Finite Group Databasebeta

← #6499427

Subgroups — #6499427

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#34625C211yesmaximal, characteristic, complemented
3#33283C214yesmaximal, complemented
4#30211C214yesmaximal, complemented
5#29356C212yesmaximal, complemented
6#29355C212yesmaximal, complemented
7#29349C212yesmaximal, complemented
8#29346C212yesmaximal, complemented
9#29120C214yesmaximal, complemented
10#29119C214yesmaximal, complemented
11#21841C211yesmaximal, characteristic, complemented
12#21833C211yesmaximal, characteristic, complemented
13#21809C211yesmaximal, characteristic
14#21807C211yesmaximal, characteristic
15#20470C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2862C2212yes—
21#2858C2212yes—
22#2667—21no—
23#2666—22no—
24#2665—21no—
25#2660C2212yes—
26#2647C2212yes—
27#2640C2212yes—
28#2640C2212yes—
29#2629C2212yes—
30#2628C2212yes—
31#2563—22no—
32#2547—21no—
33#2547—21no—
34#2504—21no—
35#2504—21no—
36#2436—22no—
37#2402—21no—
38#2399—21no—
39#2396—22no—
40#2311C2214yes—
41C4×C4⋊Q8C2212yes—
42C4×#125C2212yes—
43#2285C2211yescharacteristic
44#2283C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2074C2214yes—
47#2073C2214yes—
48#2072C2214yes—
49#2071C2214yes—
50#2026C2214yes—
51#2025C2214yes—
52#2024C2214yes—
53#2023C2214yes—
54#1993C2214yes—
55#1990C2214yes—
56#1988C2212yes—
57#1988C2212yes—
58#1988C2212yes—
59#1988C2212yes—
60#1982C2211yescharacteristic
61#1982C2211yescharacteristic
62#1982C2212yes—
63#1981C2211yescharacteristic
64#1981C2211yescharacteristic
65#1981C2212yes—
66#1959C2214yes—
67#1958C2214yes—
68#1936C2214yes—
69#1936C2214yes—
70#1935C2214yes—
71#1935C2214yes—
72#1934C2214yes—
73#1934C2214yes—
74#1933C2214yes—
75#1933C2214yes—
76#1824C2214yes—
77#1823C2212yes—
78#1823C2212yes—
79#1788C2214yes—
80#1788C2212yes—
81#1788C2212yes—
82#1767C2212yes—
83C4×C4⋊C8C2212yes—
84#1750C2214yes—
85#1746C2212yes—
86#1726C2212yes—
87C22×#41C2312yes—
88C23×Q8—21no—
89#532—24no—
90#532—24no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2312yes—
99C2×#127C2311yescharacteristic
100C2×#125C2312yes—
101C2×#124—22no—
102C2×#124—22no—
103C2×#124C2312yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×C4×Q8—22no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×#117—21no—
110C2×#117C2312yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2312yes—
113C22×C42C2311yescharacteristic
114#432—24no—
115#431—24no—
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2312yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129C2×#97C2312yes—
130C2×#97C2314yes—
131C2×C4×C8C2312yes—
132C2×C4×C8C2314yes—
133#400—22no—
134#400—22no—
135#400—22no—
136#399—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#398—22no—
142#398—22no—
143#397—22no—
144#397—22no—
145#397—22no—
146#397—22no—
147#396—22no—
148#396—22no—
149#395—22no—
150#395—22no—
151#395—22no—
152#393—22no—
153#393—24no—
154#393—24no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2314yes—
159#391C2314yes—
160#390—22no—
161#390—22no—
162#390C2314yes—
163#390C2314yes—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#387—22no—
170#387—22no—
171#386—24no—
172#386—24no—
173#384C2312yes—
174#384C2312yes—
175#383—21no—
176#382—22no—
177#382—22no—
178#382—21no—
179#382C2312yes—
180#382C2312yes—
181#381—21no—
182#380—21no—
183#380—24no—
184#380—24no—
185#380C2311yescharacteristic
186#380C2311yescharacteristic
187#380C2312yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4D411yescharacteristic
194C4×C4⋊C4D411yescharacteristic
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2311yescharacteristic
198C4×C22⋊C4—21no—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D411yescharacteristic
201C4×C22⋊C4D411yescharacteristic
202#376—21no—
203#376C2311yescharacteristic
204#376C2311yescharacteristic
205C43C2312yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—44no—
219C2×#41—44no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—44no—
