Finite Group Databasebeta

← #381106

Subgroups — #381106

Order 512.

1548 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#11204C212yesmaximal, complemented
5#11198C212yesmaximal, complemented
6#11044C212yesmaximal, complemented
7#10967C212yesmaximal
8#10962C212yesmaximal, complemented
9#10913C212yesmaximal, complemented
10C22×#123C412yes—
11C22×#121C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#423C2212yes—
14C22×#100C2214yes—
15C2×#407C2212yes—
16C2×#407C2212yes—
17C2×C4×M16C2212yes—
18#2854—21no—
19#2854—21no—
20#2854—21no—
21#2854—21no—
22#2854—21no—
23#2854—21no—
24#2854—21no—
25#2854—21no—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#388C412yes—
31C2×#388C412yes—
32C2×#386C412yes—
33C2×#380C2211yescharacteristic
34#1797—21no—
35#1797—21no—
36#1797—21no—
37#1797—21no—
38#1749—21no—
39#1749—21no—
40C4×C22⋊C8—21no—
41C4×C22⋊C8—21no—
42C2×#343C2214yes—
43C2×#343C2214yes—
44C2×#343C2214yes—
45#1728C2212yes—
46#1728C2212yes—
47C23×D4C2×C412yes—
48C23×D4C2311yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16C2314yes—
53C22×M16C2314yes—
54C2×#123—22no—
55C2×#123—22no—
56C2×#123—22no—
57C2×#123—22no—
58C2×#123—48no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—44no—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#423—44no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×#100—28no—
92C2×#100—44no—
93C2×#100—44no—
94#407—28no—
95#407—44no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C4×M16—44no—
101C2×#97—22no—
102C2×C4×C8—22no—
103#390—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380—21no—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C412yes—
138#343—28no—
139#343—44no—
140#343—44no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145C25C22×C411yescharacteristic
146C25C22×C412yes—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4—44no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
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×C4C22×C411yescharacteristic
181C23×C4C22×C412yes—
182C23×C4C22×C412yes—
183C23×C4C22×C412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2411yescharacteristic
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16#4118yes—
201C22×C8—24no—
202C22×C8—24no—
203#123—48no—
204#123—48no—
205#123—48no—
206#123—48no—
207#121—44no—
208#121—44no—
209#121—44no—
210#121—44no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
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×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C4⋊C8—44no—
274C4⋊C8—44no—
275#100—416no—
276C22⋊C8—28no—
277C22⋊C8—28no—
278C22⋊C8—44no—
279C22⋊C8—44no—
280#97—44no—
281C4×C8—44no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24C2×C22⋊C411yescharacteristic
304C24C2×C22⋊C411yescharacteristic
305C24C23×C411yescharacteristic
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—816no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
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—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
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—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—416no—
383C22×C4—416no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4#10014yescomplemented
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4114yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C42−1+412yes—
411M16—28no—
412M16—28no—
413M16—44no—
414M16—44no—
415M16—44no—
416M16—44no—
417M16—44no—
418M16—44no—
419C2×C8—28no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—44no—
423C2×C8—44no—
424C2×C8—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—28no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—48no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23C2×#10012yes—
502C23#41312yes—
503C23C22×C22⋊C411yescharacteristic
504C23C4×#4112yes—
505C23#51812yes—
506C23#52012yes—
507D4—88no—
508D4—88no—
509D4—84no—
510D4—82no—
511D4—82no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4C2×#10012yes—
570C2×C4C2×#10012yes—
571C2×C4#41312yes—
572C2×C4#41312yes—
573C2×C4#52512yes—
574C2×C4#53212yes—
575C2×C4#53412yes—
576C2×C4#53412yes—
577C2×C4#53511yescharacteristic
578C2×C4#53611yescharacteristic
579C2×C4#53712yes—
580C2×C4#53814yes—
581C2×C4#53914yes—
582C2×C4#54014yes—
583C2×C4#54214yes—
584C8—48no—
585C8—84no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—416no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—168no—
615C22#229711yescharacteristic
616C22#230811yescharacteristic
617C22#231111yescharacteristic
618C22#233514yes—
619C22C22×#10011yescharacteristic
620C22C2×#41311yescharacteristic
621C22C2×#41311yescharacteristic
622C22C2×#41311yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4#263014yes—
637C4#266112yes—
638C4#266312yes—
639C2—24no—
640C2—42no—
641C2—42no—
642C2—162no—
643C2#1428711yescharacteristic
644C2#2142411yescharacteristic
645C2#2142611yescharacteristic
646C2#2175211yescharacteristic
647C2#2175411yescharacteristic
648C2#2187811yescharacteristic
649C2#2188011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
3. C4×#407C2semidirect
4. #11204C2semidirect
5. #11198C2semidirect
6. #11044C2semidirect
8. #10962C2semidirect
9. #10913C2semidirect
394. C22×C4#100semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
#206262
C4×#4072
#112042
#111982
#110442
#109672
#109622
#109132