Finite Group Databasebeta

← #6407841

Subgroups — #6407841

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#52299C211yesmaximal, characteristic
2#23381C211yesmaximal, characteristic, complemented
3#23380C211yesmaximal, characteristic, complemented
4#23359C211yesmaximal, characteristic, complemented
5#23334C213yesmaximal, complemented
6#23331C213yesmaximal, complemented
7#23318C213yesmaximal, complemented
8#23313C213yesmaximal, complemented
9#21684C211yesmaximal, characteristic, complemented
10#21670C211yesmaximal, characteristic, complemented
11#21668C211yesmaximal, characteristic, complemented
12#21565C213yesmaximal, complemented
13#21490C213yesmaximal, complemented
14#21488C213yesmaximal, complemented
15#21445C213yesmaximal, complemented
16C22×#131C2211yescharacteristic
17C2×#436C2213yes—
18C2×#436C2213yes—
19C2×#436C2213yes—
20C2×#435C2213yes—
21C2×C4×M16C2213yes—
22#2801C411yescharacteristic
23#2801C411yescharacteristic
24#2760C411yescharacteristic
25#2752C413yes—
26#2742C413yes—
27#2703C413yes—
28#2703C413yes—
29#2675C411yescharacteristic
30#2401C2211yescharacteristic
31#2401C2211yescharacteristic
32#2397C2211yescharacteristic
33#2388C2213yes—
34C4×#123C2213yes—
35C4×#121C2213yes—
36C4×#121C2213yes—
37#1955C2213yes—
38#1955C2213yes—
39#1955C2213yes—
40#1954C2213yes—
41#1940C2213yes—
42#1939C2213yes—
43#1924C2213yes—
44#1924C2213yes—
45#1924C2213yes—
46#1924C2213yes—
47#1924C2213yes—
48#1923C2213yes—
49#1923C2213yes—
50#1923C2213yes—
51#1917C2213yes—
52#1917C2213yes—
53#1917C2213yes—
54#1916C2213yes—
55#1870C2213yes—
56#1870C2213yes—
57#1870C2213yes—
58#1870C2213yes—
59#1870C2213yes—
60#1870C2213yes—
61#1869C2213yes—
62#1869C2213yes—
63#1851C2213yes—
64#1851C2213yes—
65#1850C2213yes—
66#1850C2213yes—
67#1843C2211yescharacteristic
68#1840C2211yescharacteristic
69#1840C2211yescharacteristic
70#1817C2213yes—
71#1797C2213yes—
72#1797C2213yes—
73#1780C2213yes—
74#1771C2213yes—
75C8×C4⋊C4C2213yes—
76#1753C2213yes—
77#1753C2213yes—
78#1753C2213yes—
79C8×C22⋊C4C2213yes—
80#1726C2211yescharacteristic
81C22×#41C2311yescharacteristic
82C2×#131—46no—
83C22×M16C2313yes—
84C22×M16C2311yescharacteristic
85C22×M16C2311yescharacteristic
86C22×M16C2313yes—
87C22×M16C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×#123C2×C413yes—
92C2×#121C2×C413yes—
93C2×#121C2×C413yes—
94C2×C4×D4C2313yes—
95C2×C4×D4C2313yes—
96C2×C4×D4C2313yes—
97C2×C4×D4C2313yes—
98C22×C42C2313yes—
99#436—46no—
100#436—46no—
101#436—46no—
102#435—46no—
103C2×C4⋊C8C2313yes—
104C2×C4⋊C8C2313yes—
105C2×C4⋊C8C2313yes—
106C2×C4⋊C8C2313yes—
107C2×C4⋊C8C2313yes—
108C2×C4⋊C8C2313yes—
109C2×C22⋊C8C2313yes—
110C2×C22⋊C8C2313yes—
