Finite Group Databasebeta

← #6856531

Subgroups — #6856531

Order 512.

1259 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#34951C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic
4#32324C214yesmaximal
5#31468C214yesmaximal
6#29867C212yesmaximal
7#29286C211yesmaximal, characteristic
8#29275C212yesmaximal
9#29222C212yesmaximal
10#29218C211yesmaximal, characteristic
11#28792C212yesmaximal
12#27730C212yesmaximal
13#26281C212yesmaximal
14#24995C212yesmaximal
15#22821C211yesmaximal, characteristic
16#22113C212yesmaximal
17C4×#475C211yesmaximal, characteristic
18C2×#477C2212yes—
19C2×#475C2211yescharacteristic
20#2868C2211yescharacteristic
21#2862C2212yes—
22#2585—22no—
23#2582—21no—
24#2582C2211yescharacteristic
25#2571C2212yes—
26#2570C2214yes—
27#2568—21no—
28#2568C2211yescharacteristic
29#2552C2211yescharacteristic
30#2546C2211yescharacteristic
31#2511—21no—
32#2511—22no—
33#2509C2212yes—
34#2453C2212yes—
35#2446—22no—
36#2446—21no—
37#2402—22no—
38#2365—21no—
39#2355—21no—
40#2355C2212yes—
41C4×C4⋊Q8—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#125C2212yes—
46#2295—21no—
47#2295C2212yes—
48C42×Q8C2211yescharacteristic
49C42×Q8C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2101C2214yes—
53#2100C2214yes—
54#2086C2214yes—
55#2085C2214yes—
56#2085C2214yes—
57#2074C2214yes—
58#2074C2214yes—
59#2059C2214yes—
60#1991C2214yes—
61#1990C2212yes—
62#1988C2211yescharacteristic
63#1988C2212yes—
64#1988C2212yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1984C2212yes—
69#1948C2211yescharacteristic
70#1947C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1942C2214yes—
75#1933C2214yes—
76#1928C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1919C2212yes—
80#1918C2214yes—
81#1916C2214yes—
82#1865C2212yes—
83#1842C2211yescharacteristic
84#1835C2214yes—
85#1834C2212yes—
86#1831C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125C2312yes—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111#477—28no—
112#475—24no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2312yes—
116C2×#106C2312yes—
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2312yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2312yes—
125C2×C4×C8C2312yes—
126C2×C4×C8C2314yes—
127#400—24no—
128#400—22no—
129#400—22no—
130#400—24no—
131#400—22no—
132#400—22no—
133#398—24no—
134#398—22no—
135#398—22no—
136#398—22no—
137#398—22no—
138#391—21no—
139#391—22no—
140#391—21no—
141#391—21no—
142#391—21no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391C2312yes—
151#391C2312yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—21no—
159#389—21no—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#387—24no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—24no—
173#387—24no—
174#387—22no—
175#387—22no—
176#384—21no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#382—22no—
184#382C2312yes—
185#382C2314yes—
186#382C2314yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D412yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209#376—22no—
210C43—21no—
211C43C2312yes—
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#336C2314yes—
227#336C2314yes—
228#336C2312yes—
229#336C2314yes—
230C22×Q8—22no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8C2×D411yescharacteristic
234C22×Q8C2411yescharacteristic
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2414yes—
238C4⋊Q8—22no—
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—24no—
244C4⋊Q8—44no—
245C4⋊Q8—44no—
246C4⋊Q8—82no—
247#125—24no—
248#125—42no—
249C4×Q8—24no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—42no—
257C4×Q8—41no—
258C4×Q8—44no—
259C4×Q8—44no—
260C4×Q8—84no—
261C4×Q8#4114yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2411yescharacteristic
299C2×C42—21no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—24no—
307C2×C42—21no—
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2412yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#107—28no—
322#107—44no—
323#106—24no—
324#106—24no—
325C4⋊C8—22no—
326C4⋊C8—24no—
327C4⋊C8—24no—
328C4⋊C8—22no—
329C4⋊C8—42no—
330#103—24no—
331#103—28no—
332#103—42no—
333C4×C8—24no—
334C4×C8—44no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—24no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—42no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8C2×#4112yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2511yescharacteristic
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—81no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—84no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4C2×#4112yes—
452C4⋊C42−1+412yes—
453C42—24no—
454C42—24no—
455C42—22no—
456C42—22no—
457C42—24no—
458C42—21no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—21no—
464C42—42no—
465C42—42no—
466C42—41no—
467C42—42no—
468C42—42no—
469C42—44no—
470C42—44no—
471C42—42no—
472C42—42no—
473C42—81no—
474C42—84no—
475C42#13514yes—
476C42C2×#4112yes—
477C422−1+412yes—
478C422−1+412yes—
479C422−1+412yes—
480C23C23×D411yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—44no—
484Q8—41no—
485Q8—41no—
486Q8—88no—
487Q8—84no—
488Q8—84no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4#53612yes—
538C2×C4#53612yes—
539C2×C4#54812yes—
540C2×C4#54812yes—
541C2×C4C2×#13512yes—
542C2×C4C2×#13512yes—
543C2×C4C2×#13512yes—
544C2×C4#57512yes—
545C2×C4#57812yes—
546C2×C4C22×#4111yescharacteristic
547C2×C4C2×2−1+411yescharacteristic
548C2×C4C2×2−1+411yescharacteristic
549C2×C4C2×2−1+411yescharacteristic
550C2×C4C2×2−1+411yescharacteristic
551C2×C4#58511yescharacteristic
552C8—44no—
553C8—42no—
554C8—42no—
555C22C2×#53611yescharacteristic
556C22C2×#54811yescharacteristic
557C22#348911yescharacteristic
558C22C22×#13511yescharacteristic
559C22C2×#57511yescharacteristic
560C22C2×#57811yescharacteristic
561C22#358511yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—82no—
574C4—84no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—81no—
579C4#323314yes—
580C4#336414yes—
581C4#350711yescharacteristic
582C4#350911yescharacteristic
583C4#355511yescharacteristic
584C4#355611yescharacteristic
585C2#3972411yescharacteristic
586C2C2×#323311yescharacteristic
587C2#6089311yescharacteristic
588C2C2×#336411yescharacteristic
589C2#6132411yescharacteristic
590C2#6210211yescharacteristic
591C2#6213311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×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
#365982
#349512
#331982
#323242
#314682
#298672
#292862
#292752
#292222
#292182
#287922
#277302
#262812
#249952
#228212
#221132
C4×#4752