Finite Group Databasebeta

← #7545353

Subgroups — #7545353

Order 512.

1453 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic
3#32084C214yesmaximal
4#28748C211yesmaximal, characteristic
5#28748C211yesmaximal, characteristic
6#28638C212yesmaximal
7#27749C212yesmaximal
8#26758C214yesmaximal
9#26532C212yesmaximal
10#26322C212yesmaximal
11#25651C212yesmaximal
12#25278C212yesmaximal
13#23217C212yesmaximal
14#23149C211yesmaximal, characteristic
15#22542C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#477C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#487C2212yes—
20C2×#487C2212yes—
21C2×#477C2211yescharacteristic
22C2×#477C2211yescharacteristic
23C2×C8×Q8C2212yes—
24#2868C2211yescharacteristic
25#2859—21no—
26#2806—22no—
27#2584—22no—
28#2582—22no—
29#2568C2212yes—
30#2567C2212yes—
31#2565—22no—
32#2552—21no—
33#2552C2212yes—
34#2546C2212yes—
35#2546C2212yes—
36#2507C2214yes—
37#2506—22no—
38#2506—22no—
39#2369C2212yes—
40#2365—21no—
41#2365C2212yes—
42#2365C2212yes—
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2095C2214yes—
50#2072C2214yes—
51#2068C2214yes—
52#2068C2214yes—
53#2043C2214yes—
54#2043C2214yes—
55#2043C2214yes—
56#2043C2214yes—
57#1987C2211yescharacteristic
58#1987C2211yescharacteristic
59#1984C2212yes—
60#1981C2212yes—
61#1981C2214yes—
62#1981C2212yes—
63#1981C2212yes—
64#1981C2212yes—
65#1956C2212yes—
66#1956C2214yes—
67#1956C2212yes—
68#1947C2211yescharacteristic
69#1947C2211yescharacteristic
70#1939C2212yes—
71#1933C2214yes—
72#1933C2214yes—
73#1928C2214yes—
74#1928C2214yes—
75#1921C2212yes—
76#1921C2212yes—
77#1916C2214yes—
78#1863C2214yes—
79#1863C2214yes—
80#1863C2212yes—
81#1863C2212yes—
82#1803C2211yescharacteristic
83#1803C2214yes—
84#1803C2211yescharacteristic
85C4×#107C2212yes—
86C4×#107C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#103C2214yes—
92C4×#103C2212yes—
93#557—28no—
94#557—44no—
95C2×C4⋊Q8D411yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2314yes—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—22no—
103C2×#125—24no—
104C2×#125—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114#487—28no—
115#487—44no—
116#477—24no—
117#477—24no—
118C8×Q8—44no—
119C2×#107C2312yes—
120C2×#107C2312yes—
121C2×#107C2312yes—
122C2×#107C2312yes—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2314yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—24no—
144#398—24no—
145#398—24no—
146#398—24no—
147#398—24no—
148#398—24no—
149#395—24no—
150#391—22no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391—24no—
155#391D412yes—
156#391C2314yes—
157#391C2314yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—24no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—24no—
171#389D412yes—
172#389D412yes—
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#389C2314yes—
177#387—24no—
178#387—24no—
179#387—24no—
180#384—22no—
181#384—21no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384D411yescharacteristic
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2314yes—
189#382—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C43—21no—
207C43C2311yescharacteristic
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2314yes—
218#336C2312yes—
219#336C2312yes—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8C2×D412yes—
223C22×Q8C2414yes—
224C22×Q8C2411yescharacteristic
225C22×C8C2412yes—
226C22×C8C2412yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C4⋊Q8—28no—
230C4⋊Q8—22no—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—22no—
234C4⋊Q8—24no—
235C4⋊Q8—22no—
236C4⋊Q8—24no—
237C4⋊Q8—42no—
238C4⋊Q8—42no—
239C4⋊Q8—44no—
240C4⋊Q8Q1618yes—
241C4⋊Q8#4114yes—
242#125—48no—
