Finite Group Databasebeta

← #7563378

Subgroups — #7563378

Order 512.

1791 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Q16C212yesmaximal
2#59832C211yesmaximal, characteristic
3Q8×C4⋊Q8C211yesmaximal, characteristic
4#30957C214yesmaximal
5#28813C212yesmaximal
6#28689C211yesmaximal, characteristic
7#28511C211yesmaximal, characteristic
8#26466C212yesmaximal
9#26381C214yesmaximal
10#26046C212yesmaximal
11#25841C212yesmaximal
12#25471C212yesmaximal
13#24333C212yesmaximal
14#23149C211yesmaximal, characteristic
15#22998C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×Q82C2212yes—
20Q8×Q16—216no—
21C2×#500C2212yes—
22C2×#500C2214yes—
23C2×#477C2214yes—
24C2×#477C2212yes—
25C2×#477C2214yes—
26C2×#477C2212yes—
27C2×#477C2211yescharacteristic
28C2×#462C2211yescharacteristic
29C2×C8×Q8C2212yes—
30C2×C4×Q16C2212yes—
31C2×C4×Q16C2214yes—
32#2868—21no—
33#2868C2211yescharacteristic
34#2734—22no—
35#2582C2212yes—
36#2567—22no—
37#2567C2212yes—
38#2551—22no—
39#2551—22no—
40#2551—21no—
41#2551C2212yes—
42#2551C2212yes—
43#2551C2212yes—
44#2546—22no—
45#2451—22no—
46#2451C2214yes—
47#2365—21no—
48#2365C2212yes—
49Q8×C4⋊C4C2212yes—
50Q8×C4⋊C4C2212yes—
51C4×C4⋊Q8—21no—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C42×Q8C2211yescharacteristic
56#2095C2214yes—
57#2087C2214yes—
58#2083C2214yes—
59#2068C2214yes—
60#2057C2214yes—
61#2057C2214yes—
62#2057C2214yes—
63#2024C2214yes—
64#1990C2212yes—
65#1987C2211yescharacteristic
66#1984C2212yes—
67#1981C2211yescharacteristic
68#1956C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1947C2211yescharacteristic
72#1947C2211yescharacteristic
73#1942C2214yes—
74#1939C2212yes—
75#1933C2214yes—
76#1928C2212yes—
77#1921C2212yes—
78#1916C2214yes—
79#1878C2214yes—
80#1878C2212yes—
81#1863C2214yes—
82#1863C2212yes—
83#1863C2214yes—
84#1863C2214yes—
85#1845C2212yes—
86#1803C2211yescharacteristic
87#1803C2211yescharacteristic
88C4×#107C2212yes—
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#103C2212yes—
94C22×Q16C2312yes—
95Q82—28no—
96Q82—28no—
97Q82—44no—
98Q82—44no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8D411yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2314yes—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124#500—28no—
125#500—216no—
126#477—216no—
127#477—28no—
128#477—216no—
129#477—44no—
130#477D418yes—
131#462Q818yescomplemented
132C8×Q8—28no—
133C4×Q16—28no—
134C4×Q16—216no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157C2×C4×C8C2312yes—
158#398—24no—
159#398—24no—
160#395—24no—
161#395—24no—
162#395—24no—
163#395—24no—
164#395—24no—
165#391—22no—
166#391—22no—
167#391—21no—
168#391—24no—
169#391D412yes—
170#391C2312yes—
171#391C2314yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—24no—
181#389—24no—
182#389D412yes—
183#389D412yes—
184#389C2312yes—
185#389C2314yes—
186#389C2314yes—
187#389C2314yes—
188#387—24no—
189#387—24no—
190#384—22no—
191#384—22no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384D411yescharacteristic
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C43—21no—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#336C2314yes—
226#336C2314yes—
227#336C2312yes—
228#336C2312yes—
229C22×Q8—21no—
230C22×Q8—24no—
231C22×Q8—22no—
232C22×Q8—24no—
233C22×Q8C2×D412yes—
234C22×Q8C2412yes—
235C22×Q8C2411yescharacteristic
236C2×Q16—24no—
237C2×Q16—24no—
238C2×Q16C2×Q818yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C4⋊Q8—28no—
244C4⋊Q8—28no—
245C4⋊Q8—24no—
246C4⋊Q8—22no—
247C4⋊Q8—24no—
248C4⋊Q8—22no—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251C4⋊Q8—48no—
252C4⋊Q8—416no—
253C4⋊Q8—48no—
254C4⋊Q8—416no—
255C4⋊Q8—82no—
256C4⋊Q8—84no—
257C4⋊Q8—84no—
258C4⋊Q8—82no—
259C4⋊Q8—84no—
260C4⋊Q8Q1618yescomplemented
261C4⋊Q8C2×D414yes—
