Finite Group Databasebeta

← #204234

Subgroups — #204234

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2324C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic, complemented
3#20534C211yesmaximal, characteristic, complemented
4C2×#1466C212yesmaximal, complemented
5C2×#1464C212yesmaximal, complemented
6#9653C212yesmaximal, complemented
7#9652C212yesmaximal, complemented
8#9651C212yesmaximal, complemented
9#9649C212yesmaximal, complemented
10C22×#121C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C2×#432C2212yes—
14C2×#432C2212yes—
15#2324—24no—
16#2324—24no—
17C2×#390C411yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389C411yescharacteristic
21C2×#388C2211yescharacteristic
22C2×#388C2211yescharacteristic
23C2×#386C411yescharacteristic
24C2×#386C411yescharacteristic
25C2×#385C411yescharacteristic
26C2×#385C411yescharacteristic
27C2×#380C411yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1940—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1771—24no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1466—28no—
36#1464—28no—
37C2×#324C2214yes—
38C2×#323C2214yes—
39C2×#323C2214yes—
40C2×#323C2214yes—
41C23×D4C2×C411yescharacteristic
42C23×D4C2×C411yescharacteristic
43C23×D4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C2×#121—24no—
48C2×#121—24no—
49C2×#121—24no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—24no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
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⋊C4C2311yescharacteristic
75C22×C22⋊C4C2311yescharacteristic
76C22×C22⋊C4C2311yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#432—24no—
82#432—24no—
83#432—24no—
84#432D418yescomplemented
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C22⋊C8C2314yes—
90C2×C22⋊C8C2314yes—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97—22no—
94C2×C4×C8—22no—
95#390—24no—
96#390—22no—
97#390—22no—
98#390—22no—
99#390—22no—
100#389—24no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#386—24no—
106#386—24no—
107#385—24no—
108#385—24no—
109#380—24no—
110C4×C4⋊C4D414yes—
111C4×C4⋊C4D414yes—
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#323—28no—
124#323—28no—
125#323—28no—
126C25C22×C411yescharacteristic
127C25C22×C411yescharacteristic
128C25C22×C411yescharacteristic
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4—81no—
140C22×D4—81no—
141C22×D4—81no—
142C22×D4—81no—
143C22×D4M1614yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—42no—
153C23×C4—81no—
154C23×C4—81no—
155C23×C4—81no—
156C23×C4—81no—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
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×C4C2411yescharacteristic
169C22×C8—24no—
170C22×C8—24no—
171#121—232no—
172C4×D4—44no—
173C4×D4—44no—
174C4×D4—44no—
175C4×D4—44no—
176#117—44no—
177#117—44no—
178#117—44no—
179#117—44no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4C22⋊C414yes—
196C2×C4⋊C4C22⋊C414yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
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—22no—
222C2×C22⋊C4—81no—
223C2×C22⋊C4—81no—
224C2×C22⋊C4—81no—
225C2×C22⋊C4—81no—
226C2×C22⋊C4—81no—
227C2×C22⋊C4—81no—
228C2×C22⋊C4M1614yes—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—81no—
232C2×C42—81no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D414yes—
239C2×C42C2×D412yes—
240C2×C42#4114yes—
241C4⋊C8—28no—
242C4⋊C8—44no—
243C4⋊C8—82no—
244C4⋊C8—82no—
245C22⋊C8—24no—
246C22⋊C8—24no—
247C22⋊C8—24no—
248C22⋊C8C2×D418yes—
249#97—28no—
250#97—44no—
251#97—82no—
252C4×C8—82no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95C22⋊C414yes—
258#95C22⋊C414yes—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24C2×M1612yes—
284C24C2×M1612yes—
285C24C2×M1612yes—
286C24C23×C411yescharacteristic
287C2×D4—28no—
288C2×D4—28no—
289C2×D4—28no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—84no—
299C2×D4—84no—
300C2×D4—84no—
301C2×D4—84no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—28no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—48no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—48no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
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—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C4×D414yes—
361C22×C4C4×D414yes—
362C22×C4C2×M1612yes—
363C22×C4C2×M1612yes—
364C22×C4C2×M1612yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C2×#4112yes—
369C2×C8—28no—
370C2×C8—44no—
371C2×C8—44no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—44no—
375C4⋊C4—28no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—44no—
407C42#12114yes—
408C42#12114yes—
409C42#12414yes—
410C42#12714yes—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23C22×C22⋊C411yescharacteristic
467C23C2×C4×D412yes—
468C23C22×M1611yescharacteristic
469C23C22×M1611yescharacteristic
470C23#56812yes—
471D4—416no—
472D4—88no—
473D4—88no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
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—22no—
496C2×C4—24no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—416no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—88no—
521C2×C4—88no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#39014yes—
529C2×C4#39014yes—
530C2×C4#40714yes—
531C2×C4C2×#12112yes—
532C2×C4C2×#12411yescharacteristic
533C2×C4C2×#12711yescharacteristic
534C8—48no—
535C8—84no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22—84no—
557C22—84no—
558C22—84no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22C2×#39011yescharacteristic
572C22C2×#40711yescharacteristic
573C22#287911yescharacteristic
574C22#288311yescharacteristic
575C22#288411yescharacteristic
576C22#288811yescharacteristic
577C22#288811yescharacteristic
578C22D4×M1612yes—
579C22D4×M1612yes—
580C4—24no—
581C4—24no—
582C4—22no—
583C4—21no—
584C4—21no—
585C4—24no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C2—22no—
595C2—22no—
596C2—82no—
597C2—82no—
598C2—82no—
599C2#1736711yescharacteristic
600C2#1737211yescharacteristic
601C2#1740811yescharacteristic
602C2#1740911yescharacteristic
603C2#2119011yescharacteristic
604C2#2121411yescharacteristic
605C2#2121911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21955C2semidirect
3. #20534C2semidirect
4. C2×#1466C2semidirect
5. C2×#1464C2semidirect
6. #9653C2semidirect
7. #9652C2semidirect
8. #9651C2semidirect
9. #9649C2semidirect
84. #432D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23242
#219552
#205342
C2×#14662
C2×#14642
#96532
#96522
#96512
#96492