Finite Group Databasebeta

← #7537597

Subgroups — #7537597

Order 512.

1387 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#59869C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#32602C214yesmaximal
4#28717C211yesmaximal, characteristic
5#28511C211yesmaximal, characteristic
6#28401C212yesmaximal
7#27749C212yesmaximal
8#26322C214yesmaximal
9#26092C212yesmaximal
10#25278C212yesmaximal
11#24682C212yesmaximal
12#24632C212yesmaximal
13#23247C211yesmaximal, characteristic
14#23063C212yesmaximal
15#22808C212yesmaximal
16C4×#462C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×#487C2212yes—
19C2×#477C2212yes—
20C2×#477C2212yes—
21C2×#462C2211yescharacteristic
22C2×#462C2211yescharacteristic
23C2×C4×Q16C2212yes—
24C2×C4×Q16C2212yes—
25#2868C2211yescharacteristic
26#2859—21no—
27#2809—22no—
28#2582C2212yes—
29#2571—21no—
30#2568C2214yes—
31#2567C2212yes—
32#2552C2212yes—
33#2550—22no—
34#2546C2212yes—
35#2546C2212yes—
36#2508—22no—
37#2508—22no—
38#2507—22no—
39#2451—22no—
40#2369—21no—
41#2369C2212yes—
42#2355C2212yes—
43#2355C2212yes—
44#2355C2212yes—
45Q8×C4⋊C4C2212yes—
46C4×C4⋊Q8—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51#2095C2214yes—
52#2095C2214yes—
53#2072C2214yes—
54#2072C2214yes—
55#2068C2214yes—
56#2056C2214yes—
57#2043C2214yes—
58#2016C2214yes—
59#1990C2212yes—
60#1984C2214yes—
61#1981C2214yes—
62#1981C2211yescharacteristic
63#1981C2211yescharacteristic
64#1969C2214yes—
65#1956C2212yes—
66#1956C2214yes—
67#1947C2211yescharacteristic
68#1947C2214yes—
69#1947C2211yescharacteristic
70#1939C2212yes—
71#1939C2212yes—
72#1933C2214yes—
73#1928C2214yes—
74#1928C2214yes—
75#1928C2214yes—
76#1928C2212yes—
77#1928C2212yes—
78#1921C2214yes—
79#1878C2212yes—
80#1878C2212yes—
81#1845C2212yes—
82#1803C2211yescharacteristic
83#1803C2212yes—
84#1803C2212yes—
85#1803C2211yescharacteristic
86C4×#107C2214yes—
87C8×C4⋊C4C2212yes—
88C8×C4⋊C4C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
93C22×Q16C2312yes—
94C22×Q16C2312yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8D411yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2314yes—
101C2×C4⋊Q8C2312yes—
102C2×#125—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110#487—44no—
111#477—28no—
112#477—44no—
113#462—24no—
114#462—24no—
115C4×Q16—28no—
116C4×Q16—44no—
117C2×#107C2314yes—
118C2×#107C2312yes—
119C2×#107C2314yes—
120C2×#107C2312yes—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2311yescharacteristic
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2314yes—
141#400—24no—
142#400—24no—
143#398—24no—
144#398—24no—
145#398—24no—
146#398—24no—
147#395—24no—
148#391—22no—
149#391—24no—
150#391—24no—
151#391D412yes—
152#391C2312yes—
153#391C2312yes—
154#391C2314yes—
155#391C2314yes—
156#391C2314yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389D412yes—
173#389D412yes—
174#389C2314yes—
175#387—24no—
176#387—24no—
177#387—24no—
178#384—21no—
179#384—21no—
180#384D411yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2314yes—
184#384C2312yes—
185#384C2312yes—
186#382—24no—
187#382—22no—
188#382—22no—
189#382—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C43—21no—
209C43C2311yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2312yes—
217#336C2314yes—
218#336C2312yes—
219#336C2314yes—
220C22×Q8—22no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×Q8C2412yes—
224C22×Q8C2411yescharacteristic
225C2×Q16—24no—
226C2×Q16—24no—
227C2×Q16—24no—
228C2×Q16—42no—
229C2×Q16—42no—
230C2×Q16—42no—
