Finite Group Databasebeta

← #205433

Subgroups — #205433

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3210C211yesmaximal, characteristic
2#21891C211yesmaximal, characteristic
3#21837C211yesmaximal, characteristic, complemented
4#20545C211yesmaximal, characteristic, complemented
5C2×#1574C211yesmaximal, characteristic, complemented
6C2×#1550C211yesmaximal, characteristic, complemented
7C2×#1540C211yesmaximal, characteristic
8#9747C211yesmaximal, characteristic, complemented
9#9745C211yesmaximal, characteristic, complemented
10#9744C211yesmaximal, characteristic, complemented
11#9742C211yesmaximal, characteristic, complemented
12#9737C211yesmaximal, characteristic, complemented
13#9735C211yesmaximal, characteristic, complemented
14#9733C211yesmaximal, characteristic, complemented
15#9732C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3210—24no—
19#3210C418yescomplemented
20C2×#494C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#464—21no—
25C2×#464—21no—
26C2×#452—21no—
27C2×#452—21no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32#2283C2211yescharacteristic
33#1991—22no—
34#1988C2211yescharacteristic
35#1988C2211yescharacteristic
36#1988C2211yescharacteristic
37#1988C2211yescharacteristic
38#1987C2211yescharacteristic
39#1987C2211yescharacteristic
40#1982C2211yescharacteristic
41#1982C2211yescharacteristic
42#1895—22no—
43#1825—22no—
44#1804—22no—
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1768—21no—
50#1768—21no—
51#1752—21no—
52C8×C22⋊C4—21no—
53#1574—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1550—24no—
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61#1540—24no—
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×Q16—21no—
70C22×Q16—21no—
71C22×SD16—21no—
72C22×SD16—21no—
73C22×M16—21no—
74C23×C8—21no—
75#533—22no—
76#533—22no—
77#533—22no—
78#533C2×C414yes—
79#532—22no—
80#532—22no—
81#532—22no—
82#532C2×C414yes—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#125—21no—
88C2×#124—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×#121—21no—
92C2×#121—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×#117—21no—
98C22×C42C2311yescharacteristic
99#494—22no—
100#494C2×C414yes—
101#492—22no—
102#492C2×C414yes—
103#487—22no—
104#487C2×C414yes—
105#486—22no—
106#486C2×C414yes—
107#464—28no—
108#464—28no—
109#452—28no—
110#452—28no—
111#431—22no—
112#431—22no—
113#431C2×C414yes—
114#431D414yes—
115#423—22no—
116#423—22no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8D414yes—
121C2×C4⋊C8D411yescharacteristic
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×#103—21no—
130C2×#103—21no—
131C2×#102—21no—
132C2×#102—21no—
133C2×#102—22no—
134C2×#102—22no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8D411yescharacteristic
137C2×C22⋊C8D411yescharacteristic
138C2×#97—21no—
139C2×#97—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#383—21no—
146#382D412yes—
147#381—21no—
148#380D412yes—
149C4×C4⋊C4C2311yescharacteristic
150C4×C4⋊C4C2311yescharacteristic
151C4×C22⋊C4—21no—
152#376—21no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163#335—22no—
164#335—22no—
165#334—22no—
166#334—22no—
167#333—22no—
168#333—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#332—22no—
173#331—22no—
174#331—22no—
175C2×#41—24no—
176C2×#41—42no—
177C2×#41—81no—
178C2×#41—81no—
179C2×#41—81no—
180C2×#41—81no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—41no—
188C23×C4—41no—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4#4111yescharacteristic
192C2×Q16—24no—
193C2×Q16—24no—
194C2×Q16—81no—
195C2×Q16—81no—
196C2×Q16—81no—
197C2×Q16—81no—
198C2×SD16—24no—
199C2×SD16—24no—
200C2×SD16—81no—
201C2×SD16—81no—
202C2×SD16—81no—
203C2×SD16—81no—
204C2×M16—42no—
205C2×M16—82no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—24no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—81no—
214C22×C8—81no—
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8C22×C412yes—
229C4⋊Q8C22×C412yes—
230C4⋊Q8C22×C412yes—
231#127—21no—
232#127—21no—
233#127—21no—
234#127C22×C412yes—
235#125—24no—
236#125—42no—
237#124—24no—
238#124—42no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—42no—
242C22⋊Q8—42no—
243#121—24no—
244#121—24no—
245#121—42no—
246#121—42no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—42no—
