Finite Group Databasebeta

← #6633266

Subgroups — #6633266

Order 512.

1607 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37387C211yesmaximal, characteristic, complemented
2#35182C211yesmaximal, characteristic
3#32906C212yesmaximal, complemented
4#31664C214yesmaximal, complemented
5#31644C214yesmaximal, complemented
6#30760C214yesmaximal, complemented
7#30744C214yesmaximal, complemented
8#29717C212yesmaximal
9#29344C212yesmaximal, complemented
10#25735C212yesmaximal
11#23944C212yesmaximal
12#23943C212yesmaximal, complemented
13#21919C211yesmaximal, characteristic, complemented
14C2×#541C2212yes—
15C2×#536C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#497C2212yes—
18C2×#494C2212yes—
19C2×#462C2214yes—
20C2×#452C2214yes—
21C2×#451C2214yes—
22#2857C2212yes—
23#2853—22no—
24#2726—22no—
25#2726—22no—
26#2722—22no—
27#2722—22no—
28#2623—22no—
29#2621C2212yes—
30#2565C2214yes—
31#2546C2214yes—
32#2448C2214yes—
33#2446C2214yes—
34#2423—22no—
35#2421—22no—
36#2317C2211yescharacteristic
37#2311C2211yescharacteristic
38#2310C2211yescharacteristic
39#2297C2211yescharacteristic
40#2296C2211yescharacteristic
41#2101C2214yes—
42#2072C2212yes—
43#2071C2212yes—
44#2056C2214yes—
45#2055C2214yes—
46#2043C2214yes—
47#2026C2214yes—
48#2026C2212yes—
49#2025C2212yes—
50#2016C2212yes—
51#2016C2212yes—
52#2016C2214yes—
53#2016C2214yes—
54#2015C2214yes—
55#2013C2212yes—
56#2013C2212yes—
57#1960C2212yes—
58#1958C2212yes—
59#1935C2214yes—
60#1933C2214yes—
61#1917C2214yes—
62#1916C2214yes—
63#1880C2214yes—
64#1878C2214yes—
65#1878C2214yes—
66#1878C2214yes—
67#1878C2214yes—
68#1878C2214yes—
69#1873C2214yes—
70#1872C2214yes—
71#1818C2212yes—
72#1803C2214yes—
73#1797C2212yes—
74#1750C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8C2311yescharacteristic
77C22×Q16C2314yes—
78C22×Q16C2314yes—
79C22×Q16C2314yes—
80C22×SD16C2314yes—
81#541—48no—
82#536—44no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×#126—24no—
88C2×#125—24no—
89C2×#125C2312yes—
90C2×#124C2312yes—
91C2×#123—22no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8C2314yes—
95C2×C4×Q8—22no—
96C2×C4×Q8—44no—
97C2×C4×Q8C2314yes—
98C2×C4×Q8C2314yes—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C42—21no—
105C22×C42C2311yescharacteristic
106#497—44no—
107#494—44no—
108#462—48no—
109#452—48no—
110#451—48no—
111C2×C4⋊C8C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2312yes—
122C2×C4×C8C2312yes—
123#400—24no—
124#399—24no—
125#398—24no—
126#398—24no—
127#398—22no—
128#398—22no—
129#398—24no—
130#398—24no—
131#398—24no—
132#397—22no—
133#397—22no—
134#397—24no—
135#391—22no—
136#391C2314yes—
137#391C2314yes—
138#391C2314yes—
139#391C2312yes—
140#391C2312yes—
141#390C2312yes—
142#390C2312yes—
143#389—24no—
144#389—24no—
145#389—22no—
146#389—22no—
147#389D412yes—
148#389D412yes—
149#389C2314yes—
150#388—24no—
151#388—24no—
152#387—24no—
153#387—24no—
154#387—24no—
155#387—24no—
156#386—21no—
157#386—21no—
158#386—22no—
159#385—22no—
160#385D412yes—
161#385D412yes—
162#384—21no—
163#384C2312yes—
164#382—21no—
165#382C2312yes—
166#381—22no—
167#380—22no—
168#380—22no—
169#380—22no—
170C4×C4⋊C4C2314yes—
171C4×C22⋊C4—22no—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#336C2312yes—
177#336C2312yes—
178#336C2312yes—
179#336C2312yes—
180#336C2314yes—
181#336C2314yes—
182C2×#41—24no—
183C2×#41—42no—
184C2×#41—42no—
185C22×Q8—24no—
186C22×Q8—21no—
187C22×Q8—24no—
188C22×Q8—42no—
189C22×Q8C2414yes—
190C22×Q8C2411yescharacteristic
191C22×Q8C2412yes—
192C22×D4—21no—
193C22×D4C2411yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2411yescharacteristic
200C2×Q16—28no—
201C2×Q16—28no—
202C2×Q16—44no—
203C2×Q16—44no—
204C2×Q16—44no—
205C2×Q16—44no—
206C2×Q16—44no—
207C2×Q16—44no—
