Finite Group Databasebeta

← #227571

Subgroups — #227571

Order 512.

804 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23066C211yesmaximal, characteristic
2#23004C211yesmaximal, characteristic, complemented
3Q8×#102C211yesmaximal, characteristic
4#22430C211yesmaximal, characteristic
5C2×#1649C211yesmaximal, characteristic
6#9834C211yesmaximal, characteristic, complemented
7#9573C211yesmaximal, characteristic, complemented
8#9570C211yesmaximal, characteristic, complemented
9#9227C211yesmaximal, characteristic
10#9218C211yesmaximal, characteristic, complemented
11#9058C211yesmaximal, characteristic, complemented
12#8896C211yesmaximal, characteristic, complemented
13#8893C211yesmaximal, characteristic
14#8208C211yesmaximal, characteristic
15#8203C211yesmaximal, characteristic, complemented
16C2×D4×Q8C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×#462C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#441—21no—
22C2×#436C2211yescharacteristic
23C2×#419—21no—
24#2365C2211yescharacteristic
25#2354C2211yescharacteristic
26Q8×C4⋊C4C2211yescharacteristic
27#1990—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1988—21no—
31#1988C2211yescharacteristic
32#1981C2211yescharacteristic
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1936—21no—
38#1935—21no—
39#1926—21no—
40#1921—21no—
41#1894—21no—
42#1878—21no—
43#1873—21no—
44#1865—21no—
45#1834C2211yescharacteristic
46#1803C2211yescharacteristic
47#1802—21no—
48#1802C2211yescharacteristic
49#1787—21no—
50#1776—21no—
51#1770C2211yescharacteristic
52#1770C2211yescharacteristic
53C4×#103C2211yescharacteristic
54C4×#102—21no—
55C4×#102C2211yescharacteristic
56#1649C418yescomplemented
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C23×Q8—21no—
69C22×Q16—21no—
70C22×M16—21no—
71D4×Q8—22no—
72D4×Q8—22no—
73D4×Q8—41no—
74D4×Q8C2×C414yes—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2311yescharacteristic
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—41no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4C2311yescharacteristic
85C2×#117—21no—
86#500—41no—
87#500C2×C414yes—
88#474—22no—
89#474C2×C414yes—
90#462—22no—
91#462C2×C414yes—
92C8×Q8—22no—
93C8×Q8—41no—
94#441—44no—
95#436—22no—
96#436—41no—
97C2×#107—21no—
98C2×#107—41no—
99C2×#106—21no—
100C2×#106—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106#419—44no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×#103D411yescharacteristic
116C2×#103D411yescharacteristic
117C2×#102—21no—
118C2×#102—21no—
119C2×#102D411yescharacteristic
120C2×#102D411yescharacteristic
121C2×C22⋊C8—21no—
122C2×#97—21no—
123C2×#97—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#391—21no—
129#390—21no—
130#389—21no—
131#389—21no—
132#389—41no—
133#388—41no—
134#387—41no—
135#384—21no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#380—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—41no—
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#340D411yescharacteristic
159#336—21no—
160#336—21no—
161#332—41no—
162#332C2×C414yes—
163Q8⋊C8C2×C414yes—
164D4⋊C8C2×C414yes—
165#322—41no—
166C22×Q8—24no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—41no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—81no—
174C22×Q8—81no—
175C22×Q8C2×D411yescharacteristic
176C22×D4C2×D411yescharacteristic
177C23×C4—21no—
178C23×C4—41no—
179C2×Q16—24no—
180C2×Q16—42no—
181C2×Q16—42no—
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×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C4⋊Q8—24no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—42no—
202C4⋊Q8C22×C412yes—
203C4⋊Q8C22×C412yes—
204C22⋊Q8—24no—
205C22⋊Q8—44no—
206C22⋊Q8—44no—
207C22⋊Q8—42no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—42no—
211C4×Q8—42no—
212C4×Q8—81no—
213C4×Q8—81no—
214C4×Q8C22×C412yes—
215C4×Q8#4112yes—
216C4×D4—24no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—42no—
220C4×D4C22×C412yes—
221C4×D4#4112yes—
222#117—42no—
223#117—81no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42C2411yescharacteristic
256#107—24no—
257#107—44no—
258#106—42no—
259#106—42no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—42no—
