Finite Group Databasebeta

← #7545394

Subgroups — #7545394

Order 512.

1415 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic
3#32100C214yesmaximal
4#28748C211yesmaximal, characteristic
5#28748C211yesmaximal, characteristic
6#28646C212yesmaximal
7#27757C212yesmaximal
8#26772C214yesmaximal
9#26531C212yesmaximal
10#26317C212yesmaximal
11#25647C212yesmaximal
12#25278C212yesmaximal
13#23221C212yesmaximal
14#23153C211yesmaximal, characteristic
15#22547C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#475C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#487C2212yes—
20C2×#487C2212yes—
21C2×#477C2211yescharacteristic
22C2×#475C2211yescharacteristic
23C2×#446C2212yes—
24#2868C2211yescharacteristic
25#2859—21no—
26#2809—22no—
27#2571C2212yes—
28#2568—22no—
29#2568C2212yes—
30#2565—22no—
31#2546—21no—
32#2546C2212yes—
33#2546C2212yes—
34#2546C2212yes—
35#2508—22no—
36#2508C2214yes—
37#2506—22no—
38#2506—22no—
39#2369C2212yes—
40#2365C2212yes—
41#2365C2212yes—
42#2355—21no—
43#2355C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2097C2214yes—
50#2074C2214yes—
51#2068C2214yes—
52#2068C2214yes—
53#2043C2214yes—
54#2043C2214yes—
55#2043C2214yes—
56#2043C2214yes—
57#1988C2211yescharacteristic
58#1987C2211yescharacteristic
59#1984C2212yes—
60#1981C2212yes—
61#1981C2214yes—
62#1981C2212yes—
63#1981C2212yes—
64#1981C2212yes—
65#1960C2214yes—
66#1956C2212yes—
67#1956C2212yes—
68#1948C2211yescharacteristic
69#1947C2211yescharacteristic
70#1939C2212yes—
71#1935C2214yes—
72#1935C2214yes—
73#1928C2214yes—
74#1927C2214yes—
75#1921C2212yes—
76#1919C2212yes—
77#1917C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1865C2212yes—
81#1863C2212yes—
82#1805C2214yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85C4×#107C2212yes—
86C4×#106C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93#557—28no—
94#557—44no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8D411yescharacteristic
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2314yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—22no—
104C2×#125—24no—
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113#487—28no—
114#487—44no—
115#477—24no—
116#475—24no—
117#446—44no—
118C2×#107C2312yes—
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×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142#400—24no—
143#400—24no—
144#400—24no—
145#398—24no—
146#398—24no—
147#398—24no—
148#398—24no—
149#391—22no—
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—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389D412yes—
169#389D412yes—
170#389C2312yes—
171#389C2312yes—
172#389C2314yes—
173#389C2314yes—
174#387—24no—
175#387—24no—
176#387—24no—
177#387—24no—
178#384—21no—
179#384—21no—
180#384—21no—
181#384—21no—
182#384D411yescharacteristic
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#382—22no—
186#382—22no—
187#382—24no—
188#382C2314yes—
189C4×C4⋊C4—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—24no—
231C4⋊Q8—22no—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8—24no—
236C4⋊Q8—22no—
237C4⋊Q8—24no—
238C4⋊Q8—42no—
239C4⋊Q8—42no—
240C4⋊Q8—44no—
241C4⋊Q8—84no—
242C4⋊Q8#4114yes—
243#125—48no—
244#125—416no—
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—44no—
256C4×Q8—82no—
257C4×Q8#4114yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2411yescharacteristic
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2412yes—
293C2×C42—21no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—24no—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2414yes—
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311#107—24no—
312#107—42no—
313#106—24no—
314#106—42no—
315C4⋊C8—22no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—44no—
320#103—24no—
321#103—24no—
322#103—28no—
323#103—28no—
324#103—42no—
325#103—42no—
326#103—44no—
327#103—44no—
328#103—44no—
329#103—44no—
330#103—44no—
331#103—44no—
332#97—44no—
333C4×C8—24no—
334C4×C8—24no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—22no—
339#95—24no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—22no—
345#95—24no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95C2×D414yes—
352#95C2×D414yes—
353C2×Q8—24no—
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—42no—
370C2×Q8—42no—
371C2×Q8—81no—
372C2×Q8—81no—
373C2×Q8C2×#4112yes—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4C22×D414yes—
384C22×C4C22×D414yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2511yescharacteristic
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—
402C2×C8—22no—
403C2×C8—22no—
404C4⋊C4—24no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—42no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—82no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4C2×#4114yes—
445C4⋊C4C2×#4112yes—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—24no—
451C42—21no—
452C42—21no—
453C42—41no—
454C42—48no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—44no—
459C42—42no—
460C42—44no—
461C42—42no—
462C42—84no—
463C42#13714yes—
464C42C2×#4112yes—
465C42C2×#4112yes—
466C42H(5,𝔽2)12yes—
467C422−1+412yes—
468C23C23×D411yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—48no—
472Q8—41no—
473Q8—41no—
474Q8—44no—
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—84no—
479Q8—84no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4#54612yes—
523C2×C4#54612yes—
524C2×C4#54812yes—
525C2×C4#54812yes—
526C2×C4C2×#13712yes—
527C2×C4C2×#13712yes—
528C2×C4C2×#13712yes—
529C2×C4#57712yes—
530C2×C4#57812yes—
531C2×C4C22×#4111yescharacteristic
532C2×C4C22×#4111yescharacteristic
533C2×C4C2×H(5,𝔽2)11yescharacteristic
534C2×C4C2×2−1+411yescharacteristic
535C2×C4#58511yescharacteristic
536C2×C4#58511yescharacteristic
537C8—42no—
538C8—42no—
539C8—42no—
540C8—42no—
541C22C2×#54611yescharacteristic
542C22C2×#54811yescharacteristic
543C22#348711yescharacteristic
544C22C22×#13711yescharacteristic
545C22C2×#57711yescharacteristic
546C22C2×#57811yescharacteristic
547C22#358611yescharacteristic
548C4—22no—
549C4—22no—
550C4—22no—
551C4—44no—
552C4—42no—
553C4—44no—
554C4—42no—
555C4—44no—
556C4—41no—
557C4—81no—
558C4—82no—
559C4—84no—
560C4—82no—
561C4—81no—
562C4—82no—
563C4#332914yes—
564C4#337414yes—
565C4#348011yescharacteristic
566C4#350711yescharacteristic
567C4#351711yescharacteristic
568C4#355411yescharacteristic
569C2#5907811yescharacteristic
570C2C2×#332911yescharacteristic
571C2#6121011yescharacteristic
572C2C2×#337411yescharacteristic
573C2#6134011yescharacteristic
574C2#6189311yescharacteristic
575C2#6193411yescharacteristic
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
#598692
#365602
#321002
#287482
#286462
#277572
#267722
#265312
#263172
#256472
#252782
#232212
#231532
#225472
C4×#4772
C4×#4752