Finite Group Databasebeta

← #214238

Subgroups — #214238

Order 512.

1023 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34947C211yesmaximal, characteristic
2#21373C211yesmaximal, characteristic
3#20721C211yesmaximal, characteristic
4#9075C212yesmaximal, complemented
5#9071C212yesmaximal, complemented
6#8941C212yesmaximal, complemented
7#8938C212yesmaximal, complemented
8#8706C212yesmaximal, complemented
9#8314C212yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2568C411yescharacteristic
17#2567C411yescharacteristic
18#2566—21no—
19#2565—21no—
20#2564—21no—
21#2564—21no—
22#2563C411yescharacteristic
23#2542C411yescharacteristic
24#2417C412yes—
25#2417C412yes—
26#2396—21no—
27#2396—21no—
28C4×C22⋊Q8C2212yes—
29C4×C22⋊Q8C2212yes—
30#2297—21no—
31#2285—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1984—21no—
34#1954—21no—
35#1869—21no—
36#1869—21no—
37#1818—21no—
38#1797C2212yes—
39#1797C2212yes—
40#1780—21no—
41#1750—21no—
42#1750—21no—
43C4×C22⋊C8C2212yes—
44C4×C22⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C22×#41D411yescharacteristic
54C23×C8—21no—
55#533—41no—
56#533—41no—
57#533—41no—
58#533—41no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8C2×C412yes—
64C2×C22⋊Q8C2×C412yes—
65C2×C22⋊Q8C2×C412yes—
66C4×#41—41no—
67C4×#41—41no—
68C4×#41—41no—
69C4×#41—41no—
70C2×C4×Q8—21no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8C2312yes—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×#117—21no—
81C2×#117D411yescharacteristic
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C22×C4⋊C4C2×C412yes—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—41no—
90#432—41no—
91#431—41no—
92#431—41no—
93#431—41no—
94#431—41no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—41no—
109C2×C4×C8—41no—
110C2×C4×C8C2312yes—
111C2×C4×C8C2312yes—
112#400—21no—
113#400—21no—
114#399—21no—
115#399—21no—
116#398—22no—
117#398—22no—
118#396—22no—
119#396—22no—
120#395—21no—
121#395—21no—
122#393—21no—
123#393—21no—
124#391—21no—
125#391—21no—
126#391C2×C412yes—
127#391C2×C412yes—
128#390—21no—
129#390—21no—
130#389C2×C412yes—
131#389C2×C412yes—
132#387—22no—
133#387—22no—
134#386—22no—
135#386—22no—
136#385—22no—
137#385—22no—
138#384—21no—
139#384—21no—
140#382—21no—
141#382—21no—
142#380—22no—
143#380—22no—
144#380C2×C412yes—
145#380C2×C412yes—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4C2×C411yescharacteristic
151C4×C4⋊C4C2×C411yescharacteristic
152C4×C4⋊C4C2312yes—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—22no—
155C2×#95—21no—
156#336—22no—
157#336—22no—
158#336—22no—
159#336—22no—
160Q8⋊C8—42no—
161Q8⋊C8—42no—
162Q8⋊C8—42no—
163Q8⋊C8—42no—
164C2×#41—24no—
165C2×#41—24no—
166C2×#41—41no—
167C2×#41—41no—
168C22×Q8—21no—
169C22×Q8C22⋊C411yescharacteristic
170C22×Q8C22×C412yes—
171C22×D4C22⋊C411yescharacteristic
172C22×D4C22⋊C411yescharacteristic
173C22×D4C22⋊C411yescharacteristic
174C23×C4—22no—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—44no—
189C22×C8—42no—
190C4⋊Q8—41no—
191C4⋊Q8—41no—
192C4⋊Q8—41no—
193C4⋊Q8—41no—
194#125—42no—
195#125—42no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—42no—
213C4×Q8—42no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—41no—
226#117—41no—
227#117—41no—
228#117—41no—
229#117—41no—
230#117—41no—
231#117—41no—
232#117—42no—
233#117—42no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C22⋊C411yescharacteristic
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—44no—
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2411yescharacteristic
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—42no—
298C22⋊C8—44no—
299C22⋊C8—44no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—44no—
304C22⋊C8—44no—
305#97—42no—
306C4×C8—22no—
307C4×C8—22no—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—42no—
311C4×C8—42no—
312#95—21no—
313#95—21no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323C24C2×C22⋊C411yescharacteristic
324C24C2×C22⋊C411yescharacteristic
325C24C2×C22⋊C411yescharacteristic
326#41—48no—
327#41—48no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—41no—
337C2×Q8—42no—
338C2×Q8—82no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—82no—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4#13514yes—
396C22×C4#13714yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C2×C8—24no—
401C2×C8—42no—
402C2×C8—48no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4#13114yes—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—42no—
466C42—42no—
467C42—44no—
468C42—42no—
469C42—42no—
470C42H(5,𝔽2)11yescharacteristic
471C42H(5,𝔽2)11yescharacteristic
472C42H(5,𝔽2)11yescharacteristic
473C42H(5,𝔽2)11yescharacteristic
474C23—21no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—21no—
483C23—22no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23#41612yes—
488C23#41912yes—
489C23C22×C22⋊C411yescharacteristic
490Q8—44no—
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—84no—
496D4—44no—
497D4—44no—
498D4—44no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
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—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—88no—
553C2×C4#41612yes—
554C2×C4#41912yes—
555C2×C4#51811yescharacteristic
556C2×C4#51811yescharacteristic
557C2×C4#53411yescharacteristic
558C2×C4#53411yescharacteristic
559C2×C4#53411yescharacteristic
560C2×C4#53411yescharacteristic
561C2×C4C2×#13112yes—
562C2×C4#56812yes—
563C2×C4C2×#13512yes—
564C2×C4C2×#13712yes—
565C8—44no—
566C8—82no—
567C8—82no—
568C8—82no—
569C22—22no—
570C22—22no—
571C22—24no—
572C22—24no—
573C22—41no—
574C22—41no—
575C22—42no—
576C22—42no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—42no—
582C22—42no—
583C22—41no—
584C22#230811yescharacteristic
585C22#288012yes—
586C22C2×#41611yescharacteristic
587C22C2×#41911yescharacteristic
588C22#290112yes—
589C4—22no—
590C4—22no—
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—44no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4#297911yescharacteristic
605C4#298011yescharacteristic
606C4#298112yes—
607C4#319012yes—
608C4#319012yes—
609C2—42no—
610C2—42no—
611C2—81no—
612C2#1750812yes—
613C2#1751412yes—
614C2#2174511yescharacteristic
615C2#2176211yescharacteristic
616C2#2177511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9075C2semidirect
5. #9071C2semidirect
6. #8941C2semidirect
7. #8938C2semidirect
8. #8706C2semidirect
9. #8314C2semidirect

Maximal subgroups

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

subgroupindex
#349472
#213732
#207212
#90752
#90712
#89412
#89382
#87062
#83142