Finite Group Databasebeta

← #370651

Subgroups — #370651

Order 512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic
2C4⋊C4×M16C211yesmaximal, characteristic
3#13848C211yesmaximal, characteristic
4#11370C212yesmaximal
5#11226C212yesmaximal
6#10886C212yesmaximal
7#10852C211yesmaximal, characteristic
8#10851C212yesmaximal
9#10851C211yesmaximal, characteristic
10#10840C211yesmaximal, characteristic
11#10701C211yesmaximal, characteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2212yes—
14C2×C4×M16C2211yescharacteristic
15#2571C412yescomplemented
16#2570C412yescomplemented
17#2570C412yescomplemented
18#2569—21no—
19#2569—21no—
20#2552C412yescomplemented
21#2442—22no—
22#2402—21no—
23#2402—21no—
24C4×#125—22no—
25#2296C2212yes—
26#2285C2211yescharacteristic
27#2285C2212yes—
28C4×#117C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1771—22no—
34#1770—21no—
35C8×C4⋊C4—21no—
36C2×#339C2212yes—
37C2×#339C2212yes—
38C2×#339C2211yescharacteristic
39C2×#339C2211yescharacteristic
40C2×#339C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46#1728C2212yes—
47#1444—21no—
48#1442—21no—
49#1441—22no—
50C4×#95C2212yes—
51C22×M16C2312yes—
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C2×#125—22no—
55C2×#117C2311yescharacteristic
56C2×#117C2311yescharacteristic
57C2×#117C2312yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D412yes—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66#423—22no—
67#423Q814yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C4×M16—22no—
71C4×M16—24no—
72C4×M16—42no—
73C4×M16Q814yes—
74C2×#97—21no—
75C2×#97—22no—
76C2×C4×C8—21no—
77C2×C4×C8—22no—
78#400—22no—
79#400—22no—
80#400—22no—
81#400—22no—
82#398—22no—
83#398—22no—
84#398—22no—
85#398—22no—
86#389—22no—
87#389—22no—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#385—22no—
97#385—22no—
98#384C2×C412yes—
99#384C2×C412yes—
100#384C2×C412yes—
101#382—22no—
102#382C2×C411yescharacteristic
103#382C2×C411yescharacteristic
104#382C2×C412yes—
105#382C2×C412yes—
106#380—21no—
107#380—22no—
108#380—22no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—42no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130#376—21no—
131#376—22no—
132#376—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95D411yescharacteristic
140C2×#95D411yescharacteristic
141C43—21no—
142#339—22no—
143#339—22no—
144#339—24no—
145#339—24no—
146#339—22no—
147#339—22no—
148#339—42no—
149#339—42no—
150#339—42no—
151#339—42no—
152#336—22no—
153#336—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—22no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—21no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—21no—
175C2×M16C2×Q812yes—
176C2×M16C2×Q814yes—
177C2×M16C2×Q812yes—
178C2×M16#4112yes—
179C2×M16#4112yes—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183#125—48no—
184#117—21no—
185#117—21no—
186#117—21no—
187#117—22no—
188#117—21no—
189#117—22no—
190#117—21no—
191#117—21no—
192#117—22no—
193#117—21no—
194#117—21no—
195#117—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42C22×C412yes—
249C2×C42C22×C412yes—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×Q812yes—
257C2×C42C2×Q812yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262C4⋊C8—24no—
263C4⋊C8—24no—
264#97—24no—
265#97—44no—
266C4×C8—24no—
267C4×C8—44no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—22no—
276#95—21no—
277#95—21no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—42no—
286#95—42no—
287#95—42no—
288#95C22⋊C412yes—
289#95C22⋊C412yes—
290#95C22⋊C411yescharacteristic
291#95C22⋊C411yescharacteristic
292#95C22⋊C411yescharacteristic
293#95C22⋊C411yescharacteristic
294C24C22×D411yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C412yes—
334C22×C4#12412yes—
