Finite Group Databasebeta

← #7500080

Subgroups — #7500080

Order 512.

1387 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#59869C211yesmaximal, characteristic, complemented
2#39490C211yesmaximal, characteristic
3#32679C212yesmaximal, complemented
4#32645C212yesmaximal, complemented
5#30896C214yesmaximal, complemented
6#30757C214yesmaximal
7#30042C212yesmaximal, complemented
8#29368C212yesmaximal
9#27444C214yesmaximal, complemented
10#27210C214yesmaximal
11#25945C212yesmaximal
12#25643C212yesmaximal
13#23366C211yesmaximal, characteristic, complemented
14C2×#556C2211yescharacteristic
15C2×#554C2211yescharacteristic
16C2×#497C2212yes—
17C2×#491C2212yes—
18C2×#462C2212yes—
19C2×#461C2212yes—
20#2862C2211yescharacteristic
21#2809—21no—
22#2809—21no—
23#2689—24no—
24#2621—21no—
25#2571—21no—
26#2568C2212yes—
27#2568C2211yescharacteristic
28#2568C2212yes—
29#2567C2211yescharacteristic
30#2546C2214yes—
31#2508—24no—
32#2503—21no—
33#2499—21no—
34#2453C2214yes—
35#2447C2214yes—
36#2369—21no—
37#2355C2214yes—
38#2317—21no—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41#2293C2211yescharacteristic
42#2100C2212yes—
43#2094C2212yes—
44#2072C2214yes—
45#2066C2214yes—
46#2056C2214yes—
47#2043C2214yes—
48#2043C2212yes—
49#2026C2214yes—
50#2024C2214yes—
51#2016C2214yes—
52#2016C2214yes—
53#2016C2214yes—
54#2015C2214yes—
55#2013C2212yes—
56#1989C2212yes—
57#1985C2212yes—
58#1985C2212yes—
59#1983C2212yes—
60#1971C2214yes—
61#1935C2214yes—
62#1927C2214yes—
63#1927C2214yes—
64#1920C2214yes—
65#1917C2214yes—
66#1917C2214yes—
67#1880C2214yes—
68#1878C2214yes—
69#1878C2214yes—
70#1878C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1865C2214yes—
74#1839C2212yes—
75#1763C2212yes—
76#1762C2212yes—
77C22×Q16C2314yes—
78C22×Q16C2314yes—
79C22×SD16C2314yes—
80#556—42no—
81#556—42no—
82#554—42no—
83#554—42no—
84C2×C4⋊Q8D411yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×#126—22no—
91C2×#125C2312yes—
92C2×#124C2312yes—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—22no—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2314yes—
98C2×C4×Q8C2311yescharacteristic
99C2×C4×Q8C2311yescharacteristic
100C2×C4×D4C2311yescharacteristic
101C2×C4×D4C2312yes—
102C2×#117—21no—
103C2×#117—21no—
104C22×C4⋊C4—21no—
105#497—44no—
106#491—44no—
107#462—44no—
108#461—44no—
109C2×#106C2314yes—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2312yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×#97C2312yes—
121C2×#97C2312yes—
122#400—22no—
123#400—22no—
124#400—22no—
125#400—22no—
126#400—22no—
127#400—24no—
128#398—24no—
129#398—24no—
130#398—24no—
131#397—24no—
132#397—24no—
133#395—22no—
134#391—21no—
135#391—21no—
136#391—22no—
137#391—22no—
138#391—22no—
139#391D411yescharacteristic
140#391D411yescharacteristic
141#391C2314yes—
142#391C2314yes—
143#391C2314yes—
144#391C2314yes—
145#391C2314yes—
146#390C2312yes—
147#389—24no—
148#389—24no—
149#389—24no—
150#389—22no—
151#389—22no—
152#389D414yes—
153#389C2314yes—
154#389C2312yes—
155#388—22no—
156#387—24no—
157#387—24no—
158#387—24no—
159#385—24no—
160#384—21no—
161#384D411yescharacteristic
162#384C2312yes—
163#382—21no—
164#382—22no—
165#382—22no—
166#382—24no—
167#382—21no—
168#382—21no—
169#381—24no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2311yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2314yes—
177C4×C22⋊C4—21no—
178C43C2311yescharacteristic
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2314yes—
184#336C2312yes—
185#336C2312yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C22×Q8—21no—
190C22×Q8C2×D411yescharacteristic
191C22×Q8C2×D411yescharacteristic
192C22×Q8C2412yes—
193C22×Q8C2414yes—
194C22×D4C2411yescharacteristic
