Finite Group Databasebeta

← #188467

Subgroups — #188467

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34236C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3C2×#1505C211yesmaximal, characteristic
4C2×#1502C211yesmaximal, characteristic
5#13783C211yesmaximal, characteristic, complemented
6#8577C212yesmaximal, complemented
7#8575C212yesmaximal, complemented
8#8573C212yesmaximal, complemented
9#8571C212yesmaximal, complemented
10#8021C211yesmaximal, characteristic, complemented
11#8013C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2369C411yescharacteristic, complemented
16#2369C411yescharacteristic, complemented
17#2366C411yescharacteristic, complemented
18#2366C411yescharacteristic, complemented
19#2361—21no—
20#2361—21no—
21#2357C412yescomplemented
22#2349C412yescomplemented
23#2339—22no—
24#2337—22no—
25#2319—21no—
26C4×#121C2212yes—
27C4×#121C2212yes—
28C2×#389—21no—
29#2283C2211yescharacteristic
30#2275—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1940—22no—
35#1939—21no—
36#1939—21no—
37#1797C2212yes—
38#1797C2212yes—
39#1505—24no—
40#1502—24no—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45#1436—22no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48#1434C2212yes—
49#1294C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2211yescharacteristic
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×#121C2312yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C412yes—
71C2×C4×D4C2×C412yes—
72C2×C4×D4C2×C412yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#432—22no—
82#432—22no—
83#431—22no—
84#431—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C22⋊C8C2311yescharacteristic
90C2×C22⋊C8C2311yescharacteristic
91C2×C22⋊C8C2312yes—
92C2×#97—21no—
93C2×C4×C8—21no—
94#391—22no—
95#391—22no—
96#390—21no—
97#390—21no—
98#390C2×C412yes—
99#390C2×C412yes—
100#389—28no—
101#389—21no—
102#389—21no—
103#389C2×C412yes—
104#389C2×C412yes—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#385—22no—
114#385—22no—
115#383—21no—
116#382C2×C411yescharacteristic
117#382C2×C411yescharacteristic
118#381—21no—
119#381—22no—
120#380—22no—
121#380C2311yescharacteristic
122#380C2311yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D412yes—
136C4×C4⋊C4D411yescharacteristic
137C4×C22⋊C4—21no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141#376—21no—
142#376—22no—
143#376—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95D412yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153#336—22no—
154#336—22no—
155#327—24no—
156#327—24no—
157#327—24no—
158#327—24no—
159C2×#41—24no—
160C2×#41—24no—
161C2×#41—81no—
162C2×#41—81no—
163C2×#41—81no—
164C2×#41—81no—
165C22×Q8—21no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4C22×C412yes—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2411yescharacteristic
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C4⋊Q8—22no—
185C4⋊Q8C4⋊C414yescomplemented
186#127—22no—
187#127C4⋊C414yescomplemented
188#124—24no—
189#124—24no—
190#121—24no—
191#121—24no—
192#121—22no—
193#121—22no—
194#121C2×D414yes—
195#121C2×Q814yes—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—82no—
202C4×D4—82no—
203C4×D4—82no—
204C4×D4—82no—
205C4×D4—82no—
206C4×D4—82no—
207C4×D4C4⋊C418yescomplemented
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C42—24no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—82no—
249C2×C42—81no—
250C2×C42—81no—
251C2×C42C22×C412yes—
252C2×C42C22×C412yes—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×Q812yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C4⋊C8—42no—
264C4⋊C8—42no—
265C4⋊C8—82no—
266C4⋊C8—82no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—22no—
270C22⋊C8—21no—
271C22⋊C8—21no—
272C22⋊C8—22no—
273#97—42no—
274C4×C8—42no—
275#95—28no—
276#95—21no—
277#95—22no—
278#95—22no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—22no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—22no—
291#95—21no—
292#95—21no—
293#95—22no—
294#95—42no—
295#95—82no—
296#95—82no—
297#95C22⋊C412yes—
298#95C22⋊C412yes—
299#95C22⋊C412yes—
300#95C22⋊C412yes—
301C24—22no—
302C24C22×D411yescharacteristic
303#41—84no—
304#41—84no—
305#41—84no—
306#41—84no—
307C2×Q8—24no—
308C2×Q8—81no—
309C2×Q8—81no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—22no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—81no—
324C2×D4—81no—
325C2×D4C2×C4⋊C414yes—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—44no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4#12112yes—
386C22×C4#12112yes—
387C22×C4C22⋊Q812yes—
388C22×C4C22⋊Q812yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×Q811yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—42no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—42no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—48no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—88no—
419C4⋊C4#10414yes—
420C4⋊C4C2×C4⋊C414yes—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—28no—
424C22⋊C4—44no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—48no—
439C42—42no—
440C42—42no—
441C42—82no—
442C42—82no—
443C42—84no—
444C42C2×C4⋊C412yes—
445C42C2×C4⋊C412yes—
446C42#11712yes—
447C42#11712yes—
448C42#12112yes—
449C42#12112yes—
450C42C22⋊Q812yes—
451C42C22⋊Q812yes—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—24no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23C22×C22⋊C411yescharacteristic
464C23C2×#12111yescharacteristic
465C23C2×C22⋊Q811yescharacteristic
466C23#57611yescharacteristic
467C23#57611yescharacteristic
468C23#57711yescharacteristic
469C23#57711yescharacteristic
470Q8—84no—
471D4—44no—
472D4—44no—
473D4—44no—
474D4—44no—
475D4—44no—
476D4—44no—
477D4—48no—
478D4—84no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—88no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—164no—
552C2×C4—164no—
553C2×C4#38012yes—
554C2×C4#38012yes—
555C2×C4#38012yes—
556C2×C4#38012yes—
557C2×C4C2×#10414yes—
558C2×C4C2×#10412yes—
559C2×C4#42112yes—
560C2×C4C22×C4⋊C411yescharacteristic
561C2×C4C2×#11711yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×C22⋊Q811yescharacteristic
564C8—81no—
565C8—81no—
566C8—82no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22C2×#38011yescharacteristic
587C22#288311yescharacteristic
588C22#288411yescharacteristic
589C22#289711yescharacteristic
590C22#289711yescharacteristic
591C22C22×#10411yescharacteristic
592C22C2×#42111yescharacteristic
593C22#304111yescharacteristic
594C22#304211yescharacteristic
595C22#308111yescharacteristic
596C22#308111yescharacteristic
597C4—24no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—42no—
606C4—44no—
607C4—42no—
608C4—42no—
609C4—82no—
610C4—84no—
611C4—82no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4#180614yes—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—84no—
622C2#1754711yescharacteristic
623C2#1755011yescharacteristic
624C2#1759911yescharacteristic
625C2#1760211yescharacteristic
626C2#2063411yescharacteristic
627C2#2065011yescharacteristic
628C2C2×#180611yescharacteristic
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. #13783C2semidirect
6. #8577C2semidirect
7. #8575C2semidirect
8. #8573C2semidirect
9. #8571C2semidirect
10. #8021C2semidirect
11. #8013C2semidirect
15. #2369C4semidirect
16. #2369C4semidirect
17. #2366C4semidirect
18. #2366C4semidirect
21. #2357C4semidirect
22. #2349C4semidirect
185. C4⋊Q8C4⋊C4semidirect
187. #127C4⋊C4semidirect
207. 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×#15052
C2×#15022
#137832
#85772
#85752
#85732
#85712
#80212
#80132