Finite Group Databasebeta

← #6635528

Subgroups — #6635528

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37386C211yesmaximal, characteristic
2#35182C211yesmaximal, characteristic
3#32906C212yesmaximal, complemented
4#31952C214yesmaximal, complemented
5#31940C214yesmaximal, complemented
6#31006C214yesmaximal, complemented
7#30990C214yesmaximal, complemented
8#29759C212yesmaximal, complemented
9#28322C212yesmaximal, complemented
10#27985C212yesmaximal, complemented
11#24016C212yesmaximal
12#23928C212yesmaximal
13#21911C211yesmaximal, characteristic
14C2×#536C2212yes—
15C2×#536C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#485C2214yes—
18C2×#462C2214yes—
19C2×#451C2212yes—
20C2×#451C2212yes—
21C2×#451C2214yes—
22#2857C2212yes—
23#2853—22no—
24#2734—22no—
25#2724—22no—
26#2717—22no—
27#2717—22no—
28#2623—22no—
29#2621C2212yes—
30#2499C2214yes—
31#2448—22no—
32#2446—22no—
33#2434C2214yes—
34#2423C2214yes—
35#2421C2214yes—
36#2317C2211yescharacteristic
37#2311C2211yescharacteristic
38#2310C2211yescharacteristic
39#2297C2211yescharacteristic
40#2296C2211yescharacteristic
41#2097C2214yes—
42#2084C2212yes—
43#2083C2212yes—
44#2074C2214yes—
45#2072C2214yes—
46#2068C2214yes—
47#2066C2214yes—
48#2043C2212yes—
49#2043C2212yes—
50#2036C2214yes—
51#2035C2214yes—
52#2016C2214yes—
53#2001C2214yes—
54#2001C2214yes—
55#1971C2214yes—
56#1969C2214yes—
57#1960C2212yes—
58#1956C2212yes—
59#1924C2214yes—
60#1923C2214yes—
61#1878C2214yes—
62#1878C2214yes—
63#1873C2214yes—
64#1873C2212yes—
65#1873C2212yes—
66#1872C2214yes—
67#1854C2214yes—
68#1836C2212yes—
69#1835C2212yes—
70#1803C2214yes—
71#1797C2212yes—
72#1787C2214yes—
73#1780C2212yes—
74#1750C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8C2311yescharacteristic
77C23×Q8C2312yes—
78C22×Q16C2314yes—
79C22×Q16C2314yes—
80#536—44no—
81#536—48no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#124—21no—
87C2×#124—22no—
88C2×#123—22no—
89C2×#123—22no—
90C2×#123—22no—
91C2×C22⋊Q8C2314yes—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×C22⋊Q8C2314yes—
95C2×C4×Q8—22no—
96C2×C4×Q8—44no—
97C2×C4×Q8C2314yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2312yes—
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C42—21no—
105C22×C42C2311yescharacteristic
106#485—48no—
107#462—48no—
108#451—48no—
109#451—44no—
110#451—44no—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×#97C2312yes—
127C2×C4×C8C2312yes—
128#400—24no—
129#399—24no—
130#398—24no—
131#398—24no—
132#397—24no—
133#397—24no—
134#397—24no—
135#397—24no—
136#395—24no—
137#395—24no—
138#391—21no—
139#391—22no—
140#391D412yes—
141#391C2314yes—
142#391C2314yes—
143#391C2314yes—
144#390—21no—
145#390D412yes—
146#389—22no—
147#389—22no—
148#389—22no—
149#389C2312yes—
150#389C2312yes—
151#388—24no—
152#388—24no—
153#387—24no—
154#387—24no—
155#387—24no—
156#387—24no—
157#386—22no—
158#386D412yes—
159#386D412yes—
160#385—21no—
161#385—21no—
162#385—22no—
163#385—24no—
164#385—24no—
165#384C2312yes—
166#384C2312yes—
167#382C2312yes—
168#382C2312yes—
169#381—22no—
170#380—22no—
171#380—22no—
172#380C2314yes—
173C4×C4⋊C4—22no—
174C4×C22⋊C4C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#336C2312yes—
180#336C2312yes—
181#336C2314yes—
182C2×#41—24no—
183C2×#41—42no—
184C2×#41—42no—
185C22×Q8—24no—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8C2×D411yescharacteristic
195C22×Q8C2414yes—
196C22×Q8C2412yes—
197C22×D4C2×D412yes—
198C22×D4C2×D411yescharacteristic
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—22no—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C2×Q16—28no—
207C2×Q16—28no—
208C2×Q16—44no—
209C2×Q16—44no—
210C2×Q16—44no—
