Finite Group Databasebeta

← #188536

Subgroups — #188536

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic, complemented
3C2×#1514C211yesmaximal, characteristic
4C2×#1505C211yesmaximal, characteristic
5#13964C211yesmaximal, characteristic, complemented
6#8577C212yesmaximal, complemented
7#8575C212yesmaximal, complemented
8#8573C212yesmaximal, complemented
9#8571C212yesmaximal, complemented
10#8022C211yesmaximal, characteristic, complemented
11#8018C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520—21no—
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16#2367—22no—
17#2366C412yescomplemented
18#2365C412yescomplemented
19#2354C412yescomplemented
20#2352—22no—
21D4×C4⋊C4C412yescomplemented
22#2325—21no—
23#2325—21no—
24#2311—21no—
25C4×#121C2212yes—
26C4×#121C2212yes—
27#2285C2211yescharacteristic
28C2×#383C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1940—22no—
32#1940—22no—
33#1797C2212yes—
34#1797C2212yes—
35#1514—24no—
36#1505—24no—
37C2×#327C2212yes—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2212yes—
41#1441—21no—
42#1441—21no—
43#1436—21no—
44#1436—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47#1434C2211yescharacteristic
48#1434C2211yescharacteristic
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C22×#41—21no—
54#532—22no—
55#532—22no—
56#532D414yes—
57#532Q814yes—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#127C2×C411yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121C2312yes—
63#520—44no—
64#520—44no—
65C2×C4×Q8—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C412yes—
71C2×C4×D4C2×C412yes—
72C2×C4×D4C2×C412yes—
73C2×#117—21no—
74C2×#117—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#432—22no—
85#432—22no—
86#432—22no—
87#432—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2312yes—
94C2×#97—21no—
95C2×#97—21no—
96#391—22no—
97#390—22no—
98#390C2×C412yes—
99#390C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#385—22no—
113#385—22no—
114#384C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116#383—22no—
117#383—22no—
118#382—21no—
119#380—21no—
120#380—22no—
121#380—22no—
122#380C2312yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D412yes—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133#376—22no—
134#376—22no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95D411yescharacteristic
145C2×#95D411yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150#336—22no—
151#336—22no—
152#327—24no—
153#327—24no—
154#327—24no—
155#327—24no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—42no—
159C2×#41—42no—
160C2×#41—42no—
161C2×#41—42no—
162C22×Q8—21no—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C22×C412yes—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2411yescharacteristic
181C22×C8—22no—
182C22×C8—22no—
183C4⋊Q8—22no—
184C4⋊Q8C4⋊C414yescomplemented
185#127—22no—
186#127C4⋊C414yescomplemented
187#124—24no—
188#124—24no—
189#121—24no—
190#121—24no—
191#121—22no—
192#121—22no—
193#121C2×D414yes—
194#121C2×Q814yes—
195C4×Q8—48no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—48no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—82no—
205C4×D4—82no—
206C4×D4C4⋊C418yescomplemented
207#117—48no—
208#117—42no—
209#117—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—81no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22⋊C412yes—
231C2×C4⋊C4C22⋊C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C22⋊C4—22no—
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×C22⋊C4—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—81no—
251C2×C42—81no—
252C2×C42C22×C412yes—
253C2×C42C22×C412yes—
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×Q812yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8—82no—
267C4⋊C8—82no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272#97—42no—
273#97—42no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—81no—
296#95—81no—
297#95C22⋊C412yes—
298#95C22⋊C412yes—
299#95C22⋊C412yes—
300#95C22⋊C412yes—
301C24—22no—
302C24C22×D411yescharacteristic
303#41—48no—
304#41—48no—
305#41—48no—
306#41—48no—
307C2×Q8—24no—
308C2×Q8—42no—
309C2×Q8—42no—
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—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—28no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—44no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C412yes—
396C22×C4#12112yes—
397C22×C4#12112yes—
398C22×C4C22⋊Q812yes—
399C22×C4C22⋊Q812yes—
400C22×C4AGL(1,ℤ/8)12yes—
401C22×C4AGL(1,ℤ/8)12yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q811yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4C2×C4⋊C414yes—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—28no—
434C22⋊C4—44no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—44no—
447C42—44no—
448C42—48no—
449C42—44no—
450C42C2×C4⋊C412yes—
451C42C2×C4⋊C412yes—
452C42#11712yes—
453C42#11712yes—
454C42#12112yes—
455C42#12112yes—
456C42C22⋊Q812yes—
457C42C22⋊Q812yes—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23C22×C22⋊C411yescharacteristic
470C23C2×#12111yescharacteristic
471C23C2×C22⋊Q811yescharacteristic
472C23C2×AGL(1,ℤ/8)11yescharacteristic
473C23C2×AGL(1,ℤ/8)11yescharacteristic
474C23#57511yescharacteristic
475C23#57511yescharacteristic
476Q8—48no—
477D4—44no—
478D4—44no—
479D4—48no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—44no—
484D4—84no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—416no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—164no—
565C2×C4—164no—
566C2×C4#38012yes—
567C2×C4#38012yes—
568C2×C4#38012yes—
569C2×C4#38012yes—
570C2×C4#41814yes—
571C2×C4C22×C4⋊C411yescharacteristic
572C2×C4C2×#11711yescharacteristic
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×C22⋊Q811yescharacteristic
575C8—82no—
576C8—82no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C2×#38011yescharacteristic
597C22#288311yescharacteristic
598C22#288411yescharacteristic
599C22C2×#41811yescharacteristic
600C22#289611yescharacteristic
601C22#290111yescharacteristic
602C22#290111yescharacteristic
603C22#303611yescharacteristic
604C22#303611yescharacteristic
605C22#307711yescharacteristic
606C22#307711yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—24no—
616C4—44no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—161no—
625C4—161no—
626C2—21no—
627C2—21no—
628C2—21no—
629C2—21no—
630C2—84no—
631C2#1762011yescharacteristic
632C2#1762411yescharacteristic
633C2#1762911yescharacteristic
634C2#1763111yescharacteristic
635C2#2063411yescharacteristic
636C2#2064611yescharacteristic
637C2#2065511yescharacteristic
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. #21955C2semidirect
5. #13964C2semidirect
6. #8577C2semidirect
7. #8575C2semidirect
8. #8573C2semidirect
9. #8571C2semidirect
10. #8022C2semidirect
11. #8018C2semidirect
17. #2366C4semidirect
18. #2365C4semidirect
19. #2354C4semidirect
21. D4×C4⋊C4C4semidirect
184. C4⋊Q8C4⋊C4semidirect
186. #127C4⋊C4semidirect
206. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#219552
C2×#15142
C2×#15052
#139642
#85772
#85752
#85732
#85712
#80222
#80182