Finite Group Databasebeta

← #6469385

Subgroups — #6469385

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#29394C214yesmaximal, complemented
4#29277C212yesmaximal, complemented
5#28206C212yesmaximal, complemented
6#27648C212yesmaximal
7#27614C212yesmaximal
8#24302C212yesmaximal
9#24064C212yesmaximal
10#23835C212yesmaximal, complemented
11#23816C212yesmaximal, complemented
12C4×#477C214yesmaximal, complemented
13C4×#484C214yesmaximal
14#20777C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8—21no—
18C2×#484C2214yes—
19C2×#477C2214yes—
20C4×C4⋊Q8—21no—
21C4×C4⋊Q8—21no—
22C4×C4⋊Q8—21no—
23C4×C4⋊Q8—21no—
24C4×C4⋊Q8C2212yes—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2212yes—
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2212yes—
29C4×C4⋊Q8C2212yes—
30C4×C4⋊Q8C2211yescharacteristic
31C4×C4⋊Q8C2211yescharacteristic
32C4×#125—21no—
33C4×#125—21no—
34C4×C22⋊Q8—21no—
35C4×C22⋊Q8—21no—
36C4×C22⋊Q8C2212yes—
37C4×C22⋊Q8C2212yes—
38#2296—21no—
39#2295—21no—
40#2295C2212yes—
41C2×#391—21no—
42C2×#389—21no—
43C2×#389C2212yes—
44#2285C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C42×Q8—21no—
47C42×Q8C2212yes—
48C4×#117—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C43C2211yescharacteristic
52#2100C2214yes—
53#2083C2214yes—
54#2083C2214yes—
55#2072C2214yes—
56#2057C2214yes—
57#2057C2214yes—
58#2043C2214yes—
59#2024C2214yes—
60#1987C2214yes—
61#1947C2214yes—
62#1928C2218yes—
63#1921C2218yes—
64#1921C2218yes—
65#1872C2214yes—
66#1863C2218yes—
67#1852C2214yes—
68#1839C2212yes—
69#1831C2212yes—
70#1824C2214yes—
71#1817C2212yes—
72#1803C2214yes—
73#1797C2212yes—
74C4×#107C2218yes—
75C4×C4⋊C8C2212yes—
76C4×#103C2218yes—
77C4×C22⋊C8C2212yes—
78#1744C2212yes—
79C23×Q8—21no—
80#533—28no—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#125—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8C2312yes—
91C2×C4×Q8—21no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8—48no—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×#117—21no—
103C2×#117—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C42—22no—
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#484—216no—
119#477—216no—
120C2×#107C2318yes—
121C2×#107C2318yes—
122C2×C4⋊C8C2314yes—
123C2×#103C2318yes—
124C2×#103C2318yes—
125C2×C22⋊C8C2314yes—
126C2×C4×C8C2314yes—
127#391—22no—
128#391—22no—
129#391—22no—
130#391—22no—
131#391—21no—
132#391—21no—
133#391C2312yes—
134#391C2312yes—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—21no—
146#389—21no—
147#389C2314yes—
148#389C2314yes—
149#389C2312yes—
150#389C2312yes—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#385—22no—
156#385—22no—
157#384—21no—
158#384—21no—
159#384C2312yes—
160#384C2312yes—
161#384C2311yescharacteristic
162#384C2311yescharacteristic
163#382—21no—
164#380—21no—
165#380C2312yes—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192#376—22no—
193C2×#95—22no—
194C2×#95—22no—
195C43—22no—
196C43D411yescharacteristic
197C43D411yescharacteristic
198C43D411yescharacteristic
199C43D411yescharacteristic
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203C43C2311yescharacteristic
204#340C2318yes—
205#340C2318yes—
206#340C2318yes—
207#340C2318yes—
208#336C2314yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212C22×Q8—28no—
213C22×Q8—21no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8C2412yes—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2411yescharacteristic
228C22×C8C2418yes—
