Finite Group Databasebeta

← #362824

Subgroups — #362824

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4#11226C214yesmaximal, complemented
5#10886C214yesmaximal, complemented
6#10733C212yesmaximal
7#10733C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#407C2212yes—
12C2×C4×M16C2212yes—
13#2296—21no—
14#2296—21no—
15#2296—21no—
16#2296—21no—
17#2296C412yescomplemented
18#2296C412yescomplemented
19#2296C2212yes—
20#2296C2212yes—
21C2×#389C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#389C411yescharacteristic, complemented
24C2×#389C411yescharacteristic, complemented
25C2×#385—21no—
26C2×#385—21no—
27C2×#380—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C22⋊C4C2212yes—
32#1749—21no—
33#1749—21no—
34C4×C22⋊C8—21no—
35C4×C22⋊C8—21no—
36C2×#339C2218yes—
37C2×#339C2218yes—
38#1728C2212yes—
39C4⋊#95—21no—
40#1434—21no—
41C22⋊#95—21no—
42C4×#95C2212yes—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16C2314yes—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117—28no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—44no—
53C2×#117C2×C412yes—
54C2×#117C2×C412yes—
55C2×#117C2312yes—
56C2×#117C2312yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42—44no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2314yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#407—24no—
86#407Q818yes—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C4×M16—44no—
90C2×#97—22no—
91C2×C4×C8—22no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#385—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#380—28no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—44no—
126C4×C22⋊C4—28no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135#339—216no—
136#339—48no—
137#336—22no—
138#336—22no—
139C25C2×D411yescharacteristic
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×Q814yes—
171C23×C4#4114yes—
172C23×C4C2411yescharacteristic
173C2×M16—24no—
174C2×M16—28no—
175C2×M16—28no—
176C2×M16C2×Q818yes—
177C22×C8—24no—
178#117—28no—
179#117—28no—
180#117—24no—
181#117—24no—
182#117—24no—
183#117—28no—
184#117—44no—
185#117—44no—
186#117—44no—
187#117#4118yes—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C412yes—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C42—28no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—28no—
232C2×C42—28no—
233C2×C42—28no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—44no—
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C22⋊C8—28no—
262C22⋊C8—28no—
263#97—44no—
264C4×C8—44no—
265#95—28no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—44no—
275#95—44no—
276#95—44no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—41no—
282C24—41no—
283C24C2×C22⋊C411yescharacteristic
284C24C2×C22⋊C411yescharacteristic
285C24#12412yes—
286C24C4⋊Q812yes—
287C24C22×D411yescharacteristic
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—216no—
300C22×C4—216no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—28no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—416no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—416no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#10414yes—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C4×Q814yes—
372C22×C4C4×Q814yes—
373C22×C4C22⋊Q814yes—
374C22×C4C22⋊Q814yes—
375C22×C4#12412yes—
376C22×C4C4⋊Q812yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×Q812yes—
380C22×C4C2×#4114yes—
381C22×C4C2×#4112yes—
382C22×C4H(5,𝔽2)14yes—
383M16—216no—
384M16—48no—
385C2×C8—28no—
386C2×C8—44no—
387C2×C8—44no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—416no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4#11714yes—
406C4⋊C4#11714yes—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—416no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C42—28no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—28no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—44no—
431C42—44no—
432C42—416no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—24no—
443C23—24no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—24no—
449C23—24no—
450C23—21no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23#39112yes—
458C23#39112yes—
459C23C2×#10412yes—
460C23#42112yes—
461C23C22×C22⋊C411yescharacteristic
462C23#52512yes—
463C23C2×#12411yescharacteristic
464C23C2×C4⋊Q811yescharacteristic
465C23#53412yes—
466C2×C4—28no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—28no—
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—21no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4#39112yes—
532C2×C4#39112yes—
533C2×C4C2×#10412yes—
534C2×C4C2×#10412yes—
535C2×C4#42112yes—
536C2×C4#42112yes—
537C2×C4C2×#11712yes—
538C2×C4C2×C4×Q812yes—
539C2×C4#51812yes—
540C2×C4C2×C22⋊Q812yes—
541C2×C4#52512yes—
542C2×C4#53412yes—
543C2×C4#55318yes—
544C2×C4#55418yes—
545C8—48no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—48no—
564C22—48no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22#198614yes—
570C22#228011yescharacteristic
571C22C2×#39111yescharacteristic
572C22#230811yescharacteristic
573C22#258312yes—
574C22#259112yes—
575C22C22×#10411yescharacteristic
576C22C2×#42111yescharacteristic
577C22C2×#42111yescharacteristic
578C22C2×#42111yescharacteristic
579C4—22no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—44no—
586C4—44no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—88no—
592C4—82no—
593C4—82no—
594C4#234514yes—
595C4#257914yes—
596C2—22no—
597C2—22no—
598C2—24no—
599C2—42no—
600C2#1392311yescharacteristic
601C2#2074711yescharacteristic
602C2#2074811yescharacteristic
603C2C2×#198611yescharacteristic
604C2#2130611yescharacteristic
605C2#2177311yescharacteristic
606C2#2177411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11226C2semidirect
5. #10886C2semidirect
17. #2296C4semidirect
18. #2296C4semidirect
21. C2×#389C4semidirect
22. C2×#389C4semidirect
23. C2×#389C4semidirect
24. C2×#389C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C4×#4072
C4×#3802
#112262
#108862
#107332