Finite Group Databasebeta

← #225081

Subgroups — #225081

Order 512.

656 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22643C211yesmaximal, characteristic, complemented
2#22531C211yesmaximal, characteristic
3#21387C211yesmaximal, characteristic
4#9558C211yesmaximal, characteristic, complemented
5#9554C211yesmaximal, characteristic, complemented
6#9546C211yesmaximal, characteristic, complemented
7#9546C211yesmaximal, characteristic, complemented
8#8850C211yesmaximal, characteristic, complemented
9#8846C211yesmaximal, characteristic, complemented
10#8708C211yesmaximal, characteristic, complemented
11#8598C211yesmaximal, characteristic, complemented
12#8315C211yesmaximal, characteristic
13#8207C211yesmaximal, characteristic, complemented
14#8203C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C4×#117C2211yescharacteristic
21#1989C411yescharacteristic
22#1989C411yescharacteristic
23#1983C411yescharacteristic
24#1983C411yescharacteristic
25#1955—21no—
26#1954—21no—
27#1954C2211yescharacteristic
28#1954C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1924—21no—
32#1924—21no—
33#1875C411yescharacteristic
34#1875C411yescharacteristic
35#1873C411yescharacteristic
36#1873C411yescharacteristic
37#1839—21no—
38#1798—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1770C2211yescharacteristic
42#1770C2211yescharacteristic
43#1767—21no—
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic
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—
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C22×C42C2311yescharacteristic
84#446—22no—
85#446—41no—
86C8×Q8—22no—
87C8×Q8—41no—
88#431—41no—
89#431—41no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
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⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8C2×C411yescharacteristic
116C2×C22⋊C8C2×C411yescharacteristic
117C2×C22⋊C8D411yescharacteristic
118C2×C22⋊C8D411yescharacteristic
119C2×#97—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136C43—21no—
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
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—22no—
157Q8⋊C8—22no—
158D4⋊C8—41no—
159D4⋊C8—41no—
160#322—41no—
161C82—41no—
162C22×Q8C22×C411yescharacteristic
163C22×D4—21no—
164C22×D4C22×C411yescharacteristic
165C23×C4—21no—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4111yescharacteristic
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183#126—42no—
184#126—42no—
185#125—22no—
186#125—41no—
187#124—22no—
188#124—41no—
189#123—42no—
190#123—42no—
191C22⋊Q8—22no—
192C22⋊Q8—41no—
193#121—22no—
194#121—41no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8#4112yes—
198C4×Q8#4112yes—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—42no—
204C4×D4—42no—
205#117—22no—
206#117—42no—
207#117—42no—
208#117—41no—
209#117—41no—
210#117—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—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×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42C2411yescharacteristic
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257#103—22no—
258#103—41no—
259#103—41no—
260#103—41no—
261#102—22no—
262#102—41no—
263#102—41no—
264#102—41no—
265C22⋊C8—22no—
266C22⋊C8—22no—
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—
276#97—42no—
277#97—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—42no—
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#11712yes—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—81no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
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—41no—
335C22×C4—41no—
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
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—22no—
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—41no—
372C2×C8—41no—
373C2×C8—41no—
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#13512yes—
389C2×C8#13512yes—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
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#11712yes—
406C4⋊C4#13112yes—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C42—21no—
416C42—21no—
417C42—22no—
418C42—22no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—81no—
431C42C2×#4111yescharacteristic
432C42C2×#4111yescharacteristic
433C42H(5,𝔽2)11yescharacteristic
434C42H(5,𝔽2)11yescharacteristic
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23C2×C4×D411yescharacteristic
450Q8—22no—
451Q8—22no—
452Q8—41no—
453Q8—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—84no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—41no—
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—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—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4C2×#11711yescharacteristic
506C2×C4#51811yescharacteristic
507C2×C4#54611yescharacteristic
508C2×C4#54611yescharacteristic
509C2×C4#55111yescharacteristic
510C2×C4#55111yescharacteristic
511C2×C4C2×#13111yescharacteristic
512C2×C4#56811yescharacteristic
513C2×C4#56811yescharacteristic
514C2×C4#56811yescharacteristic
515C2×C4C2×#13511yescharacteristic
516C2×C4C2×#13511yescharacteristic
517C2×C4#57711yescharacteristic
518C2×C4#57711yescharacteristic
519C8—22no—
520C8—22no—
521C8—42no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—81no—
529C8—81no—
530C8—81no—
531C8—81no—
532C8—81no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22#234211yescharacteristic
548C22#293011yescharacteristic
549C22#293311yescharacteristic
550C22C4×#13511yescharacteristic
551C22#295211yescharacteristic
552C22#295511yescharacteristic
553C22#295611yescharacteristic
554C4—21no—
555C4—21no—
556C4—41no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4#292311yescharacteristic
567C4#292511yescharacteristic
568C4#298011yescharacteristic
569C4#298111yescharacteristic
570C4#330111yescharacteristic
571C4#330111yescharacteristic
572C4#330411yescharacteristic
573C4#330411yescharacteristic
574C2—81no—
575C2—161no—
576C2#1793211yescharacteristic
577C2#1795511yescharacteristic
578C2#1799611yescharacteristic
579C2#1803111yescharacteristic
580C2#2255711yescharacteristic
581C2#2257711yescharacteristic
582C2#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. #9558C2semidirect
5. #9554C2semidirect
6. #9546C2semidirect
7. #9546C2semidirect
8. #8850C2semidirect
9. #8846C2semidirect
10. #8708C2semidirect
11. #8598C2semidirect
13. #8207C2semidirect
14. #8203C2semidirect
15. #8148C2semidirect

Maximal subgroups

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

subgroupindex
#226432
#225312
#213872
#95582
#95542
#95462
#88502
#88462
#87082
#85982
#83152
#82072
#82032
#81482