Finite Group Databasebeta

← #359006

Subgroups — #359006

Order 512.

2058 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#441C211yesmaximal, characteristic, complemented
2C2×#2323C212yesmaximal
3C2×#2001C211yesmaximal, characteristic
4C2×C4×#103C212yesmaximal, complemented
5C2×#1750C211yesmaximal, characteristic
6#11327C211yesmaximal, characteristic, complemented
7#11095C211yesmaximal, characteristic
8#11067C212yesmaximal, complemented
9#11024C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10703C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C4×Q8C2212yes—
15C22×#117C2211yescharacteristic
16C2×#441—24no—
17C2×#441—28no—
18C22×#106C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2323—28no—
27#2323—28no—
28C2×#391C2212yes—
29C2×#389—22no—
30C2×#389C2212yes—
31C2×#388—22no—
32C2×#388—22no—
33C2×#387—22no—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2212yes—
36#2001C418yescomplemented
37C4×#103—28no—
38#1750—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C23×Q8C2312yes—
47C24×C4C2311yescharacteristic
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16C2×C414yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8C2×C418yes—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8D418yes—
69C2×#117—24no—
70C2×#117—24no—
71C2×#117—22no—
72C2×#117—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4D412yes—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#441—232no—
88C2×#106—22no—
89C2×#106—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103C2×C418yes—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8C2×C414yes—
105C2×#97—24no—
106C2×#97D414yes—
107C2×C4×C8—22no—
108C2×C4×C8—24no—
109#391—24no—
110#391—24no—
111#389—216no—
112#389—24no—
113#389—24no—
114#388—216no—
115#388—216no—
116#387—216no—
117#380—24no—
118#380—24no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4D418yes—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126#340—22no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340C2×C414yes—
131#336—22no—
132#336—22no—
133C25C2×D411yescharacteristic
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8C22×C414yes—
146C22×Q8C2×D414yes—
147C22×Q8#4114yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4C22×C412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4C2411yescharacteristic
166C2×Q16—28no—
167C2×Q16—216no—
168C2×Q16—28no—
169C2×Q16—216no—
170C2×Q16—28no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8C22×C412yes—
186C22×C8C22×C412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8#4112yes—
190C22×C8#4112yes—
191C22⋊Q8—216no—
192C22⋊Q8—216no—
193C22⋊Q8—216no—
194C22⋊Q8—216no—
195C4×Q8—216no—
196C4×Q8—216no—
197C4×Q8—48no—
198C4×Q8—48no—
199#117—44no—
200#117—48no—
201#117—44no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C42—28no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—28no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—82no—
250C2×C42—82no—
251C2×C42C2×D414yes—
252C2×C42C2×D412yes—
253C2×C42C2×D414yes—
254C2×C42#4112yes—
255#106—28no—
256#103—216no—
257#103—216no—
258#103—216no—
259C22⋊C8—28no—
260C22⋊C8—28no—
261#97—28no—
262C4×C8—44no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268C24—22no—
269C24—21no—
270C24—22no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24C4×D412yes—
277C24C22×D411yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8C4×D418yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—42no—
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—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4C4×D412yes—
368C22×C4#12014yes—
369C22×C4#12014yes—
370C22×C4#12114yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C2×#4112yes—
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381Q16—232no—
382C2×C8—24no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—24no—
392C2×C8—22no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8C4×D414yes—
402C4⋊C4—232no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—416no—
416C4⋊C4—88no—
417C4⋊C4C4×D418yes—
418C22⋊C4—232no—
419C22⋊C4—48no—
420C22⋊C4—88no—
421C42—41no—
422C42—42no—
423C42—41no—
424C42—416no—
425C42—88no—
426C42#12014yes—
427C42#12118yes—
428C42#12714yes—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—22no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—42no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23C2×C4×D411yescharacteristic
451C23C2×#12012yes—
452C23C2×#12111yescharacteristic
453C23#52512yes—
454C23#53211yescharacteristic
455C23#57712yes—
456C23#57712yes—
457C23#57812yes—
458C23#57812yes—
459Q8—28no—
460Q8—28no—
461Q8—44no—
462Q8—44no—
463Q8—44no—
464Q8—44no—
465Q8—44no—
466Q8—44no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#39218yes—
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C2×C4×D412yes—
522C2×C4C2×C4×D412yes—
523C2×C4C4×#4111yescharacteristic
524C2×C4C2×#12011yescharacteristic
525C2×C4C2×#12112yes—
526C2×C4#52512yes—
527C2×C4#52511yescharacteristic
528C2×C4#52512yes—
529C2×C4#52512yes—
530C2×C4#52511yescharacteristic
531C2×C4#52512yes—
532C2×C4#52911yescharacteristic
533C2×C4#52912yes—
534C2×C4C2×#12711yescharacteristic
535C2×C4#53211yescharacteristic
536C2×C4#53212yes—
537C2×C4#53212yes—
538C8—24no—
539C8—42no—
540C8—42no—
541C8—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—44no—
546C22—44no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22C4×#12012yes—
552C22C4×#12111yescharacteristic
553C22C2×#39211yescharacteristic
554C22#238812yes—
555C22#239211yescharacteristic
556C22#294712yes—
557C22#295011yescharacteristic
558C22#295211yescharacteristic
559C22#300912yes—
560C22#301112yes—
561C22#301312yes—
562C22#301312yes—
563C22#306312yes—
564C22#306312yes—
565C22#312012yes—
566C22#312112yes—
567C22#312512yes—
568C22#312712yes—
569C22#317311yescharacteristic
570C22#317711yescharacteristic
571C22#317812yes—
572C4—28no—
573C4—42no—
574C4—48no—
575C4—48no—
576C4—84no—
577C4—82no—
578C4—84no—
579C4C4×#12011yescharacteristic
580C4C4×#12112yes—
581C4C4×#12711yescharacteristic
582C4#238811yescharacteristic
583C4#239212yes—
584C4#239711yescharacteristic
585C2—82no—
586C2C4×#39211yescharacteristic
587C2#2146112yes—
588C2#2146312yes—
589C2#2153011yescharacteristic
590C2#2154211yescharacteristic
591C2#2339412yes—
592C2#2358112yes—
593C2#2358612yes—
594C2#2371312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#441C2semidirect
4. C2×C4×#103C2semidirect
6. #11327C2semidirect
8. #11067C2semidirect
9. #11024C2semidirect
11. #10703C2semidirect
36. #2001C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4412
C2×#23232
C2×#20012
C2×C4×#1032
C2×#17502
#113272
#110952
#110672
#110242
#109702
#107032