Finite Group Databasebeta

← #6586736

Subgroups — #6586736

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#33310C211yesmaximal, characteristic
4#33198C211yesmaximal, characteristic
5#33196C211yesmaximal, characteristic
6#28836C212yesmaximal
7#28542C212yesmaximal
8#28274C212yesmaximal
9#26089C212yesmaximal
10#26088C212yesmaximal
11#25454C212yesmaximal
12#25453C212yesmaximal
13#25061C212yesmaximal
14#25060C212yesmaximal
15#23163C212yesmaximal
16#22814C212yesmaximal
17#22729C212yesmaximal
18#21712C211yesmaximal, characteristic
19C42×Q16C211yesmaximal, characteristic
20C2×#498C2212yes—
21C2×#498C2212yes—
22C2×#477C2212yes—
23C2×#475C2212yes—
24C2×#446C2212yes—
25C2×C4×Q16C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C4×Q16C2211yescharacteristic
28#2868C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862—21no—
31#2859—21no—
32#2582C2212yes—
33#2582C2212yes—
34#2571C2212yes—
35#2570—22no—
36#2568C2212yes—
37#2567C2212yes—
38#2552C2212yes—
39#2550—22no—
40#2511—22no—
41#2507—22no—
42#2451—22no—
43#2446—22no—
44#2369C2212yes—
45#2365C2212yes—
46#2355C2212yes—
47#2355C2212yes—
48#2355C2212yes—
49Q8×C4⋊C4C2212yes—
50C4×C4⋊Q8—21no—
51C4×C4⋊Q8—21no—
52C4×C4⋊Q8C2211yescharacteristic
53C42×Q8C2211yescharacteristic
54C42×Q8C2211yescharacteristic
55#2097C2214yes—
56#2096C2214yes—
57#2085C2214yes—
58#2084C2214yes—
59#2066C2214yes—
60#2066C2214yes—
61#1988C2212yes—
62#1987C2212yes—
63#1984C2212yes—
64#1971C2214yes—
65#1949C2214yes—
66#1948C2214yes—
67#1948C2212yes—
68#1948C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1944C2214yes—
72#1939C2212yes—
73#1939C2212yes—
74#1927C2214yes—
75#1927C2214yes—
76#1920C2214yes—
77#1919C2214yes—
78#1847C2214yes—
79#1844C2212yes—
80#1844C2212yes—
81#1844C2212yes—
82#1844C2212yes—
83#1842C2211yescharacteristic
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1776C2214yes—
89C4×#107C2211yescharacteristic
90C4×#107C2211yescharacteristic
91C4×#107C2211yescharacteristic
92#1770C2212yes—
93#1770C2212yes—
94C4×#103C2211yescharacteristic
95C4×#103C2211yescharacteristic
96C4×#103C2211yescharacteristic
97C4×#103C2211yescharacteristic
98C4×#103C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C42×C8C2211yescharacteristic
101C22×Q16C2311yescharacteristic
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8D411yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#125—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119#498—28no—
120#498—44no—
121#477—44no—
122#475—44no—
123#446—28no—
124C4×Q16—24no—
125C4×Q16—24no—
126C4×Q16Q818yescomplemented
127C2×#107C2311yescharacteristic
128C2×#107C2311yescharacteristic
129C2×#107C2314yes—
130C2×#107C2311yescharacteristic
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2311yescharacteristic
141C2×#103C2311yescharacteristic
142C2×#103C2311yescharacteristic
143C2×#103C2311yescharacteristic
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2311yescharacteristic
147C2×#103C2311yescharacteristic
148C2×#97C2314yes—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155#400—24no—
156#400—24no—
157#398—24no—
158#398—24no—
159#398—24no—
160#395—24no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391—24no—
168#391D412yes—
169#391C2314yes—
170#391C2314yes—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—24no—
180#389D412yes—
181#389D412yes—
182#389C2314yes—
183#389C2314yes—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384—21no—
192#384D411yescharacteristic
193#384C2314yes—
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#382—22no—
200#382—22no—
201#382C2314yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C43—21no—
223C43—21no—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2311yescharacteristic
230#340C2311yescharacteristic
231#340C2311yescharacteristic
232#340C2311yescharacteristic
233#340C2311yescharacteristic
