Finite Group Databasebeta

← #229933

Subgroups — #229933

Order 512.

733 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23696C211yesmaximal, characteristic
2#23084C211yesmaximal, characteristic, complemented
3#23083C211yesmaximal, characteristic, complemented
4C2×C8×D8C211yesmaximal, characteristic, complemented
5C2×#1572C211yesmaximal, characteristic, complemented
6#9663C211yesmaximal, characteristic, complemented
7#9594C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9542C211yesmaximal, characteristic, complemented
10#8943C211yesmaximal, characteristic, complemented
11#8940C211yesmaximal, characteristic, complemented
12#8845C211yesmaximal, characteristic, complemented
13#8845C211yesmaximal, characteristic, complemented
14#8291C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C4×#121C2211yescharacteristic
23C4×#121C2211yescharacteristic
24#2086C411yescharacteristic
25#2075C411yescharacteristic
26#2067C411yescharacteristic
27#2067C411yescharacteristic
28#2054C411yescharacteristic
29#2033C411yescharacteristic
30#2033C411yescharacteristic
31#1999C411yescharacteristic
32#1955—21no—
33#1954—21no—
34#1954C2211yescharacteristic
35#1954C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1924—21no—
39#1923—21no—
40#1869—21no—
41#1869—21no—
42C4×#107C2211yescharacteristic
43#1770C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1752—21no—
48C8×C22⋊C4—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C8×D8—24no—
52#1572—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C23×C8—21no—
71C23×C8—21no—
72C2×#121C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C4×D8—22no—
80C4×D8—22no—
81C8×D4—22no—
82C8×D4—22no—
83C8×D4—41no—
84C8×D4—41no—
85#432—22no—
86#432—41no—
87#431—22no—
88#431—22no—
89#431—22no—
90#431—41no—
91C2×#107C2×C411yescharacteristic
92C2×#107C2×C411yescharacteristic
93C2×#107C2×C411yescharacteristic
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8C2×C411yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
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⋊C8C2×C411yescharacteristic
118C2×C22⋊C8C2×C411yescharacteristic
119C2×C22⋊C8D411yescharacteristic
120C2×C22⋊C8D411yescharacteristic
121C2×#97—21no—
122C2×#97—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#390C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#388—21no—
136#388—21no—
137#386—21no—
138#386—21no—
139#380C2×C411yescharacteristic
140#380C2×C411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336D411yescharacteristic
157#336D411yescharacteristic
158#335—22no—
159#335D414yescomplemented
160#334—22no—
161#334—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8—41no—
165D4⋊C8—41no—
166#322—22no—
167C82—22no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C22×C411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4#4111yescharacteristic
177C2×D8—22no—
178C2×D8—22no—
179C2×D8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—81no—
191C22×C8—81no—
192C22×C8C22×C411yescharacteristic
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200#121—22no—
201#121—22no—
202#121—22no—
203#121—41no—
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—81no—
213C4×D4—81no—
214C4×D4—81no—
215C4×D4—81no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
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×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×C42C22×C411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42C2411yescharacteristic
244#107—22no—
245#107—22no—
246#107—41no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—81no—
256C4⋊C8—81no—
257C4⋊C8C2×D412yes—
258C4⋊C8C2×D412yes—
259#102—22no—
260#102—22no—
261#102—22no—
262#102—22no—
263#102—22no—
264#102—41no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271C22⋊C8—42no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—81no—
277C22⋊C8—81no—
278#97—42no—
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—42no—
291C4×C8—81no—
292C4×C8C2×D412yes—
293C4×C8#4112yes—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300C24—21no—
301C24—21no—
302C24C4×D411yescharacteristic
303C2×D4—24no—
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—42no—
315C2×D4—42no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4—81no—
319C2×D4#13112yes—
320C2×D4#13112yes—
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—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
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—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361D8—28no—
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—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—22no—
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—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—41no—
399C2×C8—41no—
400C2×C8—84no—
401C2×C8—84no—
402C2×C8C4×D412yes—
403C2×C8C4×D412yes—
404C2×C8#13512yes—
405C2×C8#13512yes—
406C4⋊C4—21no—
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—21no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4#13112yes—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—81no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42C22×D411yescharacteristic
443C42C2×#4111yescharacteristic
444C42H(5,𝔽2)11yescharacteristic
445C422−1+411yescharacteristic
446C23—21no—
447C23—22no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23C2×C4×D411yescharacteristic
468D4—22no—
469D4—22no—
470D4—22no—
471D4—22no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—84no—
477D4—84no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
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—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
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×C4C2×C4×D411yescharacteristic
526C2×C4#52011yescharacteristic
527C2×C4D4211yescharacteristic
528C2×C4#54711yescharacteristic
529C2×C4#54811yescharacteristic
530C2×C4#55311yescharacteristic
531C2×C4C2×#13111yescharacteristic
532C2×C4C2×#13111yescharacteristic
533C2×C4C2×#13111yescharacteristic
534C2×C4#56811yescharacteristic
535C2×C4C2×#13511yescharacteristic
536C2×C4C2×#13511yescharacteristic
537C2×C4#57611yescharacteristic
538C2×C4#57611yescharacteristic
539C8—22no—
540C8—22no—
541C8—22no—
542C8—22no—
543C8—42no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—42no—
548C8—42no—
549C8—41no—
550C8—81no—
551C8—81no—
552C8—81no—
553C22—41no—
554C22—42no—
555C22—41no—
556C22—42no—
557C22—41no—
558C22—42no—
559C22—41no—
560C22—42no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22—41no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22#236111yescharacteristic
575C22#292911yescharacteristic
576C22#293311yescharacteristic
577C22C4×#13511yescharacteristic
578C22#294811yescharacteristic
579C22#295511yescharacteristic
580C22#295611yescharacteristic
581C4—21no—
582C4—21no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4#292911yescharacteristic
593C4#293511yescharacteristic
594C4#299211yescharacteristic
595C4#299411yescharacteristic
596C4#328011yescharacteristic
597C4#328711yescharacteristic
598C4#333911yescharacteristic
599C4#334111yescharacteristic
600C2—81no—
601C2—82no—
602C2—82no—
603C2#1790411yescharacteristic
604C2#1792211yescharacteristic
605C2#1799611yescharacteristic
606C2#1802611yescharacteristic
607C2#2299111yescharacteristic
608C2#2302011yescharacteristic
609C2#2302711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#1999C4
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
2. #23084C2semidirect
3. #23083C2semidirect
4. C2×C8×D8C2semidirect
5. C2×#1572C2semidirect
6. #9663C2semidirect
7. #9594C2semidirect
8. #9546C2semidirect
9. #9542C2semidirect
10. #8943C2semidirect
11. #8940C2semidirect
12. #8845C2semidirect
13. #8845C2semidirect
14. #8291C2semidirect
15. #8148C2semidirect
159. #335D4semidirect

Maximal subgroups

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

subgroupindex
#236962
#230842
#230832
C2×C8×D82
C2×#15722
#96632
#95942
#95462
#95422
#89432
#89402
#88452
#82912
#81482