Finite Group Databasebeta

← #225072

Subgroups — #225072

Order 512.

665 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22643C211yesmaximal, characteristic, complemented
2#22474C211yesmaximal, characteristic
3#21387C211yesmaximal, characteristic
4#9559C211yesmaximal, characteristic, complemented
5#9555C211yesmaximal, characteristic, complemented
6#9547C211yesmaximal, characteristic, complemented
7#9547C211yesmaximal, characteristic, complemented
8#8849C211yesmaximal, characteristic, complemented
9#8845C211yesmaximal, characteristic, complemented
10#8708C211yesmaximal, characteristic, complemented
11#8598C211yesmaximal, characteristic, complemented
12#8315C211yesmaximal, characteristic
13#8208C211yesmaximal, characteristic, complemented
14#8204C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C4×#117C2211yescharacteristic
21#1989C411yescharacteristic
22#1989C411yescharacteristic
23#1984C2211yescharacteristic
24#1984C2211yescharacteristic
25#1983C411yescharacteristic
26#1983C411yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1924—21no—
30#1924—21no—
31#1875C411yescharacteristic
32#1875C411yescharacteristic
33#1873C411yescharacteristic
34#1873C411yescharacteristic
35#1870—21no—
36#1869—21no—
37#1839—21no—
38#1818—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1770C2211yescharacteristic
42#1770C2211yescharacteristic
43#1767—21no—
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1729—21no—
51C2×#334C2211yescharacteristic
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—
68C23×C8—21no—
69#529—22no—
70#529—22no—
71#529—41no—
72#529—41no—
73C2×#126—21no—
74C2×#125C2×C411yescharacteristic
75C2×#124C2×C411yescharacteristic
76C2×#123—21no—
77C2×C22⋊Q8C2×C411yescharacteristic
78C2×#121C2×C411yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4C2311yescharacteristic
82C2×#117—21no—
83C2×#117C2×C411yescharacteristic
84C2×#117C2×C411yescharacteristic
85C22×C42C2311yescharacteristic
86#436—22no—
87#436—41no—
88C8×D4—22no—
89C8×D4—41no—
90#431—41no—
91#431—41no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×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⋊C8C2×C411yescharacteristic
114C2×C22⋊C8C2×C411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C2×C22⋊C8D411yescharacteristic
117C2×#97—21no—
118C2×#97—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132C43—21no—
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#336—21no—
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#336D411yescharacteristic
153#336D411yescharacteristic
154#334—41no—
155#334—41no—
156Q8⋊C8—41no—
157Q8⋊C8—41no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160#322—41no—
161#322—41no—
162C22×Q8C22×C411yescharacteristic
163C22×D4—21no—
164C22×D4C22×C411yescharacteristic
165C23×C4—21no—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4111yescharacteristic
169C2×M16—42no—
170C2×M16—42no—
171C2×M16—81no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8—81no—
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C22×C411yescharacteristic
184C22×C8C22×C411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189#126—42no—
190#126—42no—
191#125—22no—
192#125—41no—
193#124—22no—
194#124—41no—
195#123—42no—
196#123—42no—
197C22⋊Q8—22no—
198C22⋊Q8—41no—
199#121—22no—
200#121—41no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4#4112yes—
210C4×D4#4112yes—
211#117—22no—
212#117—42no—
213#117—42no—
214#117—41no—
215#117—41no—
216#117—41no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
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×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42C2411yescharacteristic
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261#103—22no—
262#103—41no—
263#103—41no—
264#103—41no—
265#102—22no—
266#102—41no—
267#102—41no—
268#102—41no—
269C22⋊C8—41no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272C22⋊C8—41no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—81no—
277C22⋊C8—81no—
278#97—81no—
279#97—81no—
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—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296C24—21no—
297C24C4×D411yescharacteristic
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—81no—
312C2×D4#11712yes—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
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—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4111yescharacteristic
348M16—84no—
349C2×C8—22no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
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—41no—
375C2×C8—41no—
376C2×C8—41no—
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—84no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4#11712yes—
408C4⋊C4#13112yes—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—81no—
433C42C2×#4111yescharacteristic
434C42C2×#4111yescharacteristic
435C42H(5,𝔽2)11yescharacteristic
436C42H(5,𝔽2)11yescharacteristic
437C23—22no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C2×C4×D411yescharacteristic
452Q8—41no—
453Q8—41no—
454Q8—41no—
455Q8—41no—
456D4—22no—
457D4—22no—
458D4—41no—
459D4—41no—
460D4—84no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
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—81no—
506C2×C4—81no—
507C2×C4C2×#11711yescharacteristic
508C2×C4#51811yescharacteristic
509C2×C4#54611yescharacteristic
510C2×C4#54611yescharacteristic
511C2×C4#55111yescharacteristic
512C2×C4#55111yescharacteristic
513C2×C4C2×#13111yescharacteristic
514C2×C4#56811yescharacteristic
515C2×C4#56811yescharacteristic
516C2×C4#56811yescharacteristic
517C2×C4#57511yescharacteristic
518C2×C4#57511yescharacteristic
519C2×C4#57711yescharacteristic
520C2×C4#57711yescharacteristic
521C8—42no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—81no—
531C8—81no—
532C8—81no—
533C8—81no—
534C8—81no—
535C22—41no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—41no—
542C22—42no—
543C22—41no—
544C22—41no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22#234211yescharacteristic
550C22#293011yescharacteristic
551C22#293311yescharacteristic
552C22#294311yescharacteristic
553C22#295211yescharacteristic
554C22#295511yescharacteristic
555C22#295611yescharacteristic
556C4—21no—
557C4—21no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—41no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4#292311yescharacteristic
569C4#292511yescharacteristic
570C4#298011yescharacteristic
571C4#298111yescharacteristic
572C4#330711yescharacteristic
573C4#330711yescharacteristic
574C4#331111yescharacteristic
575C4#331111yescharacteristic
576C2—81no—
577C2—82no—
578C2#1793711yescharacteristic
579C2#1795511yescharacteristic
580C2#1800211yescharacteristic
581C2#1803111yescharacteristic
582C2#2255711yescharacteristic
583C2#2256811yescharacteristic
584C2#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. #22643C2semidirect
4. #9559C2semidirect
5. #9555C2semidirect
6. #9547C2semidirect
7. #9547C2semidirect
8. #8849C2semidirect
9. #8845C2semidirect
10. #8708C2semidirect
11. #8598C2semidirect
13. #8208C2semidirect
14. #8204C2semidirect
15. #8149C2semidirect

Maximal subgroups

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

subgroupindex
#226432
#224742
#213872
#95592
#95552
#95472
#88492
#88452
#87082
#85982
#83152
#82082
#82042
#81492