Finite Group Databasebeta

← #6490662

Subgroups — #6490662

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35271C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic, complemented
3#33150C212yesmaximal, complemented
4#33107C212yesmaximal
5#30163C214yesmaximal, complemented
6#29360C212yesmaximal, complemented
7#29327C212yesmaximal, complemented
8#29295C212yesmaximal, complemented
9#28292C212yesmaximal, complemented
10#27645C212yesmaximal, complemented
11#24607C212yesmaximal, complemented
12#24603C212yesmaximal, complemented
13#24305C212yesmaximal, complemented
14#22122C214yesmaximal, complemented
15#20814C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C2×#481C2212yes—
18C2×#481C2212yes—
19C2×#478C2214yes—
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2856C2211yescharacteristic
23#2854C2212yes—
24#2851C2211yescharacteristic
25#2647C2211yescharacteristic
26#2644C2212yes—
27#2641C2211yescharacteristic
28#2531—22no—
29#2526—22no—
30#2444—22no—
31#2425—22no—
32#2330—22no—
33#2330—22no—
34#2330C2212yes—
35#2330C2212yes—
36#2330C2212yes—
37#2330C2212yes—
38#2330C2211yescharacteristic
39#2328C2211yescharacteristic
40#2319C2212yes—
41C4×#121C2212yes—
42C2×#389—22no—
43C2×#389C2212yes—
44C2×#387—22no—
45C2×#384C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×#376C2211yescharacteristic
48#2088C2214yes—
49#2088C2214yes—
50#2084C2214yes—
51#2071C2214yes—
52#2067C2214yes—
53#2065C2214yes—
54#2059C2214yes—
55#2025C2214yes—
56#1989C2214yes—
57#1989C2214yes—
58#1982C2214yes—
59#1949C2214yes—
60#1948C2214yes—
61#1948C2214yes—
62#1943C2214yes—
63#1941C2214yes—
64#1936C2214yes—
65#1934C2214yes—
66#1926C2214yes—
67#1926C2214yes—
68#1920C2214yes—
69#1918C2214yes—
70#1874C2212yes—
71#1874C2212yes—
72#1853C2212yes—
73#1852C2212yes—
74#1843C2212yes—
75#1832C2212yes—
76#1824C2212yes—
77#1824C2212yes—
78#1802C2214yes—
79#1797C2212yes—
80#1797C2212yes—
81#1768C2212yes—
82#1768C2212yes—
83C2×#127C2311yescharacteristic
84C2×#125—28no—
85C2×#125—22no—
86C2×#125C2312yes—
87C2×#125C2311yescharacteristic
88C2×#125C2312yes—
89C2×#123—22no—
90C2×#121—22no—
91C2×#121C2312yes—
92C2×C4×D4—22no—
93C2×C4×D4C2312yes—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C42—22no—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103#481—44no—
104#481—44no—
105#478—216no—
106#432—42no—
107#431—42no—
108C2×#107C2314yes—
109C2×#106C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2314yes—
113C2×C4⋊C8C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×C22⋊C8C2312yes—
121C2×C22⋊C8C2312yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124#400—22no—
125#398—22no—
126#397—22no—
127#397—22no—
128#397—22no—
129#396—22no—
130#394—22no—
131#394—22no—
132#390—22no—
133#390C2312yes—
134#390C2312yes—
135#390C2312yes—
136#390C2312yes—
137#389—22no—
138#389—22no—
139#389—24no—
140#389—24no—
141#389—24no—
142#389—24no—
143#389—22no—
144#389C2312yes—
145#389C2312yes—
146#389C2312yes—
147#389C2314yes—
148#389C2314yes—
149#389C2312yes—
150#388—22no—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#387—22no—
156#387—24no—
157#387—24no—
158#387—24no—
159#387—24no—
160#387—22no—
161#386—22no—
162#386—22no—
163#386—22no—
164#386—22no—
165#385—22no—
166#384—22no—
167#384C2312yes—
168#384C2312yes—
169#384C2311yescharacteristic
170#382C2311yescharacteristic
171#381—22no—
172#380—22no—
173#380C2312yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—22no—
180#376D412yes—
181#376D412yes—
182#376C2311yescharacteristic
183#376C2311yescharacteristic
184#376C2311yescharacteristic
185#376C2311yescharacteristic
186C2×#95—22no—
187C2×#95—22no—
188C2×#95D412yes—
