Finite Group Databasebeta

← #188504

Subgroups — #188504

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34236C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3C2×#1517C211yesmaximal, characteristic
4C2×#1514C211yesmaximal, characteristic
5#14356C211yesmaximal, characteristic, complemented
6#8577C212yesmaximal, complemented
7#8575C212yesmaximal, complemented
8#8573C212yesmaximal, complemented
9#8571C212yesmaximal, complemented
10#8025C211yesmaximal, characteristic, complemented
11#8017C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2369C412yescomplemented
16#2366C412yescomplemented
17#2361—22no—
18#2357C411yescharacteristic, complemented
19#2357C411yescharacteristic, complemented
20#2349C411yescharacteristic, complemented
21#2349C411yescharacteristic, complemented
22#2339—21no—
23#2339—21no—
24#2337—22no—
25#2319—21no—
26C4×#121C2212yes—
27C4×#121C2212yes—
28#2283C2211yescharacteristic
29C2×#383C2211yescharacteristic
30C2×#382C2211yescharacteristic
31#2275—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1940—21no—
34#1940—21no—
35#1939—22no—
36#1797C2212yes—
37#1797C2212yes—
38#1517—24no—
39#1514—24no—
40C2×#327C2211yescharacteristic
41C2×#327C2212yes—
42C2×#327C2211yescharacteristic
43C2×#327C2211yescharacteristic
44C2×#327C2212yes—
45C2×#327C2211yescharacteristic
46#1444—21no—
47#1442—21no—
48#1441—22no—
49C4⋊#95C2212yes—
50#1434C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55C22×#41—21no—
56#532—22no—
57#532—22no—
58#532D414yes—
59#532Q814yes—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#127C2×C411yescharacteristic
62C2×#124—21no—
63C2×#121—21no—
64C2×#121C2311yescharacteristic
65C2×#121C2311yescharacteristic
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C412yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#432—22no—
80#432—22no—
81#431—22no—
82#431—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2311yescharacteristic
89C2×C22⋊C8C2311yescharacteristic
90C2×#97—21no—
91C2×C4×C8—21no—
92#391—22no—
93#391—22no—
94#390—22no—
95#390C2×C412yes—
96#390C2×C412yes—
97#389—22no—
98#389C2×C412yes—
99#389C2×C412yes—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#385—22no—
109#385—22no—
110#383—21no—
111#383—22no—
112#383—22no—
113#382—22no—
114#382—22no—
115#382C2×C411yescharacteristic
116#382C2×C411yescharacteristic
117#381—21no—
118#381—21no—
119#381—21no—
120#380—21no—
121#380—21no—
122#380C2312yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D412yes—
132C4×C4⋊C4D411yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138#376—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95D411yescharacteristic
148C2×#95D411yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152#336—22no—
153#336—22no—
154#327—22no—
155#327—22no—
156#327—22no—
157#327—22no—
158#327—24no—
159#327—24no—
160C2×#41—24no—
161C2×#41—24no—
162C2×#41—82no—
163C2×#41—82no—
164C22×Q8—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4C22×C411yescharacteristic
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C4⋊Q8—22no—
187C4⋊Q8C4⋊C414yescomplemented
188#127—22no—
189#127C4⋊C414yescomplemented
190#124—24no—
191#124—24no—
192#121—24no—
193#121—24no—
194#121—21no—
195#121—21no—
196#121—21no—
197#121—21no—
198#121C2×D412yes—
199#121C2×D412yes—
200#121C2×Q812yes—
201#121C2×Q812yes—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—48no—
207C4×D4—82no—
208C4×D4—82no—
209C4×D4—82no—
210C4×D4—82no—
211C4×D4—82no—
212C4×D4—82no—
213C4×D4C4⋊C414yescomplemented
214C4×D4C4⋊C414yescomplemented
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—82no—
247C2×C42C22×C412yes—
248C2×C42C22×C412yes—
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×Q812yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8—82no—
262C4⋊C8—82no—
263C22⋊C8—22no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—22no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269#97—42no—
270C4×C8—42no—
271#95—21no—
272#95—22no—
273#95—22no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—21no—
286#95—21no—
287#95—44no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—82no—
292#95—82no—
293#95C22⋊C411yescharacteristic
294#95C22⋊C411yescharacteristic
295#95C22⋊C411yescharacteristic
296#95C22⋊C411yescharacteristic
297#95C22⋊C412yes—
298#95C22⋊C412yes—
299C24—21no—
300C24—21no—
301C24C22×D411yescharacteristic
302#41—88no—
303#41—84no—
304#41—84no—
305C2×Q8—24no—
306C2×Q8—82no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—44no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—82no—
325C2×D4C2×C4⋊C412yes—
326C2×D4C2×C4⋊C412yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C412yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4C22⋊Q812yes—
382C22×C4C22⋊Q812yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×Q811yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—41no—
397C2×C8—41no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—84no—
410C4⋊C4—88no—
411C4⋊C4C2×C4⋊C412yes—
412C4⋊C4C2×C4⋊C412yes—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—44no—
432C42—82no—
433C42—82no—
434C42C2×C4⋊C412yes—
435C42C2×C4⋊C412yes—
436C42#11712yes—
437C42#11712yes—
438C42#12112yes—
439C42#12112yes—
440C42C22⋊Q812yes—
441C42C22⋊Q812yes—
442C23—21no—
443C23—21no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23C22×C22⋊C411yescharacteristic
456C23C2×#12111yescharacteristic
457C23C2×C22⋊Q811yescharacteristic
458C23#57612yes—
459C23#57712yes—
460Q8—84no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4—48no—
466D4—44no—
467D4—44no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—84no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
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—22no—
496C2×C4—22no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—164no—
535C2×C4—164no—
536C2×C4#38012yes—
537C2×C4#38012yes—
538C2×C4#38012yes—
539C2×C4#38012yes—
540C2×C4#42114yes—
541C2×C4C22×C4⋊C411yescharacteristic
542C2×C4C2×#11711yescharacteristic
543C2×C4C2×#12111yescharacteristic
544C2×C4C2×C22⋊Q811yescharacteristic
545C8—82no—
546C8—81no—
547C8—81no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22C2×#38011yescharacteristic
567C22#288311yescharacteristic
568C22#288411yescharacteristic
569C22#289712yes—
570C22C2×#42112yes—
571C22#304111yescharacteristic
572C22#304211yescharacteristic
573C22#308112yes—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—24no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—82no—
589C4—82no—
590C4—161no—
591C4—162no—
592C4—161no—
593C2—21no—
594C2—21no—
595C2—22no—
596C2—82no—
597C2—82no—
598C2#1759712yes—
599C2#1760212yes—
600C2#2063411yescharacteristic
601C2#2065011yescharacteristic
602C2#2065411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #21953C2semidirect
5. #14356C2semidirect
6. #8577C2semidirect
7. #8575C2semidirect
8. #8573C2semidirect
9. #8571C2semidirect
10. #8025C2semidirect
11. #8017C2semidirect
15. #2369C4semidirect
16. #2366C4semidirect
18. #2357C4semidirect
19. #2357C4semidirect
20. #2349C4semidirect
21. #2349C4semidirect
187. C4⋊Q8C4⋊C4semidirect
189. #127C4⋊C4semidirect
213. C4×D4C4⋊C4semidirect
214. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342362
#219532
C2×#15172
C2×#15142
#143562
#85772
#85752
#85732
#85712
#80252
#80172