Finite Group Databasebeta

← #6506584

Subgroups — #6506584

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37386C211yesmaximal, characteristic
2#35639C211yesmaximal, characteristic
3#29341C212yesmaximal, complemented
4#29338C212yesmaximal, complemented
5#27985C212yesmaximal, complemented
6#27716C214yesmaximal, complemented
7#27708C214yesmaximal, complemented
8#27283C214yesmaximal, complemented
9#27268C214yesmaximal, complemented
10#25912C212yesmaximal, complemented
11#21771C212yesmaximal
12#20913C212yesmaximal
13#20532C211yesmaximal, characteristic
14C2×#536C2212yes—
15C22×#125C2211yescharacteristic
16C2×#520C2211yescharacteristic
17C2×#491C2214yes—
18C2×#419C2212yes—
19C22×#103C2212yes—
20C2×#408C2212yes—
21#2857C2212yes—
22#2734—22no—
23#2724—22no—
24#2717—22no—
25#2717—22no—
26#2621C2212yes—
27#2579—22no—
28#2499C2214yes—
29#2487—22no—
30#2454—22no—
31#2453—22no—
32#2442C2214yes—
33#2441C2214yes—
34#2434C2214yes—
35#2317C2211yescharacteristic
36#2316C2211yescharacteristic
37#2311C2211yescharacteristic
38#2297C2211yescharacteristic
39#2296C2211yescharacteristic
40#2101C2214yes—
41#2100C2212yes—
42#2100C2212yes—
43#2074C2214yes—
44#2072C2214yes—
45#2043C2212yes—
46#2043C2214yes—
47#2043C2212yes—
48#2026C2214yes—
49#2026C2214yes—
50#1992C2214yes—
51#1960C2212yes—
52#1956C2212yes—
53#1935C2214yes—
54#1935C2214yes—
55#1935C2214yes—
56#1933C2214yes—
57#1894C2214yes—
58#1892C2214yes—
59#1873C2214yes—
60#1873C2214yes—
61#1873C2214yes—
62#1872C2214yes—
63#1872C2214yes—
64#1872C2214yes—
65#1836C2212yes—
66#1835C2212yes—
67#1825C2214yes—
68#1787C2214yes—
69#1753C2214yes—
70#1752C2214yes—
71#1750C2212yes—
72#1749C2212yes—
73#1736C2212yes—
74#1735C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8C2312yes—
77C22×M16C2312yes—
78C23×C8C2312yes—
79#536—28no—
80#536—28no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×#125—22no—
85C2×#125—44no—
86C2×#125C2314yes—
87C2×#124—22no—
88C2×#123—22no—
89C2×#123—22no—
90C2×C22⋊Q8C2314yes—
91C2×C22⋊Q8C2314yes—
92#520—44no—
93C2×C4×Q8—21no—
94C2×C4×D4—22no—
95C2×C4×D4—21no—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#491—48no—
108#419—44no—
109C2×#103—44no—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118#408—28no—
119C2×C22⋊C8C2314yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2312yes—
122C2×#97C2314yes—
123C2×C4×C8C2312yes—
124#400—24no—
125#399—24no—
126#398—24no—
127#398—24no—
128#397—24no—
129#397—24no—
130#397—24no—
131#396—24no—
132#395—24no—
133#395—24no—
134#391—21no—
135#391D412yes—
136#391C2314yes—
137#391C2314yes—
138#390—21no—
139#390D412yes—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—24no—
145#389C2314yes—
146#389C2312yes—
147#389C2312yes—
148#387—24no—
149#387—24no—
150#387—24no—
151#386—22no—
152#386D412yes—
153#386D412yes—
154#385—21no—
155#385—21no—
156#385—22no—
157#385—24no—
158#385—24no—
159#385—24no—
160#384C2312yes—
161#384C2312yes—
162#382C2312yes—
163#382C2312yes—
164#380—22no—
165#380—22no—
166#380—22no—
167#380—24no—
168#380C2314yes—
169C4×C4⋊C4—22no—
170C4×C22⋊C4C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#336C2314yes—
180#336C2312yes—
181#336C2312yes—
182C2×#41—42no—
183C2×#41—41no—
184C2×#41—41no—
185C2×#41#4118yes—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8C2×D411yescharacteristic
194C22×Q8C2414yes—
195C22×D4C2×D411yescharacteristic
196C22×D4C2×D412yes—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C2×M16—24no—
206C2×M16—42no—
207C2×M16—42no—
208C22×C8—24no—
209C22×C8—42no—
210C22×C8—42no—
