Finite Group Databasebeta

← #7567261

Subgroups — #7567261

Order 512.

1433 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59875C211yesmaximal, characteristic
2Q8×C4⋊Q8C211yesmaximal, characteristic
3#32244C214yesmaximal
4#29218C211yesmaximal, characteristic
5#28717C211yesmaximal, characteristic
6#28647C212yesmaximal
7#27823C212yesmaximal
8#27570C214yesmaximal
9#26279C212yesmaximal
10#26022C212yesmaximal
11#25936C212yesmaximal
12#24818C212yesmaximal
13#23153C211yesmaximal, characteristic
14#22763C212yesmaximal
15#22525C212yesmaximal
16C4×#475C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#487C2212yes—
20C2×#475C2211yescharacteristic
21C2×#462C2211yescharacteristic
22C2×#446C2212yes—
23C2×C4×Q16C2212yes—
24#2868—21no—
25#2868C2211yescharacteristic
26#2810—22no—
27#2568—21no—
28#2568C2212yes—
29#2568C2212yes—
30#2567—22no—
31#2567C2212yes—
32#2551—22no—
33#2551—22no—
34#2551C2212yes—
35#2551C2212yes—
36#2514—22no—
37#2514—22no—
38#2514C2214yes—
39#2355—21no—
40#2355C2212yes—
41#2355C2212yes—
42Q8×C4⋊C4C2212yes—
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2101C2214yes—
50#2100C2214yes—
51#2085C2214yes—
52#2072C2214yes—
53#2066C2214yes—
54#2066C2214yes—
55#2024C2214yes—
56#2024C2214yes—
57#1993C2214yes—
58#1990C2212yes—
59#1990C2212yes—
60#1988C2211yescharacteristic
61#1988C2214yes—
62#1987C2212yes—
63#1987C2212yes—
64#1984C2212yes—
65#1981C2211yescharacteristic
66#1948C2211yescharacteristic
67#1947C2211yescharacteristic
68#1939C2212yes—
69#1935C2214yes—
70#1935C2214yes—
71#1928C2212yes—
72#1919C2212yes—
73#1919C2214yes—
74#1918C2214yes—
75#1917C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1878C2212yes—
79#1865C2212yes—
80#1834C2212yes—
81#1805C2214yes—
82#1803C2212yes—
83#1803C2212yes—
84#1803C2211yescharacteristic
85#1803C2211yescharacteristic
86C4×#106C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93C22×Q16C2312yes—
94Q82—28no—
95Q82—44no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8D411yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115#487—44no—
116#475—24no—
117#462—24no—
118#446—44no—
119C4×Q16—28no—
120C2×#107C2312yes—
121C2×#106C2312yes—
122C2×#106C2312yes—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2314yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139#401—24no—
140#400—24no—
141#400—24no—
142#400—24no—
143#395—24no—
144#395—24no—
145#395—24no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—21no—
152#391D412yes—
153#391C2312yes—
154#391C2314yes—
155#391C2314yes—
156#389—22no—
157#389—22no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—24no—
162#389—24no—
163#389D412yes—
164#389D412yes—
165#389C2312yes—
166#389C2314yes—
167#389C2314yes—
168#387—24no—
169#387—24no—
170#387—24no—
171#384—22no—
172#384—22no—
173#384—22no—
174#384—21no—
175#384—21no—
176#384D411yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2312yes—
181#383—24no—
182#382—22no—
183#382—22no—
184#382C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C43—21no—
203C43C2311yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2312yes—
207#340C2312yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2312yes—
217#336C2314yes—
218#336C2312yes—
219#336C2314yes—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8—22no—
223C22×Q8C2×D412yes—
224C22×Q8C2412yes—
225C22×Q8C2411yescharacteristic
226C2×Q16—24no—
227C2×Q16—24no—
228C2×Q16—24no—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232C22×C8C2412yes—
233C4⋊Q8—28no—
234C4⋊Q8—24no—
235C4⋊Q8—24no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—24no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—44no—
