Finite Group Databasebeta

← #225165

Subgroups — #225165

Order 512.

665 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22633C211yesmaximal, characteristic, complemented
2#22531C211yesmaximal, characteristic
3#21402C211yesmaximal, characteristic
4#9556C211yesmaximal, characteristic, complemented
5#9556C211yesmaximal, characteristic, complemented
6#9546C211yesmaximal, characteristic, complemented
7#9542C211yesmaximal, characteristic, complemented
8#8850C211yesmaximal, characteristic, complemented
9#8846C211yesmaximal, characteristic, complemented
10#8711C211yesmaximal, characteristic, complemented
11#8595C211yesmaximal, characteristic, complemented
12#8315C211yesmaximal, characteristic
13#8209C211yesmaximal, characteristic, complemented
14#8201C211yesmaximal, characteristic, complemented
15#8144C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2283C2211yescharacteristic
21#1991C411yescharacteristic
22#1991C411yescharacteristic
23#1958C411yescharacteristic
24#1958C411yescharacteristic
25#1955—21no—
26#1954—21no—
27#1954C2211yescharacteristic
28#1954C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1924—21no—
32#1923—21no—
33#1893C411yescharacteristic
34#1893C411yescharacteristic
35#1892C411yescharacteristic
36#1892C411yescharacteristic
37#1832—21no—
38#1798—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1770C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C4×#103C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1750—21no—
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1746—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
67C23×C8C2×C411yescharacteristic
68C23×C8C2×C411yescharacteristic
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—
83C22×C42C2311yescharacteristic
84#446—22no—
85#446—41no—
86C8×Q8—22no—
87C8×Q8—41no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#103C2×C411yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
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
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—41no—
118C2×C4×C8—41no—
119C2×C4×C8C2×C411yescharacteristic
120C2×C4×C8C2×C411yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#383—21no—
128#382C2×C411yescharacteristic
129#382C2×C411yescharacteristic
130#381—21no—
131#380C2×C411yescharacteristic
132#380C2×C411yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136#376—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—
161#322—41no—
162C22×Q8C22×C411yescharacteristic
163C22×D4—21no—
164C22×D4C22×C411yescharacteristic
165C23×C4—21no—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4111yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—41no—
179C22×C8—41no—
180C22×C8—41no—
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#4112yes—
204C4×Q8#4112yes—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—42no—
210C4×D4—42no—
211#117—42no—
212#117—42no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42C2411yescharacteristic
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—81no—
252C4⋊C8—81no—
253#103—22no—
254#103—22no—
255#103—22no—
256#103—41no—
257#102—22no—
258#102—22no—
259#102—22no—
260#102—41no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263C22⋊C8—42no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266C22⋊C8—81no—
267C22⋊C8—81no—
268C22⋊C8—81no—
269C22⋊C8—81no—
270#97—42no—
271#97—42no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—41no—
285C4×C8—41no—
286C4×C8—42no—
287C4×C8—42no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—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#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—22no—
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—21no—
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—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—44no—
370C2×C8—44no—
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—41no—
389C2×C8#13112yes—
390C2×C8#13112yes—
391C2×C8#13112yes—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
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—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4#11712yes—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—81no—
417C42—21no—
418C42—21no—
419C42—22no—
420C42—22no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42C2×#4111yescharacteristic
430C42C2×#4111yescharacteristic
431C42H(5,𝔽2)11yescharacteristic
432C42H(5,𝔽2)11yescharacteristic
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23#44412yes—
447C23C2×C4×D411yescharacteristic
448Q8—22no—
449Q8—22no—
450Q8—41no—
451Q8—41no—
452D4—41no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—84no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
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×C4C2×#11711yescharacteristic
504C2×C4#51811yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4#54611yescharacteristic
507C2×C4#55111yescharacteristic
508C2×C4#55111yescharacteristic
509C2×C4C2×#13111yescharacteristic
510C2×C4C2×#13111yescharacteristic
511C2×C4C2×#13111yescharacteristic
512C2×C4#56811yescharacteristic
513C2×C4#57511yescharacteristic
514C2×C4#57511yescharacteristic
515C2×C4#57711yescharacteristic
516C2×C4#57711yescharacteristic
517C8—22no—
518C8—22no—
519C8—22no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—81no—
526C8—81no—
527C8—81no—
528C8—81no—
529C8—81no—
530C8—81no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22#234211yescharacteristic
546C22#293111yescharacteristic
547C22#293511yescharacteristic
548C22#294311yescharacteristic
549C22#295211yescharacteristic
550C22C2×#44411yescharacteristic
551C22#295711yescharacteristic
552C4—21no—
553C4—21no—
554C4—41no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—42no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4#292211yescharacteristic
565C4#292511yescharacteristic
566C4#297611yescharacteristic
567C4#297711yescharacteristic
568C4#331711yescharacteristic
569C4#331711yescharacteristic
570C4#332111yescharacteristic
571C4#332111yescharacteristic
572C2—42no—
573C2—161no—
574C2#1796311yescharacteristic
575C2#1797411yescharacteristic
576C2#1807411yescharacteristic
577C2#1809311yescharacteristic
578C2#2256011yescharacteristic
579C2#2256511yescharacteristic
580C2#2258611yescharacteristic
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. #22633C2semidirect
4. #9556C2semidirect
5. #9556C2semidirect
6. #9546C2semidirect
7. #9542C2semidirect
8. #8850C2semidirect
9. #8846C2semidirect
10. #8711C2semidirect
11. #8595C2semidirect
13. #8209C2semidirect
14. #8201C2semidirect
15. #8144C2semidirect

Maximal subgroups

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

subgroupindex
#226332
#225312
#214022
#95562
#95462
#95422
#88502
#88462
#87112
#85952
#83152
#82092
#82012
#81442