Finite Group Databasebeta

← #204218

Subgroups — #204218

Order 512.

1363 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2661C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic, complemented
3#20524C211yesmaximal, characteristic, complemented
4C2×#1471C212yesmaximal, complemented
5C2×#1468C212yesmaximal, complemented
6#9653C212yesmaximal, complemented
7#9652C212yesmaximal, complemented
8#9651C212yesmaximal, complemented
9#9649C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×#121C2211yescharacteristic
12C2×#407C2212yes—
13C22×C22⋊C8C2212yes—
14#2661—24no—
15#2661—24no—
16C2×#398C411yescharacteristic
17C2×#397C411yescharacteristic
18C2×#394C411yescharacteristic
19C2×#394C411yescharacteristic
20C2×#394C411yescharacteristic
21C2×#392C411yescharacteristic
22C2×#392C411yescharacteristic
23C2×#392C411yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#388C2211yescharacteristic
27C2×#388C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1850—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1753—24no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1471—28no—
36#1468—28no—
37C2×#324C2214yes—
38C2×#323C2214yes—
39C2×#323C2214yes—
40C2×#323C2214yes—
41C23×D4C2×C411yescharacteristic
42C23×D4C2×C411yescharacteristic
43C23×D4C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C23×D4C2×C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C23×C8—22no—
50C2×#123—22no—
51C2×#123—22no—
52C2×#123—24no—
53C2×#123—24no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—24no—
57C2×#121—24no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2311yescharacteristic
76C22×C22⋊C4C2311yescharacteristic
77C22×C22⋊C4C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82#407—24no—
83#407D418yescomplemented
84C2×C22⋊C8—22no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8C2314yes—
89C2×C22⋊C8C2314yes—
90C2×#97—22no—
91C2×C4×C8—22no—
92#398—24no—
93#397—24no—
94#394—24no—
95#394—24no—
96#394—24no—
97#392—24no—
98#392—24no—
99#392—24no—
100#390—22no—
101#390—22no—
102#390—22no—
103#390—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108C4×C22⋊C4D414yes—
109C4×C22⋊C4D414yes—
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#324—28no—
121#323—28no—
122#323—28no—
123#323—28no—
124C25C22×C411yescharacteristic
125C25C22×C411yescharacteristic
126C25C22×C411yescharacteristic
127C25C22×C411yescharacteristic
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—42no—
140C22×D4—81no—
141C22×D4—81no—
142C22×D4—81no—
143C22×D4—81no—
144C22×D4—81no—
145C22×D4—81no—
146C22×D4—81no—
147C22×D4—81no—
148C22×D4—81no—
149C22×D4—81no—
150C22×D4—81no—
151C22×D4—81no—
152C23×C4—81no—
153C23×C4—81no—
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×C4C22×C411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D414yes—
167C23×C4#4114yes—
168C23×C4C2411yescharacteristic
169C2×M16—28no—
170C2×M16—82no—
171C2×M16—82no—
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—44no—
175C22×C8—82no—
176C22×C8—82no—
177#123—44no—
178#123—44no—
179#123—44no—
180#123—44no—
181#121—44no—
182#121—44no—
183#121—44no—
184#121—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
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—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—81no—
226C2×C22⋊C4—81no—
227C2×C22⋊C4—81no—
228C2×C22⋊C4—81no—
229C2×C22⋊C4—81no—
230C2×C22⋊C4—81no—
231C2×C22⋊C4C22⋊C414yes—
232C2×C22⋊C4C22⋊C414yes—
233C2×C42—81no—
234C2×C42—81no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C4⋊C8—82no—
240C4⋊C8—82no—
241C22⋊C8—28no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C22⋊C8—44no—
246C22⋊C8C2×D418yes—
247#97—82no—
248C4×C8—82no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95C22⋊C414yes—
254#95C22⋊C414yes—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24C2×C22⋊C412yes—
286C24C4×D414yes—
287C24C23×C411yescharacteristic
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—82no—
306C2×D4—88no—
307C2×D4—88no—
308C2×D4—88no—
309C2×D4—88no—
310C2×D4—88no—
311C2×D4—88no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
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—84no—
352C22×C4—84no—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C4×D414yes—
359C22×C4#12114yes—
360C22×C4#12114yes—
361C22×C4#12414yes—
362C22×C4#12714yes—
363C22×C4#13114yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C2×#4112yes—
368M16—88no—
369C2×C8—28no—
370C2×C8—44no—
371C2×C8—44no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—88no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C22⋊C4—28no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—42no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—28no—
419C23—44no—
420C23—44no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23#39014yes—
468C23C22×C22⋊C411yescharacteristic
469C23C2×C4×D412yes—
470C23C2×#13112yes—
471C23#56812yes—
472D4—88no—
473D4—88no—
474D4—88no—
475D4—88no—
476D4—88no—
477D4—88no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#39014yes—
527C2×C4#40814yes—
528C2×C4C2×#12112yes—
529C2×C4C2×#12411yescharacteristic
530C2×C4C2×#12711yescharacteristic
531C8—48no—
532C8—84no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—88no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—164no—
569C22—164no—
570C22—164no—
571C22C2×#39011yescharacteristic
572C22C2×#40811yescharacteristic
573C22#287911yescharacteristic
574C22#288311yescharacteristic
575C22#288311yescharacteristic
576C22#288811yescharacteristic
577C22#288811yescharacteristic
578C22#293514yes—
579C4—22no—
580C4—21no—
581C4—21no—
582C4—24no—
583C4—24no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C2—24no—
593C2—44no—
594C2—161no—
595C2—161no—
596C2—161no—
597C2#1738611yescharacteristic
598C2#1738911yescharacteristic
599C2#1740511yescharacteristic
600C2#1740811yescharacteristic
601C2#2118911yescharacteristic
602C2#2121311yescharacteristic
603C2#2121911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2661C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21930C2semidirect
3. #20524C2semidirect
4. C2×#1471C2semidirect
5. C2×#1468C2semidirect
6. #9653C2semidirect
7. #9652C2semidirect
8. #9651C2semidirect
9. #9649C2semidirect
83. #407D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26612
#219302
#205242
C2×#14712
C2×#14682
#96532
#96522
#96512
#96492