Finite Group Databasebeta

← #6537925

Subgroups — #6537925

Order 512.

1391 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#34335C211yesmaximal, characteristic
3#29989C212yesmaximal
4#29983C211yesmaximal, characteristic
5#29277C211yesmaximal, characteristic
6#29272C212yesmaximal
7#29223C212yesmaximal
8#29219C211yesmaximal, characteristic
9#26454C214yesmaximal
10#26441C212yesmaximal
11#26034C214yesmaximal
12#26023C212yesmaximal
13#23783C212yesmaximal
14#23777C211yesmaximal, characteristic
15#22743C212yesmaximal
16#22736C211yesmaximal, characteristic
17#22525C212yesmaximal
18C2×Q82C2211yescharacteristic
19C2×#446C2212yes—
20#2868C2211yescharacteristic
21#2859C2212yes—
22#2582—21no—
23#2567—21no—
24#2552C2212yes—
25#2550C2214yes—
26#2507—22no—
27#2451—22no—
28#2369—21no—
29#2369C2211yescharacteristic
30#2369C2212yes—
31#2365—21no—
32#2365—22no—
33#2355C2211yescharacteristic
34#2355C2212yes—
35#2355C2212yes—
36#2351—21no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4C2211yescharacteristic
39#2331—21no—
40#2331—22no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45#2295C2211yescharacteristic
46#2295C2212yes—
47#2276—21no—
48C42×Q8C2211yescharacteristic
49#2100C2212yes—
50#2100C2211yescharacteristic
51#2099C2212yes—
52#2099C2212yes—
53#2099C2212yes—
54#2096C2212yes—
55#2095C2211yescharacteristic
56#2088C2214yes—
57#2087C2212yes—
58#2068C2214yes—
59#2068C2212yes—
60#2059C2214yes—
61#2058C2212yes—
62#2057C2211yescharacteristic
63#2057C2212yes—
64#2026C2214yes—
65#2024C2212yes—
66#2024C2211yescharacteristic
67#1993C2214yes—
68#1990C2212yes—
69#1989C2214yes—
70#1989C2214yes—
71#1988C2214yes—
72#1988C2214yes—
73#1987C2212yes—
74#1949C2214yes—
75#1948C2212yes—
76#1947C2214yes—
77#1947C2212yes—
78#1940C2214yes—
79#1939C2212yes—
80#1921C2214yes—
81#1918C2214yes—
82#1918C2214yes—
83#1836C2214yes—
84#1834C2212yes—
85#1831C2211yescharacteristic
86#1831C2212yes—
87#1805C2214yes—
88#1803C2212yes—
89#1776C2214yes—
90C4×#107C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95Q82—24no—
96Q82Q818yescomplemented
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#125C2314yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117#446—28no—
118C2×#107C2312yes—
119C2×#107C2314yes—
120C2×#107C2312yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2312yes—
127C2×#103C2314yes—
128C2×#103C2312yes—
129C2×#97C2314yes—
130C2×C4×C8C2312yes—
131#398—24no—
132#398—24no—
133#395—24no—
134#391—22no—
135#391—22no—
136#391—22no—
137#391—21no—
138#391—21no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—24no—
143#391C2311yescharacteristic
144#391C2312yes—
145#391C2311yescharacteristic
146#389—22no—
147#389—21no—
148#389—22no—
149#389—21no—
150#389—22no—
151#389—22no—
152#389—21no—
153#389—22no—
154#389—21no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—24no—
160#389C2312yes—
161#389C2311yescharacteristic
162#389C2312yes—
163#389C2314yes—
164#389C2311yescharacteristic
165#389C2312yes—
166#387—24no—
167#387—22no—
168#387—21no—
169#387—22no—
170#387—21no—
171#387—22no—
172#387—22no—
173#387—22no—
174#384—22no—
175#384—21no—
176#384—21no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2312yes—
181#384C2312yes—
182#382—21no—
183#382—22no—
184#382C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4D411yescharacteristic
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209#376—22no—
210#376—21no—
211C43C2312yes—
212C43C2311yescharacteristic
213#340C2312yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#336C2311yescharacteristic
227#336C2312yes—
228#336C2312yes—
