Finite Group Databasebeta

← #7569932

Subgroups — #7569932

Order 512.

1543 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59828C211yesmaximal, characteristic
2Q8×C4⋊Q8C211yesmaximal, characteristic
3#32134C214yesmaximal
4#29218C211yesmaximal, characteristic
5#28791C212yesmaximal
6#28687C211yesmaximal, characteristic
7#28677C212yesmaximal
8#27298C214yesmaximal
9#26444C212yesmaximal
10#26280C212yesmaximal
11#26108C212yesmaximal
12#25837C212yesmaximal
13#23149C211yesmaximal, characteristic
14#22964C212yesmaximal
15Q8×#103C212yesmaximal
16C4×#475C211yesmaximal, characteristic
17C4×#475C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#557—21no—
20C2×#498C2212yes—
21C2×#475C2211yescharacteristic
22C2×#475C2211yescharacteristic
23C2×C8×Q8C2212yes—
24#2868—21no—
25#2868C2211yescharacteristic
26#2726—22no—
27#2582—21no—
28#2567—22no—
29#2567C2212yes—
30#2552C2212yes—
31#2551—22no—
32#2551—22no—
33#2551C2212yes—
34#2551C2212yes—
35#2546C2212yes—
36#2506C2214yes—
37#2453—22no—
38#2446—22no—
39#2365C2212yes—
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C42×Q8C2211yescharacteristic
48#2100C2214yes—
49#2099C2214yes—
50#2088C2214yes—
51#2088C2214yes—
52#2084C2214yes—
53#2084C2214yes—
54#2058C2214yes—
55#2058C2214yes—
56#1990C2212yes—
57#1990C2212yes—
58#1990C2214yes—
59#1988C2211yescharacteristic
60#1988C2211yescharacteristic
61#1987C2212yes—
62#1987C2212yes—
63#1987C2214yes—
64#1984C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1948C2211yescharacteristic
68#1948C2211yescharacteristic
69#1941C2214yes—
70#1941C2214yes—
71#1939C2212yes—
72#1920C2214yes—
73#1920C2214yes—
74#1919C2212yes—
75#1919C2212yes—
76#1916C2214yes—
77#1865C2212yes—
78#1865C2212yes—
79#1865C2214yes—
80#1865C2214yes—
81#1834C2212yes—
82#1803C2211yescharacteristic
83#1803C2211yescharacteristic
84#1803C2214yes—
85C4×#106C2212yes—
86C4×#106C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#103C2212yes—
92C4×#103C2214yes—
93Q82—28no—
94Q82—44no—
95#557—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8D411yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125—22no—
109C2×#125—24no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120#498—44no—
121#475—24no—
122#475Q818yescomplemented
123C8×Q8—44no—
124C2×#106C2312yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×#106C2312yes—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2314yes—
142#400—24no—
143#398—24no—
144#398—24no—
145#398—24no—
146#395—24no—
147#395—24no—
148#395—24no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391D412yes—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—24no—
160#389—21no—
161#389—21no—
162#389—24no—
163#389—24no—
164#389D412yes—
165#389D412yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#389C2314yes—
170#389C2314yes—
171#389C2314yes—
172#387—24no—
173#387—24no—
174#384—22no—
175#384—22no—
176#384—22no—
177#384—21no—
178#384—21no—
179#384D411yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2314yes—
183#384C2312yes—
184#384C2312yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C43—21no—
201C43C2311yescharacteristic
202#340C2312yes—
203#340C2312yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2312yes—
208#340C2314yes—
209#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2314yes—
215#340C2312yes—
216#336C2314yes—
217#336C2314yes—
218#336C2312yes—
219#336C2312yes—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—24no—
223C22×Q8—22no—
224C22×Q8C2×D412yes—
225C22×Q8C2411yescharacteristic
226C22×C8C2412yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C4⋊Q8—28no—
231C4⋊Q8—24no—
232C4⋊Q8—22no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—44no—
237C4⋊Q8—48no—
238C4⋊Q8—416no—
239C4⋊Q8—48no—
240C4⋊Q8—82no—
241C4⋊Q8—84no—
242C4⋊Q8—82no—
243C4⋊Q8—84no—
244C4⋊Q8SD1618yes—
