Finite Group Databasebeta

← #6644676

Subgroups — #6644676

Order 512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35087C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3#33008C212yesmaximal
4#32439C214yesmaximal, complemented
5#32367C214yesmaximal, complemented
6#31659C214yesmaximal, complemented
7#31533C214yesmaximal, complemented
8#29642C212yesmaximal
9#29112C211yesmaximal, characteristic
10#28304C211yesmaximal, characteristic
11#28121C212yesmaximal
12#27964C211yesmaximal, characteristic
13#27919C211yesmaximal, characteristic
14#27653C212yesmaximal
15#21914C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#484C2212yes—
19C2×#484C2212yes—
20C2×#483C2212yes—
21C2×#451C2212yes—
22#2857C2211yescharacteristic
23#2855C2212yes—
24#2851C2211yescharacteristic
25#2655—22no—
26#2568—22no—
27#2568—22no—
28#2568C2214yes—
29#2568C2214yes—
30#2514C2212yes—
31#2506C2212yes—
32#2493—22no—
33#2473—22no—
34#2453C2212yes—
35#2446C2212yes—
36C2×#400—22no—
37#2336—22no—
38C4×C22⋊Q8C2212yes—
39C2×#391—22no—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45#2101C2214yes—
46#2097C2214yes—
47#2095C2212yes—
48#2085C2214yes—
49#2074C2214yes—
50#2074C2212yes—
51#2068C2214yes—
52#2057C2214yes—
53#2057C2214yes—
54#2056C2214yes—
55#2026C2212yes—
56#2024C2214yes—
57#2024C2214yes—
58#2016C2212yes—
59#1991C2212yes—
60#1969C2214yes—
61#1960C2212yes—
62#1949C2214yes—
63#1935C2214yes—
64#1928C2214yes—
65#1917C2214yes—
66#1916C2214yes—
67#1878C2214yes—
68#1878C2214yes—
69#1878C2214yes—
70#1865C2214yes—
71#1865C2214yes—
72#1847C2212yes—
73#1835C2212yes—
74#1824C2212yes—
75#1824C2212yes—
76#1823C2212yes—
77#1817C2212yes—
78#1805C2214yes—
79#1797C2212yes—
80#1787C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×Q8—21no—
84C23×Q8C2311yescharacteristic
85C22×Q16C2314yes—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2314yes—
90C2×#125—22no—
91C2×#125C2311yescharacteristic
92C2×C22⋊Q8C2312yes—
93C2×C4×Q8—22no—
94C2×C4×Q8—44no—
95C2×C4×Q8C2314yes—
96C2×C4×Q8C2312yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4D412yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C42—21no—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#484—44no—
108#484—44no—
109#483—44no—
110#451—44no—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2312yes—
123C2×#97C2312yes—
124C2×C4×C8C2312yes—
125#401—22no—
126#400—22no—
127#400—24no—
128#400—24no—
129#400—24no—
130#400—24no—
131#400—22no—
132#400—22no—
133#400—22no—
134#398—22no—
135#398—22no—
136#398—22no—
137#391—24no—
138#391—24no—
139#391—24no—
140#391—24no—
141#391—21no—
142#391—21no—
143#391—22no—
144#391C2312yes—
145#391C2314yes—
146#391C2312yes—
147#391C2312yes—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#389—22no—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#385—22no—
167#385—22no—
168#385—22no—
169#384C2311yescharacteristic
170#383—22no—
171#382—21no—
172#382—21no—
173#382—22no—
174#382C2314yes—
175#382C2312yes—
176#382C2312yes—
177#382C2311yescharacteristic
178#381—22no—
179#380—22no—
180#380C2312yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C22⋊C4—22no—
188C2×#95—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—24no—
208C22×Q8—42no—
209C22×Q8—44no—
210C22×Q8—48no—
211C22×Q8C2412yes—
212C22×Q8C2412yes—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2411yescharacteristic
222C2×Q16—28no—
223C2×Q16—44no—
224C2×Q16—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—48no—
228C4⋊Q8—48no—
229C4⋊Q8—42no—
230#125—42no—
231#125—48no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C4×Q8—48no—
235C4×Q8—48no—
236C4×Q8—44no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D412yes—
280C2×C42C2412yes—
281C2×C42C2412yes—
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284#107—44no—
285#107—44no—
286#106—44no—
287C4⋊C8—44no—
288#103—28no—
289#103—44no—
290#103—44no—
291#103—44no—
292#103—44no—
293#103—44no—
294C22⋊C8—24no—
295C22⋊C8—24no—
296#97—42no—
297C4×C8—42no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95C2×D412yes—
322#95C2×D412yes—
323#95C2×D412yes—
324#95C2×D412yes—
325C24C22×D411yescharacteristic
326C2×Q8—24no—
327C2×Q8—22no—
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—416no—
335C2×Q8—42no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—42no—
339C2×Q8—48no—
340C2×Q8—816no—
341C2×Q82−1+414yes—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D412yes—
378C22×C4H(5,𝔽2)14yes—
379C22×C4C2511yescharacteristic
380Q16—88no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C42—24no—
415C42—24no—
416C42—22no—
417C42—22no—
418C42—24no—
419C42—41no—
420C42—41no—
421C42—42no—
422C42—48no—
423C42—44no—
424C42—82no—
425C23—21no—
426C23—21no—
427C23—22no—
428C23#53512yes—
429C23#57814yes—
430C23C23×D411yescharacteristic
431Q8—28no—
432Q8—48no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—816no—
437C2×C4—28no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—28no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—48no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4#53512yes—
486C2×C4#57514yes—
487C2×C4C2×H(5,𝔽2)12yes—
488C2×C4C2×2−1+412yes—
489C2×C4#58512yes—
490C8—44no—
491C8—44no—
492C22—24no—
493C22—22no—
494C22—24no—
495C22—22no—
496C22#322314yes—
497C22C2×#53511yescharacteristic
498C22#348511yescharacteristic
499C22#348611yescharacteristic
500C22C2×#57512yes—
501C22C2×#57812yes—
502C4—24no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—44no—
510C4—82no—
511C4—82no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—82no—
517C4#353514yes—
518C2—24no—
519C2#3711811yescharacteristic
520C2C2×#322311yescharacteristic
521C2#6079911yescharacteristic
522C2#6173012yes—
523C2#6184212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
4. #32439C2semidirect
5. #32367C2semidirect
6. #31659C2semidirect
7. #31533C2semidirect

Maximal subgroups

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

subgroupindex
#350872
C2×#25682
#330082
#324392
#323672
#316592
#315332
#296422
#291122
#283042
#281212
#279642
#279192
#276532
#219142