Finite Group Databasebeta

← #6559757

Subgroups — #6559757

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#36497C211yesmaximal, characteristic
3#30308C212yesmaximal
4#30304C211yesmaximal, characteristic
5#29319C211yesmaximal, characteristic
6#29317C212yesmaximal
7#28818C214yesmaximal
8#28816C212yesmaximal
9#28798C212yesmaximal
10#28795C214yesmaximal
11#28694C211yesmaximal, characteristic
12#28692C212yesmaximal
13#23838C211yesmaximal, characteristic
14#23835C212yesmaximal
15#22739C212yesmaximal
16#22727C212yesmaximal
17#20572C211yesmaximal, characteristic
18C2×#501C2214yes—
19C2×#501C2212yes—
20C2×C8×Q8C2212yes—
21#2868C2211yescharacteristic
22#2864C2212yes—
23#2859C2211yescharacteristic
24#2859C2212yes—
25#2582—21no—
26#2570C2212yes—
27#2567—21no—
28#2552C2211yescharacteristic
29#2552C2212yes—
30#2551C2211yescharacteristic
31#2550C2214yes—
32#2550C2212yes—
33#2520—22no—
34#2507—22no—
35#2462—22no—
36#2453—21no—
37#2451—22no—
38#2451—21no—
39#2370—21no—
40#2369—21no—
41#2365—21no—
42#2355C2211yescharacteristic
43#2355C2211yescharacteristic
44#2355C2212yes—
45Q8×C4⋊C4—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#125C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51#2085C2212yes—
52#2084C2211yescharacteristic
53#2084C2212yes—
54#2084C2212yes—
55#2083C2212yes—
56#2083C2212yes—
57#2083C2212yes—
58#2083C2214yes—
59#2083C2211yescharacteristic
60#2059C2212yes—
61#2059C2214yes—
62#2059C2214yes—
63#2058C2211yescharacteristic
64#2057C2211yescharacteristic
65#2057C2212yes—
66#2057C2214yes—
67#2057C2214yes—
68#1993C2212yes—
69#1992C2214yes—
70#1949C2214yes—
71#1949C2214yes—
72#1949C2214yes—
73#1949C2214yes—
74#1948C2212yes—
75#1948C2212yes—
76#1947C2212yes—
77#1947C2212yes—
78#1944C2214yes—
79#1943C2214yes—
80#1943C2214yes—
81#1846C2214yes—
82#1845C2212yes—
83#1835C2214yes—
84#1834C2212yes—
85C4×#107C2214yes—
86C4×#107C2214yes—
87C4×#107C2212yes—
88C4×#106C2212yes—
89#1770C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92#1767C2211yescharacteristic
93#1744C2212yes—
94#1744C2211yescharacteristic
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#125—21no—
102C2×#125—21no—
103C2×#125C2311yescharacteristic
104C2×#125C2314yes—
105C2×C4×Q8—21no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111#501—48no—
112#501—44no—
113C8×Q8Q8116yescomplemented
114C2×#107C2312yes—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#107C2312yes—
120C2×#106C2312yes—
121C2×#106C2312yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#97C2312yes—
127C2×#97C2314yes—
128C2×C4×C8C2312yes—
129C2×C4×C8C2314yes—
130#400—22no—
131#398—22no—
132#398—24no—
133#398—22no—
134#398—24no—
135#398—22no—
136#395—24no—
137#395—22no—
138#395—22no—
139#391—22no—
140#391—21no—
141#391—21no—
142#391—22no—
143#391—21no—
144#391—22no—
145#391—22no—
146#391—24no—
147#389—22no—
148#389—21no—
149#389—21no—
150#389—22no—
151#389—22no—
152#389—21no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—22no—
158#389—21no—
159#389—24no—
160#389—22no—
161#389C2311yescharacteristic
162#389C2311yescharacteristic
163#389C2312yes—
164#389C2314yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#387—24no—
170#387—22no—
171#387—22no—
172#387—21no—
173#387—21no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#384—22no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2311yescharacteristic
189#382—21no—
190#382C2311yescharacteristic
191#382C2312yes—
192#382C2314yes—
193#382C2312yes—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C43C2311yescharacteristic
215C43C2312yes—
216#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#336C2312yes—
