Finite Group Databasebeta

← #229938

Subgroups — #229938

Order 512.

784 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23696C211yesmaximal, characteristic
2#22986C211yesmaximal, characteristic, complemented
3#22985C211yesmaximal, characteristic, complemented
4C2×#1580C211yesmaximal, characteristic, complemented
5C2×#1570C211yesmaximal, characteristic, complemented
6#9665C211yesmaximal, characteristic, complemented
7#9595C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9542C211yesmaximal, characteristic, complemented
10#8944C211yesmaximal, characteristic, complemented
11#8939C211yesmaximal, characteristic, complemented
12#8845C211yesmaximal, characteristic, complemented
13#8845C211yesmaximal, characteristic, complemented
14#8288C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#435—21no—
18C2×C8×D4—21no—
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#2086C411yescharacteristic
27#2075C411yescharacteristic
28#2067C411yescharacteristic
29#2067C411yescharacteristic
30#2054C411yescharacteristic
31#2033C411yescharacteristic
32#2033C411yescharacteristic
33#1999C411yescharacteristic
34#1954C2211yescharacteristic
35#1954C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1924—21no—
39#1923—21no—
40#1870—21no—
41#1869—21no—
42#1850—21no—
43#1850—21no—
44C4×#107C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1580—24no—
52#1570—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×M16—21no—
71C23×C8—21no—
72C23×C8—21no—
73C23×C8—21no—
74C2×#121C2×C411yescharacteristic
75C2×#121C2×C411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C4×D8—22no—
82C4×D8—22no—
83#435—44no—
84C8×D4—22no—
85C8×D4—22no—
86C8×D4—41no—
87C8×D4—41no—
88C8×D4—44no—
89#431—22no—
90#431—22no—
91C2×#107C2×C411yescharacteristic
92C2×#107C2×C411yescharacteristic
93C2×#107C2×C411yescharacteristic
94#423—22no—
95#423—41no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—41no—
102C2×C4⋊C8C2×C411yescharacteristic
103C2×C4⋊C8C2×C411yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8C2×C411yescharacteristic
122C2×C22⋊C8C2×C411yescharacteristic
123C2×C22⋊C8D411yescharacteristic
124C2×C22⋊C8D411yescharacteristic
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#388—21no—
138#388—21no—
139#386—21no—
140#386—21no—
141#380C2×C411yescharacteristic
142#380C2×C411yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336D411yescharacteristic
159#336D411yescharacteristic
160#334—22no—
161#334—22no—
162#334—22no—
163#334D414yescomplemented
164D4⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168#322—22no—
169#322—22no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4111yescharacteristic
179C2×D8—22no—
180C2×D8—22no—
181C2×D8—22no—
182C2×M16—42no—
183C2×M16—42no—
184C2×M16—81no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8—81no—
198C22×C8—81no—
199C22×C8—81no—
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208#121—22no—
209#121—22no—
210#121—22no—
211#121—41no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42#4111yescharacteristic
251C2×C42C2411yescharacteristic
252#107—22no—
253#107—22no—
254#107—41no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—42no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8C2×D412yes—
268C4⋊C8C2×D412yes—
269#102—22no—
270#102—22no—
271#102—22no—
272#102—22no—
273#102—22no—
274#102—41no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284C22⋊C8—81no—
285C22⋊C8—81no—
286#97—42no—
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—21no—
297C4×C8—42no—
298C4×C8C2×D412yes—
299C4×C8#4112yes—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306C24—21no—
307C24—21no—
308C24C4×D411yescharacteristic
309C2×D4—24no—
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—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—81no—
325C2×D4#13112yes—
326C2×D4#13112yes—
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—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
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×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4AGL(1,ℤ/8)12yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4111yescharacteristic
367D8—28no—
368M16—84no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—21no—
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—21no—
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—41no—
409C2×C8—84no—
410C2×C8—84no—
411C2×C8—84no—
412C2×C8C4×D412yes—
413C2×C8C4×D412yes—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4#13112yes—
428C22⋊C4—24no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—81no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42C22×D411yescharacteristic
451C42C2×#4111yescharacteristic
452C42H(5,𝔽2)11yescharacteristic
453C422−1+411yescharacteristic
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23#44212yes—
475C23C2×C4×D411yescharacteristic
476D4—22no—
477D4—22no—
478D4—22no—
479D4—22no—
480D4—48no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—84no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
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—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
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—81no—
532C2×C4#44212yes—
533C2×C4C2×C4×D411yescharacteristic
534C2×C4#52011yescharacteristic
535C2×C4D4211yescharacteristic
536C2×C4#54711yescharacteristic
537C2×C4#54811yescharacteristic
538C2×C4#55311yescharacteristic
539C2×C4C2×#13111yescharacteristic
540C2×C4C2×#13111yescharacteristic
541C2×C4C2×#13111yescharacteristic
542C2×C4#56811yescharacteristic
543C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
544C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
545C2×C4#57711yescharacteristic
546C2×C4#57711yescharacteristic
547C8—22no—
548C8—22no—
549C8—42no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—42no—
557C8—42no—
558C8—81no—
559C8—81no—
560C8—81no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22#236111yescharacteristic
583C22#292911yescharacteristic
584C22#293311yescharacteristic
585C22C2×#44211yescharacteristic
586C22#295211yescharacteristic
587C22#295511yescharacteristic
588C22#295611yescharacteristic
589C4—21no—
590C4—21no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4#292911yescharacteristic
601C4#293511yescharacteristic
602C4#299211yescharacteristic
603C4#299411yescharacteristic
604C4#326511yescharacteristic
605C4#326511yescharacteristic
606C4#334511yescharacteristic
607C4#334511yescharacteristic
608C2—42no—
609C2—82no—
610C2—82no—
611C2#1790711yescharacteristic
612C2#1792711yescharacteristic
613C2#1800111yescharacteristic
614C2#1802911yescharacteristic
615C2#2299111yescharacteristic
616C2#2302111yescharacteristic
617C2#2302911yescharacteristic
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. #22986C2semidirect
3. #22985C2semidirect
4. C2×#1580C2semidirect
5. C2×#1570C2semidirect
6. #9665C2semidirect
7. #9595C2semidirect
8. #9546C2semidirect
9. #9542C2semidirect
10. #8944C2semidirect
11. #8939C2semidirect
12. #8845C2semidirect
13. #8845C2semidirect
14. #8288C2semidirect
15. #8149C2semidirect
163. #334D4semidirect

Maximal subgroups

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

subgroupindex
#236962
#229862
#229852
C2×#15802
C2×#15702
#96652
#95952
#95462
#95422
#89442
#89392
#88452
#82882
#81492