—28no—
293#107—44no—
294#106—44no—
295C4⋊C8—24no—
296C4⋊C8#4118yes—
297#102—28no—
298#102—28no—
299#102—44no—
300#102—44no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303#97—24no—
304#97—24no—
305#97—42no—
306C4×C8—24no—
307C4×C8—24no—
308C4×C8—42no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—22no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95C2×D412yes—
324#95C2×D412yes—
325#95C2×D412yes—
326#95C2×D412yes—
327C24—22no—
328C24C22×D411yescharacteristic
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—84no—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4—82no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C2511yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8C2×#4114yes—
380C2×C8C2×#4114yes—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—82no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—84no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—24no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—84no—
419C22⋊C4—82no—
420C42—24no—
421C42—24no—
422C42—22no—
423C42—21no—
424C42—21no—
425C42—22no—
426C42—22no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—82no—
431C42—84no—
432C42C2×#4114yes—
433C422−1+412yes—
434C422−1+412yes—
435C23—22no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23C23×D411yescharacteristic
448Q8—88no—
449Q8—164no—
450D4—44no—
451D4—44no—
452D4—88no—
453D4—164no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—84no—
491C2×C4—164no—
492C2×C4#53212yes—
493C2×C4#53212yes—
494C2×C4#53614yes—
495C2×C4#57614yes—
496C2×C4C22×#4112yes—
497C2×C4C2×2−1+411yescharacteristic
498C2×C4C2×2−1+411yescharacteristic
499C2×C4#58512yes—
500C8—24no—
501C8—24no—
502C8—42no—
503C8—42no—
504C22—41no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—42no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22C2×#53211yescharacteristic
517C22C2×#53611yescharacteristic
518C22#344911yescharacteristic
519C22C2×#57612yes—
520C22#358612yes—
521C4—22no—
522C4—24no—
523C4—22no—
524C4—24no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—82no—
531C4—84no—
532C4—82no—
533C4—84no—
534C4—82no—
535C4#351512yes—
536C4#351812yes—
537C2—81no—
538C2—162no—
539C2#3520111yescharacteristic
540C2#6071411yescharacteristic
541C2#6072611yescharacteristic
542C2#6088812yes—
543C2#6097612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
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. #35283C2semidirect
3. #33284C2semidirect
4. #33261C2semidirect
5. #30365C2semidirect
6. #30356C2semidirect
7. #29518C2semidirect
8. #29514C2semidirect
9. #29226C2semidirect
10. #29209C2semidirect
11. #29145C2semidirect
12. #28508C2semidirect
14. #27885C2semidirect
15. #24712C2semidirect
16. #22127C2semidirect
17. #20468C2semidirect

Maximal subgroups

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

subgroupindex
#352872
#352832
#332842
#332612
#303652
#303562
#295182
#295142
#292262
#292092
#291452
#285082
#279012
#278852
#247122
#221272
#204682