Finite Group Databasebeta

← #206031

Subgroups — #206031

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#1810C211yesmaximal, characteristic
3#20385C211yesmaximal, characteristic
4#17578C212yesmaximal
5#17397C212yesmaximal
6#9817C214yesmaximal, complemented
7#9814C214yesmaximal, complemented
8C23×Q16C411yescharacteristic, complemented
9C24×C8C2211yescharacteristic
10C22×C22⋊Q8C2212yes—
11C2×#467—21no—
12C2×#467—21no—
13C2×#467—21no—
14C2×#467—21no—
15C2×#467—21no—
16C2×#467—21no—
17C2×#467—21no—
18C2×#467—21no—
19C2×#467C412yescomplemented
20C2×#467C412yescomplemented
21C2×#467C2212yes—
22C2×#467C2212yes—
23C22×#108—21no—
24C22×#107C411yescharacteristic, complemented
25C22×#103C412yescomplemented
26C2×#407C2212yes—
27C2×#407C2211yescharacteristic
28C2×#405—21no—
29#2280C2211yescharacteristic
30#1810—21no—
31#1810—21no—
32#1810C2212yes—
33#1810C2212yes—
34#1752—21no—
35#1752C2212yes—
36C8×C22⋊C4—21no—
37C8×C22⋊C4C2212yes—
38C2×#341—21no—
39#1729—21no—
40#1520—24no—
41#1511—24no—
42#1508C2218yes—
43#1472—24no—
44#1472C2218yes—
45#1463—24no—
46C23×Q8C2×C412yes—
47C24×C4C2311yescharacteristic
48C22×Q16—21no—
49C22×Q16—21no—
50C22×Q16—21no—
51C22×Q16—21no—
52C22×Q16—44no—
53C22×Q16C2×C412yes—
54C22×Q16C2×C412yes—
55C22×M16—21no—
56C22×M16—22no—
57C23×C8—44no—
58C23×C8C2×C411yescharacteristic
59C23×C8C2×C411yescharacteristic
60C23×C8C2×C411yescharacteristic
61C23×C8C2×C411yescharacteristic
62C23×C8D411yescharacteristic
63C23×C8D411yescharacteristic
64C23×C8D411yescharacteristic
65C23×C8D411yescharacteristic
66C23×C8D411yescharacteristic
67C23×C8D411yescharacteristic
68C23×C8D411yescharacteristic
69C23×C8D411yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8C2×C414yes—
83C2×C22⋊Q8C2314yes—
84C2×#117—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4C2×C412yes—
87C22×C22⋊C4—22no—
88#467—22no—
89#467—22no—
90#467—22no—
91#467—22no—
92#467—22no—
93#467—22no—
94#467—22no—
95#467—22no—
96#467—22no—
97#467—22no—
98#467—22no—
99#467—22no—
100#467—22no—
101#467—22no—
102#467—22no—
103#467—22no—
104C2×#108—22no—
105C2×#108—22no—
106C2×#108—44no—
107C2×#107—21no—
108C2×#107—21no—
109C2×#107—21no—
110C2×#107—21no—
111C2×#107C2×C412yes—
112C2×#107C2×C412yes—
113C2×#103—22no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103—22no—
117C2×#103C2×C414yes—
118C2×#103C2×C414yes—
119#407—21no—
120#407—21no—
121#407—22no—
122#407—22no—
123#407—22no—
124#407D412yes—
125#407D412yes—
126#407C2314yes—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—22no—
130C2×C22⋊C8C2312yes—
131#405—44no—
132C2×#97—22no—
133C2×C4×C8—22no—
134#381D412yes—
135#380—21no—
136#380D412yes—
137#379—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4C2312yes—
140#341—22no—
141#341—22no—
142#336—22no—
143#336—22no—
144#328—216no—
145#324—216no—
146C25#4111yescharacteristic
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8C22×C414yes—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4#4111yescharacteristic
170C23×C4#4111yescharacteristic
171C23×C4#4111yescharacteristic
172C23×C4C2411yescharacteristic
173C2×Q16—22no—
174C2×Q16—22no—
175C2×Q16—22no—
176C2×Q16—22no—
177C2×Q16—44no—
178C2×Q16—44no—
179C2×Q16—44no—
180C2×Q16—44no—
181C2×Q16—44no—
182C2×Q16—44no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—24no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16—42no—
189C2×M16—44no—
190C2×M16—44no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—22no—
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×C8—42no—
206C22×C8—42no—
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×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C22×C411yescharacteristic
