Finite Group Databasebeta

← #218967

Subgroups — #218967

Order 512.

699 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22139C211yesmaximal, characteristic, complemented
2#22133C211yesmaximal, characteristic, complemented
3C4×#466C211yesmaximal, characteristic
4#9545C211yesmaximal, characteristic, complemented
5#9541C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9536C211yesmaximal, characteristic, complemented
8#9074C211yesmaximal, characteristic, complemented
9#9074C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8847C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic, complemented
14#8596C211yesmaximal, characteristic, complemented
15#7849C211yesmaximal, characteristic, complemented
16C2×#466C411yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C4×#121C2211yescharacteristic
24C4×#121C2211yescharacteristic
25#1970C411yescharacteristic
26#1955—21no—
27#1954—21no—
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1942C411yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1934C411yescharacteristic
34#1934C411yescharacteristic
35#1924—21no—
36#1923—21no—
37#1893C411yescharacteristic
38#1893C411yescharacteristic
39#1870—21no—
40#1869—21no—
41#1851—21no—
42#1850—21no—
43#1808C411yescharacteristic
44#1797C2211yescharacteristic
45#1797C2211yescharacteristic
46C4×#107C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×#102C2211yescharacteristic
50C4×#102C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C22×M16—21no—
69C23×C8—21no—
70C23×C8C2×C411yescharacteristic
71C23×C8C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#466—24no—
80C4×D8—41no—
81C4×D8—41no—
82#436—41no—
83#436—41no—
84C8×D4—41no—
85C8×D4—41no—
86#431—41no—
87#431—41no—
88#431—41no—
89#431—41no—
90C2×#107C2×C411yescharacteristic
91C2×#107C2×C411yescharacteristic
92C2×#107C2×C411yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8D411yescharacteristic
118C2×C22⋊C8D411yescharacteristic
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8—41no—
122C2×C4×C8—41no—
123C2×C4×C8C2×C411yescharacteristic
124C2×C4×C8C2×C411yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#390C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#388—21no—
136#388—21no—
137#386—21no—
138#386—21no—
139#380C2×C411yescharacteristic
140#380C2×C411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#335—41no—
158#335—41no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161D4⋊C8—41no—
162D4⋊C8—41no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C22×D4C22×C411yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4#4111yescharacteristic
172C2×D8—22no—
173C2×D8—22no—
174C2×D8—41no—
175C2×M16—42no—
176C2×M16—81no—
177C2×M16—81no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—41no—
189C22×C8—41no—
190C22×C8—42no—
191C22×C8—81no—
192C22×C8—81no—
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201#121—22no—
202#121—22no—
203#121—22no—
204#121M1614yescomplemented
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—81no—
214C4×D4—81no—
215C4×D4—81no—
216C4×D4—81no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245#107—22no—
246#107—22no—
247#107—41no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—41no—
257C4⋊C8—41no—
258C4⋊C8—42no—
259C4⋊C8—81no—
260C4⋊C8—81no—
261#102—22no—
262#102—22no—
263#102—22no—
264#102—22no—
265#102—41no—
266#102—41no—
267C22⋊C8—41no—
268C22⋊C8—41no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271C22⋊C8—42no—
272C22⋊C8—81no—
273C22⋊C8—81no—
274C22⋊C8—81no—
275C22⋊C8—81no—
276C22⋊C8—81no—
277C22⋊C8—81no—
278#97—81no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
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—41no—
292C4×C8—41no—
293C4×C8—81no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300C24—21no—
301C24—21no—
302C24C4×D411yescharacteristic
303C2×D4—24no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—42no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4—81no—
319C2×D4—81no—
320C2×D4C2×M1612yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
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×C4C4×D411yescharacteristic
354C22×C4C4×D411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C2×M1612yes—
357C22×C4#13112yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361D8—44no—
362M16—84no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—44no—
386C2×C8—44no—
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—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—84no—
406C2×C8#13112yes—
407C2×C8#13112yes—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—81no—
420C4⋊C4—81no—
421C4⋊C4C2×M1612yes—
422C22⋊C4—24no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—42no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C22⋊C4—81no—
433C22⋊C4—81no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42H(5,𝔽2)11yescharacteristic
445C42H(5,𝔽2)11yescharacteristic
446C42H(5,𝔽2)11yescharacteristic
447C422−1+411yescharacteristic
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23#43512yes—
469C23C2×C4×D411yescharacteristic
470D4—22no—
471D4—22no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—84no—
479D4—84no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4#43512yes—
527C2×C4#51811yescharacteristic
528C2×C4#52011yescharacteristic
529C2×C4#53411yescharacteristic
530C2×C4#53711yescharacteristic
531C2×C4#53811yescharacteristic
532C2×C4#53911yescharacteristic
533C2×C4C22×M1611yescharacteristic
534C2×C4C2×#13111yescharacteristic
535C2×C4C2×#13111yescharacteristic
536C2×C4C2×#13111yescharacteristic
537C2×C4#57611yescharacteristic
538C2×C4#57611yescharacteristic
539C2×C4#57611yescharacteristic
540C2×C4#57811yescharacteristic
541C8—22no—
542C8—22no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—81no—
550C8—81no—
551C8—81no—
552C8—81no—
553C8—81no—
554C8—81no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22#232511yescharacteristic
577C22C2×#43511yescharacteristic
578C22#293111yescharacteristic
579C22#294811yescharacteristic
580C22#295111yescharacteristic
581C22#295711yescharacteristic
582C22#295711yescharacteristic
583C4—22no—
584C4—21no—
585C4—21no—
586C4—41no—
587C4—41no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4#297411yescharacteristic
595C4#297711yescharacteristic
596C4#299011yescharacteristic
597C4#299411yescharacteristic
598C4#319911yescharacteristic
599C4#320311yescharacteristic
600C4#325911yescharacteristic
601C4#326211yescharacteristic
602C2—42no—
603C2—82no—
604C2—161no—
605C2#1787711yescharacteristic
606C2#1788111yescharacteristic
607C2#1797011yescharacteristic
608C2#1797311yescharacteristic
609C2#2217211yescharacteristic
610C2#2219611yescharacteristic
611C2#2220611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#466C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22139C2semidirect
2. #22133C2semidirect
4. #9545C2semidirect
5. #9541C2semidirect
6. #9540C2semidirect
7. #9536C2semidirect
8. #9074C2semidirect
9. #9074C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8847C2semidirect
13. #8843C2semidirect
14. #8596C2semidirect
15. #7849C2semidirect
204. #121M16semidirect

Maximal subgroups

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

subgroupindex
#221392
#221332
C4×#4662
#95452
#95412
#95402
#95362
#90742
#89512
#89472
#88472
#88432
#85962
#78492