Finite Group Databasebeta

← #369008

Subgroups — #369008

Order 512.

1404 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34953C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic
3C2×#1995C211yesmaximal, characteristic
4C2×#1986C211yesmaximal, characteristic
5#21303C211yesmaximal, characteristic
6#11226C212yesmaximal
7#11226C212yesmaximal
8#10879C212yesmaximal
9#10878C212yesmaximal
10#10852C211yesmaximal, characteristic
11#10701C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2584C414yescomplemented
17#2583—21no—
18#2583—21no—
19#2567C412yescomplemented
20#2546C412yescomplemented
21#2502—21no—
22#2434—21no—
23#2399—22no—
24C4×C4⋊Q8—22no—
25#2296C2212yes—
26#2296C2212yes—
27#2295—24no—
28#2295—24no—
29C2×#391—21no—
30C2×#389—21no—
31C2×#387—21no—
32C2×#387—21no—
33C4×#117C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×#376C2211yescharacteristic
38C2×#376C2211yescharacteristic
39C2×C43C2211yescharacteristic
40#1995—24no—
41#1986—24no—
42#1985—22no—
43#1984—22no—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51C2×#339C2212yes—
52#1728C2212yes—
53#1728C2212yes—
54C23×Q8D411yescharacteristic
55C22×M16C2312yes—
56C22×M16C2312yes—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#125C2×C412yes—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8C2×C412yes—
74C2×C4×Q8C2×C412yes—
75C2×C4×Q8D412yes—
76C2×C4×Q8D412yes—
77C2×#117C2312yes—
78C2×#117C2312yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#423—22no—
95#423—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C4×M16—22no—
99C4×M16—22no—
100C2×#97—21no—
101C2×C4×C8—21no—
102#398—22no—
103#398—22no—
104#398—24no—
105#395—24no—
106#395—22no—
107#395—22no—
108#391—28no—
109#391—22no—
110#391—22no—
111#389—28no—
112#389—24no—
113#389—24no—
114#389C2×C414yes—
115#389C2×C414yes—
116#389C2×C414yes—
117#389C2×C414yes—
118#387—28no—
119#387—28no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#384—22no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4D414yes—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138#376—22no—
139#376—22no—
140#376—22no—
141#376D414yes—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—21no—
149C43—21no—
150C43—22no—
151C43—22no—
152#339—24no—
153#339—24no—
154#339—24no—
155#339—24no—
156#339—24no—
157#339—24no—
158#339—24no—
159#339Q818yescomplemented
160#336—24no—
161#336—24no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—44no—
167C22×Q8C22⋊C412yes—
168C22×Q8C22⋊C412yes—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2411yescharacteristic
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16C2×Q814yes—
192C22×C8—22no—
193C22×C8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198C4⋊Q8—48no—
199#125—24no—
200#125—24no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—42no—
214C4×Q8—42no—
215C4×Q8—42no—
216C4×Q8—42no—
217#117—22no—
218#117—22no—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—22no—
223#117C2×Q814yes—
224#117#4114yes—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C22⋊C412yes—
246C2×C4⋊C4C22⋊C412yes—
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C22⋊C412yes—
250C2×C4⋊C4C22⋊C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C414yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42C22×C412yes—
284C2×C42C22×C412yes—
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×Q814yes—
298C2×C42#4112yes—
299C2×C42#4112yes—
300C4⋊C8—42no—
301C4⋊C8—42no—
302#97—42no—
303C4×C8—42no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317C24C22×D411yescharacteristic
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—44no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—82no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
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×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C412yes—
371C22×C4#12012yes—
372C22×C4#12012yes—
373C22×C4#12312yes—
374C22×C4#12312yes—
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×Q812yes—
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4112yes—
383C22×C42−1+412yes—
384C22×C42−1+412yes—
385M16—24no—
386M16—24no—
387M16—42no—
388M16—42no—
389M16—42no—
390M16—42no—
391M16—42no—
392M16—42no—
393C2×C8—24no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—48no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4C4×Q818yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—22no—
439C42—22no—
440C42—24no—
441C42—24no—
442C42—48no—
443C42—48no—
444C42—44no—
445C42—42no—
446C42—42no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—42no—
452C42—82no—
453C42C4×D412yes—
454C42C4×D412yes—
455C42C4×D412yes—
456C42C4×D412yes—
457C42#12112yes—
458C42#12112yes—
459C42#12112yes—
460C42#12112yes—
461C42C4⋊Q814yescomplemented
462C23—22no—
463C23—22no—
464C23C22×C22⋊C411yescharacteristic
465C23C2×#12011yescharacteristic
466C23C2×#12311yescharacteristic
467C23#54812yes—
468C23D4×Q812yes—
469Q8—48no—
470Q8—48no—
471Q8—84no—
472Q8—84no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4#38612yes—
522C2×C4#38612yes—
523C2×C4#38612yes—
524C2×C4#38612yes—
525C2×C4#42114yes—
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×Q812yes—
529C2×C4#51912yes—
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×C4⋊Q811yescharacteristic
533C2×C4#53211yescharacteristic
534C2×C4#53312yes—
535C2×C4#53611yescharacteristic
536C2×C4#53611yescharacteristic
537C2×C4#54112yes—
538C2×C4#54712yes—
539C2×C4#54812yes—
540C2×C4D4×Q812yes—
541C2×C4D4×Q812yes—
542C2×C4#55612yes—
543C2×C4#55712yes—
544C8—44no—
545C8—82no—
546C22—24no—
547C22—24no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22C2×#38611yescharacteristic
553C22Q8×C22⋊C412yes—
554C22#243211yescharacteristic
555C22#243411yescharacteristic
556C22#244612yes—
557C22C2×#42112yes—
558C22#290112yes—
559C4—24no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—44no—
565C4—44no—
566C4—48no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4C4×C4⋊Q811yescharacteristic
575C4#232711yescharacteristic
576C4#236512yes—
577C4#242912yes—
578C4#254511yescharacteristic
579C4#254611yescharacteristic
580C2—21no—
581C2—21no—
582C2—22no—
583C2#1402411yescharacteristic
584C2#2097911yescharacteristic
585C2#2098011yescharacteristic
586C2#2252914yes—
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 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2584C4semidirect
19. #2567C4semidirect
20. #2546C4semidirect
159. #339Q8semidirect
461. C42C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#349532
C2×#22952
C2×#19952
C2×#19862
#213032
#112262
#108792
#108782
#108522
#107012