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—42no—
214C4⋊Q8—42no—
215C4⋊Q8—44no—
216C4⋊Q8—48no—
217#125—28no—
218#125—44no—
219#125—48no—
220#124—48no—
221#123—48no—
222#123—48no—
223C22⋊Q8—28no—
224C22⋊Q8—28no—
225C22⋊Q8—28no—
226C22⋊Q8—28no—
227C4×Q8—44no—
228C4×D4—44no—
229C4×D4—48no—
230#117—24no—
231#117—24no—
232#117—24no—
233#117—41no—
234#117—41no—
235#117—41no—
236#117—41no—
237#117—42no—
238#117—42no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2411yescharacteristic
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2411yescharacteristic
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×D414yes—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—21no—
282C2×C42—42no—
283C2×C42C2×D412yes—
284C2×C42C2412yes—
285C2×C42C2412yes—
286#103—28no—
287#103—28no—
288#103—44no—
289#103—44no—
290#103—44no—
291#103—44no—
292#103—44no—
293#103—44no—
294C22⋊C8—28no—
295C22⋊C8—28no—
296#97—42no—
297#97—44no—
298C4×C8—42no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95C2×D414yes—
310#95C2×D414yes—
311C24C22×D412yes—
312C24C22×D411yescharacteristic
313#41—48no—
314#41—48no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—44no—
326C2×Q8—82no—
327C2×Q8—82no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×Q8#12414yes—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—42no—
335C2×D4#12414yes—
336C2×D4#12418yes—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—84no—
364C22×C4C22×D414yes—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4114yes—
370C22×C4C2×#4114yes—
371C22×C4C2×#4114yes—
372C22×C4C2511yescharacteristic
373M16—84no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—84no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—84no—
406C4⋊C4H(5,𝔽2)14yes—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—84no—
417C42—24no—
418C42—22no—
419C42—22no—
420C42—42no—
421C42—44no—
422C42—42no—
423C42—42no—
424C42—44no—
425C23—24no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—21no—
430C23—22no—
431C23—21no—
432C23—22no—
433C23C2×#12412yes—
434C23C2×#12414yes—
435C23C23×D411yescharacteristic
436Q8—44no—
437Q8—44no—
438Q8—44no—
439Q8—44no—
440Q8—44no—
441Q8—88no—
442D4—44no—
443D4—48no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—84no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—88no—
484C2×C4—84no—
485C2×C4—164no—
486C2×C4C2×#12414yes—
487C2×C4C2×#12412yes—
488C2×C4C22×#4112yes—
489C2×C4C2×H(5,𝔽2)12yes—
490C2×C4#58511yescharacteristic
491C2×C4#58511yescharacteristic
492C8—44no—
493C8—44no—
494C22—24no—
495C22—22no—
496C22—28no—
497C22—24no—
498C22—44no—
499C22—42no—
500C22—41no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22C22×#12411yescharacteristic
505C22#348411yescharacteristic
506C22#348511yescharacteristic
507C22#358612yes—
508C22#358612yes—
509C4—22no—
510C4—24no—
511C4—44no—
512C4—42no—
513C4—44no—
514C4—42no—
515C4—84no—
516C4—82no—
517C4—81no—
518C4—84no—
519C4—81no—
520C4—82no—
521C4#350512yes—
522C4#350512yes—
523C2—44no—
524C2—42no—
525C2#3537711yescharacteristic
526C2#6062712yes—
527C2#6164911yescharacteristic
528C2#6165211yescharacteristic
529C2#6179812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29341C2semidirect
4. #29338C2semidirect
5. #27985C2semidirect
6. #27716C2semidirect
7. #27708C2semidirect
8. #27283C2semidirect
9. #27268C2semidirect
10. #25912C2semidirect

Maximal subgroups

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

subgroupindex
#373862
#356392
#293412
#293382
#279852
#277162
#277082
#272832
#272682
#259122
#217712
#209132
#205322