Finite Group Databasebeta

← #218549

Subgroups — #218549

Order 512.

727 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22260C211yesmaximal, characteristic, complemented
2#22008C211yesmaximal, characteristic, complemented
3C4×#459C211yesmaximal, characteristic
4#9674C211yesmaximal, characteristic, complemented
5#9659C211yesmaximal, characteristic, complemented
6#9657C211yesmaximal, characteristic, complemented
7#9603C211yesmaximal, characteristic, complemented
8#9556C211yesmaximal, characteristic, complemented
9#9554C211yesmaximal, characteristic, complemented
10#8947C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8845C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic, complemented
14#8565C211yesmaximal, characteristic, complemented
15#8264C211yesmaximal, characteristic, complemented
16C2×#459C411yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C4×#127C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1982C411yescharacteristic
24#1954C2211yescharacteristic
25#1954C2211yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1947C411yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1925C411yescharacteristic
32#1925C411yescharacteristic
33#1923—21no—
34#1923—21no—
35#1879C411yescharacteristic
36#1879C411yescharacteristic
37#1869—21no—
38#1869—21no—
39#1869—21no—
40#1869—21no—
41#1831C2211yescharacteristic
42#1817—21no—
43#1802C411yescharacteristic
44#1797—21no—
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C22×D8C2×C411yescharacteristic
69C23×C8—21no—
70C23×C8—21no—
71C2×#127C2×C411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C22×C42—21no—
80#459C818yescomplemented
81C4×D8—41no—
82C4×D8—41no—
83C4×D8—41no—
84C4×D8—41no—
85C8×D4—41no—
86C8×D4—41no—
87C8×D4—41no—
88C8×D4—41no—
89C2×#107C2×C411yescharacteristic
90C2×#107C2×C411yescharacteristic
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×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×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⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#390—21no—
126#390—21no—
127#390C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#384C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C43C2311yescharacteristic
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#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#336D411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152#336D411yescharacteristic
153#335—41no—
154#334—41no—
155D4⋊C8—41no—
156D4⋊C8—41no—
157D4⋊C8—41no—
158D4⋊C8—41no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161#322—41no—
162C82—41no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C22×D4C22×C411yescharacteristic
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C2×D8—41no—
174C2×D8—41no—
175C2×D8—41no—
176C2×D8—41no—
177C2×D8C2×C814yescomplemented
178C2×D8C2×C814yescomplemented
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—81no—
188C22×C8—81no—
189C22×C8—81no—
190C22×C8—81no—
191C22×C8C22×C411yescharacteristic
192C22×C8C22×C411yescharacteristic
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199#127—22no—
200#127C2×C814yes—
201C4×D4—22no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—81no—
217C4×D4—81no—
218C4×D4—81no—
219C4×D4—81no—
220C4×D4C2×C814yescomplemented
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—81no—
241C2×C42—81no—
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42C2411yescharacteristic
249#107—41no—
250#107—41no—
251C4⋊C8—22no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—42no—
261C4⋊C8—81no—
262C4⋊C8—81no—
263C4⋊C8C2×C814yescomplemented
264#102—41no—
265#102—41no—
266#102—41no—
267#102—41no—
268#102—41no—
269#102—41no—
270#102C2×C814yescomplemented
271#102C2×C814yescomplemented
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—81no—
275C22⋊C8—81no—
276C22⋊C8—81no—
277C22⋊C8—81no—
278C22⋊C8—81no—
279C22⋊C8—81no—
280C22⋊C8—81no—
281C22⋊C8—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—
294C4×C8—81no—
295#95—21no—
296#95—21no—
297#95—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—81no—
319C2×D4—81no—
320C2×D4—81no—
321C2×D4—81no—
322C2×D4C22×C812yes—
323C2×D4C22×C812yes—
324C2×D4C22×C812yes—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
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×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4C4×D411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361D8—44no—
362D8—44no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—22no—
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—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
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—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—84no—
404C2×C8—84no—
405C2×C8C22×C812yes—
406C2×C8C22×C812yes—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4C22×C812yes—
419C22⋊C4—24no—
420C22⋊C4—42no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—81no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—81no—
442C42—81no—
443C42C22×C812yes—
444C42#13112yes—
445C42H(5,𝔽2)11yescharacteristic
446C42H(5,𝔽2)11yescharacteristic
447C42H(5,𝔽2)11yescharacteristic
448C422−1+411yescharacteristic
449C23—22no—
450C23—22no—
451C23—22no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23C2×C4×D411yescharacteristic
471D4—22no—
472D4—22no—
473D4—22no—
474D4—22no—
475D4—22no—
476D4—22no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—84no—
484D4—84no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
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—84no—
513C2×C4—84no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4C8×D412yes—
527C2×C4C8×D412yes—
528C2×C4#51811yescharacteristic
529C2×C4#52011yescharacteristic
530C2×C4#53511yescharacteristic
531C2×C4#53711yescharacteristic
532C2×C4#54011yescharacteristic
533C2×C4#54211yescharacteristic
534C2×C4C23×C811yescharacteristic
535C2×C4C2×#13111yescharacteristic
536C2×C4#56811yescharacteristic
537C2×C4#56811yescharacteristic
538C2×C4#57611yescharacteristic
539C2×C4#57611yescharacteristic
540C2×C4#57611yescharacteristic
541C2×C4#57711yescharacteristic
542C8—22no—
543C8—22no—
544C8—42no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—81no—
552C8—81no—
553C8—81no—
554C8—81no—
555C8—81no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22#232411yescharacteristic
578C22C2×C8×D411yescharacteristic
579C22#293011yescharacteristic
580C22#294611yescharacteristic
581C22#294811yescharacteristic
582C22#295611yescharacteristic
583C22#295711yescharacteristic
584C4—21no—
585C4—21no—
586C4—22no—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—42no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4#297211yescharacteristic
596C4#298111yescharacteristic
597C4#298811yescharacteristic
598C4#299311yescharacteristic
599C4#322011yescharacteristic
600C4#322411yescharacteristic
601C4#325711yescharacteristic
602C4#325911yescharacteristic
603C2—82no—
604C2—82no—
605C2—82no—
606C2#1785211yescharacteristic
607C2#1785411yescharacteristic
608C2#1794811yescharacteristic
609C2#1795011yescharacteristic
610C2#2213311yescharacteristic
611C2#2215811yescharacteristic
612C2#2216711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#459C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22260C2semidirect
2. #22008C2semidirect
4. #9674C2semidirect
5. #9659C2semidirect
6. #9657C2semidirect
7. #9603C2semidirect
8. #9556C2semidirect
9. #9554C2semidirect
10. #8947C2semidirect
11. #8947C2semidirect
12. #8845C2semidirect
13. #8843C2semidirect
14. #8565C2semidirect
15. #8264C2semidirect
80. #459C8semidirect
177. C2×D8C2×C8semidirect
178. C2×D8C2×C8semidirect
220. C4×D4C2×C8semidirect
263. C4⋊C8C2×C8semidirect
270. #102C2×C8semidirect
271. #102C2×C8semidirect

Maximal subgroups

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

subgroupindex
#222602
#220082
C4×#4592
#96742
#96592
#96572
#96032
#95562
#95542
#89472
#88452
#88432
#85652
#82642