Finite Group Databasebeta

← #195874

Subgroups — #195874

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×M16C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic
4C2×#1491C212yesmaximal, complemented
5#9077C214yesmaximal, complemented
6#9071C214yesmaximal, complemented
7#7904C212yesmaximal, complemented
8C23×M16—21no—
9C22×C22⋊Q8C411yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#432C2212yes—
13C2×C4×M16—21no—
14C2×C4×M16—21no—
15C2×C4×M16—28no—
16C2×C4×M16C2212yes—
17C2×C4×M16C2212yes—
18C22×#97—21no—
19C22×C4×C8C2212yes—
20C4×C22⋊Q8—21no—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8C412yes—
25C4×C22⋊Q8C412yes—
26C4×C22⋊Q8C2212yes—
27C4×C22⋊Q8C2212yes—
28C2×#391C411yescharacteristic
29C2×#389C411yescharacteristic
30C2×#387—21no—
31C2×#385—21no—
32C2×#380C411yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4—21no—
36#1817—21no—
37#1797C2212yes—
38#1797C2212yes—
39#1779—21no—
40#1749—21no—
41#1749—21no—
42#1491—28no—
43C2×Q8⋊C8C2218yescomplemented
44#1279C2218yescomplemented
45C23×Q8C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C22×M16—28no—
50C22×M16—28no—
51C22×M16—28no—
52C22×M16—22no—
53C22×M16—22no—
54C23×C8—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—28no—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—24no—
63C2×C4×Q8—24no—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2312yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—24no—
75C22×C42—24no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#432—24no—
91#432—24no—
92C2×C4⋊C8C2314yes—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C4×M16—28no—
97C4×M16—28no—
98C4×M16—24no—
99C4×M16—28no—
100C4×M16D418yes—
101C2×#97—28no—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—22no—
105C2×C4×C8—22no—
106C2×C4×C8—28no—
107C2×C4×C8C2314yes—
108#391—21no—
109#391—21no—
110#391C2×C412yes—
111#391C2×C412yes—
112#389—21no—
113#389—21no—
114#389C2×C412yes—
115#389C2×C412yes—
116#387—22no—
117#387—22no—
118#387—22no—
119#387—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#380—21no—
125#380—21no—
126#380C2×C412yes—
127#380C2×C412yes—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4C2312yes—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145#336—22no—
146#336—22no—
147Q8⋊C8—216no—
148C25C22⋊C411yescharacteristic
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8C22×C412yes—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C2×M16—264no—
187C2×M16—28no—
188C2×M16—28no—
189C2×M16—28no—
190C2×M16—28no—
191C2×M16—28no—
192C2×M16—28no—
193C22×C8—28no—
194C22×C8—28no—
195C22×C8—28no—
196C22×C8—24no—
197C22×C8—24no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—24no—
201C22⋊Q8—24no—
202C22⋊Q8—24no—
203C22⋊Q8M1618yescomplemented
204C4×Q8—28no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—42no—
208C4×Q8—44no—
209C4×Q8—42no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—44no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D414yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4114yes—
274C2×C42C2411yescharacteristic
275C4⋊C8—24no—
276C4⋊C8—24no—
277C22⋊C8—44no—
278C22⋊C8—44no—
279C22⋊C8—44no—
280#97—28no—
281#97—28no—
282#97—44no—
283C4×C8—28no—
284C4×C8—24no—
285C4×C8C2×D418yes—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—41no—
301C24—41no—
302C24#10414yes—
303C24C2×C22⋊C411yescharacteristic
304C24C2×C22⋊C411yescharacteristic
305C24C2×C22⋊C411yescharacteristic
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—416no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8C2×M1614yescomplemented
314C22×C4—216no—
315C22×C4—216no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4#10414yes—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C4×D414yes—
395C22×C4#12012yes—
396C22×C4#12114yes—
397C22×C4#12312yes—
398C22×C4C2×M1614yes—
399C22×C4#13714yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403M16—232no—
404M16—232no—
405C2×C8—232no—
406C2×C8—28no—
407C2×C8—28no—
408C2×C8—28no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—22no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4C2×M1614yescomplemented
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C42—22no—
431C42—22no—
432C42—21no—
433C42—21no—
434C42—22no—
435C42—22no—
436C42—21no—
437C42—21no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—44no—
445C42—44no—
446C42—44no—
447C42—44no—
448C42C4×D414yes—
449C42#12012yes—
450C42#12114yes—
451C42#12312yes—
452C42C22×D412yes—
453C42C2×#4112yes—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—28no—
468C23—28no—
469C23—21no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23#38612yes—
476C23#40712yes—
477C23#41912yes—
478C23C2×#10412yes—
479C23C2×#10412yes—
480C23C2×#10412yes—
481C23C2×#10412yes—
482C23C22×C22⋊C411yescharacteristic
483Q8—28no—
484Q8—48no—
485Q8—44no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4#38612yes—
546C2×C4#38612yes—
547C2×C4#38612yes—
548C2×C4#40712yes—
549C2×C4#41912yes—
550C2×C4C2×#10412yes—
551C2×C4C2×#10412yes—
552C2×C4C2×C4×D412yes—
553C2×C4C2×#12011yescharacteristic
554C2×C4C2×#12112yes—
555C2×C4C2×#12311yescharacteristic
556C2×C4C22×M1612yes—
557C2×C4#56812yes—
558C2×C4C2×#13712yes—
559C2×C4#57512yes—
560C8—216no—
561C8—84no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—416no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22#149114yescomplemented
583C22#189614yes—
584C22C2×#38611yescharacteristic
585C22C2×#40711yescharacteristic
586C22#287911yescharacteristic
587C22C2×#41911yescharacteristic
588C22#289611yescharacteristic
589C22C22×#10411yescharacteristic
590C22C22×#10411yescharacteristic
591C4—22no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—21no—
596C4—21no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—44no—
601C4—41no—
602C4—44no—
603C4—41no—
604C4—82no—
605C4—82no—
606C4D4×M1614yes—
607C4#305614yes—
608C2—24no—
609C2—24no—
610C2—42no—
611C2C2×#149111yescharacteristic
612C2#1744711yescharacteristic
613C2#1750111yescharacteristic
614C2#1750211yescharacteristic
615C2#2089811yescharacteristic
616C2#2097211yescharacteristic
617C2C2×#189611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1491C2semidirect
5. #9077C2semidirect
6. #9071C2semidirect
7. #7904C2semidirect
43. C2×Q8⋊C8C22semidirect
44. #1279C22semidirect
203. C22⋊Q8M16semidirect
313. C2×Q8C2×M16semidirect
425. C4⋊C4C2×M16semidirect
582. C22#1491semidirect

Maximal subgroups

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

subgroupindex
C22×C4×M162
C2×C4×C22⋊Q82
#207152
C2×#14912
#90772
#90712
#79042