Finite Group Databasebeta

← #162236

Subgroups — #162236

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#7698C214yesmaximal, complemented
5#7678C214yesmaximal, complemented
6#7576C212yesmaximal, complemented
7#7572C212yesmaximal, complemented
8C22×#123—21no—
9C22×#121C411yescharacteristic
10C23×C42C2211yescharacteristic
11C22×C4⋊C8—21no—
12C2×#407—21no—
13C22×C22⋊C8—21no—
14C2×C4×M16—21no—
15#2854—21no—
16#2854—21no—
17#2854—21no—
18#2854—21no—
19#2854C412yes—
20#2854C412yes—
21#2854C2212yes—
22#2854C2212yes—
23C2×#390C411yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C411yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388—21no—
28C2×#388—21no—
29C2×#386—21no—
30C2×#380C411yescharacteristic
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yescomplemented
35#1749—21no—
36#1749C2212yescomplemented
37C4×C22⋊C8—21no—
38C4×C22⋊C8C2212yes—
39#1728C2212yescomplemented
40C2×#336C2212yescomplemented
41#1271C2218yescomplemented
42#1270C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C22×M16—22no—
49C23×C8—22no—
50C2×#123—22no—
51C2×#123—22no—
52C2×#123—22no—
53C2×#123—22no—
54C2×#123—48no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—44no—
58C2×#121C2×C412yes—
59C2×#121C2×C412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—44no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—44no—
87#407—28no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—44no—
93C4×M16—44no—
94C2×#97—22no—
95C2×C4×C8—22no—
96#390—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390C2×C412yes—
101#390C2×C412yes—
102#390C2×C412yes—
103#390C2312yes—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389C2×C412yes—
109#389C2×C412yes—
110#389C2×C412yes—
111#389C2312yes—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#380—21no—
125#380—21no—
126#380C2×C412yes—
127#380C2×C412yes—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131#336—22no—
132#336—22no—
133#336C2314yes—
134#336C2314yes—
135C25—21no—
136C25C22⋊C411yescharacteristic
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—24no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4C22×C412yes—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C2×M16—28no—
179C2×M16—44no—
180C2×M16—44no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—44no—
184C22×C8—44no—
185C22×C8—44no—
186#123—48no—
187#123—48no—
188#123—48no—
189#123—48no—
190#121—44no—
191#121—44no—
192#121—44no—
193#121—44no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2411yescharacteristic
256C4⋊C8—44no—
257C4⋊C8—44no—
258C22⋊C8—28no—
259C22⋊C8—28no—
260C22⋊C8—44no—
261C22⋊C8—44no—
262#97—44no—
263C4×C8—44no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24#10414yes—
285C24C2×C22⋊C411yescharacteristic
286C24C2×C22⋊C411yescharacteristic
287C24C2×C22⋊C411yescharacteristic
288C2×D4—24no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—88no—
301C2×D4—816no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
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—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
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—22no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—416no—
358C22×C4—416no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4#9914yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4#13114yes—
381C22×C4AGL(1,ℤ/8)14yes—
382C22×C4AGL(1,ℤ/8)12yes—
383C22×C4#13712yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387M16—416no—
388C2×C8—28no—
389C2×C8—44no—
390C2×C8—44no—
391C2×C8—44no—
392C2×C8—44no—
393C2×C8—44no—
394C2×C8—88no—
395C4⋊C4—24no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—28no—
449C23—28no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—84no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23—84no—
468C23—84no—
469C23#40812yes—
470C23C2×#9912yes—
471C23#41712yes—
472C23#41812yes—
473C23C2×#10412yes—
474C23C2×#10412yes—
475C23C2×#10412yes—
476C23C22×C22⋊C411yescharacteristic
477D4—84no—
478D4—88no—
479D4—88no—
480D4—84no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—21no—
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—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4#40812yes—
541C2×C4C2×#9912yes—
542C2×C4C2×#9912yes—
543C2×C4#41312yes—
544C2×C4#41712yes—
545C2×C4#41812yes—
546C2×C4#42112yes—
547C2×C4C2×#13112yes—
548C2×C4C2×AGL(1,ℤ/8)12yes—
549C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
550C2×C4C2×#13711yescharacteristic
551C8—48no—
552C8—84no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—416no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—168no—
581C22#148314yes—
582C22#150714yes—
583C22C2×#40811yescharacteristic
584C22C22×#9911yescharacteristic
585C22C2×#41311yescharacteristic
586C22C2×#41811yescharacteristic
587C22C2×#41711yescharacteristic
588C22C22×#10411yescharacteristic
589C22C2×#42111yescharacteristic
590C4—22no—
591C4—22no—
592C4—21no—
593C4—21no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4#146814yes—
606C2—24no—
607C2—24no—
608C2—42no—
609C2—162no—
610C2C2×#146811yescharacteristic
611C2C2×#148311yescharacteristic
612C2#1747811yescharacteristic
613C2C2×#150711yescharacteristic
614C2#1759211yescharacteristic
615C2#1765011yescharacteristic
616C2#1774511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1797C2semidirect
3. C4×#407C2semidirect
4. #7698C2semidirect
5. #7678C2semidirect
6. #7576C2semidirect
7. #7572C2semidirect
34. #1797C22semidirect
36. #1749C22semidirect
39. #1728C22semidirect
40. C2×#336C22semidirect
41. #1271C22semidirect
42. #1270C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#17972
C4×#4072
#76982
#76782
#75762
#75722