242C4⋊Q8—48no—
243C4⋊Q8—416no—
244C4⋊Q8—82no—
245C4⋊Q8—84no—
246C4⋊Q8—82no—
247C4⋊Q8—84no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—24no—
251C4×Q8—22no—
252C4×Q8—24no—
253C4×Q8—22no—
254C4×Q8—28no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—44no—
258C4×Q8—48no—
259C4×Q8—44no—
260C4×Q8#4114yes—
261C4×Q8#4114yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—24no—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2414yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314#107—24no—
315#106—24no—
316#106—24no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—44no—
322#103—24no—
323#103—24no—
324#103—24no—
325#103—24no—
326#103—24no—
327#103—24no—
328#103—44no—
329#103—44no—
330#97—44no—
331C4×C8—24no—
332C4×C8—24no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—24no—
337#95—24no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—22no—
345#95—24no—
346#95—24no—
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—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—48no—
364C2×Q8—44no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×Q8—84no—
371C2×Q8—84no—
372C2×Q8C2×#4112yes—
373C2×Q8C2×#4112yes—
374C2×Q82−1+412yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4C22×D414yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D414yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2511yescharacteristic
393Q16—48no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—82no—
441C4⋊C4—84no—
442C4⋊C4—82no—
443C4⋊C4—84no—
444C4⋊C4C2×#4112yes—
445C4⋊C4C2×#4112yes—
446C4⋊C42−1+412yes—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—24no—
453C42—24no—
454C42—21no—
455C42—21no—
456C42—41no—
457C42—48no—
458C42—42no—
459C42—44no—
460C42—44no—
461C42—42no—
462C42—42no—
463C42—84no—
464C42#13714yes—
465C42C2×#4112yes—
466C42C2×#4112yes—
467C42H(5,𝔽2)12yes—
468C422−1+412yes—
469C23C23×D411yescharacteristic
470Q8—22no—
471Q8—22no—
472Q8—22no—
473Q8—48no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—84no—
479Q8—164no—
480Q8—168no—
481Q8#55614yes—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4#54612yes—
525C2×C4#54612yes—
526C2×C4#54812yes—
527C2×C4#54812yes—
528C2×C4C2×#13712yes—
529C2×C4C2×#13712yes—
530C2×C4C2×#13712yes—
531C2×C4#57712yes—
532C2×C4#57812yes—
533C2×C4C22×#4111yescharacteristic
534C2×C4C22×#4111yescharacteristic
535C2×C4C2×H(5,𝔽2)11yescharacteristic
536C2×C4C2×2−1+411yescharacteristic
537C2×C4C2×2−1+411yescharacteristic
538C2×C4C2×2−1+411yescharacteristic
539C8—42no—
540C8—42no—
541C8—42no—
542C8—42no—
543C22C2×#54611yescharacteristic
544C22C2×#54811yescharacteristic
545C22#348611yescharacteristic
546C22C22×#13711yescharacteristic
547C22C2×#57711yescharacteristic
548C22C2×#57811yescharacteristic
549C22#358511yescharacteristic
550C4—22no—
551C4—22no—
552C4—22no—
553C4—24no—
554C4—44no—
555C4—42no—
556C4—42no—
557C4—41no—
558C4—42no—
559C4—44no—
560C4—84no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4#332814yes—
566C4#336914yes—
567C4C2×#55611yescharacteristic
568C4#350911yescharacteristic
569C4#351111yescharacteristic
570C4#355211yescharacteristic
571C2#5913411yescharacteristic
572C2C2×#332811yescharacteristic
573C2#6120611yescharacteristic
574C2C2×#336911yescharacteristic
575C2#6134811yescharacteristic
576C2#6183011yescharacteristic
577C2#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

Maximal subgroups

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

subgroupindex
#598752
Q8×C4⋊Q82
#322442
#292182
#287172
#286472
#278232
#275702
#262792
#260222
#259362
#248182
#231532
#227632
#225252
C4×#4752
C4×#4622