111C2×C22⋊C8C2313yes—
112C2×C22⋊C8C2313yes—
113C2×C22⋊C8C2313yes—
114C2×C22⋊C8C2313yes—
115C2×C22⋊C8C2313yes—
116C2×C22⋊C8C2313yes—
117C2×C22⋊C8C2313yes—
118C2×C22⋊C8C2313yes—
119C2×C22⋊C8C2313yes—
120C2×C22⋊C8C2313yes—
121C4×M16—46no—
122C2×#97C2313yes—
123C2×#97C2313yes—
124C2×#97C2313yes—
125C2×C4×C8C2313yes—
126C2×C4×C8C2313yes—
127C2×C4×C8C2313yes—
128#401C2×C411yescharacteristic
129#401C2×C411yescharacteristic
130#400C2×C413yes—
131#400C2×C413yes—
132#399C2×C413yes—
133#399C2×C411yescharacteristic
134#397C2×C413yes—
135#397C2×C413yes—
136#396C2×C411yescharacteristic
137#396C2×C411yescharacteristic
138#395C2×C413yes—
139#394C2×C413yes—
140#394C2×C413yes—
141#394C2×C413yes—
142#393C2×C413yes—
143#392C2×C413yes—
144#392C2×C413yes—
145#392C2×C413yes—
146#392C2×C411yescharacteristic
147#392C2×C413yes—
148#390C2×C413yes—
149#390C2×C413yes—
150#389C2×C413yes—
151#389C2×C413yes—
152#388C2×C413yes—
153#388C2×C413yes—
154#388C2×C413yes—
155#388C2×C413yes—
156#388C2×C413yes—
157#388C2×C413yes—
158#387C2×C413yes—
159#387C2×C413yes—
160#386C2×C413yes—
161#386C2×C413yes—
162#386C2×C413yes—
163#386C2×C413yes—
164#386C2×C413yes—
165#385C2×C413yes—
166#381C2×C413yes—
167#380C2×C413yes—
168#380C2×C413yes—
169C4×C4⋊C4C2313yes—
170C4×C4⋊C4C2313yes—
171C4×C22⋊C4C2313yes—
172C4×C22⋊C4C2313yes—
173C4×C22⋊C4C2313yes—
174C4×C22⋊C4C2313yes—
175#376C2311yescharacteristic
176#336C2313yes—
177#336C2313yes—
178#336C2313yes—
179#336C2313yes—
180#336C2313yes—
181#336C2313yes—
182#336C2313yes—
183#336C2313yes—
184#336C2313yes—
185#336C2313yes—
186#336C2313yes—
187#336C2313yes—
188C2×#41—43no—
189C2×#41—43no—
190C22×Q8C22×C411yescharacteristic
191C22×D4C22×C413yes—
192C22×D4C22×C413yes—
193C22×D4C22×C413yes—
194C22×D4C22×C413yes—
195C22×D4C22×C411yescharacteristic
196C22×D4C22×C411yescharacteristic
197C22×D4C22×C411yescharacteristic
198C23×C4C22×C413yes—
199C23×C4C22×C413yes—
200C23×C4C2413yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204#131—416no—
205C2×M16—26no—
206C2×M16—26no—
207C2×M16—26no—
208C2×M16—26no—
209C2×M16—26no—
210C2×M16—43no—
211C2×M16—43no—
212C2×M16—43no—
213C2×M16—43no—
214C22×C8—26no—
215C22×C8—43no—
216C22×C8—43no—
217C22×C8C2411yescharacteristic
218C22×C8C2411yescharacteristic
219C22×C8C2411yescharacteristic
220C22×C8C2413yes—
221C22×C8C2413yes—
222C22×C8C2413yes—
223C22×C8C2413yes—
224C22×C8C2411yescharacteristic
225#123—46no—
226#121—46no—
227#121—46no—
228C4×D4—43no—
229C4×D4—43no—
230C4×D4—43no—
231C4×D4—43no—
232C4×D4—43no—
233C4×D4—43no—
234C4×D4—43no—
235C4×D4—43no—
236C2×C4⋊C4C22×C413yes—
237C2×C4⋊C4C22×C413yes—
238C2×C4⋊C4C22×C413yes—
239C2×C4⋊C4C22×C413yes—
