Finite Group Databasebeta

← #6856570

Subgroups — #6856570

Order 512.

1273 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#34950C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic
4#32334C214yesmaximal
5#31480C214yesmaximal
6#29867C212yesmaximal
7#29287C211yesmaximal, characteristic
8#29273C212yesmaximal
9#29224C212yesmaximal
10#29218C211yesmaximal, characteristic
11#28799C212yesmaximal
12#27738C212yesmaximal
13#26279C212yesmaximal
14#24997C212yesmaximal
15#22825C211yesmaximal, characteristic
16#22119C212yesmaximal
17C4×#475C211yesmaximal, characteristic
18C2×#477C2212yes—
19C2×#475C2211yescharacteristic
20#2868C2211yescharacteristic
21#2862C2212yes—
22#2586—22no—
23#2582—21no—
24#2582C2211yescharacteristic
25#2571C2212yes—
26#2570C2214yes—
27#2568—21no—
28#2567C2211yescharacteristic
29#2567C2211yescharacteristic
30#2551C2211yescharacteristic
31#2511—22no—
32#2507—21no—
33#2507C2212yes—
34#2451—21no—
35#2451C2212yes—
36#2446—22no—
37#2399—22no—
38#2365—21no—
39#2355—21no—
40#2355C2212yes—
41C4×C4⋊Q8—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46#2295—21no—
47#2295C2212yes—
48C42×Q8C2211yescharacteristic
49C42×Q8C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2088C2214yes—
55#2085C2214yes—
56#2085C2214yes—
57#2074C2214yes—
58#2074C2214yes—
59#2058C2214yes—
60#1993C2214yes—
61#1990C2212yes—
62#1988C2211yescharacteristic
63#1988C2212yes—
64#1987C2212yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1984C2212yes—
69#1948C2211yescharacteristic
70#1948C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1943C2214yes—
75#1935C2214yes—
76#1927C2214yes—
77#1919C2212yes—
78#1919C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1917C2214yes—
82#1865C2212yes—
83#1842C2211yescharacteristic
84#1836C2214yes—
85#1834C2212yes—
86#1831C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90C4×#106C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4×Q8—21no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—21no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112#477—28no—
113#475—24no—
114C2×#107C2314yes—
115C2×#106C2312yes—
116C2×#106C2312yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2312yes—
123C2×#103C2312yes—
124C2×#103C2314yes—
125C2×#103C2312yes—
126C2×#97C2314yes—
127C2×C4×C8C2312yes—
128#400—24no—
129#400—24no—
130#398—24no—
131#398—22no—
132#398—22no—
133#398—22no—
134#398—22no—
135#395—22no—
136#395—22no—
137#395—22no—
138#395—22no—
139#391—21no—
140#391—22no—
141#391—21no—
142#391—22no—
143#391—21no—
144#391—22no—
145#391—21no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391C2312yes—
152#391C2312yes—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—22no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389C2312yes—
164#389C2314yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#387—22no—
171#387—22no—
172#387—24no—
173#387—22no—
174#387—22no—
175#387—24no—
176#387—24no—
177#384—21no—
178#384—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#382—22no—
186#382C2314yes—
187#382C2314yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D412yes—
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#340C2314yes—
216#340C2312yes—
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—22no—
233C22×Q8—21no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2411yescharacteristic
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2414yes—
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—24no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—24no—
246C4⋊Q8—42no—
247C4⋊Q8—44no—
248C4⋊Q8—44no—
249C4⋊Q8—82no—
250C4×Q8—24no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—42no—
258C4×Q8—41no—
259C4×Q8—44no—
260C4×Q8—84no—
261C4×Q8—82no—
262C4×Q8—84no—
263C4×Q8#4114yes—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2411yescharacteristic
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—24no—
310C2×C42—21no—
311C2×C42C2×D412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2412yes—
320C2×C42C2411yescharacteristic
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—28no—
325#106—24no—
326#106—24no—
327#106—44no—
328C4⋊C8—22no—
329C4⋊C8—24no—
330C4⋊C8—24no—
331C4⋊C8—22no—
332C4⋊C8—42no—
333#103—24no—
334#103—28no—
335#103—42no—
336#97—44no—
337C4×C8—24no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—24no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—81no—
380C2×Q8—82no—
381C2×Q8—81no—
382C2×Q8C2×#4112yes—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2511yescharacteristic
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—81no—
449C4⋊C4—84no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4C2×#4112yes—
457C4⋊C42−1+412yes—
458C42—24no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—24no—
463C42—21no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—42no—
470C42—42no—
471C42—41no—
472C42—42no—
473C42—42no—
474C42—44no—
475C42—42no—
476C42—42no—
477C42—44no—
478C42—81no—
479C42—84no—
480C42#13714yes—
481C42C2×#4112yes—
482C422−1+412yes—
483C422−1+412yes—
484C422−1+412yes—
485C23C23×D411yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—44no—
489Q8—41no—
490Q8—41no—
491Q8—88no—
492Q8—84no—
493Q8—88no—
494Q8—84no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4#53612yes—
544C2×C4#53612yes—
545C2×C4#54812yes—
546C2×C4#54812yes—
547C2×C4C2×#13712yes—
548C2×C4C2×#13712yes—
549C2×C4C2×#13712yes—
550C2×C4#57512yes—
551C2×C4#57712yes—
552C2×C4C22×#4111yescharacteristic
553C2×C4C2×2−1+411yescharacteristic
554C2×C4C2×2−1+411yescharacteristic
555C2×C4C2×2−1+411yescharacteristic
556C2×C4C2×2−1+411yescharacteristic
557C2×C4#58511yescharacteristic
558C8—44no—
559C8—42no—
560C8—42no—
561C22C2×#53611yescharacteristic
562C22C2×#54811yescharacteristic
563C22#348911yescharacteristic
564C22C22×#13711yescharacteristic
565C22C2×#57511yescharacteristic
566C22C2×#57711yescharacteristic
567C22#358511yescharacteristic
568C4—22no—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—41no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—82no—
580C4—84no—
581C4—82no—
582C4—82no—
583C4—81no—
584C4—82no—
585C4#323714yes—
586C4#336914yes—
587C4#350711yescharacteristic
588C4#350911yescharacteristic
589C4#355511yescharacteristic
590C4#355611yescharacteristic
591C2#3972411yescharacteristic
592C2C2×#323711yescharacteristic
593C2#6089211yescharacteristic
594C2C2×#336911yescharacteristic
595C2#6132411yescharacteristic
596C2#6211411yescharacteristic
597C2#6214411yescharacteristic
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
#349502
#331982
#323342
#314802
#298672
#292872
#292732
#292242
#292182
#287992
#277382
#262792
#249972
#228252
#221192
C4×#4752