Finite Group Databasebeta

← #188292

Subgroups — #188292

Order 512.

2838 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊C4C211yesmaximal, characteristic
2#20626C212yesmaximal, complemented
3C2×#1462C212yesmaximal
4#8559C214yesmaximal, complemented
5#8557C214yesmaximal, complemented
6#7909C212yesmaximal, complemented
7C25×C4C411yescharacteristic
8C23×C22⋊C4C411yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10C23×C42C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#423—22no—
13C2×#407C2214yes—
14C2×C4×C22⋊C4—22no—
15C2×C4×C22⋊C4—22no—
16C2×C4×C22⋊C4—22no—
17C2×C4×C22⋊C4—22no—
18C2×C4×C22⋊C4C412yes—
19C2×C4×C22⋊C4C412yes—
20C2×C4×C22⋊C4C2212yes—
21C2×C4×C22⋊C4C2212yes—
22C22×#95C411yescharacteristic
23C22×#95C2211yescharacteristic
24#1797—22no—
25#1797—22no—
26#1797C2214yescomplemented
27#1797C2214yescomplemented
28#1728—22no—
29#1462—28no—
30C2×#323C2218yes—
31#1280C2218yescomplemented
32C26C2×C411yescharacteristic
33C24×C4—21no—
34C24×C4—21no—
35C24×C4—21no—
36C24×C4—21no—
37C24×C4—28no—
38C24×C4—44no—
39C24×C4—44no—
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C412yes—
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C24×C4C2311yescharacteristic
48C22×M16—24no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—24no—
54C22×C22⋊C4—28no—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4D414yes—
59C22×C22⋊C4Q814yes—
60C22×C22⋊C4C2312yes—
61C22×C42—21no—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42—21no—
65C22×C42—44no—
66C22×C42—44no—
67C22×C42C2×C412yes—
68C22×C42C2×C412yes—
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42D412yes—
72C22×C42D412yes—
73C22×C42D412yes—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#423—48no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80#407—28no—
81#407—28no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8C2318yes—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—24no—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—24no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95C2×C412yes—
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2312yes—
100#336—24no—
101#336—24no—
102#323—216no—
103C25—21no—
104C25—21no—
105C25—22no—
106C25—24no—
107C25—24no—
108C25—42no—
109C25—42no—
110C25—42no—
111C25—42no—
112C25C4⋊C414yes—
113C25C22×C411yescharacteristic
114C25C22×C412yes—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—28no—
119C23×C4—28no—
120C23×C4—28no—
121C23×C4—28no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—464no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C412yes—
165C23×C4C22⋊C412yes—
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C4⋊C414yes—
169C23×C4M1618yescomplemented
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C412yes—
174C23×C4C22×C412yes—
175C23×C4C22×C412yes—
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×Q812yes—
182C23×C4#4114yes—
183C23×C4C2411yescharacteristic
184C2×M16—48no—
185C2×M16—48no—
186C2×M16—48no—
187C22×C8—28no—
188C2×C22⋊C4—216no—
189C2×C22⋊C4—216no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4C4⋊C418yes—
204C2×C22⋊C4M1618yescomplemented
205C2×C22⋊C4C2×D414yes—
206C2×C22⋊C4C2×Q814yes—
207C2×C42—28no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—44no—
215C2×C42—44no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42—44no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—44no—
225C2×C42C22⋊C412yes—
226C2×C42C22⋊C412yes—
227C2×C42C22⋊C412yes—
228C2×C42C22⋊C412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C4⋊C8—48no—
234C4⋊C8—48no—
235C22⋊C8—28no—
236C22⋊C8—28no—
237C22⋊C8—48no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—216no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—24no—
258C24—24no—
259C24—216no—
260C24—42no—
261C24—42no—
262C24—48no—
263C24—48no—
264C24—42no—
265C24—42no—
266C24—416no—
267C24—42no—
268C24—42no—
269C24—416no—
270C24—416no—
271C24—416no—
272C24—48no—
273C24—48no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24C2×C22⋊C411yescharacteristic
281C24C2×C22⋊C411yescharacteristic
