Finite Group Databasebeta

← #482089

Subgroups — #482089

Order 512.

1092 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3050C211yesmaximal, characteristic
2C22×C22⋊C16C211yesmaximal, characteristic, complemented
3C2×#1810C211yesmaximal, characteristic, complemented
4#11893C214yesmaximal, complemented
5#11892C212yesmaximal, complemented
6#11891C212yesmaximal, complemented
7#11709C214yesmaximal, complemented
8C22×#135C411yescharacteristic
9C24×C8C2211yescharacteristic
10C2×#525—21no—
11C23×C16—21no—
12#3050—21no—
13#3050—21no—
14#3050—21no—
15#3050—21no—
16#3050C412yes—
17#3050C412yes—
18#3050C2212yes—
19#3050C2212yes—
20C2×#467C411yescharacteristic
21C2×#467C2211yescharacteristic
22C2×#466C411yescharacteristic
23C2×#466C2211yescharacteristic
24C2×#465—21no—
25C2×#465—21no—
26C22×#108—21no—
27C2×#427C411yescharacteristic
28C2×#416—21no—
29C2×#407—21no—
30C2×C22⋊C16—21no—
31C2×C22⋊C16—21no—
32C2×C22⋊C16—21no—
33C2×C22⋊C16C2212yes—
34C2×C22⋊C16C2212yescomplemented
35C2×C22⋊C16C2212yescomplemented
36#1810—21no—
37#1810—21no—
38#1810C2212yes—
39#1810C2212yescomplemented
40C2×#341C2212yescomplemented
41#1350C2218yescomplemented
42#1349C2218yescomplemented
43C22×#41—21no—
44C24×C4D411yescharacteristic
45C2×#135—21no—
46C2×#135—21no—
47C2×#135—44no—
48C2×#135C2×C412yes—
49C2×#135C2×C412yes—
50C22×Q16C2×C411yescharacteristic
51C22×SD16—21no—
52C22×D8C2×C411yescharacteristic
53C22×M16—22no—
54C23×C8—44no—
55C23×C8C2×C411yescharacteristic
56C23×C8C2×C411yescharacteristic
57C23×C8C2×C411yescharacteristic
58C23×C8C2×C411yescharacteristic
59C23×C8D411yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8D411yescharacteristic
62C23×C8D411yescharacteristic
63C23×C8D411yescharacteristic
64C23×C8D411yescharacteristic
65C23×C8D411yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69#525—22no—
70#525—22no—
71#525—22no—
72#525—22no—
73#525—41no—
74#525—41no—
75#525—41no—
76#525—41no—
77C2×#123—41no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×#121—21no—
81C2×#121—21no—
82C2×#120—41no—
83C2×C4×D4—41no—
84C2×C4×D4—41no—
85C2×#117—21no—
86C22×C16—22no—
87C22×C16—22no—
88C22×C16—22no—
89C22×C16—44no—
90#467—21no—
91#467—21no—
92#467—21no—
93#467—21no—
94#467C2×C412yes—
95#467C2×C412yes—
96#467C2×C412yes—
97#467C2312yes—
98#466—21no—
99#466—21no—
100#466—21no—
101#466—21no—
102#466C2×C412yes—
103#466C2×C412yes—
104#466C2×C412yes—
105#466C2312yes—
106#465—22no—
107#465—22no—
108#465—22no—
109#465—22no—
110#465—22no—
111#465—22no—
112#465—22no—
113#465—22no—
114C2×#108—22no—
115C2×#108—22no—
116C2×#108—44no—
117#427—21no—
118#427—21no—
119#427C2×C412yes—
120#427C2×C412yes—
121C2×#107C2×C411yescharacteristic
122C2×#107C2×C411yescharacteristic
123C2×#106—21no—
124#416—22no—
125#416—22no—
126#416—22no—
127#416—22no—
128C2×#103—21no—
129C2×#103—21no—
130C2×#102—21no—
131C2×#102—21no—
132#407—22no—
133#407—22no—
134#407—41no—
135#407—41no—
136C2×C22⋊C8—41no—
137C2×C22⋊C8—41no—
138C22⋊C16—22no—
139C22⋊C16—22no—
140C22⋊C16—22no—
141C22⋊C16C2314yes—
142#341—22no—
143#341C2314yes—
144C25SD1611yescharacteristic
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—44no—
148C2×#41—44no—
149C2×#41—41no—
150C2×#41—41no—
151C22×Q8—21no—
152C22×D4—21no—
153C22×D4—41no—
