Finite Group Databasebeta

← #205250

Subgroups — #205250

Order 512.

1052 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3182C211yesmaximal, characteristic
2#21390C211yesmaximal, characteristic
3#21311C211yesmaximal, characteristic, complemented
4#20480C211yesmaximal, characteristic, complemented
5C2×#1565C211yesmaximal, characteristic, complemented
6C2×#1554C211yesmaximal, characteristic, complemented
7C2×#1551C211yesmaximal, characteristic
8#9747C211yesmaximal, characteristic, complemented
9#9745C211yesmaximal, characteristic, complemented
10#9744C211yesmaximal, characteristic, complemented
11#9739C211yesmaximal, characteristic, complemented
12#9737C211yesmaximal, characteristic, complemented
13#9735C211yesmaximal, characteristic, complemented
14#9732C211yesmaximal, characteristic, complemented
15#9730C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3182—24no—
19#3182C418yescomplemented
20C2×#500C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#468—21no—
25C2×#468—21no—
26C2×#440—21no—
27C2×#440—21no—
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1989—22no—
34#1988C2211yescharacteristic
35#1988C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2211yescharacteristic
38#1987C2211yescharacteristic
39#1987C2211yescharacteristic
40#1982C2211yescharacteristic
41#1982C2211yescharacteristic
42#1875—22no—
43#1770C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×#102—22no—
48#1753—21no—
49#1753—21no—
50#1745—21no—
51#1735—22no—
52C4×#97—21no—
53#1565—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1554—24no—
59#1551—24no—
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#333C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×SD16—21no—
70C22×SD16—21no—
71C22×M16D411yescharacteristic
72C22×M16D411yescharacteristic
73C23×C8—21no—
74#533—22no—
75#533—22no—
76#533—22no—
77#533C2×C414yes—
78#532—22no—
79#532—22no—
80#532—22no—
81#532C2×C414yes—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#125—21no—
87C2×#124—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×#121—21no—
91C2×#121—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×#117—21no—
97C2×#117D412yes—
98C22×C42C2311yescharacteristic
99#500—22no—
100#500C2×C414yes—
101#492—22no—
102#492C2×C414yes—
103#487—22no—
104#487C2×C414yes—
105#486—22no—
106#486C2×C414yes—
107#468—28no—
108#468—28no—
109#440—28no—
110#440—28no—
111#432—22no—
112#432D414yes—
113#431—22no—
114#431—22no—
115C2×#107—21no—
116C2×#107—21no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#103—21no—
127C2×#103—21no—
128C2×#102—21no—
129C2×#102—21no—
130C2×#102—22no—
131C2×#102—22no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C4×M16C2×C414yes—
135C4×M16D414yes—
136C2×#97—21no—
137C2×#97—21no—
138C2×#97—21no—
139C2×#97D411yescharacteristic
140C2×#97D411yescharacteristic
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C4×C4⋊C4D412yes—
147C4×C4⋊C4C2311yescharacteristic
148C4×C4⋊C4C2311yescharacteristic
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151#376—21no—
152C43—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#333—22no—
170#333—22no—
171#332—22no—
172#332—22no—
173#331—22no—
174#331—22no—
175C2×#41—24no—
176C2×#41—42no—
177C2×#41—84no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—41no—
185C23×C4—41no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4#4111yescharacteristic
189C2×SD16—24no—
190C2×SD16—24no—
191C2×SD16—42no—
192C2×SD16—42no—
193C2×SD16—42no—
194C2×SD16—42no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—41no—
198C2×M16—41no—
199C2×M16C22⋊C414yes—
200C2×M16C22⋊C414yes—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—44no—
208C22×C8—42no—
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8C22×C412yes—
223C4⋊Q8C22×C412yes—
224C4⋊Q8C22×C412yes—
225#127—21no—
226#127—21no—
227#127—21no—
228#127C22×C412yes—
229#125—24no—
230#125—42no—
231#124—24no—
232#124—42no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C22⋊Q8—42no—
236C22⋊Q8—42no—
237#121—24no—
238#121—24no—
239#121—42no—
240#121—42no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—42no—
244C4×Q8—42no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—42no—
248C4×D4—42no—
