Finite Group Databasebeta

← #225006

Subgroups — #225006

Order 512.

821 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22603C211yesmaximal, characteristic, complemented
2#22469C211yesmaximal, characteristic
3C4×#417C211yesmaximal, characteristic
4#9559C211yesmaximal, characteristic, complemented
5#9557C211yesmaximal, characteristic, complemented
6#9548C211yesmaximal, characteristic, complemented
7#9547C211yesmaximal, characteristic, complemented
8#8845C211yesmaximal, characteristic, complemented
9#8843C211yesmaximal, characteristic, complemented
10#8707C211yesmaximal, characteristic, complemented
11#8597C211yesmaximal, characteristic, complemented
12#8312C211yesmaximal, characteristic
13#8204C211yesmaximal, characteristic, complemented
14#8202C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#432—21no—
20C2×#417C411yescharacteristic
21C2×#417C411yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1984C2211yescharacteristic
25#1984C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1917—21no—
29#1917—21no—
30#1869—21no—
31#1869—21no—
32#1817—21no—
33#1817—21no—
34#1797C2211yescharacteristic
35#1797C2211yescharacteristic
36#1770C2211yescharacteristic
37#1770C2211yescharacteristic
38#1763C411yescharacteristic
39#1763C411yescharacteristic
40#1762C411yescharacteristic
41#1762C411yescharacteristic
42C4×#103C2211yescharacteristic
43C4×#103C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×C22⋊C8C2211yescharacteristic
47C4×C22⋊C8C2211yescharacteristic
48#1736C411yescharacteristic
49#1736C411yescharacteristic
50C2×#336—21no—
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×#41C2×C411yescharacteristic
68C22×M16C2×C411yescharacteristic
69C22×M16C2×C411yescharacteristic
70C23×C8—21no—
71C23×C8—21no—
72C4×#41—22no—
73C4×#41—22no—
74C4×#41—41no—
75C4×#41—41no—
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×#117—21no—
83C2×#117C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C8×D4—22no—
90C8×D4—22no—
91C8×D4—41no—
92C8×D4—41no—
93#432—44no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100#417—24no—
101#417—24no—
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8D411yescharacteristic
116C2×C22⋊C8D411yescharacteristic
117C4×M16—41no—
118C4×M16—41no—
119C2×#97—21no—
120C2×#97—21no—
121C2×#97C2×C411yescharacteristic
122C2×#97C2×C411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C2×#95—21no—
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336D411yescharacteristic
156#336D411yescharacteristic
157#335—41no—
158#334—41no—
159Q8⋊C8—41no—
160Q8⋊C8—41no—
161D4⋊C8—22no—
162D4⋊C8—22no—
163#322—41no—
164#322—41no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—41no—
169C2×#41—41no—
170C2×#41M1614yes—
171C22×Q8C22×C411yescharacteristic
172C22×D4—21no—
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—41no—
186C2×M16—41no—
187C2×M16—41no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—81no—
199C22×C8—81no—
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C4×Q8—22no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—41no—
214C4×D4—22no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—41no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4#4112yes—
223C4×D4#4112yes—
224#117—22no—
225#117—42no—
226#117—42no—
227#117—41no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42C2411yescharacteristic
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—42no—
273C4⋊C8—42no—
274#103—22no—
275#103—22no—
276#103—22no—
277#103—22no—
278#102—22no—
279#102—22no—
280#102—22no—
281#102M1614yes—
282C22⋊C8—22no—
283C22⋊C8—41no—
284C22⋊C8—42no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288C22⋊C8—81no—
289#97—41no—
290#97—41no—
291#97—42no—
292#97—42no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309C24—21no—
310C24C4×D411yescharacteristic
311#41—28no—
312#41—44no—
313#41—44no—
314#41—44no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8C2×M1612yes—
321C2×D4—24no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4#11712yes—
328C2×D4C2×M1612yes—
329C2×D4C2×M1612yes—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C2×M1612yes—
373C22×C4#13112yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4111yescharacteristic
377M16—44no—
378M16—44no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—22no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
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—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—84no—
420C2×C8—84no—
421C2×C8C2×M1612yes—
422C2×C8AGL(1,ℤ/8)12yes—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4#11712yes—
439C4⋊C4C2×M1612yes—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42C2×#4111yescharacteristic
461C42C2×#4111yescharacteristic
462C42H(5,𝔽2)11yescharacteristic
463C42H(5,𝔽2)11yescharacteristic
464C23—22no—
465C23—21no—
466C23—22no—
467C23—21no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#43512yes—
478C23C2×C4×D411yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—41no—
482Q8—41no—
483D4—22no—
484D4—22no—
485D4—22no—
486D4—48no—
487D4#43214yescomplemented
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—48no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
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—81no—
533C2×C4—81no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4#43512yes—
538C2×C4C2×#11711yescharacteristic
539C2×C4#51811yescharacteristic
540C2×C4#54611yescharacteristic
541C2×C4#54611yescharacteristic
542C2×C4#55111yescharacteristic
543C2×C4#55111yescharacteristic
544C2×C4C22×M1611yescharacteristic
545C2×C4C22×M1611yescharacteristic
546C2×C4C2×#13111yescharacteristic
547C2×C4#56811yescharacteristic
548C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
549C2×C4#57511yescharacteristic
550C2×C4#57711yescharacteristic
551C2×C4#57811yescharacteristic
552C8—22no—
553C8—22no—
554C8—42no—
555C8—41no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—41no—
561C8—81no—
562C8—81no—
563C8—81no—
564C8—81no—
565C8—81no—
566C22—22no—
567C22—24no—
568C22—22no—
569C22—22no—
570C22—41no—
571C22—42no—
572C22—41no—
573C22—42no—
574C22—41no—
575C22—42no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22#234211yescharacteristic
581C22C2×#43511yescharacteristic
582C22D4×M1611yescharacteristic
583C22C4×AGL(1,ℤ/8)11yescharacteristic
584C22#294711yescharacteristic
585C22#295511yescharacteristic
586C22#295611yescharacteristic
587C4—22no—
588C4—21no—
589C4—21no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4C2×#43211yescharacteristic
600C4#292411yescharacteristic
601C4#297311yescharacteristic
602C4#297411yescharacteristic
603C4#330611yescharacteristic
604C4#331511yescharacteristic
605C4#332311yescharacteristic
606C4#333011yescharacteristic
607C2—42no—
608C2—44no—
609C2#1787111yescharacteristic
610C2#1787611yescharacteristic
611C2#1804611yescharacteristic
612C2#1805111yescharacteristic
613C2#2255411yescharacteristic
614C2#2258211yescharacteristic
615C2#2258811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22603C2semidirect
4. #9559C2semidirect
5. #9557C2semidirect
6. #9548C2semidirect
7. #9547C2semidirect
8. #8845C2semidirect
9. #8843C2semidirect
10. #8707C2semidirect
11. #8597C2semidirect
13. #8204C2semidirect
14. #8202C2semidirect
15. #8147C2semidirect
487. D4#432semidirect

Maximal subgroups

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

subgroupindex
#226032
#224692
C4×#4172
#95592
#95572
#95482
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