240C2×C4⋊C4C22×C413yes—
241C2×C4⋊C4C22×C413yes—
242C2×C4⋊C4C22×C413yes—
243C2×C4⋊C4C22×C413yes—
244C2×C22⋊C4C22×C413yes—
245C2×C22⋊C4C22×C413yes—
246C2×C22⋊C4C22×C413yes—
247C2×C22⋊C4C22×C413yes—
248C2×C22⋊C4C22×C413yes—
249C2×C22⋊C4C22×C413yes—
250C2×C22⋊C4C22×C413yes—
251C2×C22⋊C4C22×C413yes—
252C2×C22⋊C4C22×C413yes—
253C2×C22⋊C4C22×C413yes—
254C2×C22⋊C4C22×C413yes—
255C2×C22⋊C4C22×C413yes—
256C2×C22⋊C4C22×C413yes—
257C2×C22⋊C4C22×C413yes—
258C2×C22⋊C4C22×C413yes—
259C2×C22⋊C4C22×C413yes—
260C2×C42—43no—
261C2×C42—43no—
262C2×C42C22×C413yes—
263C2×C42C22×C413yes—
264C2×C42C2413yes—
265C2×C42C2413yes—
266C2×C42C2413yes—
267C4⋊C8—43no—
268C4⋊C8—43no—
269C4⋊C8—43no—
270C4⋊C8—43no—
271C4⋊C8—43no—
272C4⋊C8—43no—
273C22⋊C8—43no—
274C22⋊C8—43no—
275C22⋊C8—43no—
276C22⋊C8—43no—
277C22⋊C8—43no—
278C22⋊C8—43no—
279C22⋊C8—43no—
280C22⋊C8—43no—
281C22⋊C8—43no—
282C22⋊C8—43no—
283C22⋊C8—43no—
284C22⋊C8—43no—
285#97—26no—
286#97—43no—
287#97—43no—
288C4×C8—26no—
289C4×C8—26no—
290C4×C8—43no—
291#95C22×C411yescharacteristic
292#95C22×C413yes—
293#95C22×C413yes—
294#95C22×C413yes—
295#95C22×C413yes—
296#95C22×C413yes—
297#95C22×C413yes—
298#95C22×C413yes—
299#95C22×C413yes—
300#95C22×C413yes—
301#95C22×C413yes—
302#95C22×C413yes—
303#95C22×C413yes—
304#95C22×C413yes—
305C24C23×C411yescharacteristic
306C24C23×C411yescharacteristic
307C24C23×C411yescharacteristic
308C24C23×C413yes—
309#41—44no—
310#41—412no—
311C2×Q8—43no—
312C2×D4—26no—
313C2×D4—43no—
314C2×D4—43no—
315C2×D4—43no—
316C2×D4—43no—
317C2×D4—43no—
318C2×D4—43no—
319C2×D4—43no—
320C2×D4—43no—
321C2×D4—43no—
322C2×D4—43no—
323C2×D4—43no—
324C2×D4—43no—
325C2×D4—43no—
326C2×D4—43no—
327C22×C4—23no—
328C22×C4—23no—
329C22×C4—23no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—23no—
334C22×C4—23no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—23no—
339C22×C4—26no—
340C22×C4—43no—
341C22×C4—43no—
342C22×C4—43no—
343C22×C4—43no—
344C22×C4—43no—
345C22×C4C23×C413yes—
346C22×C4C23×C413yes—
347C22×C4C23×C413yes—
348C22×C4C23×C413yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C23×C411yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C23×C413yes—
353C22×C4C23×C413yes—
354C22×C4C23×C413yes—
355C22×C4C2511yescharacteristic
356M16—412no—
357M16—44no—
358M16—412no—
359M16—44no—
360M16—44no—
361C2×C8—23no—
362C2×C8—23no—
363C2×C8—23no—
364C2×C8—23no—
365C2×C8—23no—
366C2×C8—23no—
367C2×C8—23no—
368C2×C8—23no—
369C2×C8—23no—
370C2×C8—23no—
371C2×C8—23no—
372C2×C8—23no—
373C2×C8—23no—
374C2×C8—23no—
375C2×C8—23no—
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—44no—
379C2×C8H(5,𝔽2)16yes—
380C4⋊C4—26no—
381C4⋊C4—26no—
382C4⋊C4—43no—