249#117—24no—
250#117—24no—
251#117—42no—
252#117—42no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—41no—
272C2×C42C22×C412yes—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42#4111yescharacteristic
278C2×C42#4112yes—
279C2×C42C2411yescharacteristic
280#107—24no—
281#107—24no—
282C4⋊C8—21no—
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—42no—
293C4⋊C8C2×D412yes—
294C4⋊C8C2×D412yes—
295#103—24no—
296#103—24no—
297#102—24no—
298#102—24no—
299#102—44no—
300#102—44no—
301C22⋊C8—24no—
302C22⋊C8—42no—
303#97—24no—
304#97—42no—
305#97—42no—
306#97C22⋊C414yes—
307#97C22⋊C414yes—
308C4×C8—42no—
309C4×C8—42no—
310C4×C8—42no—
311C4×C8C22×C412yes—
312C4×C8C22×C412yes—
313C4×C8C2×D412yes—
314C4×C8C2×D412yes—
315#95—22no—
316#95—21no—
317#95—21no—
318C24—41no—
319C24#12111yescharacteristic
320#41—88no—
321#41—88no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—44no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—82no—
348C2×D4—84no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—44no—
362C22×C4—84no—
363C22×C4C2×C22⋊C412yes—
364C22×C4C4×D412yes—
365C22×C4C4×D412yes—
366C22×C4#12112yes—
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12111yescharacteristic
370C22×C4#12412yes—
371C22×C4#12712yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375SD16—48no—
376SD16—48no—
377M16—28no—
378M16—44no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—84no—
400C2×C8C2×C22⋊C412yes—
401C2×C8C2×C22⋊C412yes—
402C2×C8C2×C22⋊C412yes—
403C2×C8C2×C22⋊C412yes—
404C2×C8#13512yes—
405C2×C8#13512yes—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4#13714yes—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C42—44no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—44no—
434C42—82no—
435C42C2×C22⋊C412yes—
436C42C4×D412yes—
437C42C4×D412yes—
438C42#12112yes—
439C42#12412yes—
440C42#12712yes—
441C42C23×C411yescharacteristic
442C42C22×D411yescharacteristic
443C42C22×D411yescharacteristic
444C42C2×#4111yescharacteristic
445C23—21no—
446C23—21no—
447C23—22no—
448C23—22no—
449C23—42no—
450C23—81no—
451C23—81no—
452C23—82no—
453C23—82no—
454C23#39012yes—
455C23C2×#12111yescharacteristic
456Q8—42no—
457Q8—42no—
458Q8—82no—
459Q8—82no—
460Q8—82no—
461Q8—82no—
462Q8—81no—
463Q8—81no—
464D4—42no—
465D4—42no—
466D4—44no—
467D4—82no—
468D4—82no—
469D4—88no—
470D4—81no—
471D4—81no—
472D4—81no—
473D4—81no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—88no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4#39012yes—
503C2×C4#39012yes—
504C2×C4#39012yes—
505C2×C4C22×C22⋊C411yescharacteristic
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×#12411yescharacteristic
510C2×C4C2×#12711yescharacteristic
511C2×C4C2×#13511yescharacteristic
512C2×C4C2×#13511yescharacteristic
513C2×C4C2×#13712yes—
514C2×C4#57511yescharacteristic
515C2×C4#57511yescharacteristic
516C2×C4#57512yes—
517C8—22no—
518C8—22no—
519C8—22no—
520C8—42no—
521C8—42no—
522C8—81no—
523C8—81no—
524C8—81no—
525C8—81no—
526C8#41614yes—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—162no—
536C22—162no—
537C22C2×#39011yescharacteristic
538C22#303511yescharacteristic
539C22#303511yescharacteristic
540C22#305612yes—
541C22#306812yes—
542C4—22no—
543C4—21no—
544C4—21no—
545C4—22no—
546C4—22no—
547C4—44no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4C2×#41611yescharacteristic
553C4#289611yescharacteristic
554C4C4×#13712yes—
555C4#303511yescharacteristic
556C4#303511yescharacteristic
557C4#315211yescharacteristic
558C4#315311yescharacteristic
559C2—42no—
560C2—162no—
561C2#1917912yes—
562C2#1918312yes—
563C2#2123711yescharacteristic
564C2#2124911yescharacteristic
565C2#2125711yescharacteristic
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. #21311C2semidirect
4. #20480C2semidirect
5. C2×#1565C2semidirect
6. C2×#1554C2semidirect
8. #9747C2semidirect
9. #9745C2semidirect
10. #9744C2semidirect
11. #9739C2semidirect
12. #9737C2semidirect
13. #9735C2semidirect
14. #9732C2semidirect
15. #9730C2semidirect
19. #3182C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31822
#213902
#213112
#204802
C2×#15652
C2×#15542
C2×#15512
#97472
#97452
#97442
#97392
#97372
#97352
#97322
#97302