Finite Group Databasebeta

← #7500090

Subgroups — #7500090

Order 512.

1443 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic, complemented
2#38954C211yesmaximal, characteristic
3#32670C212yesmaximal, complemented
4#32644C212yesmaximal, complemented
5#30880C214yesmaximal, complemented
6#30757C214yesmaximal
7#30042C212yesmaximal, complemented
8#29369C212yesmaximal
9#27441C214yesmaximal, complemented
10#27191C214yesmaximal
11#25929C212yesmaximal
12#25643C212yesmaximal
13#23365C211yesmaximal, characteristic, complemented
14C2×#556C2211yescharacteristic
15C2×D4×Q8C2211yescharacteristic
16C2×#497C2212yes—
17C2×#487C2212yes—
18C2×#462C2212yes—
19C2×#461C2212yes—
20#2862C2211yescharacteristic
21#2809—21no—
22#2809—21no—
23#2698—24no—
24#2621—21no—
25#2571—21no—
26#2568C2211yescharacteristic
27#2568C2212yes—
28#2568C2212yes—
29#2546C2211yescharacteristic
30#2546C2214yes—
31#2508—24no—
32#2499—21no—
33#2451C2214yes—
34#2447C2214yes—
35#2434—21no—
36#2369—21no—
37#2355C2214yes—
38#2310—21no—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41#2293C2211yescharacteristic
42#2099C2212yes—
43#2093C2212yes—
44#2072C2214yes—
45#2068C2214yes—
46#2056C2214yes—
47#2043C2214yes—
48#2043C2212yes—
49#2024C2214yes—
50#2024C2214yes—
51#2016C2214yes—
52#2016C2214yes—
53#2016C2214yes—
54#2015C2214yes—
55#2013C2212yes—
56#1988C2212yes—
57#1985C2212yes—
58#1984C2212yes—
59#1983C2212yes—
60#1969C2214yes—
61#1935C2214yes—
62#1928C2214yes—
63#1928C2214yes—
64#1918C2214yes—
65#1917C2214yes—
66#1916C2214yes—
67#1880C2214yes—
68#1878C2214yes—
69#1878C2214yes—
70#1878C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1863C2214yes—
74#1839C2212yes—
75#1763C2212yes—
76C4×#102C2212yes—
77C23×Q8—21no—
78C22×Q16C2314yes—
79C22×Q16C2314yes—
80C22×SD16C2314yes—
81#556—42no—
82#556—42no—
83D4×Q8—42no—
84D4×Q8—42no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8D411yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#126—22no—
94C2×#124C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2314yes—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×Q8C2311yescharacteristic
102C2×C4×D4C2311yescharacteristic
103C2×C4×D4C2312yes—
104C2×#117—21no—
105C2×#117—21no—
106#497—44no—
107#487—44no—
108#462—44no—
109#461—44no—
110C2×#107C2314yes—
111C2×C4⋊C8C2312yes—
112C2×C4⋊C8C2312yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102C2314yes—
121C2×#97C2312yes—
122C2×C4×C8C2312yes—
123#400—22no—
124#400—22no—
125#400—22no—
126#400—22no—
127#400—22no—
128#398—22no—
129#398—24no—
130#398—24no—
131#398—24no—
132#397—24no—
133#397—24no—
134#395—24no—
135#391—21no—
136#391—22no—
137#391—22no—
138#391—21no—
139#391—21no—
140#391—21no—
141#391—24no—
142#391D411yescharacteristic
143#391D411yescharacteristic
144#391C2314yes—
145#391C2314yes—
146#391C2314yes—
147#391C2314yes—
148#391C2314yes—
149#390C2312yes—
150#389—24no—
151#389—24no—
152#389—22no—
153#389—22no—
154#389D414yes—
155#389C2314yes—
156#389C2312yes—
157#388—22no—
158#387—24no—
159#387—24no—
160#387—24no—
161#385—24no—
162#384D411yescharacteristic
163#384C2312yes—
164#382—21no—
165#382—22no—
166#382—22no—
167#382—24no—
168#382—22no—
169#381—24no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2311yescharacteristic
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×Q8—22no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8—82no—
194C22×Q8C2×D411yescharacteristic
195C22×Q8C2×D411yescharacteristic
196C22×Q8C2412yes—
197C22×Q8C2414yes—
198C22×D4C2411yescharacteristic
199C23×C4—21no—
200C23×C4—22no—
201C2×Q16—44no—
202C2×Q16—44no—
