Finite Group Databasebeta

← #204990

Subgroups — #204990

Order 512.

1450 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
1#35019C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic, complemented
3C2×#1520C211yesmaximal, characteristic, complemented
4C2×#1519C211yesmaximal, characteristic
5C22×#328C211yesmaximal, characteristic, complemented
6C22×#327C211yesmaximal, characteristic
7#13802C211yesmaximal, characteristic, complemented
8#9716C214yesmaximal, complemented
9#9713C214yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2850C2214yes—
15#2639—22no—
16#2591—22no—
17#2561—22no—
18#2504C412yescomplemented
19#2503C412yescomplemented
20#2436C412yescomplemented
21#2435C412yescomplemented
22#2308—21no—
23#2308—21no—
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27C22×#95C2211yescharacteristic
28#1850—22no—
29#1850—22no—
30#1797C2214yescomplemented
31#1520—24no—
32#1519—24no—
33C2×#328—24no—
34C2×#328C2212yescomplemented
35C2×#328C2212yescomplemented
36C2×#328C2214yescomplemented
37C2×#327—24no—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2214yes—
41#1436—22no—
42#1434C2212yescomplemented
43C22⋊#95—21no—
44C4×#95C2212yescomplemented
45C23×Q8C2×C411yescharacteristic
46C23×D4—21no—
47C23×D4C2×C411yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8—21no—
51C23×C8—21no—
52C2×#124C2×C414yescomplemented
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2312yes—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121C2×C412yes—
68C2×#121C2312yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D411yescharacteristic
77C22×C22⋊C4D411yescharacteristic
78C22×C42—22no—
79C22×C42C2312yes—
80C2×C4⋊C8—24no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2312yes—
89#400—24no—
90#397—24no—
91#394—24no—
92#393—24no—
93#391C2×C414yescomplemented
94#390C2×C414yescomplemented
95#388—24no—
96#388—24no—
97#386—24no—
98#386—24no—
99#386—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#382C2×C414yescomplemented
104#381—22no—
105#381—22no—
106#380—22no—
107#380D414yes—
108#379—22no—
109#379—22no—
110C4×C22⋊C4—44no—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4C2×C412yes—
113C4×C22⋊C4D412yes—
114C4×C22⋊C4D412yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D414yes—
121C2×#95D412yes—
122C2×#95C2311yescharacteristic
123C2×#95C2312yes—
124C2×#95C2311yescharacteristic
125#336—24no—
126#328—24no—
127#328—24no—
128#328—28no—
129#327—24no—
130#327—24no—
131#327—28no—
132C25—21no—
133C25C2×D411yescharacteristic
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—82no—
138C22×Q8C22×C412yes—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—82no—
151C22×D4C22×C412yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—44no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—82no—
163C23×C4—82no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4C2411yescharacteristic
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8—82no—
181C22×C8—82no—
182#124—84no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193#121—24no—
194#121—24no—
195#121—22no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—42no—
200#121—42no—
201#121—42no—
202#121—42no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4C22⋊C412yes—
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—82no—
243C2×C42—82no—
244C2×C42C22×C414yes—
245C2×C42C2×D414yes—
246C4⋊C8—84no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—44no—
262#95—44no—
263#95—82no—
264#95C22⋊C414yes—
265#95C22⋊C414yes—
266#95C2×D412yes—
267#95C2×D412yes—
268#95C2×D414yes—
269C24—22no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—82no—
276C24—82no—
277C24C2×C22⋊C411yescharacteristic
278C24C2×C22⋊C411yescharacteristic
279C24#12112yes—
280C24#12112yes—
281C24#13512yes—
282C24#13512yes—
283C24AGL(1,ℤ/8)12yes—
284C24#13712yes—
285C24C22×D411yescharacteristic
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—44no—
289C2×Q8—42no—
290C2×Q8—82no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—44no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—82no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4—84no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—48no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—48no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—84no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4C2×C22⋊C414yes—
364C22×C4C2×C22⋊C414yes—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4#12112yes—
372C22×C4#12112yes—
373C22×C4#12412yes—
374C22×C4#12412yes—
375C22×C4#12712yes—
376C22×C4#12712yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—84no—
388C2×C8—84no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—84no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—44no—
406C22⋊C4—48no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C42—44no—
415C42—44no—
416C42—88no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—44no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—42no—
436C23—84no—
437C23—84no—
438C23—84no—
439C23—164no—
440C23—164no—
441C23#39012yes—
442C23#39012yes—
443C23#41612yes—
444C23#41612yes—
445C23#41712yes—
446C23#41712yes—
447C23#46314yes—
448C23#46314yes—
449C23#46414yes—
450C23#46414yes—
451C23C22×C22⋊C411yescharacteristic
452C23C2×#12111yescharacteristic
453C23C2×#12111yescharacteristic
454C23C2×#13511yescharacteristic
455C23C2×#13511yescharacteristic
456C23C2×AGL(1,ℤ/8)11yescharacteristic
457C23C2×#13711yescharacteristic
458Q8—82no—
459Q8—82no—
460Q8—84no—
461D4—82no—
462D4—82no—
463D4—84no—
464D4—88no—
465D4—82no—
466D4—82no—
467D4—82no—
468D4—82no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
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—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—816no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4#39012yes—
526C2×C4#39012yes—
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×#12411yescharacteristic
530C2×C4C2×#12711yescharacteristic
531C8—82no—
532C8—82no—
533C22—22no—
534C22—22no—
535C22—24no—
536C22—24no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—21no—
546C22—21no—
547C22—22no—
548C22—22no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—82no—
554C22—84no—
555C22—84no—
556C22—82no—
557C22—82no—
558C22—168no—
559C22#198314yes—
560C22#198314yes—
561C22C2×#39011yescharacteristic
562C22#288312yes—
563C22C2×#41611yescharacteristic
564C22C2×#41711yescharacteristic
565C22#290112yes—
566C22C2×#46311yescharacteristic
567C22C2×#46311yescharacteristic
568C22C2×#46411yescharacteristic
569C22C2×#46411yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—22no—
576C4—44no—
577C4—44no—
578C4—44no—
579C4—84no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C2—21no—
585C2—21no—
586C2—24no—
587C2—24no—
588C2—21no—
589C2—21no—
590C2—162no—
591C2#1761512yes—
592C2#1762212yes—
593C2#2121011yescharacteristic
594C2#2121311yescharacteristic
595C2C2×#198311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21930C2semidirect
3. C2×#1520C2semidirect
5. C22×#328C2semidirect
7. #13802C2semidirect
8. #9716C2semidirect
9. #9713C2semidirect
18. #2504C4semidirect
19. #2503C4semidirect
20. #2436C4semidirect
21. #2435C4semidirect
30. #1797C22semidirect
34. C2×#328C22semidirect
35. C2×#328C22semidirect
36. C2×#328C22semidirect
42. #1434C22semidirect
44. C4×#95C22semidirect
52. C2×#124C2×C4semidirect
93. #391C2×C4semidirect
94. #390C2×C4semidirect
103. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350192
#219302
C2×#15202
C2×#15192
C22×#3282
C22×#3272
#138022
#97162
#97132