383C4⋊C4—43no—
384C4⋊C4—43no—
385C4⋊C4—43no—
386C4⋊C4—43no—
387C4⋊C4—43no—
388C22⋊C4—26no—
389C22⋊C4—26no—
390C22⋊C4—26no—
391C22⋊C4—43no—
392C22⋊C4—43no—
393C22⋊C4—43no—
394C22⋊C4—43no—
395C22⋊C4—43no—
396C22⋊C4—43no—
397C22⋊C4—43no—
398C22⋊C4—43no—
399C22⋊C4—43no—
400C22⋊C4—43no—
401C22⋊C4—43no—
402C22⋊C4—43no—
403C22⋊C4—43no—
404C42—23no—
405C42—23no—
406C42—23no—
407C42—23no—
408C42—23no—
409C42—23no—
410C42—43no—
411C42—43no—
412C23—23no—
413C23—23no—
414C23—23no—
415C23—23no—
416C23—23no—
417C23—23no—
418C23—23no—
419C23—23no—
420C23—23no—
421C23—23no—
422C23—23no—
423C23—23no—
424C23—21no—
425C23—21no—
426C23—23no—
427C23—21no—
428C23C24×C411yescharacteristic
429Q8—44no—
430D4—44no—
431D4—44no—
432D4—44no—
433D4—86no—
434D4—86no—
435D4—86no—
436D4—86no—
437C2×C4—23no—
438C2×C4—23no—
439C2×C4—23no—
440C2×C4—23no—
441C2×C4—23no—
442C2×C4—23no—
443C2×C4—23no—
444C2×C4—23no—
445C2×C4—23no—
446C2×C4—23no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—23no—
450C2×C4—23no—
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—23no—
457C2×C4—23no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—43no—
464C2×C4—43no—
465C2×C4—43no—
466C2×C4—43no—
467C2×C4—43no—
468C2×C4—43no—
469C2×C4—43no—
470C2×C4—86no—
471C2×C4—86no—
472C2×C4#56816yes—
473C2×C4C2×H(5,𝔽2)13yes—
474C2×C4#58513yes—
475C8—41no—
476C8—41no—
477C8—41no—
478C8—43no—
479C8—43no—
480C8—43no—
481C8—43no—
482C8#56014yes—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—43no—
487C22—43no—
488C22—43no—
489C22—43no—
490C22—43no—
491C22—43no—
492C22—43no—
493C22—43no—
494C22—43no—
495C22—43no—
496C22—43no—
497C22—43no—
498C22—43no—
499C22C4×H(5,𝔽2)13yes—
500C22C2×#56813yes—
501C22#357311yescharacteristic
502C4—26no—
503C4—43no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—43no—
508C4—43no—
509C4—43no—
510C4—43no—
511C4—43no—
512C4C2×#56011yescharacteristic
513C4#352913yes—
514C2—81no—
515C2—81no—
516C2—81no—
517C2—83no—
518C2C4×#56011yescharacteristic
519C2#6032413yes—
520C2#6032613yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radical#52299C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23381C2semidirect
3. #23380C2semidirect
4. #23359C2semidirect
5. #23334C2semidirect
6. #23331C2semidirect
7. #23318C2semidirect
8. #23313C2semidirect
9. #21684C2semidirect
10. #21670C2semidirect
11. #21668C2semidirect
12. #21565C2semidirect
13. #21490C2semidirect
14. #21488C2semidirect
15. #21445C2semidirect

Maximal subgroups

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

subgroupindex
#522992
#233812
#233802
#233592
#233342
#233312
#233182
#233132
#216842
#216702
#216682
#215652
#214902
#214882
#214452