Finite Group Databasebeta

← #6631344

Subgroups — #6631344

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic, complemented
2#35254C211yesmaximal, characteristic
3#32751C214yesmaximal, complemented
4#32750C214yesmaximal, complemented
5#32747C214yesmaximal, complemented
6#32746C214yesmaximal, complemented
7#23669C212yesmaximal, complemented
8#23659C211yesmaximal, characteristic, complemented
9#23643C212yesmaximal, complemented
10C2×#2073C211yesmaximal, characteristic, complemented
11#23614C212yesmaximal, complemented
12#23551C211yesmaximal, characteristic, complemented
13#23486C212yesmaximal
14#23454C211yesmaximal, characteristic, complemented
15#20630C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#428C2212yes—
20C2×#427C2212yes—
21C2×#424C2212yes—
22C2×#418C2212yes—
23C22×#102C2212yes—
24C2×#408C2212yes—
25#2854C2214yes—
26#2851C2212yescomplemented
27#2851C2212yescomplemented
28#2628C2214yescomplemented
29#2624C2214yes—
30#2581—22no—
31#2580—22no—
32#2564—24no—
33#2448—22no—
34#2445—22no—
35#2418—24no—
36#2316C2212yes—
37#2316C2211yescharacteristic
38#2311C2212yes—
39#2311C2211yescharacteristic
40#2309C2211yescharacteristic
41#2296C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#382C2211yescharacteristic
44#2088C2214yescomplemented
45#2088C2214yescomplemented
46#2087C2214yescomplemented
47#2086C2214yescomplemented
48#2085C2214yescomplemented
49#2085C2214yescomplemented
50#2077C2214yescomplemented
51#2077C2214yescomplemented
52#2076C2214yescomplemented
53#2075C2214yescomplemented
54#2073C2214yes—
55#2073C2212yes—
56#2073C2212yes—
57#2067C2214yes—
58#2067C2214yes—
59#2065C2214yes—
60#2065C2214yes—
61#2059C2212yescomplemented
62#2059C2212yescomplemented
63#2058C2212yescomplemented
64#2057C2212yescomplemented
65#2034C2214yes—
66#2034C2214yes—
67#2033C2214yes—
68#2033C2214yes—
69#2025C2212yes—
70#2025C2212yes—
71#2023C2214yes—
72#1798C2212yescomplemented
73#1798C2212yescomplemented
74#1797C2214yescomplemented
75#1736C2212yes—
76#1736C2212yes—
77#1735C2212yes—
78C2×#340C2212yes—
79#1728C2211yescharacteristic
80C2×#336C2211yescharacteristic
81C22×#41C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C22×M16C2312yes—
84C23×C8C2312yes—
85#535—44no—
86#533—44no—
87C2×C4⋊Q8C2312yes—
88C2×#127—21no—
89C2×#125C2311yescharacteristic
90C2×#125C2311yescharacteristic
91C2×#124—21no—
92C2×#123—24no—
93C2×C22⋊Q8—22no—
94C2×#121C2314yes—
95C2×#120—22no—
96C4×#41—44no—
97C2×C4×Q8—22no—
98C2×C4×Q8D412yes—
99C2×C4×D4D414yes—
100C2×C4×D4D412yes—
101C2×#117C2311yescharacteristic
102C2×#117C2312yes—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—22no—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111#428—44no—
112#427—44no—
113C2×#107C2312yes—
114C2×#107C2312yes—
115C2×#107C2314yes—
116C2×#106C2312yes—
117C2×#106C2312yes—
118C2×#106C2314yes—
119#424—44no—
120C2×C4⋊C8C2314yes—
121C2×C4⋊C8C2314yes—
122#418—44no—
123C2×#102—44no—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128#408—44no—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131#400—24no—
132#399—24no—
133#399—24no—
134#397—24no—
135#395—24no—
136#394—24no—
137#394—24no—
138#393—24no—
139#391—22no—
140#391—21no—
141#390—22no—
142#390—21no—
143#390—22no—
144#390—22no—
145#390C2314yes—
146#390C2312yes—
147#390C2312yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389C2312yes—
152#389C2312yes—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#388—24no—
157#388—24no—
158#387—24no—
159#387—24no—
160#386—24no—
161#386—22no—
162#386—24no—
163#385—24no—
164#385—24no—
165#385—22no—
166#382—22no—
167#382C2312yes—
168#382C2312yes—
169#380—24no—
170#380—22no—
171#380C2314yes—
172#380C2314yes—
173#380C2314yes—
174#379—22no—
175C4×C4⋊C4—22no—
176C2×#95—22no—
177#340C2312yes—
178#340C2312yes—
179#340C2312yes—
180#340C2312yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2312yes—
188#336C2312yes—
189#336C2311yescharacteristic
190#336C2311yescharacteristic
191#336C2312yes—
192C25—21no—
193C2×#41—24no—
194C2×#41—22no—
195C2×#41—24no—
196C2×#41—22no—
197C22×Q8—22no—
198C22×Q8C2×D411yescharacteristic
199C22×D4—24no—
200C22×D4—21no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4C2×D412yes—
206C22×D4C2×D411yescharacteristic
207C22×D4C2412yes—
208C23×C4—21no—
209C23×C4—22no—
210C23×C4C2×D412yes—
211C23×C4C2×D414yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C2×M16—24no—
215C2×M16—24no—
