Finite Group Databasebeta

← #225090

Subgroups — #225090

Order 512.

687 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22629C211yesmaximal, characteristic, complemented
2#22474C211yesmaximal, characteristic
3#21388C211yesmaximal, characteristic
4#9557C211yesmaximal, characteristic, complemented
5#9555C211yesmaximal, characteristic, complemented
6#9548C211yesmaximal, characteristic, complemented
7#9547C211yesmaximal, characteristic, complemented
8#8847C211yesmaximal, characteristic, complemented
9#8845C211yesmaximal, characteristic, complemented
10#8706C211yesmaximal, characteristic, complemented
11#8600C211yesmaximal, characteristic, complemented
12#8315C211yesmaximal, characteristic
13#8208C211yesmaximal, characteristic, complemented
14#8202C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1989C411yescharacteristic
22#1989C411yescharacteristic
23#1984C2211yescharacteristic
24#1984C2211yescharacteristic
25#1983C411yescharacteristic
26#1983C411yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1917—21no—
30#1917—21no—
31#1874C411yescharacteristic
32#1874C411yescharacteristic
33#1872C411yescharacteristic
34#1872C411yescharacteristic
35#1870—21no—
36#1869—21no—
37#1818—21no—
38#1797—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1770C2211yescharacteristic
42#1770C2211yescharacteristic
43C4×#103C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1749—21no—
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—
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—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#436—22no—
88#436—41no—
89C8×D4—22no—
90C8×D4—41no—
91#431—41no—
92#431—41no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8C2×C411yescharacteristic
115C2×C22⋊C8C2×C411yescharacteristic
116C2×C22⋊C8D411yescharacteristic
117C2×C22⋊C8D411yescharacteristic
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C2×#95—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#335—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×C4—21no—
167C23×C4—21no—
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4111yescharacteristic
171C2×M16—42no—
172C2×M16—42no—
173C2×M16—81no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—42no—
181C22×C8—42no—
182C22×C8—81no—
183C22×C8C22×C411yescharacteristic
184C22×C8C22×C411yescharacteristic
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191#126—42no—
192#126—42no—
193#125—22no—
194#125—41no—
195#124—22no—
196#124—41no—
197#123—42no—
198#123—42no—
199C22⋊Q8—22no—
200C22⋊Q8—41no—
201#121—22no—
202#121—41no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4#4112yes—
212C4×D4#4112yes—
213#117—42no—
214#117—42no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—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×C42—81no—
246C2×C42—81no—
247C2×C42C22×C411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42C2411yescharacteristic
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—41no—
260C4⋊C8—41no—
261C4⋊C8—81no—
262C4⋊C8—81no—
263#103—22no—
264#103—41no—
265#103—41no—
266#103—41no—
267#102—22no—
268#102—41no—
269#102—41no—
270#102—41no—
271C22⋊C8—22no—
272C22⋊C8—41no—
273C22⋊C8—41no—
274C22⋊C8—41no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C22⋊C8—81no—
280#97—81no—
281#97—81no—
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—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298C24—21no—
299C24C4×D411yescharacteristic
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—81no—
314C2×D4#11712yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
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—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356M16—84no—
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—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
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—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—84no—
399C2×C8#13712yes—
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—21no—
409C4⋊C4—21no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4#11712yes—
418C4⋊C4#13112yes—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—81no—
438C42—81no—
439C42C2×#4111yescharacteristic
440C42C2×#4111yescharacteristic
441C42H(5,𝔽2)11yescharacteristic
442C42H(5,𝔽2)11yescharacteristic
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—22no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23#44412yes—
457C23C2×C4×D411yescharacteristic
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462D4—22no—
463D4—22no—
464D4—41no—
465D4—41no—
466D4—84no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
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—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
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—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4C2×#11711yescharacteristic
516C2×C4#51811yescharacteristic
517C2×C4#54611yescharacteristic
518C2×C4#54611yescharacteristic
519C2×C4#55111yescharacteristic
520C2×C4#55111yescharacteristic
521C2×C4C2×#13111yescharacteristic
522C2×C4#56811yescharacteristic
523C2×C4#56811yescharacteristic
524C2×C4#56811yescharacteristic
525C2×C4C2×#13711yescharacteristic
526C2×C4#57511yescharacteristic
527C2×C4#57611yescharacteristic
528C2×C4#57711yescharacteristic
529C8—22no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—81no—
538C8—81no—
539C8—81no—
540C8—81no—
541C8—81no—
542C8—81no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#234211yescharacteristic
558C22#293011yescharacteristic
559C22#293311yescharacteristic
560C22C4×#13711yescharacteristic
561C22#294611yescharacteristic
562C22C2×#44411yescharacteristic
563C22#295711yescharacteristic
564C4—21no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4#292311yescharacteristic
577C4#292511yescharacteristic
578C4#298011yescharacteristic
579C4#298111yescharacteristic
580C4#330711yescharacteristic
581C4#331411yescharacteristic
582C4#332211yescharacteristic
583C4#333111yescharacteristic
584C2—42no—
585C2—82no—
586C2#1793911yescharacteristic
587C2#1794611yescharacteristic
588C2#1801011yescharacteristic
589C2#1802011yescharacteristic
590C2#2255711yescharacteristic
591C2#2257811yescharacteristic
592C2#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. #22629C2semidirect
4. #9557C2semidirect
5. #9555C2semidirect
6. #9548C2semidirect
7. #9547C2semidirect
8. #8847C2semidirect
9. #8845C2semidirect
10. #8706C2semidirect
11. #8600C2semidirect
13. #8208C2semidirect
14. #8202C2semidirect
15. #8149C2semidirect

Maximal subgroups

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

subgroupindex
#226292
#224742
#213882
#95572
#95552
#95482
#95472
#88472
#88452
#87062
#86002
#83152
#82082
#82022
#81492