Finite Group Databasebeta

← #6555948

Subgroups — #6555948

Order 512.

1371 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#34328C211yesmaximal, characteristic
3#29985C212yesmaximal
4#29983C211yesmaximal, characteristic
5#29278C212yesmaximal
6#29276C211yesmaximal, characteristic
7#29209C212yesmaximal
8#29208C211yesmaximal, characteristic
9#26515C214yesmaximal
10#26513C212yesmaximal
11#26095C214yesmaximal
12#26093C212yesmaximal
13#23777C211yesmaximal, characteristic
14#23777C212yesmaximal
15#22734C211yesmaximal, characteristic
16#22727C212yesmaximal
17#22521C212yesmaximal
18C2×#557C2211yescharacteristic
19C2×C8×Q8C2212yes—
20#2868C2211yescharacteristic
21#2862C2212yes—
22#2582—21no—
23#2571C2212yes—
24#2570C2214yes—
25#2568—21no—
26#2511—22no—
27#2446—22no—
28#2371—22no—
29#2370C2212yes—
30#2365—21no—
31#2365—21no—
32#2365C2211yescharacteristic
33#2356—21no—
34#2356C2212yes—
35#2355—21no—
36#2355C2211yescharacteristic
37#2355C2211yescharacteristic
38#2355C2212yes—
39#2336—22no—
40#2327—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×#125C2212yes—
45#2295C2211yescharacteristic
46#2295C2212yes—
47#2276—21no—
48C42×Q8C2211yescharacteristic
49#2101C2214yes—
50#2100C2212yes—
51#2100C2211yescharacteristic
52#2099C2212yes—
53#2097C2214yes—
54#2096C2212yes—
55#2095C2211yescharacteristic
56#2087C2214yes—
57#2087C2212yes—
58#2068C2214yes—
59#2068C2212yes—
60#2058C2212yes—
61#2058C2212yes—
62#2057C2212yes—
63#2057C2212yes—
64#2057C2211yescharacteristic
65#2024C2212yes—
66#2024C2211yescharacteristic
67#1991C2214yes—
68#1990C2212yes—
69#1989C2214yes—
70#1988C2212yes—
71#1949C2214yes—
72#1949C2214yes—
73#1949C2214yes—
74#1949C2214yes—
75#1948C2212yes—
76#1948C2212yes—
77#1947C2214yes—
78#1939C2212yes—
79#1939C2214yes—
80#1920C2214yes—
81#1919C2214yes—
82#1918C2214yes—
83#1846C2214yes—
84#1845C2212yes—
85#1831C2211yescharacteristic
86#1831C2212yes—
87#1805C2214yes—
88#1803C2212yes—
89C4×#107C2212yes—
90C4×#107C2214yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×C4⋊C8C2212yes—
93C4×#103C2212yes—
94C4×#103C2214yes—
95#557—24no—
96#557Q818yescomplemented
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—22no—
104C2×#125C2312yes—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C8×Q8—28no—
116C2×#107C2312yes—
117C2×#107C2314yes—
118C2×#107C2312yes—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2312yes—
127C2×#103C2314yes—
128C2×#103C2312yes—
129C2×C4×C8C2312yes—
130C2×C4×C8C2314yes—
131#400—24no—
132#400—24no—
133#398—24no—
134#391—22no—
135#391—21no—
136#391—21no—
137#391—22no—
138#391—21no—
139#391—21no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391C2311yescharacteristic
145#391C2312yes—
146#391C2311yescharacteristic
147#389—22no—
148#389—22no—
149#389—22no—
150#389—21no—
151#389—21no—
152#389C2311yescharacteristic
153#389C2312yes—
154#389C2312yes—
155#389C2314yes—
156#389C2311yescharacteristic
157#389C2312yes—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—24no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—21no—
167#387—22no—
168#387—21no—
169#387—24no—
170#387—24no—
171#387—22no—
172#387—22no—
173#387—22no—
174#384—21no—
175#384—21no—
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2312yes—
179#384C2312yes—
180#383—22no—
181#382—22no—
182#382—21no—
183#382C2312yes—
184#382C2314yes—
185#382C2314yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D412yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210#376—21no—
211#376—22no—
212#376—22no—
213C43C2312yes—
214C43C2311yescharacteristic
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#336C2311yescharacteristic
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2411yescharacteristic
