Finite Group Databasebeta

← #7535086

Subgroups — #7535086

Order 512.

1427 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59828C211yesmaximal, characteristic
2#36598C211yesmaximal, characteristic
3#31469C214yesmaximal
4#29219C211yesmaximal, characteristic
5#28717C211yesmaximal, characteristic
6#28187C212yesmaximal
7#27816C212yesmaximal
8#27425C214yesmaximal
9#26443C212yesmaximal
10#25323C212yesmaximal
11#24997C212yesmaximal
12#24332C212yesmaximal
13#23214C212yesmaximal
14#23149C211yesmaximal, characteristic
15#22521C212yesmaximal
16C4×#475C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#487C2212yes—
20C2×#475C2211yescharacteristic
21C2×#462C2211yescharacteristic
22C2×C8×Q8C2212yes—
23C2×C4×Q16C2212yes—
24#2868C2211yescharacteristic
25#2862—21no—
26#2726—22no—
27#2582C2212yes—
28#2582C2212yes—
29#2571C2212yes—
30#2570—22no—
31#2568C2212yes—
32#2552C2212yes—
33#2546—21no—
34#2511—22no—
35#2506—22no—
36#2453C2214yes—
37#2446—22no—
38#2446—22no—
39#2365C2212yes—
40#2365C2212yes—
41#2355C2212yes—
42#2355C2212yes—
43Q8×C4⋊C4—21no—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2101C2214yes—
50#2099C2214yes—
51#2087C2214yes—
52#2074C2214yes—
53#2072C2214yes—
54#2059C2214yes—
55#2056C2214yes—
56#2026C2214yes—
57#1991C2214yes—
58#1990C2212yes—
59#1990C2212yes—
60#1989C2214yes—
61#1988C2212yes—
62#1988C2212yes—
63#1988C2211yescharacteristic
64#1984C2212yes—
65#1981C2211yescharacteristic
66#1948C2211yescharacteristic
67#1947C2211yescharacteristic
68#1939C2212yes—
69#1933C2214yes—
70#1933C2214yes—
71#1928C2212yes—
72#1927C2214yes—
73#1919C2214yes—
74#1919C2212yes—
75#1916C2214yes—
76#1878C2214yes—
77#1878C2212yes—
78#1865C2214yes—
79#1865C2212yes—
80#1834C2212yes—
81#1805C2214yes—
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1803C2211yescharacteristic
86C4×#106C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93C22×Q16C2312yes—
94#557—28no—
95#557—44no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—24no—
104C2×#125C2314yes—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2312yes—
114#487—28no—
115#475—24no—
116#462—24no—
117C8×Q8—44no—
118C4×Q16—44no—
119C2×#107C2312yes—
120C2×#106C2312yes—
121C2×#106C2312yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2314yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2314yes—
138#400—24no—
139#400—24no—
140#400—24no—
141#398—24no—
142#398—24no—
143#398—24no—
144#398—24no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—21no—
152#391D412yes—
153#391C2312yes—
154#391C2314yes—
155#391C2314yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—24no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389D412yes—
166#389D412yes—
167#389C2312yes—
168#389C2314yes—
169#389C2314yes—
170#387—24no—
171#387—24no—
172#387—24no—
173#387—24no—
174#387—24no—
175#384—22no—
176#384—21no—
177#384—21no—
178#384D411yescharacteristic
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#382—22no—
184#382—22no—
185#382C2314yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C43—21no—
203C43C2311yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C22×Q8—21no—
221C22×Q8—22no—
222C22×Q8C2×D412yes—
223C22×Q8C2412yes—
224C22×Q8C2411yescharacteristic
225C2×Q16—24no—
226C2×Q16—24no—
227C2×Q16—42no—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—24no—
237C4⋊Q8—48no—
238C4⋊Q8—44no—
239C4⋊Q8—82no—
240C4⋊Q8#4114yes—
241C4⋊Q8#4114yes—
242#125—28no—
243#125—44no—
244#125—416no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—28no—
250C4×Q8—24no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—24no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—44no—
257C4×Q8—48no—
258C4×Q8—44no—
259C4×Q8—84no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—24no—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2414yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—42no—
314#106—24no—
315#106—24no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—44no—
321#103—24no—
322#103—28no—
323#103—28no—
324#103—24no—
325#103—24no—
326#103—24no—
327#103—42no—
328#103—42no—
329C4×C8—24no—
330C4×C8—24no—
331C4×C8—44no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—22no—
341#95—24no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95C2×D414yes—
349#95C2×D414yes—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—44no—
365C2×Q8—42no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—42no—
369C2×Q8C2×#4114yes—
370C2×Q82−1+412yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4C22×D414yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D414yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2511yescharacteristic
389Q16—48no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—24no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—24no—
423C4⋊C4—42no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4C2×#4114yes—
443C4⋊C42−1+412yes—
444C42—24no—
445C42—22no—
446C42—22no—
447C42—24no—
448C42—24no—
449C42—21no—
450C42—21no—
451C42—41no—
452C42—48no—
453C42—42no—
454C42—42no—
455C42—44no—
456C42—42no—
457C42—44no—
458C42—42no—
459C42—42no—
460C42—84no—
461C42#13514yes—
462C42C2×#4112yes—
463C42C2×#4112yes—
464C42H(5,𝔽2)12yes—
465C422−1+412yes—
466C23C23×D411yescharacteristic
467Q8—24no—
468Q8—22no—
469Q8—24no—
470Q8—22no—
471Q8—41no—
472Q8—42no—
473Q8—41no—
474Q8—42no—
475Q8—48no—
476Q8—48no—
477Q8—88no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4#54612yes—
521C2×C4#54612yes—
522C2×C4#54812yes—
523C2×C4#54812yes—
524C2×C4C2×#13512yes—
525C2×C4C2×#13512yes—
526C2×C4C2×#13512yes—
527C2×C4#57712yes—
528C2×C4#57812yes—
529C2×C4C22×#4111yescharacteristic
530C2×C4C22×#4111yescharacteristic
531C2×C4C2×H(5,𝔽2)11yescharacteristic
532C2×C4C2×2−1+411yescharacteristic
533C2×C4C2×2−1+411yescharacteristic
534C2×C4C2×2−1+411yescharacteristic
535C8—42no—
536C8—42no—
537C8—42no—
538C8—42no—
539C22C2×#54611yescharacteristic
540C22C2×#54811yescharacteristic
541C22#348611yescharacteristic
542C22C22×#13511yescharacteristic
543C22C2×#57711yescharacteristic
544C22C2×#57811yescharacteristic
545C22#358611yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—44no—
550C4—44no—
551C4—42no—
552C4—41no—
553C4—42no—
554C4—44no—
555C4—44no—
556C4—81no—
557C4—82no—
558C4—84no—
559C4—82no—
560C4—81no—
561C4#330214yes—
562C4#336514yes—
563C4#348211yescharacteristic
564C4#352011yescharacteristic
565C4#352311yescharacteristic
566C4#355011yescharacteristic
567C2#5904211yescharacteristic
568C2C2×#330211yescharacteristic
569C2#6120211yescharacteristic
570C2C2×#336511yescharacteristic
571C2#6134211yescharacteristic
572C2#6182211yescharacteristic
573C2#6186911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#598282
#365982
#314692
#292192
#287172
#281872
#278162
#274252
#264432
#253232
#249972
#243322
#232142
#231492
#225212
C4×#4752
C4×#4622