282C24C2×C4⋊C412yes—
283C24C2×C4⋊C414yes—
284C24#11712yes—
285C24C2×M1614yes—
286C24C2×M1614yes—
287C24C23×C411yescharacteristic
288C22×C4—232no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—48no—
341C22×C4—41no—
342C22×C4—416no—
343C22×C4—432no—
344C22×C4—432no—
345C22×C4—416no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—432no—
351C22×C4—432no—
352C22×C4—416no—
353C22×C4—416no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—416no—
357C22×C4—416no—
358C22×C4—48no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—416no—
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—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4#9914yes—
382C22×C4#10014yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C4⋊C414yes—
391C22×C4C2×C4⋊C412yes—
392C22×C4#11712yes—
393C22×C4#12114yes—
394C22×C4#12114yes—
395C22×C4C22⋊Q814yes—
396C22×C4C22⋊Q814yes—
397C22×C4C2×M1614yes—
398C22×C4C2×M1614yes—
399C22×C4C2×M1614yes—
400C22×C4C2×M1614yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4112yes—
405M16—816no—
406C2×C8—48no—
407C2×C8—48no—
408C2×C8—48no—
409C22⋊C4—216no—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C22⋊C4—216no—
413C22⋊C4—216no—
414C42—24no—
415C42—24no—
416C42—24no—
417C42—24no—
418C42—42no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—44no—
423C42—44no—
424C42—44no—
425C42—44no—
426C23—28no—
427C23—28no—
428C23—28no—
429C23—28no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
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—24no—
447C23—24no—
448C23—216no—
449C23—216no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—48no—
464C23—48no—
465C23—464no—
466C23—464no—
467C23—44no—
468C23—48no—
469C23—44no—
470C23—48no—
471C23—44no—
472C23—48no—
473C23—48no—
474C23—44no—
475C23—48no—
476C23—48no—
477C23—44no—
478C23—48no—
479C23—48no—
480C23—48no—
481C23—48no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23#38012yes—
487C23#38012yes—
488C23C2×#9912yes—
489C23#41012yes—
490C23C2×#10012yes—
491C23#41312yes—
492C23#42318yescomplemented
493C23C22×C22⋊C411yescharacteristic
494C23C22×C4⋊C411yescharacteristic
495C23C2×#11711yescharacteristic
496C23C22×M1612yes—
497C23C22×M1612yes—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—416no—
535C2×C4—416no—
536C2×C4—416no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—416no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4#38014yes—
558C2×C4#40714yes—
559C2×C4#40712yes—
560C2×C4#40712yes—
561C2×C4C2×#9912yes—
562C2×C4C2×#9912yes—
563C2×C4C2×#10012yes—
564C2×C4C2×#10012yes—
565C2×C4C2×#12112yes—
566C2×C4C2×C22⋊Q812yes—
567C8—88no—
568C22—216no—
569C22—216no—
570C22—22no—
571C22—22no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—21no—
576C22—21no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—28no—
591C22—48no—
592C22—432no—
593C22—432no—
594C22—48no—
595C22—48no—
596C22—48no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22#179914yes—
604C22#180014yes—
605C22C2×#38011yescharacteristic
606C22C2×#40711yescharacteristic
607C22C2×#40711yescharacteristic
608C22C22×#9911yescharacteristic
609C22C2×#41011yescharacteristic
610C22C22×#10011yescharacteristic
611C22C2×#41311yescharacteristic
612C22C2×#42312yes—
613C22#292012yes—
614C4—22no—
615C4—22no—
616C4—24no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—24no—
621C4—44no—
622C4—44no—
623C4—42no—
624C4—44no—
625C4#146214yescomplemented
626C2—24no—
627C2—24no—
628C2—22no—
629C2—22no—
630C2—48no—
631C2C2×#146211yescharacteristic
632C2#1735711yescharacteristic
633C2#1736211yescharacteristic
634C2#1736411yescharacteristic
635C2#2062611yescharacteristic
636C2C2×#179911yescharacteristic
637C2C2×#180011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
4. #8559C2semidirect
5. #8557C2semidirect
6. #7909C2semidirect
26. #1797C22semidirect
27. #1797C22semidirect
31. #1280C22semidirect
169. C23×C4M16semidirect
204. C2×C22⋊C4M16semidirect
492. C23#423semidirect
625. C4#1462semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊C42
#206262
C2×#14622
#85592
#85572
#79092