Finite Group Databasebeta

← #6817407

Subgroups — #6817407

Order 512.

1776 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3231C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic
3#36134C211yesmaximal, characteristic
4#33225C211yesmaximal, characteristic, complemented
5#31119C218yesmaximal, complemented
6#29844C212yesmaximal, complemented
7#28748C211yesmaximal, characteristic, complemented
8#28687C211yesmaximal, characteristic, complemented
9#26461C214yesmaximal, complemented
10#26231C214yesmaximal
11#25904C212yesmaximal, complemented
12#25672C212yesmaximal, complemented
13C4×#500C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic
15C2×#533C2212yes—
16C2×#533C2212yes—
17#3231—216no—
18C2×#500C2211yescharacteristic
19C2×#498C2212yes—
20C2×#487C2212yes—
21C2×#487C2212yes—
22C2×#487C2212yes—
23C2×#485C2218yes—
24C2×#479C2218yes—
25C2×#431C2212yes—
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2857C2212yes—
30#2552—21no—
31#2552C2212yes—
32#2542—21no—
33#2542C2212yes—
34#2538—21no—
35#2538—21no—
36#2441—22no—
37#2441C2214yes—
38#2429—22no—
39#2429C2214yes—
40#2423—22no—
41#2423—22no—
42#2365—21no—
43#2365C2212yes—
44#2354—21no—
45#2354C2212yes—
46Q8×C4⋊C4—21no—
47Q8×C4⋊C4—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2084C2214yes—
52#2068C2218yes—
53#2043C2214yes—
54#2036C2218yes—
55#2035C2218yes—
56#2026C2218yes—
57#1990C2212yes—
58#1990C2211yescharacteristic
59#1990C2211yescharacteristic
60#1981C2214yes—
61#1956C2212yes—
62#1956C2214yes—
63#1956C2214yes—
64#1954C2212yes—
65#1948C2214yes—
66#1942C2214yes—
67#1939C2212yes—
68#1923C2214yes—
69#1873C2218yes—
70#1865C2218yes—
71#1845C2211yescharacteristic
72C4×#107C2214yes—
73C4×C4⋊C8C2211yescharacteristic
74C4×C22⋊C8C2212yes—
75C42×C8C2211yescharacteristic
76#533—28no—
77#533—28no—
78#533—44no—
79#533—44no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×#125—22no—
89C2×#125C2314yes—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8C2314yes—
95C2×C22⋊Q8C2314yes—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8C2314yes—
102C2×C4×D4D411yescharacteristic
103C2×C4×D4D411yescharacteristic
104C2×C4×D4C2311yescharacteristic
105C2×#117—22no—
106C2×#117C2314yes—
107C22×C42C2312yes—
108#500D418yescomplemented
109#498—28no—
110#487—28no—
111#487—28no—
112#487—44no—
113#485—232no—
114#479—232no—
115#431—28no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2318yes—
119C2×C4⋊C8C2314yes—
120C2×C4⋊C8C2312yes—
121C2×#103C2318yes—
122C2×#103C2318yes—
123C2×#103C2318yes—
124C2×#103C2318yes—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2312yes—
129C2×C4×C8C2312yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2311yescharacteristic
132#399—24no—
133#399—24no—
134#398—24no—
135#398—24no—
136#396—24no—
137#396—24no—
138#391—21no—
139#391—21no—
140#391—22no—
141#391C2314yes—
142#390—22no—
143#390—22no—
144#389—22no—
145#389—22no—
146#389—21no—
147#389—21no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—24no—
152#389—24no—
153#389C2312yes—
154#389C2312yes—
155#389C2314yes—
156#388—24no—
157#388—24no—
158#387—24no—
159#387—24no—
160#384—22no—
161#384—22no—
162#384—21no—
163#384—21no—
164#384—21no—
165#384—21no—
166#384—21no—
167#384—21no—
168#384C2311yescharacteristic
169#384C2311yescharacteristic
170#384C2312yes—
171#384C2312yes—
172#380—22no—
173#380—22no—
174#380—22no—
175#380C2314yes—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2311yescharacteristic
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4C2312yes—
188C43C2311yescharacteristic
189#340C2318yes—
190#340C2314yes—
191#340C2314yes—
192#336C2312yes—
193#336C2314yes—
194C22×Q8—21no—
