Finite Group Databasebeta

← #6625957

Subgroups — #6625957

Order 512.

1527 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2C4×Q82C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#33073C212yesmaximal
5#29319C212yesmaximal
6#29255C212yesmaximal
7#29206C211yesmaximal, characteristic
8#28284C211yesmaximal, characteristic
9#27021C214yesmaximal
10#26389C214yesmaximal
11#25060C212yesmaximal
12#24333C212yesmaximal
13#22971C212yesmaximal
14#22825C211yesmaximal, characteristic
15#22674C212yesmaximal
16#21506C212yesmaximal
17C4×#462C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C2×#477C2212yes—
20C2×#475C2212yes—
21C2×#462C2211yescharacteristic
22C2×#441C2214yes—
23C2×C4×Q16C2212yes—
24#2868C2211yescharacteristic
25#2859C2212yes—
26#2582C2212yes—
27#2567—21no—
28#2552—21no—
29#2550—22no—
30#2507—22no—
31#2451C2214yes—
32#2369—22no—
33#2369—21no—
34#2369—21no—
35#2369C2212yes—
36#2369C2211yescharacteristic
37#2355—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2211yescharacteristic
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2211yescharacteristic
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C42×Q8—21no—
50C42×Q8C2212yes—
51C42×Q8C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2096C2214yes—
54#2084C2214yes—
55#2068C2214yes—
56#2058C2214yes—
57#2057C2214yes—
58#2024C2214yes—
59#1990C2212yes—
60#1988C2212yes—
61#1984C2212yes—
62#1981C2212yes—
63#1981C2211yescharacteristic
64#1948C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1947C2212yes—
69#1947C2211yescharacteristic
70#1943C2214yes—
71#1943C2214yes—
72#1928C2212yes—
73#1927C2214yes—
74#1920C2214yes—
75#1918C2214yes—
76#1917C2214yes—
77#1878C2214yes—
78#1878C2212yes—
79#1865C2214yes—
80#1865C2214yes—
81#1865C2214yes—
82#1863C2214yes—
83#1844C2212yes—
84#1842C2211yescharacteristic
85#1831C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96Q82—24no—
97Q82—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2311yescharacteristic
121#477—28no—
122#475—44no—
123#462Q818yescomplemented
124#441—48no—
125C4×Q16—28no—
126C2×#107C2314yes—
127C2×#107C2312yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#398—24no—
146#398—24no—
147#395—24no—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391—24no—
155#391C2312yes—
156#391C2314yes—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—24no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#389C2314yes—
179#387—24no—
180#384—21no—
181#384—22no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#382—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C43—22no—
221C43—21no—
222C43C2312yes—
223C43C2311yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2412yes—
240C22×Q8C2411yescharacteristic
241C2×Q16—24no—
242C2×Q16—24no—
243C2×Q16C2×Q818yes—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C4⋊Q8—216no—
248C4⋊Q8—28no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—42no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8C2×Q814yes—
258C4⋊Q8#4114yes—
259#125—84no—
260C4×Q8—28no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—42no—
274C4×Q8—41no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—82no—
279C4×Q8—84no—
280C4×Q8C2×Q814yes—
281C4×Q8#4114yes—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—24no—
349#107—28no—
350#106—44no—
351#106—44no—
352#106—44no—
353C4⋊C8—24no—
354C4⋊C8—22no—
355C4⋊C8—24no—
356C4⋊C8—42no—
357C4⋊C8C2×Q814yes—
358#103—24no—
359#103—28no—
360#103—24no—
361#103—28no—
362#103—42no—
363#103—44no—
364#103C2×Q818yes—
365#97—44no—
366C4×C8—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—24no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C2×Q8—28no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—44no—
398C2×Q8—81no—
399C2×Q8—82no—
400C2×Q8—81no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8#13714yes—
405C2×Q8C22×Q814yes—
406C2×Q8C22×Q812yes—
407C2×Q8C2×#4112yes—
408C2×Q8H(5,𝔽2)12yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2511yescharacteristic
429Q16—216no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8C22×Q814yes—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—84no—
474C4⋊C4—81no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4C22×Q812yes—
484C4⋊C4C2×#4114yes—
485C4⋊C4C2×#4112yes—
486C42—28no—
487C42—24no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—24no—
492C42—22no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—22no—
499C42—21no—
500C42—42no—
501C42—41no—
502C42—42no—
503C42—44no—
504C42—44no—
505C42—44no—
506C42—42no—
507C42—44no—
508C42—42no—
509C42—81no—
510C42—82no—
511C42—84no—
512C42#13714yes—
513C42C22×Q812yes—
514C42C2×#4112yes—
515C42C2×#4112yes—
516C422−1+412yes—
517C23C23×D411yescharacteristic
518Q8—28no—
519Q8—22no—
520Q8—22no—
521Q8—22no—
522Q8—24no—
523Q8—24no—
524Q8—42no—
525Q8—42no—
526Q8—84no—
527Q8—88no—
528Q8—164no—
529Q8#55414yes—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4#52512yes—
578C2×C4#52512yes—
579C2×C4D4×Q812yes—
580C2×C4D4×Q812yes—
581C2×C4C2×#13714yes—
582C2×C4C2×#13712yes—
583C2×C4C2×#13712yes—
584C2×C4C2×#13712yes—
585C2×C4C2×#13712yes—
586C2×C4#57712yes—
587C2×C4C23×Q811yescharacteristic
588C2×C4C22×#4111yescharacteristic
589C2×C4C22×#4111yescharacteristic
590C2×C4C2×H(5,𝔽2)11yescharacteristic
591C2×C4C2×2−1+411yescharacteristic
592C2×C4#58511yescharacteristic
593C8—24no—
594C8—42no—
595C8—44no—
596C22C2×#52511yescharacteristic
597C22C2×D4×Q811yescharacteristic
598C22#348811yescharacteristic
599C22C22×#13711yescharacteristic
600C22C22×#13711yescharacteristic
601C22C2×#57711yescharacteristic
602C22#358511yescharacteristic
603C4—22no—
604C4—24no—
605C4—22no—
606C4—22no—
607C4—24no—
608C4—42no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—44no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—84no—
617C4—82no—
618C4—81no—
619C4—82no—
620C4#311314yes—
621C4#338714yes—
622C4C2×#55411yescharacteristic
623C4Q8×#4111yescharacteristic
624C4#350711yescharacteristic
625C4#352211yescharacteristic
626C2#3654411yescharacteristic
627C2C2×#311311yescharacteristic
628C2#6059711yescharacteristic
629C2C2×#338711yescharacteristic
630C2Q8×#13711yescharacteristic
631C2#6196511yescharacteristic
632C2#6200211yescharacteristic
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
123. #462Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
C4×Q822
#331962
#330732
#293192
#292552
#292062
#282842
#270212
#263892
#250602
#243332
#229712
#228252
#226742
#215062
C4×#4622