227#336C2314yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2311yescharacteristic
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8C2×D411yescharacteristic
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2414yes—
238C4⋊Q8—24no—
239C4⋊Q8—24no—
240C4⋊Q8—42no—
241C4⋊Q8—42no—
242C4⋊Q8—42no—
243C4⋊Q8—44no—
244C4⋊Q8—44no—
245C4⋊Q8—82no—
246#125—22no—
247#125—41no—
248#125—44no—
249#125—44no—
250#125—44no—
251#125—82no—
252C4×Q8—42no—
253C4×Q8—42no—
254C4×Q8—44no—
255C4×Q8—84no—
256C4×Q8—84no—
257C4×Q8C2×Q818yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
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—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2412yes—
300C2×C42—22no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42C2×D412yes—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2412yes—
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2412yes—
318C2×C42C2411yescharacteristic
319#107—28no—
320#107—24no—
321#107—28no—
322#107—42no—
323#107—44no—
324#107—44no—
325#106—24no—
326#106—24no—
327#106—44no—
328#106—44no—
329C4⋊C8C2×Q818yes—
330C4⋊C8C2×Q8116yes—
331#97—42no—
332#97—44no—
333C4×C8C2×Q818yes—
334C4×C8C2×Q8116yes—
335#95—21no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—21no—
340#95—21no—
341#95—24no—
342#95—21no—
343#95—24no—
344#95—22no—
345#95—21no—
346#95—21no—
347#95—22no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C2×Q8—24no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—42no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×Q8—82no—
371C2×Q8C4⋊Q814yes—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2511yescharacteristic
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—22no—
398C2×C8C22×Q814yes—
399C2×C8C22×Q814yes—
400C2×C8C22×Q818yes—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—84no—
437C4⋊C4—82no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4C4⋊Q818yes—
446C4⋊C4C4⋊Q814yes—
447C4⋊C4H(5,𝔽2)12yes—
448C4⋊C42−1+414yes—
449C42—22no—
450C42—22no—
451C42—24no—
452C42—22no—
453C42—24no—
454C42—22no—
455C42—21no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—21no—
460C42—42no—
461C42—42no—
462C42—44no—
463C42—42no—
464C42—42no—
465C42—82no—
466C42—84no—
467C42—82no—
468C42C22×Q818yes—
469C42C22×Q814yes—
470C23C23×D411yescharacteristic
471Q8—28no—
472Q8—48no—
473Q8—88no—
474Q8—164no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4C2×C4⋊Q814yes—
525C2×C4C2×C4⋊Q812yes—
526C2×C4C2×C4⋊Q814yes—
527C2×C4C2×C4⋊Q812yes—
528C2×C4#57512yes—
529C2×C4#57512yes—
530C2×C4#57612yes—
531C2×C4#57812yes—
532C2×C4C23×Q812yes—
533C2×C4C2×H(5,𝔽2)11yescharacteristic
534C2×C4C2×2−1+412yes—
535C2×C4#58511yescharacteristic
536C8—24no—
537C8—24no—
538C8—28no—
539C22C22×C4⋊Q811yescharacteristic
540C22#348611yescharacteristic
541C22#348811yescharacteristic
542C22C2×#57511yescharacteristic
543C22C2×#57511yescharacteristic
544C22C2×#57611yescharacteristic
545C22C2×#57811yescharacteristic
546C4—22no—
547C4—22no—
548C4—24no—
549C4—24no—
550C4—22no—
551C4—42no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—42no—
556C4—44no—
557C4—82no—
558C4—84no—
559C4—81no—
560C4—82no—
561C4—81no—
562C4—82no—
563C4—81no—
564C4#316214yes—
565C4#350212yes—
566C4#350312yes—
567C2#3593711yescharacteristic
568C2C2×#316211yescharacteristic
569C2#6066711yescharacteristic
570C2#6183511yescharacteristic
571C2#6184211yescharacteristic
572C2#6199511yescharacteristic
573C2#6199811yescharacteristic
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
113. C8×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#364972
#303082
#303042
#293192
#293172
#288182
#288162
#287982
#287952
#286942
#286922
#238382
#238352
#227392
#227272
#205722