195C23×C4—21no—
196C23×C4—22no—
197C2×Q16—44no—
198C2×Q16—44no—
199C2×Q16—44no—
200C2×Q16—44no—
201C2×Q16—44no—
202C2×Q16—44no—
203C2×SD16—28no—
204C2×SD16—44no—
205C2×SD16—44no—
206C22×C8C2414yes—
207C22×C8C2414yes—
208C4⋊Q8—24no—
209C4⋊Q8—41no—
210C4⋊Q8—41no—
211C4⋊Q8—44no—
212C4⋊Q8—41no—
213C4⋊Q8—41no—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4⋊Q8—42no—
218#126—48no—
219#125—24no—
220#125—42no—
221#124—24no—
222#124—42no—
223C22⋊Q8—48no—
224C22⋊Q8—44no—
225C22⋊Q8—44no—
226C4×Q8—24no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—44no—
232C4×Q8—44no—
233C4×Q8—42no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—24no—
237C4×D4—42no—
238#117—44no—
239#117—44no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
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—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2411yescharacteristic
283C2×C42C2412yes—
284C2×C42C2412yes—
285C2×C42C2412yes—
286#106—44no—
287C4⋊C8—42no—
288C4⋊C8—42no—
289#103—28no—
290#103—44no—
291#103—44no—
292#103—44no—
293#103—44no—
294#103—44no—
295#103—44no—
296#102—44no—
297#97—42no—
298#97—42no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—21no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95C2×D414yes—
315#95C2×D414yes—
316C24—21no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—82no—
335C2×D4—21no—
336C2×D4—24no—
337C2×D4H(5,𝔽2)12yes—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—84no—
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D412yes—
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D412yes—
357C22×C4C2511yescharacteristic
358Q16—88no—
359Q16—88no—
360SD16—88no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C4⋊C4—28no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—41no—
375C4⋊C4—44no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—48no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—82no—
391C4⋊C4—84no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4H(5,𝔽2)12yes—
397C22⋊C4—24no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—82no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—21no—
408C42—21no—
409C42—24no—
410C42—42no—
411C42—41no—
412C42—44no—
413C42—48no—
414C42—42no—
415C42—82no—
416C42—82no—
417C42#13714yes—
418C23—22no—
419C23—42no—
420C23—41no—
421C23—42no—
422C23—41no—
423C23C23×D411yescharacteristic
424Q8—44no—
425Q8—44no—
426Q8—44no—
427Q8—42no—
428Q8—44no—
429Q8—44no—
430Q8—42no—
431Q8—82no—
432Q8—82no—
433Q8—162no—
434D4—24no—
435D4—42no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—48no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—164no—
473C2×C4C2×#13712yes—
474C2×C4C2×#13714yes—
475C2×C4C2×H(5,𝔽2)11yescharacteristic
476C2×C4C2×H(5,𝔽2)11yescharacteristic
477C2×C4#58514yes—
478C8—44no—
479C8—44no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—82no—
484C22—82no—
485C22#348411yescharacteristic
486C22#348712yes—
487C22C22×#13711yescharacteristic
488C22#358611yescharacteristic
489C22#358612yes—
490C4—22no—
491C4—24no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—41no—
498C4—42no—
499C4—82no—
500C4—84no—
501C4—82no—
502C4—82no—
503C4—84no—
504C4#354512yes—
505C4#355112yes—
506C2—82no—
507C2#5208311yescharacteristic
508C2#6158611yescharacteristic
509C2#6167411yescharacteristic
510C2#6189312yes—
511C2#6194112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59869C2semidirect
3. #32679C2semidirect
4. #32645C2semidirect
5. #30896C2semidirect
7. #30042C2semidirect
9. #27444C2semidirect
13. #23366C2semidirect

Maximal subgroups

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

subgroupindex
#598692
#394902
#326792
#326452
#308962
#307572
#300422
#293682
#274442
#272102
#259452
#256432
#233662