Finite Group Databasebeta

← #230801

Subgroups — #230801

Order 512.

738 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23174C211yesmaximal, characteristic, complemented
2#22835C211yesmaximal, characteristic, complemented
3C4×#459C211yesmaximal, characteristic
4#9663C211yesmaximal, characteristic, complemented
5#9659C211yesmaximal, characteristic, complemented
6#9648C211yesmaximal, characteristic, complemented
7#9606C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic, complemented
9#9541C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8263C211yesmaximal, characteristic, complemented
13#8210C211yesmaximal, characteristic, complemented
14#8200C211yesmaximal, characteristic, complemented
15#8172C211yesmaximal, characteristic, complemented
16C2×#459C411yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#435—21no—
21C2×C8×D4C2211yescharacteristic
22C4×#127C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1982C411yescharacteristic
25#1955—21no—
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1947C411yescharacteristic
31#1925C411yescharacteristic
32#1925C411yescharacteristic
33#1923—21no—
34#1923—21no—
35#1879C411yescharacteristic
36#1879C411yescharacteristic
37#1870—21no—
38#1869—21no—
39#1831C2211yescharacteristic
40#1817—21no—
41#1802C411yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44#1767C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8—21no—
50C4×#97C2211yescharacteristic
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×#322C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C22×D8C2×C411yescharacteristic
69C22×M16—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#459—24no—
81C4×D8—41no—
82C4×D8—41no—
83C4×D8—41no—
84C4×D8—41no—
85#436—22no—
86#436—41no—
87#435—44no—
88C8×D4—22no—
89C8×D4—41no—
90C2×#107C2×C411yescharacteristic
91C2×#107C2×C411yescharacteristic
92C2×C4⋊C8C2×C411yescharacteristic
93C2×C4⋊C8C2×C411yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
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×#97—21no—
119C2×#97—21no—
120C2×#97—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#390—21no—
130#390—21no—
131#390C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#384C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C43C2311yescharacteristic
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#336C2×C411yescharacteristic
151#336C2×C411yescharacteristic
152#336D411yescharacteristic
153#336D411yescharacteristic
154#335—41no—
155#334—41no—
156D4⋊C8—22no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—41no—
161D4⋊C8—41no—
162#322—41no—
163#322—41no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C22×C411yescharacteristic
167C22×D4C22×C411yescharacteristic
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C2×D8—22no—
175C2×D8—22no—
176C2×D8—22no—
177C2×D8—22no—
178C2×D8—41no—
179C2×D8—41no—
180C2×M16—42no—
181C2×M16—42no—
182C2×M16—81no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—42no—
190C22×C8—42no—
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#127—22no—
201#127—41no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
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—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—41no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—81no—
221C4×D4—81no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—81no—
242C2×C42—81no—
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42C2411yescharacteristic
250#107—41no—
251#107—41no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263C4⋊C8#4112yes—
264#102—22no—
265#102—22no—
266#102—22no—
267#102—41no—
268#102—41no—
269#102—41no—
270#102—41no—
271#102M1614yescomplemented
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—81no—
279C22⋊C8—81no—
280C22⋊C8—81no—
281C22⋊C8—81no—
282#97—42no—
283#97—42no—
284#97—42no—
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—21no—
295C4×C8—21no—
296C4×C8—81no—
297C4×C8—81no—
298C4×C8#4112yes—
299#95—21no—
300#95—21no—
301#95—21no—
302C24—21no—
303C24—21no—
304C24—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—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—81no—
326C2×D4—81no—
327C2×D4C2×M1612yes—
328C2×D4#13112yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—42no—
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—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4111yescharacteristic
365D8—44no—
366D8—44no—
367M16—84no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
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—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—84no—
409C2×C8C2×M1612yes—
410C2×C8AGL(1,ℤ/8)12yes—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—42no—
420C4⋊C4—81no—
421C4⋊C4C2×M1612yes—
422C4⋊C4#13112yes—
423C22⋊C4—24no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—22no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—81no—
448C42—81no—
449C42C2×#4111yescharacteristic
450C42C2×#4111yescharacteristic
451C42H(5,𝔽2)11yescharacteristic
452C42H(5,𝔽2)11yescharacteristic
453C23—22no—
454C23—22no—
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—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23C2×C4×D411yescharacteristic
475D4—22no—
476D4—22no—
477D4—22no—
478D4—22no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—84no—
488D4—84no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
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—41no—
516C2×C4—41no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4#44412yes—
532C2×C4C4×#4111yescharacteristic
533C2×C4#51811yescharacteristic
534C2×C4#54611yescharacteristic
535C2×C4#55011yescharacteristic
536C2×C4#55311yescharacteristic
537C2×C4#55511yescharacteristic
538C2×C4C22×M1611yescharacteristic
539C2×C4C2×#13111yescharacteristic
540C2×C4C2×#13111yescharacteristic
541C2×C4#56811yescharacteristic
542C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
543C2×C4#57511yescharacteristic
544C2×C4#57611yescharacteristic
545C2×C4#57611yescharacteristic
546C8—22no—
547C8—22no—
548C8—42no—
549C8—42no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—42no—
557C8—81no—
558C8—81no—
559C8—81no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22#236311yescharacteristic
582C22#292911yescharacteristic
583C22D4×M1611yescharacteristic
584C22C4×AGL(1,ℤ/8)11yescharacteristic
585C22#294811yescharacteristic
586C22C2×#44411yescharacteristic
587C22#295611yescharacteristic
588C4—21no—
589C4—21no—
590C4—22no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4#296511yescharacteristic
600C4#296811yescharacteristic
601C4#297411yescharacteristic
602C4#298111yescharacteristic
603C4#330911yescharacteristic
604C4#331811yescharacteristic
605C4#339911yescharacteristic
606C4#340311yescharacteristic
607C2—82no—
608C2—82no—
609C2—161no—
610C2#1791711yescharacteristic
611C2#1792211yescharacteristic
612C2#1804811yescharacteristic
613C2#1806411yescharacteristic
614C2#2308511yescharacteristic
615C2#2310811yescharacteristic
616C2#2311411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#459C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23174C2semidirect
2. #22835C2semidirect
4. #9663C2semidirect
5. #9659C2semidirect
6. #9648C2semidirect
7. #9606C2semidirect
8. #9542C2semidirect
9. #9541C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8263C2semidirect
13. #8210C2semidirect
14. #8200C2semidirect
15. #8172C2semidirect
271. #102M16semidirect

Maximal subgroups

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

subgroupindex
#231742
#228352
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