Finite Group Databasebeta

← #190447

Subgroups — #190447

Order 512.

2782 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊C4C211yesmaximal, characteristic
2#20714C212yesmaximal, complemented
3#8698C212yesmaximal, complemented
4#8559C214yesmaximal, complemented
5#8298C212yesmaximal
6C4×#323C214yesmaximal, complemented
7C25×C4C411yescharacteristic
8C23×C22⋊C4C411yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10C23×C42C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431—22no—
13C2×C4×C22⋊C4—22no—
14C2×C4×C22⋊C4—22no—
15C2×C4×C22⋊C4—22no—
16C2×C4×C22⋊C4—22no—
17C2×C4×C22⋊C4C412yes—
18C2×C4×C22⋊C4C412yes—
19C2×C4×C22⋊C4C2212yes—
20C2×C4×C22⋊C4C2212yes—
21C22×#95C411yescharacteristic
22C22×#95C2211yescharacteristic
23#1817—22no—
24#1797—22no—
25#1797C2214yescomplemented
26#1779C2214yes—
27C4×C22⋊C8—22no—
28C4×C22⋊C8C2214yescomplemented
29C2×#323C2218yes—
30#1280C2218yescomplemented
31C26C2×C411yescharacteristic
32C24×C4—21no—
33C24×C4—21no—
34C24×C4—21no—
35C24×C4—21no—
36C24×C4—28no—
37C24×C4—44no—
38C24×C4—44no—
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C412yes—
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×C22⋊C4—22no—
48C22×C22⋊C4—22no—
49C22×C22⋊C4—24no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—28no—
53C22×C22⋊C4C818yescomplemented
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2312yes—
58C22×C42—21no—
59C22×C42—21no—
60C22×C42—21no—
61C22×C42—21no—
62C22×C42—44no—
63C22×C42—44no—
64C22×C42C2×C412yes—
65C22×C42C2×C412yes—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D412yes—
69C22×C42D412yes—
70C22×C42D412yes—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#431—48no—
75C2×C4⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8C2318yes—
79C2×C4×C8—24no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—24no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95C2×C412yes—
93C2×#95C2×C412yes—
94C2×#95C2×C412yes—
95C2×#95C2312yes—
96#336—24no—
97#336—24no—
98#323—216no—
99C25—21no—
100C25—21no—
101C25—24no—
102C25—22no—
103C25—24no—
104C25—42no—
105C25—42no—
106C25—42no—
107C25—42no—
108C25C2×C814yes—
109C25C22×C411yescharacteristic
110C25C22×C412yes—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—28no—
115C23×C4—28no—
116C23×C4—28no—
117C23×C4—28no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—464no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—44no—
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—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C412yes—
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C412yes—
164C23×C4C2×C814yes—
165C23×C4C2×C814yes—
166C23×C4M1618yes—
167C23×C4M1614yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2411yescharacteristic
179C22×C8—28no—
180C2×C22⋊C4—216no—
181C2×C22⋊C4—216no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—216no—
184C2×C22⋊C4—216no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4C2×C818yes—
194C2×C22⋊C4C2×C818yes—
195C2×C22⋊C4#4118yes—
196C2×C42—28no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—44no—
204C2×C42—44no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—44no—
208C2×C42—44no—
209C2×C42—44no—
210C2×C42—44no—
211C2×C42—44no—
212C2×C42—44no—
213C2×C42—44no—
214C2×C42C22⋊C412yes—
215C2×C42C22⋊C412yes—
216C2×C42C22⋊C412yes—
217C2×C42C22⋊C412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C4⋊C8—48no—
223C22⋊C8—216no—
224C22⋊C8—48no—
225C22⋊C8—48no—
226C4×C8—48no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—24no—
238C24—24no—
239C24—24no—
240C24—24no—
241C24—216no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—24no—
247C24—24no—
248C24—216no—
249C24—42no—
250C24—42no—
251C24—48no—
252C24—48no—
253C24—42no—
254C24—42no—
255C24—416no—
256C24—42no—
257C24—42no—
258C24—416no—
259C24—416no—
260C24—416no—