234#340C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8C2×D412yes—
240C22×Q8C2411yescharacteristic
241C22×Q8C2411yescharacteristic
242C2×Q16—22no—
243C2×Q16—22no—
244C2×Q16C2×Q814yes—
245C22×C8C2414yes—
246C22×C8C2411yescharacteristic
247C22×C8C2411yescharacteristic
248C22×C8C2411yescharacteristic
249C22×C8C2411yescharacteristic
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8—82no—
258C4⋊Q8—82no—
259#125—84no—
260C4×Q8—24no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—24no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—84no—
273C4×Q8—84no—
274C4×Q8C2×Q814yes—
275C4×Q8C2×Q814yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2414yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—22no—
333#107—22no—
334#107—44no—
335#107C2×Q814yes—
336#106—28no—
337#106—28no—
338#106—44no—
339#106—44no—
340#106—44no—
341C4⋊C8—28no—
342C4⋊C8—24no—
343C4⋊C8—24no—
344C4⋊C8—24no—
345#103—22no—
346#103—22no—
347#103—22no—
348#103—22no—
349#103—44no—
350#103—44no—
351#103C2×Q814yes—
352#103C2×Q814yes—
353#97—28no—
354C4×C8—22no—
355C4×C8—22no—
356C4×C8—22no—
357C4×C8C2×Q814yes—
358C4×C8#4114yes—
359C4×C8#4114yes—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95C2×D414yes—
377#95C2×D414yes—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8C22×Q812yes—
398C2×Q8C22×Q812yes—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4C22×D414yes—
410C22×C4C22×D414yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2511yescharacteristic
418Q16—28no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8C22×Q812yes—
429C2×C8C22×Q812yes—
430C2×C8C2×#4112yes—
431C2×C8C2×#4112yes—
432C2×C8C2×#4112yes—
433C2×C8C2×#4112yes—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4C22×Q812yes—
475C4⋊C4C22×Q812yes—
476C42—24no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—24no—
482C42—24no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—42no—
493C42—84no—
494C42—84no—
495C42#13714yes—
496C42C22×Q812yes—
497C42C2×#4112yes—
498C42C2×#4112yes—
499C42H(5,𝔽2)12yes—
500C422−1+412yes—
501C422−1+412yes—
502C23C23×D411yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—22no—
506Q8—22no—
507Q8—48no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—88no—
513Q8—164no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4#54712yes—
557C2×C4#54712yes—
558C2×C4D4×Q812yes—
559C2×C4D4×Q812yes—
560C2×C4#55012yes—
561C2×C4#55012yes—
562C2×C4C2×#13712yes—
563C2×C4C2×#13712yes—
564C2×C4C2×#13712yes—
565C2×C4#57712yes—
566C2×C4#57712yes—
567C2×C4#57712yes—
568C2×C4C23×Q811yescharacteristic
569C2×C4C22×#4111yescharacteristic
570C2×C4C22×#4111yescharacteristic
571C2×C4C2×H(5,𝔽2)11yescharacteristic
572C2×C4C2×2−1+411yescharacteristic
573C2×C4C2×2−1+411yescharacteristic
574C8—22no—
575C8—22no—
576C8—22no—
577C8—44no—
578C8#53314yes—
579C22C2×#54711yescharacteristic
580C22C2×D4×Q811yescharacteristic
581C22C2×#55011yescharacteristic
582C22C22×#13711yescharacteristic
583C22C2×#57711yescharacteristic
584C22C2×#57711yescharacteristic
585C22C2×#57711yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—42no—
590C4—42no—
591C4—44no—
592C4—44no—
593C4—42no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—84no—
598C4—81no—
599C4—81no—
600C4—84no—
601C4#335614yes—
602C4#338714yes—
603C4#340014yes—
604C4C2×#53311yescharacteristic
605C4#350311yescharacteristic
606C4#350911yescharacteristic
607C4#351111yescharacteristic
608C2#3613411yescharacteristic
609C2C2×#335611yescharacteristic
610C2#6128811yescharacteristic
611C2C2×#338711yescharacteristic
612C2#6137411yescharacteristic
613C2C2×#340011yescharacteristic
614C2#6141011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
126. C4×Q16Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
#365632
#333102
#331982
#331962
#288362
#285422
#282742
#260892
#260882
#254542
#254532
#250612
#250602
#231632
#228142
#227292
#217122
C42×Q162