226C22×C8C22×C411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C22×C8C2×D411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—24no—
243C22⋊Q8—42no—
244C22⋊Q8—42no—
245C22⋊Q8—42no—
246C22⋊Q8—42no—
247C22⋊Q8—42no—
248C22⋊Q8—42no—
249C22⋊Q8—42no—
250C22⋊Q8—42no—
251#117—42no—
252#117—42no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4C22×C414yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C42—22no—
276#108—44no—
277#108—44no—
278#107—22no—
279#107—22no—
280#107—22no—
281#107—22no—
282#103—24no—
283#103—24no—
284#103—24no—
285#103—24no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293C22⋊C8C2×D414yes—
294#97—44no—
295C4×C8—44no—
296#95—22no—
297#95C2×D414yes—
298C24—24no—
299C24—42no—
300C24—42no—
301C24C4×D411yescharacteristic
302C24C4×D411yescharacteristic
303C24#12111yescharacteristic
304C24#12111yescharacteristic
305C24#12411yescharacteristic
306C24#12411yescharacteristic
307C24C2×#4111yescharacteristic
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—84no—
319C2×Q8—84no—
320C2×Q8—84no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12411yescharacteristic
368C22×C4#12411yescharacteristic
369C22×C4#12711yescharacteristic
370C22×C4#12711yescharacteristic
371C22×C4#12711yescharacteristic
372C22×C4#12711yescharacteristic
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379Q16—44no—
380Q16—44no—
381Q16—44no—
382Q16—44no—
383M16—44no—
384M16—44no—
385M16—44no—
386M16—44no—
387M16—84no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—44no—
419C2×C8—84no—
420C2×C8#9914yes—
421C2×C8C2×C22⋊C411yescharacteristic
422C2×C8C2×C22⋊C411yescharacteristic
423C2×C8C2×C22⋊C411yescharacteristic
424C2×C8C2×C22⋊C411yescharacteristic
425C2×C8C2×C22⋊C411yescharacteristic
426C2×C8C2×C22⋊C411yescharacteristic
427C2×C8C2×C22⋊C411yescharacteristic
428C2×C8C2×C22⋊C411yescharacteristic
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C42—42no—
446C42—42no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—416no—
468C23#39011yescharacteristic
469C23#39011yescharacteristic
470C23#39011yescharacteristic
471C23#39011yescharacteristic
472C23#47314yes—
473C23C2×C4×D411yescharacteristic
474C23C2×#12111yescharacteristic
475C23C2×#12411yescharacteristic
476Q8—44no—
477Q8—44no—
478Q8—84no—
479Q8—84no—
480Q8—84no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—48no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4#39011yescharacteristic
526C2×C4#39011yescharacteristic
527C2×C4#39011yescharacteristic
528C2×C4#39011yescharacteristic
529C2×C4C2×#9912yes—
530C2×C4#41012yes—
531C2×C4C22×C22⋊C411yescharacteristic
532C2×C4C2×C4×D411yescharacteristic
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×#12711yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—44no—
542C8—84no—
543C8C2×#9912yes—
544C8C2×#9912yes—
545C22—28no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22#196412yes—
569C22#197912yes—
570C22#197912yes—
571C22C2×#39011yescharacteristic
572C22#295712yes—
573C22C2×#47312yes—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—84no—
581C4—84no—
582C4—84no—
583C4#196412yes—
584C4C22×#9911yescharacteristic
585C4C2×#41011yescharacteristic
586C2—22no—
587C2—22no—
588C2—22no—
589C2—22no—
590C2—22no—
591C2—41no—
592C2—41no—
593C2C2×#196411yescharacteristic
594C2#2124011yescharacteristic
595C2C2×#197911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9817C2semidirect
7. #9814C2semidirect
8. C23×Q16C4semidirect
19. C2×#467C4semidirect
20. C2×#467C4semidirect
24. C22×#107C4semidirect
25. C22×#103C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#18102
#203852
#175782
#173972
#98172
#98142