261C24—42no—
262C24—42no—
263C24—48no—
264C24—48no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C411yescharacteristic
271C24#11714yes—
272C24C22×C812yes—
273C24C22×C814yes—
274C24C2×M1612yes—
275C24C2×M1614yes—
276C24C23×C411yescharacteristic
277C22×C4—232no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—416no—
331C22×C4—432no—
332C22×C4—432no—
333C22×C4—416no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—432no—
339C22×C4—432no—
340C22×C4—416no—
341C22×C4—416no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—48no—
349C22×C4—416no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4C22⋊C818yes—
370C22×C4C22⋊C818yes—
371C22×C4#9914yes—
372C22×C4#10014yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4#11714yes—
381C22×C4C22×C814yes—
382C22×C4C22×C812yes—
383C22×C4C22×C812yes—
384C22×C4C2×M1614yes—
385C22×C4C2×M1614yes—
386C22×C4C2×M1612yes—
387C22×C4C2×M1612yes—
388C22×C4C22×D412yes—
389C22×C4C2×#4114yes—
390C2×C8—48no—
391C2×C8—48no—
392C2×C8—48no—
393C22⋊C4—216no—
394C22⋊C4—216no—
395C22⋊C4—216no—
396C22⋊C4—216no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—44no—
406C42—44no—
407C42—44no—
408C42—44no—
409C23—28no—
410C23—28no—
411C23—28no—
412C23—28no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—21no—
430C23—21no—
431C23—216no—
432C23—216no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—48no—
447C23—48no—
448C23—464no—
449C23—464no—
450C23—44no—
451C23—48no—
452C23—44no—
453C23—48no—
454C23—44no—
455C23—48no—
456C23—48no—
457C23—44no—
458C23—48no—
459C23—48no—
460C23—44no—
461C23—48no—
462C23—48no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—48no—
468C23—48no—
469C23C2×C22⋊C812yes—
470C23C2×C22⋊C812yes—
471C23C2×#9912yes—
472C23#41012yes—
473C23C2×#10012yes—
474C23#41312yes—
475C23#43118yescomplemented
476C23C22×C22⋊C411yescharacteristic
477C23C2×#11712yes—
478C23C23×C811yescharacteristic
479C23C22×M1611yescharacteristic
480C23C22×M1612yes—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—216no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—416no—
515C2×C4—416no—
516C2×C4—416no—
517C2×C4—44no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—48no—
533C2×C4—416no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4C2×C22⋊C814yes—
538C2×C4C2×C22⋊C814yes—
539C2×C4#40712yes—
540C2×C4#40712yes—
541C2×C4C2×#9912yes—
542C2×C4C2×#9912yes—
543C2×C4C2×#10012yes—
544C2×C4C2×#10012yes—
545C2×C4#52514yes—
546C8—88no—
547C22—216no—
548C22—216no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—21no—
553C22—21no—
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—22no—
569C22—28no—
570C22—48no—
571C22—48no—
572C22—48no—
573C22—48no—
574C22—432no—
575C22—432no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22#145914yes—
583C22#146014yes—
584C22#228011yescharacteristic
585C22C22×C22⋊C811yescharacteristic
586C22C2×#40711yescharacteristic
587C22C22×#9911yescharacteristic
588C22C2×#41011yescharacteristic
589C22C22×#10011yescharacteristic
590C22C2×#41311yescharacteristic
591C22C2×#43112yes—
592C22#292012yes—
593C4—24no—
594C4—24no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—24no—
599C4—44no—
600C4—44no—
601C4—42no—
602C4—44no—
603C4#146214yes—
604C2—22no—
605C2—22no—
606C2—24no—
607C2—24no—
608C2—48no—
609C2C2×#145911yescharacteristic
610C2C2×#146011yescharacteristic
611C2C2×#146211yescharacteristic
612C2#1735711yescharacteristic
613C2#2071411yescharacteristic
614C2#2072211yescharacteristic
615C2#2072611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. #8698C2semidirect
4. #8559C2semidirect
6. C4×#323C2semidirect
25. #1797C22semidirect
28. C4×C22⋊C8C22semidirect
30. #1280C22semidirect
53. C22×C22⋊C4C8semidirect
475. C23#431semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊C42
#207142
#86982
#85592
#82982
C4×#3232