243#125—416no—
244#125—84no—
245C4×Q8—24no—
246C4×Q8—22no—
247C4×Q8—24no—
248C4×Q8—28no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—41no—
252C4×Q8—41no—
253C4×Q8—44no—
254C4×Q8—48no—
255C4×Q8—82no—
256C4×Q8—82no—
257C4×Q8—84no—
258C4×Q8#4114yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—24no—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2414yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314#107—24no—
315#107—24no—
316#107—42no—
317#107—42no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8—44no—
323#103—24no—
324#103—24no—
325#103—28no—
326#103—28no—
327#103—42no—
328#103—42no—
329#103—44no—
330#103—44no—
331#103—44no—
332#103—44no—
333#103—44no—
334#103—44no—
335C4×C8—24no—
336C4×C8—24no—
337C4×C8—44no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—24no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D414yes—
353#95C2×D414yes—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—21no—
363C2×Q8—24no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—42no—
369C2×Q8—81no—
370C2×Q8—81no—
371C2×Q8—81no—
372C2×Q8—81no—
373C2×Q8C2×Q1618yes—
374C2×Q8C2×#4112yes—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4C22×D414yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D414yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2511yescharacteristic
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—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—42no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—84no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—84no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4C2×Q1618yes—
446C4⋊C4C2×Q1618yes—
447C4⋊C4C2×#4114yes—
448C4⋊C4C2×#4112yes—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—24no—
453C42—24no—
454C42—21no—
455C42—21no—
456C42—41no—
457C42—48no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—42no—
463C42—44no—
464C42—44no—
465C42—84no—
466C42C2×Q1614yes—
467C42C2×#4112yes—
468C42C2×#4112yes—
469C42H(5,𝔽2)12yes—
470C422−1+412yes—
471C23C23×D411yescharacteristic
472Q8—216no—
473Q8—22no—
474Q8—22no—
475Q8—41no—
476Q8—41no—
477Q8—44no—
478Q8—44no—
479Q8—44no—
480Q8—44no—
481Q8—84no—
482Q8—84no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4#54612yes—
525C2×C4#54612yes—
526C2×C4#54812yes—
527C2×C4#54812yes—
528C2×C4C22×Q1612yes—
529C2×C4C22×Q1618yes—
530C2×C4C22×Q1612yes—
531C2×C4C22×Q1612yes—
532C2×C4#57712yes—
533C2×C4#57712yes—
534C2×C4C22×#4111yescharacteristic
535C2×C4C22×#4111yescharacteristic
536C2×C4C2×H(5,𝔽2)11yescharacteristic
537C2×C4C2×2−1+411yescharacteristic
538C2×C4#58511yescharacteristic
539C2×C4#58511yescharacteristic
540C8—42no—
541C8—42no—
542C8—42no—
543C8—42no—
544C22C2×#54611yescharacteristic
545C22C2×#54811yescharacteristic
546C22#348711yescharacteristic
547C22C23×Q1611yescharacteristic
548C22C2×#57711yescharacteristic
549C22C2×#57711yescharacteristic
550C22#358611yescharacteristic
551C4—22no—
552C4—22no—
553C4—22no—
554C4—28no—
555C4—44no—
556C4—42no—
557C4—44no—
558C4—42no—
559C4—41no—
560C4—82no—
561C4—81no—
562C4—84no—
563C4—81no—
564C4—82no—
565C4—82no—
566C4#329914yes—
567C4#336314yes—
568C4#348011yescharacteristic
569C4#350711yescharacteristic
570C4#351711yescharacteristic
571C4#355411yescharacteristic
572C2#5907811yescharacteristic
573C2C2×#329911yescharacteristic
574C2#6120911yescharacteristic
575C2C2×#336311yescharacteristic
576C2#6134011yescharacteristic
577C2#6188411yescharacteristic
578C2#6193511yescharacteristic
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
#598712
#365602
#320842
#287482
#286382
#277492
#267582
#265322
#263222
#256512
#252782
#232172
#231492
#225422
C4×#4772