154C22×D4—41no—
155C22×D4—41no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—41no—
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4D811yescharacteristic
164C23×C4D811yescharacteristic
165C23×C4D811yescharacteristic
166C23×C4D811yescharacteristic
167C23×C4SD1611yescharacteristic
168C23×C4SD1611yescharacteristic
169C23×C4SD1611yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173#135—44no—
174#135—44no—
175#135—44no—
176#135—44no—
177C2×Q16—21no—
178C2×Q16—42no—
179C2×Q16C22×C412yes—
180C2×SD16—22no—
181C2×SD16—22no—
182C2×SD16—44no—
183C2×D8—21no—
184C2×D8—42no—
185C2×D8C22×C412yes—
186C2×M16—24no—
187C2×M16—42no—
188C2×M16—44no—
189C2×M16—44no—
190C22×C8—24no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—42no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8C22⋊C411yescharacteristic
206C22×C8C22⋊C411yescharacteristic
207C22×C8C22⋊C411yescharacteristic
208C22×C8C22⋊C411yescharacteristic
209C22×C8C22⋊C411yescharacteristic
210C22×C8C22⋊C411yescharacteristic
211C22×C8C22⋊C411yescharacteristic
212C22×C8C22⋊C411yescharacteristic
213C22×C8C22⋊C411yescharacteristic
214C22×C8C22⋊C411yescharacteristic
215C22×C8C22⋊C411yescharacteristic
216C22×C8C22⋊C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C22×C411yescharacteristic
222C22×C8C22×C411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C22×C8C2×D411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C22×C8C2411yescharacteristic
233#123—42no—
234#123—42no—
235#123—42no—
236#123—42no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—41no—
242C22⋊Q8—41no—
243C22⋊Q8—41no—
244C22⋊Q8—41no—
245#121—22no—
246#121—22no—
247#121—22no—
248#121—22no—
249#121—41no—
250#121—41no—
251#121—41no—
252#121—41no—
253#120—42no—
254#120—42no—
255#120—42no—
256#120—42no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—42no—
264C4×D4—42no—
265#117—22no—
266#117—22no—
267#117—41no—
268#117—41no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C42—41no—
279C2×C16—24no—
280C2×C16—44no—
281C2×C16—44no—
282C2×C16—44no—
283#108—44no—
284#108—44no—
285#107—21no—
286#107—21no—
287#107C22×C412yes—
288#107C22×C412yes—
289#106—22no—
290#106—22no—
291#103—22no—
292#103—22no—
293#103—22no—
294#103—22no—
295#102—22no—
296#102—22no—
297#102—22no—
298#102—22no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C24—24no—
304C24—42no—
305C24—42no—
306C24—41no—
307C24#10211yescharacteristic
308C24#10211yescharacteristic
309C24#10211yescharacteristic
310C24#10211yescharacteristic
311C24C2×SD1611yescharacteristic
312C24C2×SD1611yescharacteristic
313C24C2×SD1611yescharacteristic
314#41—44no—
315#41—44no—
316#41—44no—
317#41—44no—
318#41—84no—
319#41—84no—
320C2×Q8—22no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—41no—
324C2×D4—22no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—84no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—84no—
364C22×C4#10211yescharacteristic
365C22×C4#10211yescharacteristic
366C22×C4#10211yescharacteristic
367C22×C4#10211yescharacteristic
368C22×C4#10211yescharacteristic
369C22×C4#10211yescharacteristic
370C22×C4#10211yescharacteristic
371C22×C4#10211yescharacteristic
372C22×C4#10211yescharacteristic
373C22×C4#10211yescharacteristic
374C22×C4#10211yescharacteristic
