Finite Group Databasebeta

← #225222

Subgroups — #225222

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22600C211yesmaximal, characteristic, complemented
2#22474C211yesmaximal, characteristic
3#21403C211yesmaximal, characteristic
4#9559C211yesmaximal, characteristic, complemented
5#9557C211yesmaximal, characteristic, complemented
6#9548C211yesmaximal, characteristic, complemented
7#9543C211yesmaximal, characteristic, complemented
8#8847C211yesmaximal, characteristic, complemented
9#8845C211yesmaximal, characteristic, complemented
10#8710C211yesmaximal, characteristic, complemented
11#8597C211yesmaximal, characteristic, complemented
12#8315C211yesmaximal, characteristic
13#8210C211yesmaximal, characteristic, complemented
14#8204C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#433—21no—
20C2×C4×M16C2211yescharacteristic
21#2285C2211yescharacteristic
22#1992C411yescharacteristic
23#1992C411yescharacteristic
24#1984C2211yescharacteristic
25#1984C2211yescharacteristic
26#1959C411yescharacteristic
27#1959C411yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1917—21no—
31#1916—21no—
32#1895C411yescharacteristic
33#1895C411yescharacteristic
34#1894C411yescharacteristic
35#1894C411yescharacteristic
36#1870—21no—
37#1869—21no—
38#1818—21no—
39#1818—21no—
40#1797C2211yescharacteristic
41#1797C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1729—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×M16—21no—
68C22×M16C2×C411yescharacteristic
69C22×M16C2×C411yescharacteristic
70C23×C8—21no—
71#529—22no—
72#529—22no—
73#529—41no—
74#529—41no—
75C2×#126—21no—
76C2×#125C2×C411yescharacteristic
77C2×#124C2×C411yescharacteristic
78C2×#123—21no—
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×#121C2×C411yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4C2311yescharacteristic
84C2×#117—21no—
85C2×#117—21no—
86C22×C42C2311yescharacteristic
87#436—22no—
88#436—41no—
89C8×D4—22no—
90C8×D4—41no—
91#433—44no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8D411yescharacteristic
112C2×C22⋊C8D411yescharacteristic
113C4×M16—41no—
114C4×M16—41no—
115C2×#97—21no—
116C2×#97C2×C411yescharacteristic
117C2×#97C2×C411yescharacteristic
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#384C2×C411yescharacteristic
126#384C2×C411yescharacteristic
127#382—21no—
128#380—21no—
129#380C2×C411yescharacteristic
130#380C2×C411yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#336—21no—
143#336—21no—
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#336D411yescharacteristic
154#336D411yescharacteristic
155#335—41no—
156#334—41no—
157Q8⋊C8—41no—
158Q8⋊C8—41no—
159D4⋊C8—22no—
160D4⋊C8—22no—
161#322—41no—
162#322—41no—
163C22×Q8C22×C411yescharacteristic
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4#4111yescharacteristic
170C2×M16—22no—
171C2×M16—41no—
172C2×M16—41no—
173C2×M16—41no—
174C2×M16—41no—
175C2×M16—41no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—81no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—81no—
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C22×C411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196#126—42no—
197#126—42no—
198#125—22no—
199#125—41no—
200#124—22no—
201#124—41no—
202#123—42no—
203#123—42no—
204C22⋊Q8—22no—
205C22⋊Q8—41no—
206#121—22no—
207#121—41no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4#4112yes—
217C4×D4#4112yes—
218#117—42no—
219#117—42no—
220#117—42no—
221#117—42no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42C2411yescharacteristic
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264#103—22no—
265#103—41no—
266#103—41no—
267#103—41no—
268#102—22no—
269#102—41no—
270#102—41no—
271#102—41no—
272C22⋊C8—22no—
273C22⋊C8—41no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—81no—
279#97—41no—
280#97—41no—
281#97—42no—
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—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—42no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299C24—21no—
300C24C4×D411yescharacteristic
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—81no—
315C2×D4#11712yes—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
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×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4C4×D411yescharacteristic
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4111yescharacteristic
351M16—44no—
352M16—44no—
353M16—84no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
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—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—84no—
395C2×C8#13112yes—
396C2×C8AGL(1,ℤ/8)12yes—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—81no—
417C4⋊C4#11712yes—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C42H(5,𝔽2)11yescharacteristic
441C42H(5,𝔽2)11yescharacteristic
442C23—22no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23C2×C4×D411yescharacteristic
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461D4—22no—
462D4—22no—
463D4—41no—
464D4—41no—
465D4—84no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
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—81no—
511C2×C4—81no—
512C2×C4C2×#11711yescharacteristic
513C2×C4#51811yescharacteristic
514C2×C4#54611yescharacteristic
515C2×C4#54611yescharacteristic
516C2×C4#55111yescharacteristic
517C2×C4#55111yescharacteristic
518C2×C4C2×#13111yescharacteristic
519C2×C4#56811yescharacteristic
520C2×C4#56811yescharacteristic
521C2×C4#56811yescharacteristic
522C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
523C2×C4#57511yescharacteristic
524C2×C4#57711yescharacteristic
525C2×C4#57811yescharacteristic
526C8—22no—
527C8—22no—
528C8—42no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—81no—
536C8—81no—
537C8—81no—
538C8—81no—
539C8—81no—
540C22—41no—
541C22—42no—
542C22—41no—
543C22—42no—
544C22—41no—
545C22—42no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22#234211yescharacteristic
555C22#293211yescharacteristic
556C22#293511yescharacteristic
557C22C4×AGL(1,ℤ/8)11yescharacteristic
558C22#294711yescharacteristic
559C22#295511yescharacteristic
560C22#295611yescharacteristic
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4#292311yescharacteristic
574C4#292511yescharacteristic
575C4#297611yescharacteristic
576C4#297811yescharacteristic
577C4#330611yescharacteristic
578C4#331511yescharacteristic
579C4#332311yescharacteristic
580C4#333011yescharacteristic
581C2—81no—
582C2—82no—
583C2#1798311yescharacteristic
584C2#1798811yescharacteristic
585C2#1807811yescharacteristic
586C2#1808111yescharacteristic
587C2#2256111yescharacteristic
588C2#2258211yescharacteristic
589C2#2258811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×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. #22600C2semidirect
4. #9559C2semidirect
5. #9557C2semidirect
6. #9548C2semidirect
7. #9543C2semidirect
8. #8847C2semidirect
9. #8845C2semidirect
10. #8710C2semidirect
11. #8597C2semidirect
13. #8210C2semidirect
14. #8204C2semidirect
15. #8147C2semidirect

Maximal subgroups

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

subgroupindex
#226002
#224742
#214032
#95592
#95572
#95482
#95432
#88472
#88452
#87102
#85972
#83152
#82102
#82042
#81472