267C4⋊C8C22×C412yes—
268C4⋊C8C22×C412yes—
269#103—22no—
270#103—22no—
271#103—24no—
272#103—42no—
273#103—42no—
274#103—42no—
275#103—42no—
276#103—81no—
277#103—81no—
278#103—81no—
279#102—24no—
280#102—22no—
281#102—41no—
282#102—42no—
283C22⋊C8—42no—
284#97—42no—
285#97—42no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—42no—
292C4×C8C22×C412yes—
293#95—21no—
294#95—21no—
295#95—41no—
296#95—41no—
297#95—41no—
298#95—41no—
299C24—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—44no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—48no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—82no—
318C2×Q8—82no—
319C2×Q8—84no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8C4×D412yes—
323C2×Q8AGL(1,ℤ/8)12yes—
324C2×Q8AGL(1,ℤ/8)12yes—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4C4×D412yes—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—44no—
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—82no—
350C22×C4—82no—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354Q16—48no—
355M16—84no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
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—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8#13712yes—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—44no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—82no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—82no—
416C4⋊C4—161no—
417C4⋊C4C4×D412yes—
418C4⋊C4C4×D412yes—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—44no—
422C22⋊C4—82no—
423C22⋊C4—161no—
424C42—22no—
425C42—22no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—82no—
431C42—82no—
432C42—82no—
433C42C23×C411yescharacteristic
434C42C2×#4111yescharacteristic
435C42C2×#4111yescharacteristic
436C42H(5,𝔽2)11yescharacteristic
437C23—22no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23D4211yescharacteristic
445Q8—22no—
446Q8—22no—
447Q8—22no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—88no—
453Q8—81no—
454Q8—81no—
455Q8—81no—
456Q8—81no—
457Q8#44214yes—
458D4—22no—
459D4—22no—
460D4—41no—
461D4—41no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—88no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×C4×D411yescharacteristic
504C2×C4#51811yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4#54611yescharacteristic
507C2×C4#55111yescharacteristic
508C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
509C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
510C2×C4C2×#13711yescharacteristic
511C2×C4#57511yescharacteristic
512C2×C4#57611yescharacteristic
513C2×C4#57711yescharacteristic
514C2×C4#57811yescharacteristic
515C2×C4#57811yescharacteristic
516C8—22no—
517C8—42no—
518C8—42no—
519C8—42no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—81no—
524C8—81no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22#235811yescharacteristic
533C22#326411yescharacteristic
534C22#326511yescharacteristic
535C22#327611yescharacteristic
536C22#327711yescharacteristic
537C22#328911yescharacteristic
538C22#328911yescharacteristic
539C4—24no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—42no—
544C4—82no—
545C4—82no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—161no—
551C4—161no—
552C4C2×#44211yescharacteristic
553C4C4×#13711yescharacteristic
554C4#294611yescharacteristic
555C4#295011yescharacteristic
556C4#330611yescharacteristic
557C4#331211yescharacteristic
558C4#331311yescharacteristic
559C4#331811yescharacteristic
560C2—82no—
561C2#2016711yescharacteristic
562C2#2016811yescharacteristic
563C2#2019911yescharacteristic
564C2#2020011yescharacteristic
565C2#2285811yescharacteristic
566C2#2286911yescharacteristic
567C2#2289211yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23004C2semidirect
6. #9834C2semidirect
7. #9573C2semidirect
8. #9570C2semidirect
10. #9218C2semidirect
11. #9058C2semidirect
12. #8896C2semidirect
15. #8203C2semidirect
56. #1649C4semidirect

Maximal subgroups

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

subgroupindex
#230662
#230042
Q8×#1022
#224302
C2×#16492
#98342
#95732
#95702
#92272
#92182
#90582
#88962
#88932
#82082
#82032