236C22×C8C2414yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C4⋊Q8—28no—
240C4⋊Q8—22no—
241C4⋊Q8—42no—
242C4⋊Q8—44no—
243C4⋊Q8—44no—
244C4⋊Q8—82no—
245C4⋊Q8C2×Q814yes—
246#125—216no—
247#125—24no—
248#125C2×Q818yes—
249C4×Q8—28no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—22no—
253C4×Q8—42no—
254C4×Q8—44no—
255C4×Q8—44no—
256C4×Q8—44no—
257C4×Q8—84no—
258C4×Q8C2×Q818yes—
259C4×Q8C2×Q814yes—
260C4×Q8C2×Q814yes—
261C4×Q8#4114yes—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
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—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C42—22no—
298C2×C42—21no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—21no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2412yes—
318C2×C42C2411yescharacteristic
319#107—24no—
320#107—42no—
321#107—44no—
322#107—44no—
323#106—44no—
324#106—44no—
325C4⋊C8—24no—
326C4⋊C8—28no—
327C4⋊C8—28no—
328C4⋊C8—24no—
329#103—24no—
330#103—44no—
331#103—42no—
332C4×C8—28no—
333C4×C8—24no—
334#95—21no—
335#95—21no—
336#95—22no—
337#95—21no—
338#95—21no—
339#95—22no—
340#95—21no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—22no—
346#95—24no—
347#95—21no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—21no—
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—24no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—42no—
369C2×Q8—44no—
370C2×Q8—42no—
371C2×Q8—44no—
372C2×Q8—42no—
373C2×Q8—44no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8C22⋊Q814yes—
377C2×Q8C22×Q812yes—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C2511yescharacteristic
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C82−1+414yes—
407C4⋊C4—28no—
408C4⋊C4—24no—
409C4⋊C4—28no—
410C4⋊C4—24no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—84no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4C22⋊Q814yes—
444C4⋊C4C22⋊Q818yes—
445C4⋊C4C22×Q812yes—
446C4⋊C4C22×Q814yes—
447C42—28no—
448C42—24no—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—24no—
453C42—22no—
454C42—24no—
455C42—22no—
456C42—42no—
457C42—42no—
458C42—41no—
459C42—44no—
460C42—44no—
461C42—42no—
462C42—44no—
463C42—82no—
464C42—82no—
465C42—84no—
466C42#13514yes—
467C42C22×Q814yes—
468C42C22×Q812yes—
469C42C2×#4114yes—
470C42C2×#4112yes—
471C23C23×D411yescharacteristic
472Q8—28no—
473Q8—24no—
474Q8—48no—
475Q8—42no—
476Q8—88no—
477Q8—84no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4C2×C22⋊Q814yes—
527C2×C4C2×C22⋊Q812yes—
528C2×C4C2×C22⋊Q814yes—
529C2×C4C2×C22⋊Q812yes—
530C2×C4C2×#13512yes—
531C2×C4C2×#13512yes—
532C2×C4C2×#13512yes—
533C2×C4#57512yes—
534C2×C4#57812yes—
535C2×C4C23×Q811yescharacteristic
536C2×C4C22×#4111yescharacteristic
537C2×C4C2×2−1+412yes—
538C2×C4#58512yes—
539C8—24no—
540C8—44no—
541C8—42no—
542C22C22×C22⋊Q811yescharacteristic
543C22#348711yescharacteristic
544C22#349911yescharacteristic
545C22C22×#13511yescharacteristic
546C22C2×#57511yescharacteristic
547C22C2×#57811yescharacteristic
548C22#358511yescharacteristic
549C4—22no—
550C4—24no—
551C4—24no—
552C4—22no—
553C4—22no—
554C4—22no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—44no—
560C4—42no—
561C4—82no—
562C4—81no—
563C4—82no—
564C4—84no—
565C4—81no—
566C4—82no—
567C4#307614yes—
568C4#350212yes—
569C4#351612yes—
570C2#3584511yescharacteristic
571C2C2×#307611yescharacteristic
572C2#6049811yescharacteristic
573C2#6188511yescharacteristic
574C2#6190511yescharacteristic
575C2#6247311yescharacteristic
576C2#6251211yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
96. #557Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
#343282
#299852
#299832
#292782
#292762
#292092
#292082
#265152
#265132
#260952
#260932
#237772
#227342
#227272
#225212