216C2×M16—24no—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—24no—
220C22×C8C2412yes—
221C22×C8C2412yes—
222C22×C8C2414yes—
223C4⋊Q8—42no—
224C4⋊Q8—42no—
225#127—44no—
226#125—42no—
227#125—42no—
228#124—44no—
229#123—88no—
230C22⋊Q8—48no—
231#121—44no—
232#121—44no—
233#120—48no—
234C4×Q8—44no—
235C4×Q8—48no—
236C4×D4—44no—
237C4×D4—48no—
238#117—24no—
239#117—24no—
240#117—24no—
241#117—24no—
242#117—22no—
243#117—22no—
244#117—44no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—84no—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D414yes—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—44no—
285C2×C42C2×D412yes—
286C2×C42C2×D414yes—
287C2×C42C2411yescharacteristic
288C2×C42C2412yes—
289C2×C42C2411yescharacteristic
290#107—42no—
291#107—42no—
292#107—44no—
293#106—42no—
294#106—42no—
295#106—44no—
296C4⋊C8—44no—
297C4⋊C8—44no—
298#102—44no—
299#102—44no—
300#102—44no—
301#102—44no—
302C22⋊C8—44no—
303C22⋊C8—44no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314C24—22no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—82no—
319C24—82no—
320C24C22×D412yes—
321C24C22×D411yescharacteristic
322#41—48no—
323#41—48no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—44no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—48no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—84no—
347C2×D4—84no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D414yes—
372C22×C4C22×D412yes—
373C22×C4C22×D411yescharacteristic
374C22×C4H(5,𝔽2)14yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+414yes—
377C22×C42−1+412yes—
378C22×C4C2511yescharacteristic
379M16—48no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—24no—
389C2×C8—48no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—84no—
410C4⋊C4—82no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C22⋊C4—24no—
414C22⋊C4—44no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—42no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C42—24no—
427C42—24no—
428C42—22no—
429C42—22no—
430C42—44no—
431C42—44no—
432C42—42no—
433C42—42no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—88no—
447C23—84no—
448C23—164no—
449C23—162no—
450C23—162no—
451C23#53412yes—
452C23#53612yes—
453C23#53712yes—
454C23C23×D411yescharacteristic
455Q8—44no—
456Q8—88no—
457D4—48no—
458D4—44no—
459D4—44no—
460D4—44no—
461D4—84no—
462D4—168no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—84no—
495C2×C4—88no—
496C2×C4—162no—
497C2×C4—164no—
498C2×C4#53412yes—
499C2×C4#53612yes—
500C2×C4#53712yes—
501C2×C4#56218yes—
502C2×C4#56318yes—
503C2×C4C2×H(5,𝔽2)11yescharacteristic
504C2×C4C2×H(5,𝔽2)12yes—
505C2×C4C2×2−1+412yes—
506C2×C4C2×2−1+411yescharacteristic
507C8—42no—
508C8—42no—
509C8—44no—
510C22—24no—
511C22—22no—
512C22—22no—
513C22—24no—
514C22—42no—
515C22—44no—
516C22—42no—
517C22—42no—
518C22—44no—
519C22—82no—
520C22—84no—
521C22—82no—
522C22—82no—
523C22—84no—
524C22—168no—
525C22#281414yes—
526C22C2×#53411yescharacteristic
527C22C2×#53611yescharacteristic
528C22C2×#53711yescharacteristic
529C22#358412yes—
530C22#358512yes—
531C4—42no—
532C4—44no—
533C4—42no—
534C4—42no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—84no—
541C4#281414yes—
542C4C2×#56212yes—
543C4C2×#56312yes—
544C2—24no—
545C2—82no—
546C2—162no—
547C2C2×#281411yescharacteristic
548C2#6076712yes—
549C2#6087911yescharacteristic
550C2#6088311yescharacteristic
551C2#6092612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35334C2semidirect
3. #32751C2semidirect
4. #32750C2semidirect
5. #32747C2semidirect
6. #32746C2semidirect
7. #23669C2semidirect
8. #23659C2semidirect
9. #23643C2semidirect
10. C2×#2073C2semidirect
11. #23614C2semidirect
12. #23551C2semidirect
14. #23454C2semidirect
15. #20630C2semidirect
26. #2851C22semidirect
27. #2851C22semidirect
28. #2628C22semidirect
44. #2088C22semidirect
45. #2088C22semidirect
46. #2087C22semidirect
47. #2086C22semidirect
48. #2085C22semidirect
49. #2085C22semidirect
50. #2077C22semidirect
51. #2077C22semidirect
52. #2076C22semidirect
53. #2075C22semidirect
61. #2059C22semidirect
62. #2059C22semidirect
63. #2058C22semidirect
64. #2057C22semidirect
72. #1798C22semidirect
73. #1798C22semidirect
74. #1797C22semidirect

Maximal subgroups

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

subgroupindex
#353342
#352542
#327512
#327502
#327472
#327462
#236692
#236592
#236432
C2×#20732
#236142
#235512
#234862
#234542
#206302