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8C2414yes—
198C22×D4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2412yes—
202C22×C8C2412yes—
203C22×C8C2414yes—
204C22×C8C2412yes—
205C4⋊Q8—24no—
206C4⋊Q8—24no—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—28no—
212C4⋊Q8—44no—
213C4⋊Q8—44no—
214C4⋊Q8—82no—
215C4⋊Q8C2×D414yes—
216C4⋊Q8C2×D414yes—
217#125—28no—
218#125—44no—
219#125—48no—
220C22⋊Q8—28no—
221C22⋊Q8—44no—
222C22⋊Q8—48no—
223C22⋊Q8—48no—
224C22⋊Q8—48no—
225C22⋊Q8—48no—
226C22⋊Q8—84no—
227C4×Q8—28no—
228C4×Q8—44no—
229C4×Q8—48no—
230C4×Q8—48no—
231C4×Q8—48no—
232C4×Q8—44no—
233C4×Q8—44no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—24no—
237#117—28no—
238#117—44no—
239#117—48no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2412yes—
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296#107—44no—
297#107C2×D4116yes—
298#106—216no—
299C4⋊C8—28no—
300C4⋊C8—24no—
301#103—216no—
302#103—216no—
303#103—48no—
304#103—48no—
305C22⋊C8—28no—
306C22⋊C8—44no—
307C4×C8—24no—
308C4×C8—22no—
309C4×C8—24no—
310C4×C8—42no—
311C4×C8C2×D414yes—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95C2×D412yes—
321#95C2×D412yes—
322#95C2×D412yes—
323C24C22×D412yes—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—28no—
329C2×Q8—28no—
330C2×Q8—44no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C2511yescharacteristic
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8C22×D414yes—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4C22×D414yes—
410C4⋊C4C22×D414yes—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C42—24no—
419C42—24no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—48no—
426C42—48no—
427C42—44no—
428C42—44no—
429C42#13514yes—
430C42#13714yes—
431C42C22×D412yes—
432C42H(5,𝔽2)12yes—
433C422−1+414yes—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23C23×D411yescharacteristic
442Q8—48no—
443Q8—48no—
444Q8—48no—
445Q8—48no—
446Q8—88no—
447D4—44no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4#53612yes—
486C2×C4#53612yes—
487C2×C4D4212yes—
488C2×C4D4212yes—
489C2×C4C2×#13512yes—
490C2×C4C2×#13512yes—
491C2×C4C2×#13512yes—
492C2×C4C2×#13712yes—
493C2×C4C2×#13712yes—
494C2×C4C2×#13712yes—
495C2×C4#57712yes—
496C2×C4#57812yes—
497C2×C4C23×D411yescharacteristic
498C2×C4C2×H(5,𝔽2)11yescharacteristic
499C2×C4C2×2−1+412yes—
500C2×C4#58512yes—
501C8—24no—
502C8—44no—
503C8—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22C2×#53611yescharacteristic
512C22C2×D4211yescharacteristic
513C22#350011yescharacteristic
514C22C22×#13511yescharacteristic
515C22C22×#13711yescharacteristic
516C22C2×#57711yescharacteristic
517C22C2×#57811yescharacteristic
518C4—22no—
519C4—22no—
520C4—22no—
521C4—42no—
522C4—42no—
523C4—44no—
524C4—44no—
525C4—42no—
526C4—82no—
527C4—82no—
528C4—84no—
529C4—84no—
530C4#323114yescomplemented
531C4#327814yes—
532C4#328314yes—
533C4#349712yes—
534C4#355412yes—
535C2—82no—
536C2#3936311yescharacteristic
537C2C2×#323111yescharacteristic
538C2#6087811yescharacteristic
539C2C2×#327811yescharacteristic
540C2C2×#328311yescharacteristic
541C2#6253211yescharacteristic
542C2#6253611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3231C2semidirect
4. #33225C2semidirect
5. #31119C2semidirect
6. #29844C2semidirect
7. #28748C2semidirect
8. #28687C2semidirect
9. #26461C2semidirect
11. #25904C2semidirect
12. #25672C2semidirect
13. C4×#500C2semidirect
108. #500D4semidirect
530. C4#3231semidirect

Maximal subgroups

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

subgroupindex
C2×#32312
#361342
#332252
#311192
#298442
#287482
#286872
#264612
#262312
#259042
#256722
C4×#5002
#212642