208C2×Q16—44no—
209C2×SD16—44no—
210C2×SD16—44no—
211C2×SD16—44no—
212C22×C8C2414yes—
213C22×C8C2414yes—
214C4⋊Q8—24no—
215C4⋊Q8—42no—
216C4⋊Q8—44no—
217C4⋊Q8—44no—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220#126—416no—
221#125—44no—
222#125—416no—
223#124—24no—
224#124—42no—
225#123—48no—
226C22⋊Q8—28no—
227C22⋊Q8—28no—
228C22⋊Q8—44no—
229C22⋊Q8—44no—
230C4×Q8—28no—
231C4×Q8—44no—
232C4×Q8—44no—
233C4×Q8—44no—
234C4×Q8—48no—
235#117—24no—
236#117—42no—
237#117—42no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
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—24no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4C2×D412yes—
271C2×C22⋊C4C2×D412yes—
272C2×C42—21no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—42no—
276C2×C42—44no—
277C2×C42C2×D412yes—
278C2×C42C2412yes—
279C2×C42C2412yes—
280C2×C42C2412yes—
281C4⋊C8—44no—
282#103—44no—
283#103—44no—
284#103—44no—
285#103—44no—
286#103—44no—
287#103—44no—
288#103—44no—
289#102—44no—
290C22⋊C8—28no—
291#97—42no—
292C4×C8—42no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95C2×D414yes—
306#95C2×D414yes—
307C24—21no—
308C24C22×D411yescharacteristic
309#41—88no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—22no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—22no—
323C2×Q8—416no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8H(5,𝔽2)14yes—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—82no—
353C22×C4C22×D412yes—
354C22×C4C22×D414yes—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4H(5,𝔽2)14yes—
360C22×C4C2511yescharacteristic
361Q16—88no—
362Q16—88no—
363Q16—88no—
364SD16—88no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C4⋊C4—24no—
372C4⋊C4—28no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—44no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—48no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—44no—
390C4⋊C4—84no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C22⋊C4—24no—
396C22⋊C4—42no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—48no—
401C22⋊C4—44no—
402C22⋊C4—84no—
403C22⋊C4—82no—
404C42—22no—
405C42—22no—
406C42—24no—
407C42—24no—
408C42—42no—
409C42—48no—
410C42—42no—
411C42—48no—
412C23—22no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23#53512yes—
422C23C23×D411yescharacteristic
423Q8—28no—
424Q8—48no—
425Q8—44no—
426Q8—44no—
427Q8—44no—
428Q8—44no—
429Q8—44no—
430Q8—44no—
431D4—44no—
432D4—84no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—44no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—84no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4#53512yes—
471C2×C4#57814yes—
472C2×C4C2×H(5,𝔽2)12yes—
473C2×C4C2×H(5,𝔽2)12yes—
474C2×C4#58512yes—
475C8—44no—
476C8—44no—
477C22—22no—
478C22—24no—
479C22—41no—
480C22—41no—
481C22—42no—
482C22—81no—
483C22—82no—
484C22—81no—
485C22—81no—
486C22—82no—
487C22C2×#53511yescharacteristic
488C22#348411yescharacteristic
489C22#348711yescharacteristic
490C22C2×#57812yes—
491C22#358612yes—
492C4—22no—
493C4—24no—
494C4—44no—
495C4—44no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—81no—
500C4—84no—
501C4—84no—
502C4—82no—
503C4—82no—
504C4#353414yes—
505C2—42no—
506C2—161no—
507C2#3671011yescharacteristic
508C2#6083712yes—
509C2#6163411yescharacteristic
510C2#6165211yescharacteristic
511C2#6191512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37387C2semidirect
3. #32906C2semidirect
4. #31664C2semidirect
5. #31644C2semidirect
6. #30760C2semidirect
7. #30744C2semidirect
9. #29344C2semidirect
12. #23943C2semidirect
13. #21919C2semidirect

Maximal subgroups

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

subgroupindex
#373872
#351822
#329062
#316642
#316442
#307602
#307442
#297172
#293442
#257352
#239442
#239432
#219192