203C2×Q16—44no—
204C2×Q16—44no—
205C2×Q16—44no—
206C2×Q16—44no—
207C2×SD16—28no—
208C2×SD16—44no—
209C2×SD16—44no—
210C22×C8C2414yes—
211C22×C8C2414yes—
212C4⋊Q8—24no—
213C4⋊Q8—24no—
214C4⋊Q8—42no—
215C4⋊Q8—41no—
216C4⋊Q8—41no—
217C4⋊Q8—44no—
218C4⋊Q8—41no—
219C4⋊Q8—41no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C4⋊Q8—42no—
223C4⋊Q8—42no—
224C4⋊Q8—82no—
225#126—48no—
226#124—24no—
227#124—42no—
228C22⋊Q8—48no—
229C22⋊Q8—44no—
230C22⋊Q8—44no—
231C4×Q8—24no—
232C4×Q8—41no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—44no—
237C4×Q8—44no—
238C4×Q8—42no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—24no—
242C4×D4—42no—
243#117—44no—
244#117—82no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2411yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2411yescharacteristic
286C2×C42C2412yes—
287C2×C42C2412yes—
288C2×C42C2412yes—
289#107—44no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292#103—28no—
293#103—44no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298#103—44no—
299#102—44no—
300#97—42no—
301C4×C8—42no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—21no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95C2×D414yes—
318#95C2×D414yes—
319C24—21no—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—44no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—44no—
339C2×Q8—42no—
340C2×Q8—82no—
341C2×Q8—816no—
342C2×Q8#13714yes—
343C2×D4—21no—
344C2×D4—24no—
345C2×D4H(5,𝔽2)12yes—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—82no—
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D412yes—
366C22×C4C2511yescharacteristic
367Q16—88no—
368Q16—88no—
369SD16—88no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—48no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—84no—
404C4⋊C4H(5,𝔽2)12yes—
405C22⋊C4—24no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—82no—
411C42—22no—
412C42—24no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—21no—
418C42—21no—
419C42—24no—
420C42—41no—
421C42—44no—
422C42—42no—
423C42—48no—
424C42—82no—
425C42—82no—
426C23—22no—
427C23—41no—
428C23—42no—
429C23—41no—
430C23—42no—
431C23C23×D411yescharacteristic
432Q8—44no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—42no—
437Q8—44no—
438Q8—44no—
439Q8—42no—
440Q8—82no—
441Q8—88no—
442Q8—84no—
443D4—24no—
444D4—42no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—48no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—88no—
482C2×C4C2×#13712yes—
483C2×C4C2×#13714yes—
484C2×C4#57814yes—
485C2×C4C2×H(5,𝔽2)11yescharacteristic
486C2×C4C2×H(5,𝔽2)11yescharacteristic
487C2×C4#58514yes—
488C8—44no—
489C8—44no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—82no—
494C22—82no—
495C22#348411yescharacteristic
496C22#348712yes—
497C22C22×#13711yescharacteristic
498C22C2×#57812yes—
499C22#358511yescharacteristic
500C4—22no—
501C4—24no—
502C4—24no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—41no—
508C4—42no—
509C4—82no—
510C4—84no—
511C4—82no—
512C4—82no—
513C4—84no—
514C4#354512yes—
515C4#355112yes—
516C2—82no—
517C2#5208311yescharacteristic
518C2#6157011yescharacteristic
519C2#6166511yescharacteristic
520C2#6190012yes—
521C2#6191712yes—
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. #32670C2semidirect
4. #32644C2semidirect
5. #30880C2semidirect
7. #30042C2semidirect
9. #27441C2semidirect
13. #23365C2semidirect

Maximal subgroups

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

subgroupindex
#598692
#389542
#326702
#326442
#308802
#307572
#300422
#293692
#274412
#271912
#259292
#256432
#233652