245C4⋊Q8C2×Q814yes—
246#125—416no—
247#125—48no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—24no—
251C4×Q8—22no—
252C4×Q8—24no—
253C4×Q8—22no—
254C4×Q8—28no—
255C4×Q8—44no—
256C4×Q8—48no—
257C4×Q8—48no—
258C4×Q8—48no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8C2×Q814yes—
262C4×Q8#4114yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2414yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319#106—24no—
320#106—24no—
321#106—24no—
322#106—44no—
323#106—44no—
324#106C2×Q818yes—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—44no—
329C4⋊C8C2×Q814yes—
330#103—24no—
331#103—24no—
332#103—24no—
333#103C2×Q818yes—
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—22no—
345#95—24no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95C2×D414yes—
350#95C2×D414yes—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—48no—
358C2×Q8—42no—
359C2×Q8—44no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×Q8—84no—
367C2×Q8—84no—
368C2×Q8C2×SD1618yes—
369C2×Q8C22×Q812yes—
370C2×Q8C2×#4112yes—
371C2×Q82−1+412yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D414yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D414yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C2511yescharacteristic
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8C22×Q814yes—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—28no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—84no—
438C4⋊C4—82no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C4⋊C4—84no—
442C4⋊C4C2×SD1618yes—
443C4⋊C4C2×SD1618yes—
444C4⋊C4C22×Q814yes—
445C4⋊C4C22×Q812yes—
446C4⋊C4C2×#4112yes—
447C4⋊C42−1+412yes—
448C42—24no—
449C42—24no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—24no—
454C42—24no—
455C42—21no—
456C42—21no—
457C42—41no—
458C42—48no—
459C42—42no—
460C42—48no—
461C42—44no—
462C42—44no—
463C42—42no—
464C42—42no—
465C42C2×SD1614yes—
466C42C22×Q812yes—
467C42C2×#4112yes—
468C42H(5,𝔽2)12yes—
469C422−1+412yes—
470C23C23×D411yescharacteristic
471Q8—22no—
472Q8—22no—
473Q8—22no—
474Q8—216no—
475Q8—48no—
476Q8—84no—
477Q8—164no—
478Q8—168no—
479Q8#55714yescomplemented
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4#54612yes—
522C2×C4#54612yes—
523C2×C4D4×Q812yes—
524C2×C4D4×Q812yes—
525C2×C4C22×SD1618yes—
526C2×C4C22×SD1612yes—
527C2×C4C22×SD1612yes—
528C2×C4C22×SD1612yes—
529C2×C4#57712yes—
530C2×C4#57812yes—
531C2×C4C23×Q811yescharacteristic
532C2×C4C22×#4111yescharacteristic
533C2×C4C2×H(5,𝔽2)11yescharacteristic
534C2×C4C2×2−1+411yescharacteristic
535C2×C4C2×2−1+411yescharacteristic
536C2×C4C2×2−1+411yescharacteristic
537C8—24no—
538C8—42no—
539C8—42no—
540C8—42no—
541C22C2×#54611yescharacteristic
542C22C2×D4×Q811yescharacteristic
543C22#348611yescharacteristic
544C22C23×SD1611yescharacteristic
545C22C2×#57711yescharacteristic
546C22C2×#57811yescharacteristic
547C22#358511yescharacteristic
548C4—24no—
549C4—22no—
550C4—22no—
551C4—28no—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—44no—
556C4—41no—
557C4—42no—
558C4—42no—
559C4—84no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4#329814yes—
564C4Q8×SD1614yes—
565C4C2×#55711yescharacteristic
566C4#350311yescharacteristic
567C4#350911yescharacteristic
568C4#355211yescharacteristic
569C2#5914411yescharacteristic
570C2C2×#329811yescharacteristic
571C2#6120611yescharacteristic
572C2C2×Q8×SD1611yescharacteristic
573C2#6138411yescharacteristic
574C2#6182011yescharacteristic
575C2#6186911yescharacteristic
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
122. #475Q8semidirect
479. Q8#557semidirect

Maximal subgroups

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

subgroupindex
#598282
Q8×C4⋊Q82
#321342
#292182
#287912
#286872
#286772
#272982
#264442
#262802
#261082
#258372
#231492
#229642
Q8×#1032
C4×#4752