224C22×Q8—82no—
225C22×Q8—82no—
226C22×Q8C2412yes—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4C2412yes—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2414yes—
239C4⋊Q8—24no—
240C4⋊Q8—24no—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247C4⋊Q8—44no—
248C4⋊Q8—44no—
249#127—22no—
250#127—22no—
251#125—42no—
252#125—42no—
253#124—24no—
254#124—24no—
255#124—84no—
256#124—84no—
257#121—44no—
258#121—44no—
259C4×Q8—84no—
260C4×D4—44no—
261C4×D4—44no—
262#117—42no—
263#117—42no—
264#117—82no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C42—22no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42#4114yes—
312C2×C42#4114yes—
313C2×C42C2412yes—
314C2×C42C2412yes—
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—24no—
320#103—44no—
321#103—44no—
322#103—44no—
323#103—44no—
324#102—44no—
325#102—44no—
326#102—44no—
327#102—44no—
328C22⋊C8—24no—
329C22⋊C8—24no—
330#97—24no—
331#97—44no—
332C4×C8—24no—
333C4×C8—44no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—21no—
338#95—21no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—21no—
347#95—21no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D411yescharacteristic
355#95C2×D411yescharacteristic
356#95C2×D411yescharacteristic
357#95C2×D411yescharacteristic
358C24—22no—
359C24C22×D411yescharacteristic
360#41—84no—
361#41—84no—
362#41—88no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—82no—
373C2×Q8—82no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×Q8—816no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—44no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—42no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4#12414yes—
405C22×C4#12414yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2511yescharacteristic
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4—82no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C42—24no—
465C42—24no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—42no—
471C42—42no—
472C42—44no—
473C42—42no—
474C42—42no—
475C42#12414yes—
476C42#12414yes—
477C42C2×#4112yes—
478C42C2×#4112yes—
479C42C2×#4112yes—
480C42C2×#4112yes—
481C23—22no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—44no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23C2×#12412yes—
491C23C23×D411yescharacteristic
492Q8—42no—
493Q8—42no—
494Q8—44no—
495Q8—88no—
496Q8—88no—
497Q8—88no—
498D4—42no—
499D4—42no—
500D4—44no—
501D4—84no—
502D4—88no—
503D4—84no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—88no—
544C2×C4—88no—
545C2×C4C2×#12412yes—
546C2×C4C2×#12412yes—
547C2×C4C2×#12412yes—
548C2×C4C2×#12412yes—
549C2×C4C2×#12412yes—
550C2×C4#57612yes—
551C2×C4#57712yes—
552C2×C4C22×#4111yescharacteristic
553C2×C4C22×#4111yescharacteristic
554C2×C4#58512yes—
555C2×C4#58512yes—
556C8—42no—
557C8—42no—
558C8—44no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—42no—
563C22—42no—
564C22—82no—
565C22—84no—
566C22—84no—
567C22—82no—
568C22—82no—
569C22C22×#12411yescharacteristic
570C22#344912yes—
571C22C2×#57611yescharacteristic
572C22C2×#57711yescharacteristic
573C22#358612yes—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—84no—
588C4#313614yes—
589C4#351011yescharacteristic
590C4#351011yescharacteristic
591C4#351012yes—
592C2—42no—
593C2—162no—
594C2#3533811yescharacteristic
595C2C2×#313611yescharacteristic
596C2#6062011yescharacteristic
597C2#6098812yes—
598C2#6099212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34625C2semidirect
3. #33283C2semidirect
4. #30211C2semidirect
5. #29356C2semidirect
6. #29355C2semidirect
7. #29349C2semidirect
8. #29346C2semidirect
9. #29120C2semidirect
10. #29119C2semidirect
11. #21841C2semidirect
12. #21833C2semidirect
15. #20470C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#346252
#332832
#302112
#293562
#293552
#293492
#293462
#291202
#291192
#218412
#218332
#218092
#218072
#204702