Finite Group Databasebeta

← #204233

Subgroups — #204233

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic, complemented
3#20524C211yesmaximal, characteristic, complemented
4C2×#1465C212yesmaximal
5C2×#1463C212yesmaximal, complemented
6#9654C212yesmaximal, complemented
7#9652C212yesmaximal, complemented
8#9651C212yesmaximal, complemented
9#9650C212yesmaximal, complemented
10C22×#123C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#407C2212yes—
14C2×#407C2212yes—
15#2322—24no—
16#2322—24no—
17C2×#389C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#388C2211yescharacteristic
20C2×#388C2211yescharacteristic
21C2×#387C411yescharacteristic
22C2×#387C411yescharacteristic
23C2×#386C411yescharacteristic
24C2×#385C411yescharacteristic
25C2×#385C411yescharacteristic
26C2×#385C411yescharacteristic
27C2×#381C411yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1851—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1753—24no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1465—28no—
36#1463—28no—
37C2×#324C2214yes—
38C2×#324C2214yes—
39C2×#324C2214yes—
40C2×#323C2214yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48C2×#123—24no—
49C2×#123—24no—
50C2×#123—24no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
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×C42C2×C411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81#407—24no—
82#407—24no—
83#407—24no—
84#407D418yes—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8C2314yes—
88C2×C22⋊C8C2314yes—
89C2×#97—22no—
90C2×C4×C8—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#387—24no—
100#387—24no—
101#386—24no—
102#385—24no—
103#385—24no—
104#385—24no—
105#381—24no—
106C4×C22⋊C4D414yes—
107C4×C22⋊C4D414yes—
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#324—28no—
123#324—28no—
124#324—28no—
125#323—28no—
126C25C22×C411yescharacteristic
127C25C22×C411yescharacteristic
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—22no—
132C22×D4—42no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—42no—
142C23×C4—81no—
143C23×C4—81no—
144C23×C4—81no—
145C23×C4—81no—
146C23×C4—81no—
147C23×C4—81no—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D414yes—
163C23×C4#4114yes—
164C23×C4C2411yescharacteristic
165C2×M16—28no—
166C2×M16—44no—
167C2×M16—82no—
168C2×M16—82no—
169C2×M16—82no—
170C2×M16—82no—
171C22×C8—24no—
172C22×C8—24no—
173#123—232no—
174C4×D4—44no—
175C4×D4—44no—
176C4×D4—44no—
177C4×D4—44no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—81no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4M1614yescomplemented
208C2×C22⋊C4—22no—
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—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—81no—
225C2×C22⋊C4—81no—
226C2×C22⋊C4C22⋊C414yes—
227C2×C22⋊C4C22⋊C414yes—
228C2×C22⋊C4M1614yescomplemented
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—81no—
232C2×C42—81no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C4⋊C8—82no—
238C4⋊C8—82no—
239C22⋊C8—28no—
240C22⋊C8—24no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243C22⋊C8—44no—
244C22⋊C8C2×D418yes—
245#97—82no—
246C4×C8—82no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95C22⋊C414yes—
256#95C22⋊C414yes—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24C2×C22⋊C412yes—
273C24C4×D414yes—
274C24C2×M1612yes—
275C24C23×C411yescharacteristic
276C2×D4—28no—
277C2×D4—48no—
278C2×D4—48no—
279C2×D4—44no—
280C2×D4—44no—
281C22×C4—28no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—88no—
343C22×C4—88no—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C4×D414yes—
350C22×C4#12114yes—
351C22×C4#12114yes—
352C22×C4#12414yes—
353C22×C4#12714yes—
354C22×C4C2×M1612yes—
355C22×C4C2×M1612yes—
356C22×C4C2×M1612yes—
357C22×C4C2×M1612yes—
358C22×C4C2×M1612yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D412yes—
362C22×C4C2×#4112yes—
363M16—88no—
364M16—88no—
365C2×C8—28no—
366C2×C8—44no—
367C2×C8—44no—
368C2×C8—44no—
369C2×C8—44no—
370C2×C8—44no—
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—42no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—44no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—24no—
412C23—24no—
413C23—22no—
414C23—24no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—41no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23#39014yes—
439C23C22×C22⋊C411yescharacteristic
440C23C2×C4×D412yes—
441C23C22×M1611yescharacteristic
442C23C22×M1611yescharacteristic
443C23#56812yes—
444D4—416no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—416no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—88no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—88no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4#39014yes—
510C2×C4#40714yes—
511C2×C4C2×#12112yes—
512C2×C4C2×#12411yescharacteristic
513C2×C4C2×#12711yescharacteristic
514C8—48no—
515C8—84no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—48no—
528C22—48no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—84no—
537C22—84no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22C2×#39011yescharacteristic
543C22C2×#40711yescharacteristic
544C22#287911yescharacteristic
545C22#288311yescharacteristic
546C22#288311yescharacteristic
547C22#288911yescharacteristic
548C22#288911yescharacteristic
549C22D4×M1612yes—
550C22D4×M1612yes—
551C4—24no—
552C4—22no—
553C4—21no—
554C4—24no—
555C4—21no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—81no—
566C2—22no—
567C2—22no—
568C2—44no—
569C2—82no—
570C2#1736811yescharacteristic
571C2#1737011yescharacteristic
572C2#1740611yescharacteristic
573C2#1740611yescharacteristic
574C2#2119011yescharacteristic
575C2#2121311yescharacteristic
576C2#2122011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
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. #21934C2semidirect
3. #20524C2semidirect
5. C2×#1463C2semidirect
6. #9654C2semidirect
7. #9652C2semidirect
8. #9651C2semidirect
9. #9650C2semidirect
207. C2×C4⋊C4M16semidirect
228. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
#219342
#205242
C2×#14652
C2×#14632
#96542
#96522
#96512
#96502