189C2×#95D412yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4C2412yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2411yescharacteristic
211C22×C8C2414yes—
212C22×C8C2414yes—
213#127—24no—
214#125—216no—
215#125—28no—
216#125—24no—
217#125—24no—
218#125#4118yes—
219#123—84no—
220#121—24no—
221#121—24no—
222#121—84no—
223C4×D4—28no—
224C4×D4—84no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—84no—
244C2×C4⋊C4—84no—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2414yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—84no—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279#107—28no—
280#106—28no—
281#106—44no—
282#106—44no—
283C4⋊C8—28no—
284C4⋊C8—28no—
285C4⋊C8—42no—
286C4⋊C8—42no—
287#102—28no—
288#102—28no—
289#102—44no—
290#102—44no—
291C22⋊C8—24no—
292C22⋊C8—24no—
293#97—42no—
294C4×C8—42no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95C2×D412yes—
307#95C2×D412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D412yes—
311#95C2×D412yes—
312#95C2×D412yes—
313#95C2×D412yes—
314C24—22no—
315C24C22×D411yescharacteristic
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—24no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—82no—
325C2×D4—82no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—28no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C22×D411yescharacteristic
361C22×C4H(5,𝔽2)12yes—
362C22×C42−1+412yes—
363C22×C4C2511yescharacteristic
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C4⋊C4—28no—
371C4⋊C4—216no—
372C4⋊C4—216no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—82no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4C2×#4118yes—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C42—24no—
404C42—24no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—21no—
409C42—21no—
410C42—22no—
411C42—21no—
412C42—21no—
413C42—84no—
414C42—84no—
415C42—82no—
416C42C2×#4114yes—
417C422−1+414yes—
418C23—22no—
419C23—21no—
420C23—21no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23#53712yes—
429C23#57714yes—
430C23C23×D411yescharacteristic
431D4—44no—
432D4—44no—
433D4—88no—
434D4—164no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—28no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—416no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—88no—
478C2×C4—164no—
479C2×C4#52514yes—
480C2×C4#53614yes—
481C2×C4#53712yes—
482C2×C4C22×#4112yes—
483C2×C4C2×H(5,𝔽2)11yescharacteristic
484C2×C4C2×2−1+412yes—
485C2×C4C2×2−1+411yescharacteristic
486C8—44no—
487C8—44no—
488C22—24no—
489C22—22no—
490C22—24no—
491C22—22no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22#326014yes—
500C22C2×#52511yescharacteristic
501C22C2×#53611yescharacteristic
502C22C2×#53711yescharacteristic
503C22C2×#57712yes—
504C22#358512yes—
505C4—24no—
506C4—22no—
507C4—24no—
508C4—42no—
509C4—48no—
510C4—42no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4—84no—
519C4#352212yes—
520C4#352312yes—
521C2—24no—
522C2—162no—
523C2#3524711yescharacteristic
524C2#6058812yes—
525C2#6087312yes—
526C2C2×#326011yescharacteristic
527C2#6092011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2330C2semidirect
3. #33150C2semidirect
5. #30163C2semidirect
6. #29360C2semidirect
7. #29327C2semidirect
8. #29295C2semidirect
9. #28292C2semidirect
10. #27645C2semidirect
11. #24607C2semidirect
12. #24603C2semidirect
13. #24305C2semidirect
14. #22122C2semidirect
15. #20814C2semidirect

Maximal subgroups

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

subgroupindex
#352712
C2×#23302
#331502
#331072
#301632
#293602
#293272
#292952
#282922
#276452
#246072
#246032
#243052
#221222
#208142