211C2×Q16—44no—
212C22×C8C2414yes—
213C22×C8C2414yes—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8—44no—
217C4⋊Q8—44no—
218C4⋊Q8—48no—
219#124—48no—
220#124—44no—
221#123—48no—
222#123—48no—
223#123—48no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—28no—
227C22⋊Q8—28no—
228C22⋊Q8—42no—
229C22⋊Q8—42no—
230C22⋊Q8—44no—
231C22⋊Q8—44no—
232C4×Q8—28no—
233C4×Q8—44no—
234C4×Q8—48no—
235#117—24no—
236#117—42no—
237#117—42no—
238#117—44no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2411yescharacteristic
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2411yescharacteristic
260C2×C4⋊C4C2412yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4C2×D414yes—
270C2×C22⋊C4C2×D412yes—
271C2×C22⋊C4C2×D414yes—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—24no—
276C2×C42—42no—
277C2×C42—44no—
278C2×C42C2×D412yes—
279C2×C42C2412yes—
280C2×C42C2412yes—
281#107—44no—
282#106—44no—
283C4⋊C8—44no—
284#103—44no—
285#103—44no—
286#103—44no—
287#103—44no—
288#103—44no—
289#103—44no—
290#103—44no—
291#103—44no—
292C22⋊C8—28no—
293C22⋊C8—44no—
294C22⋊C8—42no—
295C22⋊C8—42no—
296#97—42no—
297C4×C8—42no—
298#95—22no—
299#95—22no—
300#95—24no—
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—88no—
314C2×Q8—24no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—416no—
328C2×Q8—42no—
329C2×Q8—44no—
330C2×Q8—42no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×Q8—88no—
334C2×Q8—88no—
335C2×Q8H(5,𝔽2)14yes—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4C22×D414yes—
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4H(5,𝔽2)14yes—
373C22×C4C2511yescharacteristic
374Q16—88no—
375Q16—88no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—42no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—84no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—84no—
404C22⋊C4—28no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—42no—
411C22⋊C4—48no—
412C22⋊C4—42no—
413C22⋊C4—48no—
414C22⋊C4—84no—
415C42—24no—
416C42—22no—
417C42—22no—
418C42—42no—
419C42—48no—
420C42—42no—
421C42—44no—
422C42—42no—
423C23—22no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—24no—
432C23#53512yes—
433C23C23×D411yescharacteristic
434Q8—28no—
435Q8—48no—
436Q8—44no—
437Q8—44no—
438Q8—44no—
439Q8—44no—
440Q8—44no—
441Q8—82no—
442Q8—88no—
443D4—82no—
444D4—84no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—28no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—44no—
461C2×C4—48no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—88no—
484C2×C4#53512yes—
485C2×C4#57814yes—
486C2×C4C2×H(5,𝔽2)12yes—
487C2×C4C2×H(5,𝔽2)12yes—
488C2×C4#58512yes—
489C8—44no—
490C8—44no—
491C22—22no—
492C22—24no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—44no—
497C22—42no—
498C22—44no—
499C22—42no—
500C22—42no—
501C22C2×#53511yescharacteristic
502C22#348411yescharacteristic
503C22#348511yescharacteristic
504C22C2×#57812yes—
505C22#358612yes—
506C4—22no—
507C4—24no—
508C4—44no—
509C4—42no—
510C4—44no—
511C4—42no—
512C4—84no—
513C4—82no—
514C4—84no—
515C4—81no—
516C4—81no—
517C4—82no—
518C4#353414yes—
519C2—42no—
520C2—82no—
521C2#3672111yescharacteristic
522C2#6083712yes—
523C2#6163411yescharacteristic
524C2#6165211yescharacteristic
525C2#6180612yes—
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. #32906C2semidirect
4. #31952C2semidirect
5. #31940C2semidirect
6. #31006C2semidirect
7. #30990C2semidirect
8. #29759C2semidirect
9. #28322C2semidirect
10. #27985C2semidirect

Maximal subgroups

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

subgroupindex
#373862
#351822
#329062
#319522
#319402
#310062
#309902
#297592
#283222
#279852
#240162
#239282
#219112