335C22×C4#12412yes—
336C22×C4C4⋊Q812yes—
337C22×C4C4⋊Q812yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×Q811yescharacteristic
341C22×C4C22×Q811yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C4H(5,𝔽2)11yescharacteristic
350C22×C42−1+411yescharacteristic
351C22×C42−1+411yescharacteristic
352M16—22no—
353M16—22no—
354M16—22no—
355M16—22no—
356M16—22no—
357M16—24no—
358M16—24no—
359M16—42no—
360M16—42no—
361M16—41no—
362M16—41no—
363M16#12514yescomplemented
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—42no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—42no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—48no—
389C4⋊C4—416no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—42no—
401C22⋊C4—41no—
402C22⋊C4—42no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—44no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—42no—
422C42—41no—
423C42—42no—
424C42—41no—
425C42—41no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—82no—
431C42C4×Q812yes—
432C42C4×Q812yes—
433C42C4×Q812yes—
434C42C4×Q812yes—
435C42C22⋊Q812yes—
436C42C22⋊Q812yes—
437C42C22⋊Q812yes—
438C42C22⋊Q812yes—
439C23—22no—
440C23—22no—
441C23C22×C22⋊C411yescharacteristic
442C23C2×#12411yescharacteristic
443C23C2×C4⋊Q811yescharacteristic
444C23#54611yescharacteristic
445C23#54711yescharacteristic
446C23#54811yescharacteristic
447C23#55011yescharacteristic
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#39112yes—
501C2×C4#39112yes—
502C2×C4#39112yes—
503C2×C4#39112yes—
504C2×C4#42114yes—
505C2×C4C2×C4×Q811yescharacteristic
506C2×C4C2×C4×Q811yescharacteristic
507C2×C4C4×#4111yescharacteristic
508C2×C4C4×#4111yescharacteristic
509C2×C4#52011yescharacteristic
510C2×C4#52011yescharacteristic
511C2×C4C2×C22⋊Q811yescharacteristic
512C2×C4C2×C22⋊Q811yescharacteristic
513C2×C4C2×#12511yescharacteristic
514C2×C4#52911yescharacteristic
515C2×C4#52911yescharacteristic
516C2×C4#53311yescharacteristic
517C2×C4#54011yescharacteristic
518C2×C4#54111yescharacteristic
519C2×C4#54411yescharacteristic
520C2×C4#54411yescharacteristic
521C2×C4#54611yescharacteristic
522C2×C4#54711yescharacteristic
523C2×C4#54811yescharacteristic
524C2×C4#55011yescharacteristic
525C2×C4#55211yescharacteristic
526C2×C4#55211yescharacteristic
527C2×C4#55311yescharacteristic
528C2×C4#55311yescharacteristic
529C2×C4#55411yescharacteristic
530C2×C4#55411yescharacteristic
531C2×C4#55711yescharacteristic
532C2×C4#55711yescharacteristic
533C8—24no—
534C8—42no—
535C8—44no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22C2×#39111yescharacteristic
545C22#233911yescharacteristic
546C22#234111yescharacteristic
547C22#248911yescharacteristic
548C22#249011yescharacteristic
549C22#255011yescharacteristic
550C22#255211yescharacteristic
551C22C2×#42112yes—
552C22#290112yes—
553C4—22no—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—24no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—84no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4C4×#12511yescharacteristic
569C4#233011yescharacteristic
570C4#237011yescharacteristic
571C4#237011yescharacteristic
572C4#244111yescharacteristic
573C4#244211yescharacteristic
574C4#245311yescharacteristic
575C4#245411yescharacteristic
576C2—21no—
577C2—21no—
578C2—21no—
579C2—21no—
580C2#1405411yescharacteristic
581C2#2130611yescharacteristic
582C2#2130711yescharacteristic
583C2#2250612yes—
584C2#2255112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2571C4semidirect
16. #2570C4semidirect
17. #2570C4semidirect
20. #2552C4semidirect
363. M16#125semidirect

Maximal subgroups

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

subgroupindex
#349482
C4⋊C4×M162
#138482
#113702
#112262
#108862
#108522
#108512
#108402
#107012