262C4⋊Q8C2×Q814yes—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—24no—
266C4×Q8—22no—
267C4×Q8—24no—
268C4×Q8—22no—
269C4×Q8—28no—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—48no—
276C4×Q8—48no—
277C4×Q8—82no—
278C4×Q8C2×D414yes—
279C4×Q8C2×Q814yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2414yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—28no—
336#107—24no—
337#107—28no—
338#107—28no—
339#107—24no—
340#107—44no—
341#107C2×D418yes—
342C4⋊C8—28no—
343C4⋊C8—22no—
344C4⋊C8—22no—
345C4⋊C8C2×D414yes—
346C4⋊C8C2×Q814yes—
347#103—24no—
348#103—24no—
349#103—28no—
350#103—24no—
351#103—24no—
352#103—44no—
353#103C2×D418yes—
354#103C2×Q818yes—
355C4×C8—28no—
356C4×C8—24no—
357C4×C8—24no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95C2×D414yes—
371#95C2×D414yes—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—44no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×Q8—84no—
391C2×Q8—84no—
392C2×Q8—84no—
393C2×Q8—84no—
394C2×Q8C2×Q1618yes—
395C2×Q8C22×D412yes—
396C2×Q8C22×Q814yes—
397C2×Q8C22×Q812yes—
398C2×Q82−1+412yes—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D414yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D414yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2511yescharacteristic
417Q16—216no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8C22×D414yes—
429C2×C8C22×Q814yes—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—84no—
465C4⋊C4—82no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4C2×Q1618yes—
469C4⋊C4C2×Q1618yes—
470C4⋊C4C22×D412yes—
471C4⋊C4C22×D414yes—
472C4⋊C4C22×Q812yes—
473C4⋊C42−1+412yes—
474C42—24no—
475C42—24no—
476C42—24no—
477C42—22no—
478C42—22no—
479C42—24no—
480C42—24no—
481C42—21no—
482C42—21no—
483C42—41no—
484C42—48no—
485C42—48no—
486C42—42no—
487C42—44no—
488C42—44no—
489C42—42no—
490C42—42no—
491C42C2×Q1614yes—
492C42C22×D412yes—
493C42C22×Q812yes—
494C42H(5,𝔽2)12yes—
495C422−1+412yes—
496C23C23×D411yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—216no—
500Q8—22no—
501Q8—24no—
502Q8—24no—
503Q8—42no—
504Q8—42no—
505Q8—84no—
506Q8—164no—
507Q8—168no—
508Q8—168no—
509Q8D4×Q814yes—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4D4212yes—
552C2×C4D4212yes—
553C2×C4D4×Q812yes—
554C2×C4D4×Q812yes—
555C2×C4C22×Q1612yes—
556C2×C4C22×Q1618yes—
557C2×C4C22×Q1612yes—
558C2×C4C22×Q1612yes—
559C2×C4#57612yes—
560C2×C4#57812yes—
561C2×C4C23×D411yescharacteristic
562C2×C4C23×Q811yescharacteristic
563C2×C4C2×H(5,𝔽2)11yescharacteristic
564C2×C4C2×2−1+411yescharacteristic
565C2×C4C2×2−1+411yescharacteristic
566C2×C4C2×2−1+411yescharacteristic
567C8—24no—
568C8—24no—
569C8—42no—
570C8—42no—
571C22C2×D4211yescharacteristic
572C22C2×D4×Q811yescharacteristic
573C22#348611yescharacteristic
574C22C23×Q1611yescharacteristic
575C22C2×#57611yescharacteristic
576C22C2×#57811yescharacteristic
577C22#358511yescharacteristic
578C4—28no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—24no—
583C4—44no—
584C4—44no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—42no—
589C4—44no—
590C4—84no—
591C4—82no—
592C4—82no—
593C4D4×Q1614yes—
594C4Q8×Q1614yescomplemented
595C4C2×D4×Q811yescharacteristic
596C4#349911yescharacteristic
597C4#350311yescharacteristic
598C4#355211yescharacteristic
599C2#5912711yescharacteristic
600C2C2×D4×Q1611yescharacteristic
601C2#6109311yescharacteristic
602C2C2×Q8×Q1611yescharacteristic
603C2#6138411yescharacteristic
604C2#6182111yescharacteristic
605C2#6187011yescharacteristic
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
131. #462Q8semidirect
260. C4⋊Q8Q16semidirect
594. C4Q8×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Q162
#598322
Q8×C4⋊Q82
#309572
#288132
#286892
#285112
#264662
#263812
#260462
#258412
#254712
#243332
#231492
#229982
C4×#4772
C4×#4622