Finite Group Databasebeta

← #6644669

Subgroups — #6644669

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35082C211yesmaximal, characteristic
2C2×#2565C211yesmaximal, characteristic, complemented
3#33005C212yesmaximal, complemented
4#32436C214yesmaximal, complemented
5#32370C214yesmaximal, complemented
6#31662C214yesmaximal, complemented
7#31526C214yesmaximal, complemented
8#29640C212yesmaximal, complemented
9#29112C211yesmaximal, characteristic, complemented
10#28305C211yesmaximal, characteristic
11#28118C212yesmaximal, complemented
12#27964C211yesmaximal, characteristic, complemented
13#27918C211yesmaximal, characteristic
14#27645C212yesmaximal, complemented
15#21914C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#481C2212yes—
19C2×#481C2212yes—
20C2×#480C2212yes—
21C2×#448C2212yes—
22#2856C2211yescharacteristic
23#2854C2212yes—
24#2851C2211yescharacteristic
25#2651—22no—
26#2565—22no—
27#2565—22no—
28#2565C2214yes—
29#2565C2214yes—
30#2513C2212yes—
31#2507C2212yes—
32#2454C2212yes—
33#2445C2212yes—
34#2443—22no—
35#2425—22no—
36C2×#398—22no—
37#2335—22no—
38C4×#121C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#387—22no—
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45#2101C2214yes—
46#2096C2212yes—
47#2092C2214yes—
48#2084C2214yes—
49#2074C2212yes—
50#2071C2214yes—
51#2068C2214yes—
52#2059C2214yes—
53#2059C2214yes—
54#2055C2214yes—
55#2026C2212yes—
56#2025C2214yes—
57#2025C2214yes—
58#2014C2212yes—
59#1991C2212yes—
60#1968C2214yes—
61#1961C2212yes—
62#1948C2214yes—
63#1935C2214yes—
64#1928C2214yes—
65#1917C2214yes—
66#1916C2214yes—
67#1880C2214yes—
68#1880C2214yes—
69#1877C2214yes—
70#1865C2214yes—
71#1865C2214yes—
72#1847C2212yes—
73#1835C2212yes—
74#1824C2212yes—
75#1824C2212yes—
76#1823C2212yes—
77#1817C2212yes—
78#1802C2214yes—
79#1797C2212yes—
80#1787C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×Q8C2311yescharacteristic
84C22×SD16C2314yes—
85C2×#127C2311yescharacteristic
86C2×#125—22no—
87C2×#125—44no—
88C2×#125C2314yes—
89C2×#125C2311yescharacteristic
90C2×#123—22no—
91C2×#123—22no—
92C2×#121—22no—
93C2×#121C2312yes—
94C2×C4×Q8—22no—
95C2×C4×Q8—44no—
96C2×C4×Q8C2314yes—
97C2×C4×D4—22no—
98C2×C4×D4C2312yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D412yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#481—44no—
112#481—44no—
113#480—44no—
114#448—44no—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×#97C2312yes—
128C2×C4×C8C2312yes—
129#399—22no—
130#398—24no—
131#398—24no—
132#398—24no—
133#398—24no—
134#398—22no—
135#398—22no—
136#397—22no—
137#397—22no—
138#396—22no—
139#394—22no—
140#394—22no—
141#391—21no—
142#391—22no—
143#391—21no—
144#391C2314yes—
145#391C2312yes—
146#391C2312yes—
147#390—22no—
148#390C2312yes—
149#390C2312yes—
150#390C2312yes—
151#390C2312yes—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#387—22no—
163#387—22no—
164#387—24no—
165#387—24no—
166#387—24no—
167#387—24no—
168#386—22no—
169#386—22no—
170#386—22no—
171#385—22no—
172#385—22no—
173#384—21no—
174#384—21no—
175#384C2314yes—
176#384C2311yescharacteristic
177#382—22no—
178#382C2312yes—
179#382C2312yes—
180#382C2311yescharacteristic
181#380—22no—
182#380C2312yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C2×#95—22no—
190C2×#95—22no—
191C2×#95—22no—
192C2×#95D412yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204C22×Q8—24no—
205C22×Q8—21no—
206C22×Q8—24no—
207C22×Q8—42no—
208C22×Q8C2412yes—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2412yes—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—22no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D412yes—
221C23×C4C2411yescharacteristic
222C2×SD16—28no—
223C2×SD16—44no—
224C2×SD16—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227#127—42no—
228#125—48no—
229#125—42no—
230#125—48no—
231#123—48no—
232#123—84no—
233#121—24no—
234#121—24no—
235#121—84no—
236C4×Q8—48no—
237C4×Q8—48no—
238C4×D4—44no—
239C4×D4—84no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42C2×D412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2412yes—
294C2×C42C2412yes—
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297#107—44no—
298#106—44no—
299#106—44no—
300C4⋊C8—44no—
301#103—28no—
302#103—44no—
303#103—44no—
304#102—44no—
305#102—44no—
306#102—44no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309#97—42no—
310C4×C8—42no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95C2×D412yes—
325#95C2×D412yes—
326#95C2×D412yes—
327#95C2×D412yes—
328C24—22no—
329C24C22×D411yescharacteristic
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—416no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q82−1+414yes—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—82no—
347C2×D4—82no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4H(5,𝔽2)14yes—
387C22×C4C2511yescharacteristic
388SD16—88no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C22⋊C4—82no—
431C42—24no—
432C42—24no—
433C42—22no—
434C42—22no—
435C42—24no—
436C42—41no—
437C42—41no—
438C42—42no—
439C42—48no—
440C42—82no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23#53512yes—
452C23#57714yes—
453C23C23×D411yescharacteristic
454Q8—28no—
455Q8—48no—
456Q8—44no—
457D4—44no—
458D4—44no—
459D4—88no—
460D4—164no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—48no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—164no—
508C2×C4#53512yes—
509C2×C4#57514yes—
510C2×C4C2×H(5,𝔽2)12yes—
511C2×C4C2×2−1+412yes—
512C2×C4#58512yes—
513C8—44no—
514C8—44no—
515C22—24no—
516C22—22no—
517C22—24no—
518C22—22no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22#322114yes—
527C22C2×#53511yescharacteristic
528C22#348511yescharacteristic
529C22#348611yescharacteristic
530C22C2×#57512yes—
531C22C2×#57712yes—
532C4—24no—
533C4—22no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4#353514yes—
547C2—24no—
548C2—162no—
549C2#3711811yescharacteristic
550C2C2×#322111yescharacteristic
551C2#6080211yescharacteristic
552C2#6172912yes—
553C2#6183612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2565C2semidirect
3. #33005C2semidirect
4. #32436C2semidirect
5. #32370C2semidirect
6. #31662C2semidirect
7. #31526C2semidirect
8. #29640C2semidirect
9. #29112C2semidirect
11. #28118C2semidirect
12. #27964C2semidirect
14. #27645C2semidirect
15. #21914C2semidirect

Maximal subgroups

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

subgroupindex
#350822
C2×#25652
#330052
#324362
#323702
#316622
#315262
#296402
#291122
#283052
#281182
#279642
#279182
#276452
#219142