Finite Group Databasebeta

← #225087

Subgroups — #225087

Order 512.

692 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22603C211yesmaximal, characteristic, complemented
2#22531C211yesmaximal, characteristic
3#21388C211yesmaximal, characteristic
4#9560C211yesmaximal, characteristic, complemented
5#9554C211yesmaximal, characteristic, complemented
6#9546C211yesmaximal, characteristic, complemented
7#9545C211yesmaximal, characteristic, complemented
8#8850C211yesmaximal, characteristic, complemented
9#8844C211yesmaximal, characteristic, complemented
10#8707C211yesmaximal, characteristic, complemented
11#8600C211yesmaximal, characteristic, complemented
12#8315C211yesmaximal, characteristic
13#8207C211yesmaximal, characteristic, complemented
14#8205C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×#432—21no—
20C2×#431C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1989C411yescharacteristic
23#1989C411yescharacteristic
24#1983C411yescharacteristic
25#1983C411yescharacteristic
26#1955—21no—
27#1954—21no—
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1917—21no—
33#1917—21no—
34#1874C411yescharacteristic
35#1874C411yescharacteristic
36#1872C411yescharacteristic
37#1872C411yescharacteristic
38#1817—21no—
39#1798—21no—
40#1797C2211yescharacteristic
41#1797C2211yescharacteristic
42#1770C2211yescharacteristic
43#1770C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67#529—22no—
68#529—22no—
69#529—41no—
70#529—41no—
71C2×#126—21no—
72C2×#125C2×C411yescharacteristic
73C2×#124C2×C411yescharacteristic
74C2×#123—21no—
75C2×C22⋊Q8C2×C411yescharacteristic
76C2×#121C2×C411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4C2311yescharacteristic
80C2×#117—21no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#446—22no—
86#446—41no—
87C8×Q8—22no—
88C8×Q8—41no—
89#432—44no—
90#431—41no—
91#431—41no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8C2×C411yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
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⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8C2×C411yescharacteristic
118C2×C22⋊C8C2×C411yescharacteristic
119C2×C22⋊C8D411yescharacteristic
120C2×C22⋊C8D411yescharacteristic
121C2×#97—21no—
122C2×#97—21no—
123C2×#97—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C2×#95—21no—
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
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—22no—
158Q8⋊C8—22no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161#322—41no—
162C82—41no—
163C22×Q8C22×C411yescharacteristic
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4#4111yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186#126—42no—
187#126—42no—
188#125—22no—
189#125—41no—
190#124—22no—
191#124—41no—
192#123—42no—
193#123—42no—
194C22⋊Q8—22no—
195C22⋊Q8—41no—
196#121—22no—
197#121—41no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8#4112yes—
201C4×Q8#4112yes—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—42no—
207C4×D4—42no—
208#117—42no—
209#117—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42C2411yescharacteristic
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260#103—22no—
261#103—41no—
262#103—41no—
263#103—41no—
264#102—22no—
265#102—41no—
266#102—41no—
267#102—41no—
268C22⋊C8—24no—
269C22⋊C8—22no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272C22⋊C8—41no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—81no—
277C22⋊C8—81no—
278C22⋊C8—81no—
279#97—42no—
280#97—42no—
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#11712yes—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—81no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
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—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—81no—
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—22no—
360C2×C8—22no—
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—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
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—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8AGL(1,ℤ/8)12yes—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4#11712yes—
417C4⋊C4#13112yes—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C42—21no—
425C42—21no—
426C42—22no—
427C42—22no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—42no—
433C42—42no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C42H(5,𝔽2)11yescharacteristic
441C42H(5,𝔽2)11yescharacteristic
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23#44312yes—
456C23C2×C4×D411yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—41no—
460Q8—41no—
461D4—41no—
462D4—41no—
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—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
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—84no—
509C2×C4—84no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4C2×#11711yescharacteristic
515C2×C4#51811yescharacteristic
516C2×C4#54611yescharacteristic
517C2×C4#54611yescharacteristic
518C2×C4#55111yescharacteristic
519C2×C4#55111yescharacteristic
520C2×C4C2×#13111yescharacteristic
521C2×C4#56811yescharacteristic
522C2×C4#56811yescharacteristic
523C2×C4#56811yescharacteristic
524C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
525C2×C4#57511yescharacteristic
526C2×C4#57711yescharacteristic
527C2×C4#57811yescharacteristic
528C8—22no—
529C8—42no—
530C8—42no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—81no—
539C8—81no—
540C8—81no—
541C8—81no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22#234211yescharacteristic
557C22#293011yescharacteristic
558C22#293311yescharacteristic
559C22C4×AGL(1,ℤ/8)11yescharacteristic
560C22#294711yescharacteristic
561C22C2×#44311yescharacteristic
562C22#295711yescharacteristic
563C4—21no—
564C4—21no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—41no—
569C4—42no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4#292311yescharacteristic
576C4#292511yescharacteristic
577C4#298011yescharacteristic
578C4#298111yescharacteristic
579C4#330811yescharacteristic
580C4#331711yescharacteristic
581C4#332711yescharacteristic
582C4#333211yescharacteristic
583C2—42no—
584C2—161no—
585C2#1794211yescharacteristic
586C2#1794711yescharacteristic
587C2#1800611yescharacteristic
588C2#1801711yescharacteristic
589C2#2255711yescharacteristic
590C2#2258311yescharacteristic
591C2#2258711yescharacteristic
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. #22603C2semidirect
4. #9560C2semidirect
5. #9554C2semidirect
6. #9546C2semidirect
7. #9545C2semidirect
8. #8850C2semidirect
9. #8844C2semidirect
10. #8707C2semidirect
11. #8600C2semidirect
13. #8207C2semidirect
14. #8205C2semidirect
15. #8148C2semidirect

Maximal subgroups

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

subgroupindex
#226032
#225312
#213882
#95602
#95542
#95462
#95452
#88502
#88442
#87072
#86002
#83152
#82072
#82052
#81482