231C22×C8C2412yes—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C4⋊Q8—24no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—24no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—44no—
242C4⋊Q8—44no—
243C4⋊Q8—82no—
244C4⋊Q8—82no—
245C4⋊Q8Q1618yescomplemented
246#125—84no—
247C4×Q8—22no—
248C4×Q8—24no—
249C4×Q8—22no—
250C4×Q8—24no—
251C4×Q8—22no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—42no—
255C4×Q8—42no—
256C4×Q8—41no—
257C4×Q8—41no—
258C4×Q8—84no—
259C4×Q8#4114yes—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2414yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—28no—
314#107—24no—
315#107—42no—
316#107—44no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—44no—
321C4⋊C8#4114yes—
322#103—24no—
323#103—24no—
324#103—28no—
325#103—24no—
326#103—24no—
327#103—42no—
328#103—42no—
329#103—42no—
330#103—42no—
331#103—44no—
332#103—44no—
333#103—44no—
334C4×C8—24no—
335C4×C8—24no—
336C4×C8—44no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95C2×D414yes—
354#95C2×D414yes—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—44no—
375C2×Q8—82no—
376C2×Q8—82no—
377C2×Q8C2×Q1618yes—
378C2×Q8C2×#4112yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4C22×D414yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D414yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2511yescharacteristic
397Q16—48no—
398Q16—48no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8C2×#4114yes—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4C2×Q1618yes—
450C4⋊C4C2×Q1618yes—
451C4⋊C4C2×#4112yes—
452C42—24no—
453C42—22no—
454C42—22no—
455C42—24no—
456C42—24no—
457C42—21no—
458C42—21no—
459C42—41no—
460C42—42no—
461C42—42no—
462C42—44no—
463C42—42no—
464C42—44no—
465C42—42no—
466C42—42no—
467C42—84no—
468C42—84no—
469C42C2×Q1614yes—
470C42C2×#4112yes—
471C42C2×#4112yes—
472C42H(5,𝔽2)12yes—
473C422−1+412yes—
474C23C23×D411yescharacteristic
475Q8—22no—
476Q8—22no—
477Q8—48no—
478Q8—41no—
479Q8—41no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—164no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4#54812yes—
531C2×C4#54812yes—
532C2×C4#55012yes—
533C2×C4#55012yes—
534C2×C4C22×Q1612yes—
535C2×C4C22×Q1618yes—
536C2×C4C22×Q1612yes—
537C2×C4C22×Q1612yes—
538C2×C4#57712yes—
539C2×C4#57812yes—
540C2×C4C22×#4111yescharacteristic
541C2×C4C22×#4111yescharacteristic
542C2×C4C2×H(5,𝔽2)11yescharacteristic
543C2×C4C2×2−1+411yescharacteristic
544C2×C4#58511yescharacteristic
545C2×C4#58511yescharacteristic
546C8—24no—
547C8—42no—
548C8—42no—
549C8—42no—
550C22C2×#54811yescharacteristic
551C22C2×#55011yescharacteristic
552C22#348511yescharacteristic
553C22C23×Q1611yescharacteristic
554C22C2×#57711yescharacteristic
555C22C2×#57811yescharacteristic
556C22#358511yescharacteristic
557C4—28no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—41no—
562C4—44no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—81no—
570C4—82no—
571C4—84no—
572C4#336314yes—
573C4#339514yes—
574C4#347511yescharacteristic
575C4#350611yescharacteristic
576C4#351811yescharacteristic
577C4#354711yescharacteristic
578C2#5905011yescharacteristic
579C2C2×#336311yescharacteristic
580C2#6134811yescharacteristic
581C2C2×#339511yescharacteristic
582C2#6142111yescharacteristic
583C2#6169811yescharacteristic
584C2#6178911yescharacteristic
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
245. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#598692
#365632
#326022
#287172
#285112
#284012
#277492
#263222
#260922
#252782
#246822
#246322
#232472
#230632
#228082
C4×#4622