229#336C2314yes—
230#336C2311yescharacteristic
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2×D411yescharacteristic
236C22×Q8C2411yescharacteristic
237C22×C8C2412yes—
238C22×C8C2414yes—
239C22×C8C2412yes—
240C4⋊Q8—216no—
241C4⋊Q8—28no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—24no—
245C4⋊Q8—44no—
246C4⋊Q8—44no—
247C4⋊Q8—82no—
248C4⋊Q8C2×Q814yes—
249C4⋊Q8C2×Q818yes—
250#125—44no—
251#125—44no—
252C4×Q8—28no—
253C4×Q8—24no—
254C4×Q8—22no—
255C4×Q8—24no—
256C4×Q8—42no—
257C4×Q8—44no—
258C4×Q8—44no—
259C4×Q8—84no—
260C4×Q8—82no—
261C4×Q8—84no—
262C4×Q8C2×Q814yes—
263C4×Q8C2×Q818yes—
264C4×Q8C2×Q814yes—
265C4×Q8#4114yes—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C42—22no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2412yes—
325C2×C42C2411yescharacteristic
326#107—24no—
327#107—28no—
328#107—42no—
329#106—44no—
330C4⋊C8—24no—
331C4⋊C8—28no—
332C4⋊C8—44no—
333C4⋊C8—42no—
334#103—24no—
335#103—42no—
336#103—44no—
337#97—28no—
338C4×C8—24no—
339#95—21no—
340#95—22no—
341#95—21no—
342#95—22no—
343#95—21no—
344#95—22no—
345#95—21no—
346#95—21no—
347#95—22no—
348#95—21no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—28no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—42no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8—82no—
375C2×Q8—81no—
376C2×Q8—82no—
377C2×Q8—81no—
378C2×Q8—82no—
379C2×Q8—82no—
380C2×Q8C22⋊Q814yes—
381C2×Q8C22×Q812yes—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—22no—
411C4⋊C4—24no—
412C4⋊C4—28no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—82no—
441C4⋊C4—81no—
442C4⋊C4—82no—
443C4⋊C4—81no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4C22⋊Q818yes—
450C4⋊C4C22⋊Q814yes—
451C4⋊C4C22×Q812yes—
452C4⋊C4C22×Q814yes—
453C4⋊C42−1+414yes—
454C42—28no—
455C42—24no—
456C42—24no—
457C42—22no—
458C42—24no—
459C42—22no—
460C42—22no—
461C42—24no—
462C42—22no—
463C42—42no—
464C42—41no—
465C42—42no—
466C42—44no—
467C42—44no—
468C42—42no—
469C42—84no—
470C42—82no—
471C42—82no—
472C42—82no—
473C42#13714yes—
474C42C22×Q814yes—
475C42C22×Q812yes—
476C42C2×#4112yes—
477C42C2×#4114yes—
478C23C23×D411yescharacteristic
479Q8—28no—
480Q8—24no—
481Q8—48no—
482Q8—42no—
483Q8—88no—
484Q8—84no—
485Q8—164no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4C2×C22⋊Q814yes—
535C2×C4C2×C22⋊Q812yes—
536C2×C4C2×C22⋊Q814yes—
537C2×C4C2×C22⋊Q812yes—
538C2×C4C2×#13712yes—
539C2×C4C2×#13712yes—
540C2×C4C2×#13712yes—
541C2×C4#57512yes—
542C2×C4#57612yes—
543C2×C4C23×Q811yescharacteristic
544C2×C4C22×#4111yescharacteristic
545C2×C4C2×2−1+412yes—
546C2×C4#58512yes—
547C8—42no—
548C8—42no—
549C8—44no—
550C22C22×C22⋊Q811yescharacteristic
551C22#348611yescharacteristic
552C22#350011yescharacteristic
553C22C22×#13711yescharacteristic
554C22C2×#57511yescharacteristic
555C22C2×#57611yescharacteristic
556C22#358611yescharacteristic
557C4—24no—
558C4—22no—
559C4—24no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—44no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—44no—
570C4—84no—
571C4—82no—
572C4—81no—
573C4—82no—
574C4—81no—
575C4#307814yes—
576C4#350212yes—
577C4#351812yes—
578C2#3571111yescharacteristic
579C2C2×#307811yescharacteristic
580C2#6049411yescharacteristic
581C2#6182911yescharacteristic
582C2#6183511yescharacteristic
583C2#6253711yescharacteristic
584C2#6258111yescharacteristic
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. Q82Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#343352
#299892
#299832
#292772
#292722
#292232
#292192
#264542
#264412
#260342
#260232
#237832
#237772
#227432
#227362
#225252