Finite Group Databasebeta

← #6644658

Subgroups — #6644658

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#33010C212yesmaximal
4#32432C214yesmaximal, complemented
5#32348C214yesmaximal, complemented
6#31640C214yesmaximal, complemented
7#31524C214yesmaximal, complemented
8#29642C212yesmaximal
9#29105C211yesmaximal, characteristic
10#28304C211yesmaximal, characteristic
11#28121C212yesmaximal
12#27961C211yesmaximal, characteristic
13#27910C211yesmaximal, characteristic
14#27648C212yesmaximal
15#21912C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#484C2212yes—
19C2×#484C2212yes—
20C2×#484C2212yes—
21C2×#451C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2646—21no—
26#2627—21no—
27#2590—22no—
28#2588—22no—
29#2567—22no—
30#2567—22no—
31#2567C2214yes—
32#2567C2214yes—
33#2511C2212yes—
34#2507C2212yes—
35#2451C2212yes—
36#2446C2212yes—
37C2×#395—21no—
38C2×#395—21no—
39#2329—21no—
40#2313—21no—
41C4×C22⋊Q8C2212yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#389—21no—
45C2×#389—21no—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2100C2214yes—
50#2095C2214yes—
51#2095C2212yes—
52#2083C2214yes—
53#2072C2214yes—
54#2072C2212yes—
55#2068C2214yes—
56#2057C2214yes—
57#2057C2214yes—
58#2056C2214yes—
59#2024C2212yes—
60#2024C2214yes—
61#2024C2214yes—
62#2016C2212yes—
63#1990C2212yes—
64#1969C2214yes—
65#1956C2212yes—
66#1947C2214yes—
67#1933C2214yes—
68#1928C2214yes—
69#1916C2214yes—
70#1916C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1878C2214yes—
74#1863C2214yes—
75#1863C2214yes—
76#1845C2212yes—
77#1834C2212yes—
78#1824C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1817C2212yes—
82#1803C2214yes—
83#1797C2212yes—
84#1787C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8—21no—
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—44no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8C2312yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—44no—
99C2×C4×Q8C2314yes—
100C2×C4×Q8C2312yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C42—21no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#484—44no—
117#484—44no—
118#484—44no—
119#451—44no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×C4⋊C8C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C4×C8C2312yes—
133C2×C4×C8C2312yes—
134#400—22no—
135#400—22no—
136#400—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#395—22no—
142#395—24no—
143#395—24no—
144#395—24no—
145#395—24no—
146#391—22no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391C2314yes—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#389—24no—
159#389—24no—
160#389—24no—
161#389—24no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#385—22no—
180#385—22no—
181#384—22no—
182#384—21no—
183#384—21no—
184#384C2314yes—
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#381—22no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380C2312yes—
194C4×C4⋊C4—22no—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8—44no—
225C22×Q8—82no—
226C22×Q8—82no—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×Q16—28no—
241C2×Q16—44no—
242C2×Q16—44no—
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—48no—
246C4⋊Q8—42no—
247C4⋊Q8—48no—
248C4⋊Q8—42no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—44no—
252C4×Q8—48no—
253C4×Q8—48no—
254C4×Q8—44no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4Q1618yescomplemented
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2412yes—
314C2×C42C2412yes—
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317#107—44no—
318#107—44no—
319#107—44no—
320C4⋊C8—44no—
321#103—28no—
322#103—44no—
323#103—44no—
324#103—44no—
325#103—44no—
326#103—44no—
327C22⋊C8—24no—
328C22⋊C8—24no—
329C4×C8—42no—
330C4×C8—42no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95C2×D412yes—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355C24C22×D411yescharacteristic
356C2×Q8—24no—
357C2×Q8—22no—
358C2×Q8—24no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—416no—
365C2×Q8—42no—
366C2×Q8—48no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—84no—
370C2×Q8—84no—
371C2×Q8—81no—
372C2×Q8—81no—
373C2×Q8—816no—
374C2×Q82−1+414yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4C2×Q1614yes—
412C22×C4C2×Q1614yes—
413C22×C4C2×Q1614yes—
414C22×C4#13714yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4H(5,𝔽2)14yes—
425C22×C4C2511yescharacteristic
426Q16—88no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—84no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—84no—
462C4⋊C4—82no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4C2×Q1618yes—
467C4⋊C4C2×Q1618yes—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C42—24no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—24no—
475C42—42no—
476C42—41no—
477C42—41no—
478C42—48no—
479C42—82no—
480C42—82no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23#53512yes—
486C23C22×Q1612yes—
487C23C2×#13712yes—
488C23C23×D411yescharacteristic
489Q8—28no—
490Q8—48no—
491Q8—44no—
492Q8—44no—
493Q8—44no—
494Q8—816no—
495Q8—162no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4#53512yes—
552C2×C4C22×Q1614yes—
553C2×C4C22×Q1612yes—
554C2×C4C22×Q1612yes—
555C2×C4C22×Q1614yes—
556C2×C4C2×#13712yes—
557C2×C4C2×#13712yes—
558C2×C4C2×H(5,𝔽2)12yes—
559C2×C4C2×2−1+412yes—
560C2×C4#58512yes—
561C8—44no—
562C8—44no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22#320914yes—
570C22C2×#53511yescharacteristic
571C22#348511yescharacteristic
572C22#348611yescharacteristic
573C22C23×Q1611yescharacteristic
574C22C22×#13711yescharacteristic
575C22#358511yescharacteristic
576C22#358511yescharacteristic
577C4—24no—
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—24no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—44no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4#353514yes—
595C2—24no—
596C2#3711811yescharacteristic
597C2C2×#320911yescharacteristic
598C2#6080611yescharacteristic
599C2#6169811yescharacteristic
600C2#6170911yescharacteristic
601C2#6182111yescharacteristic
602C2#6183011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32432C2semidirect
5. #32348C2semidirect
6. #31640C2semidirect
7. #31524C2semidirect
289. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#25672
#330102
#324322
#323482
#316402
#315242
#296422
#291052
#283042
#281212
#279612
#279102
#276482
#219122