375C22×C4#10211yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×D811yescharacteristic
383C22×C4C2×D811yescharacteristic
384C22×C4C2×D811yescharacteristic
385C22×C4C2×D811yescharacteristic
386C22×C4C2×D811yescharacteristic
387C22×C4C2×D811yescharacteristic
388C22×C4C2×SD1611yescharacteristic
389C22×C4C2×SD1611yescharacteristic
390C22×C4C2×SD1611yescharacteristic
391C22×C4C22×D411yescharacteristic
392Q16—42no—
393Q16—42no—
394SD16—44no—
395SD16—44no—
396D8—42no—
397D8—42no—
398M16—44no—
399M16—44no—
400M16—84no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—44no—
429C2×C8—84no—
430C2×C8#9914yes—
431C2×C8C2×C22⋊C411yescharacteristic
432C2×C8C2×C22⋊C411yescharacteristic
433C2×C8C2×C22⋊C411yescharacteristic
434C2×C8C2×C22⋊C411yescharacteristic
435C2×C8C2×C22⋊C411yescharacteristic
436C2×C8C2×C22⋊C411yescharacteristic
437C2×C8C23×C411yescharacteristic
438C2×C8C22×D411yescharacteristic
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—42no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C42—42no—
458C42—42no—
459C16—44no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—416no—
481C23—42no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23#35914yes—
486C23C2×#10211yescharacteristic
487C23C2×#10211yescharacteristic
488C23C2×#10211yescharacteristic
489C23C2×#10211yescharacteristic
490C23C2×#10211yescharacteristic
491C23C2×#10211yescharacteristic
492C23C22×SD1611yescharacteristic
493Q8—42no—
494Q8—42no—
495Q8—82no—
496D4—42no—
497D4—42no—
498D4—82no—
499D4—84no—
500D4—84no—
501D4—84no—
502D4—84no—
503D4—84no—
504D4—84no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
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—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—84no—
546C2×C4C2×#9912yes—
547C2×C4C2×#10211yescharacteristic
548C2×C4C2×#10211yescharacteristic
549C2×C4C2×#10211yescharacteristic
550C2×C4C2×#10211yescharacteristic
551C2×C4C2×#10211yescharacteristic
552C2×C4C2×#10211yescharacteristic
553C2×C4#42112yes—
554C2×C4C22×C22⋊C411yescharacteristic
555C2×C4C22×D811yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—21no—
562C8—84no—
563C8C2×#9912yes—
564C8C2×#9912yes—
565C22—28no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—164no—
592C22#151912yes—
593C22C2×#35912yes—
594C22C2×#35912yes—
595C22C2×#35912yes—
596C22#214812yes—
597C22C22×#10211yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4#151912yes—
608C4C22×#9911yescharacteristic
609C4C2×#42111yescharacteristic
610C2—22no—
611C2—22no—
612C2—22no—
613C2—22no—
614C2—22no—
615C2—41no—
616C2—41no—
617C2—162no—
618C2C2×#151911yescharacteristic
619C2C22×#35911yescharacteristic
620C2C2×#214811yescharacteristic
subgroupG/H
centerC2×C4C2×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C22×#102
prime radicalC2×#3050C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊C16C2semidirect
3. C2×#1810C2semidirect
4. #11893C2semidirect
5. #11892C2semidirect
6. #11891C2semidirect
7. #11709C2semidirect
34. C2×C22⋊C16C22semidirect
35. C2×C22⋊C16C22semidirect
39. #1810C22semidirect
40. C2×#341C22semidirect
41. #1350C22semidirect
42. #1349C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30502
C22×C22⋊C162
C2×#18102
#118932
#118922
#118912
#117092