250C4×Q8—42no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—42no—
254C4×D4—42no—
255#117—24no—
256#117—42no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—41no—
274C2×C42C22×C412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D412yes—
279C2×C42#4111yescharacteristic
280C2×C42#4112yes—
281C2×C42C2411yescharacteristic
282C4⋊C8—24no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—42no—
294C4⋊C8C22⋊C414yes—
295C4⋊C8C22⋊C414yes—
296C4⋊C8C2×D412yes—
297C4⋊C8C2×D412yes—
298#103—24no—
299#103—24no—
300#102—24no—
301#102—24no—
302#102—82no—
303#102—82no—
304C22⋊C8—42no—
305C22⋊C8C22⋊C414yes—
306C22⋊C8C22⋊C414yes—
307#97—81no—
308#97—81no—
309C4×C8—42no—
310C4×C8—81no—
311C4×C8—81no—
312C4×C8C22×C412yes—
313C4×C8C22×C412yes—
314C4×C8C2×D412yes—
315C4×C8C2×D412yes—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—21no—
320#95—21no—
321C24—41no—
322C24#12111yescharacteristic
323#41—84no—
324#41—84no—
325#41—84no—
326#41—84no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—82no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—44no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—42no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—42no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4—82no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—44no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4C2×C22⋊C412yes—
374C22×C4C4×D412yes—
375C22×C4C4×D412yes—
376C22×C4#12112yes—
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12412yes—
381C22×C4#12712yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385Q16—84no—
386Q16—84no—
387SD16—84no—
388SD16—84no—
389M16—84no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—84no—
413C2×C8C2×C22⋊C412yes—
414C2×C8C2×C22⋊C412yes—
415C2×C8C2×C22⋊C412yes—
416C2×C8C2×C22⋊C412yes—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—44no—
444C42C2×C22⋊C412yes—
445C42C4×D412yes—
446C42C4×D412yes—
447C42#12112yes—
448C42#12412yes—
449C42#12712yes—
450C42C23×C411yescharacteristic
451C42C22×D411yescharacteristic
452C42C22×D411yescharacteristic
453C42C2×#4111yescharacteristic
454C23—21no—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—42no—
459C23—81no—
460C23—81no—
461C23—82no—
462C23—82no—
463C23#39012yes—
464C23C2×#12111yescharacteristic
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—82no—
470Q8—82no—
471Q8—81no—
472Q8—81no—
473Q8—81no—
474Q8—81no—
475D4—42no—
476D4—42no—
477D4—44no—
478D4—82no—
479D4—82no—
480D4—88no—
481D4—81no—
482D4—81no—
483D4—81no—
484D4—81no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—88no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4#39012yes—
515C2×C4#39012yes—
516C2×C4#39012yes—
517C2×C4C22×C22⋊C411yescharacteristic
518C2×C4C2×C4×D411yescharacteristic
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×#12111yescharacteristic
521C2×C4C2×#12411yescharacteristic
522C2×C4C2×#12711yescharacteristic
523C2×C4#57611yescharacteristic
524C2×C4#57611yescharacteristic
525C2×C4#57711yescharacteristic
526C2×C4#57711yescharacteristic
527C2×C4#57712yes—
528C2×C4#57812yes—
529C8—22no—
530C8—22no—
531C8—22no—
532C8—22no—
533C8—42no—
534C8—42no—
535C8—81no—
536C8—81no—
537C8—81no—
538C8—81no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—42no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—162no—
548C22—162no—
549C22C2×#39011yescharacteristic
550C22#304511yescharacteristic
551C22#304511yescharacteristic
552C22#306312yes—
553C22#306712yes—
554C4—21no—
555C4—21no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4#289711yescharacteristic
565C4#289711yescharacteristic
566C4#294712yes—
567C4#304111yescharacteristic
568C4#304311yescharacteristic
569C4#315411yescharacteristic
570C4#315611yescharacteristic
571C2—42no—
572C2—162no—
573C2#1957712yes—
574C2#1957912yes—
575C2#2123411yescharacteristic
576C2#2124811yescharacteristic
577C2#2126311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21837C2semidirect
4. #20545C2semidirect
5. C2×#1574C2semidirect
6. C2×#1550C2semidirect
8. #9747C2semidirect
9. #9745C2semidirect
10. #9744C2semidirect
11. #9742C2semidirect
12. #9737C2semidirect
13. #9735C2semidirect
14. #9733C2semidirect
15. #9732C2semidirect
19. #3210C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32102
#218912
#218372
#205452
C2×#15742
C2×#15502
C2×#15402
#97472
#97452
#97442
#97422
#97372
#97352
#97332
#97322