229C4⋊Q8—28no—
230C4⋊Q8—28no—
231C4⋊Q8—24no—
232C4⋊Q8—28no—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8#4118yes—
236#125—28no—
237C22⋊Q8—28no—
238C22⋊Q8—24no—
239C22⋊Q8#4118yes—
240C4×Q8—28no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—28no—
244C4×Q8—28no—
245C4×Q8—44no—
246C4×Q8—48no—
247C4×Q8—48no—
248C4×Q8—48no—
249C4×Q8—44no—
250C4×Q8—48no—
251#117—28no—
252#117—44no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—48no—
304C2×C42Q1618yes—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—216no—
334#107—216no—
335C4⋊C8—28no—
336#103—216no—
337#103—216no—
338C22⋊C8—28no—
339C4×C8—28no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354C24C22×D411yescharacteristic
355C2×Q8—24no—
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—44no—
362C2×Q8—42no—
363C2×Q8—432no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—42no—
368C2×Q8C2×#4114yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4C2×Q1614yes—
399C22×C4C2×Q1614yes—
400C22×C4C2×Q1614yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2×#4114yes—
415C22×C4C2511yescharacteristic
416C2×C8—28no—
417C2×C8—28no—
418C2×C8—28no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—28no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4C2×#4118yes—
448C4⋊C4C2×#4114yes—
449C22⋊C4—28no—
450C22⋊C4—44no—
451C42—28no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—48no—
474C42—44no—
475C42—44no—
476C42—44no—
477C42—48no—
478C42—44no—
479C42—44no—
480C42C2×Q1614yes—
481C42C2×Q1614yes—
482C42C2×Q1614yes—
483C42C2×Q1614yes—
484C42AGL(1,ℤ/8)14yes—
485C42C2×#4114yes—
486C422−1+414yes—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23#52512yes—
492C23C22×Q1612yes—
493C23#57512yes—
494C23C23×D411yescharacteristic
495Q8—28no—
496Q8—48no—
497Q8—416no—
498Q8—416no—
499Q8—44no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—416no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
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—88no—
551C2×C4#52512yes—
552C2×C4#52512yes—
553C2×C4#52512yes—
554C2×C4#53612yes—
555C2×C4#53612yes—
556C2×C4C22×Q1612yes—
557C2×C4C22×Q1612yes—
558C2×C4C22×Q1612yes—
559C2×C4C22×Q1612yes—
560C2×C4C22×Q1612yes—
561C2×C4C22×Q1612yes—
562C2×C4C2×AGL(1,ℤ/8)12yes—
563C2×C4C2×AGL(1,ℤ/8)12yes—
564C2×C4C2×AGL(1,ℤ/8)12yes—
565C2×C4#57512yes—
566C2×C4C22×#4112yes—
567C2×C4C22×#4112yes—
568C2×C4C2×2−1+412yes—
569C8—48no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22#310414yes—
577C22C2×#52511yescharacteristic
578C22C2×#52511yescharacteristic
579C22C2×#53611yescharacteristic
580C22C23×Q1611yescharacteristic
581C22C22×AGL(1,ℤ/8)11yescharacteristic
582C22C2×#57511yescharacteristic
583C22C2×#57511yescharacteristic
584C4—22no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—48no—
594C4—41no—
595C4—44no—
596C4—44no—
597C4—41no—
598C4—82no—
599C4—84no—
600C4—82no—
601C4#310414yes—
602C4#323014yes—
603C4#348214yes—
604C2—24no—
605C2#3497111yescharacteristic
606C2C2×#310411yescharacteristic
607C2C2×#310411yescharacteristic
608C2#6055811yescharacteristic
609C2#6056111yescharacteristic
610C2C2×#323011yescharacteristic
611C2#6085411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29394C2semidirect
4. #29277C2semidirect
5. #28206C2semidirect
10. #23835C2semidirect
11. #23816C2semidirect
12. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C2×C4×C4⋊Q82
#293942
#292772
#282062
#276482
#276142
#243022
#240642
#238352
#238162
C4×#4772
C4×#4842
#207772