Finite Group Databasebeta

← #229442

Subgroups — #229442

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23190C211yesmaximal, characteristic, complemented
2#23001C211yesmaximal, characteristic
3#22968C211yesmaximal, characteristic, complemented
4D4×#107C211yesmaximal, characteristic, complemented
5C2×#1665C211yesmaximal, characteristic
6#9842C211yesmaximal, characteristic, complemented
7#9621C211yesmaximal, characteristic, complemented
8#9574C211yesmaximal, characteristic, complemented
9#9571C211yesmaximal, characteristic, complemented
10#8981C211yesmaximal, characteristic, complemented
11#8977C211yesmaximal, characteristic
12#8926C211yesmaximal, characteristic, complemented
13#8921C211yesmaximal, characteristic, complemented
14#8287C211yesmaximal, characteristic, complemented
15#8202C211yesmaximal, characteristic, complemented
16C2×#492C2211yescharacteristic
17C2×#474C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×#442—21no—
20C2×#440—21no—
21C2×C8×D4C2211yescharacteristic
22C22×#107—21no—
23#2366C2211yescharacteristic
24#2365C2211yescharacteristic
25D4×C4⋊C4C2211yescharacteristic
26D4×C4⋊C4C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1982C2211yescharacteristic
30#1981C2211yescharacteristic
31#1973—21no—
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1943—21no—
39#1936—21no—
40#1921—21no—
41#1880—21no—
42#1879—21no—
43#1877—21no—
44#1864—21no—
45#1864—21no—
46#1852—21no—
47#1852—21no—
48#1845C2211yescharacteristic
49#1808—21no—
50C4×#107C2211yescharacteristic
51#1770C2211yescharacteristic
52#1770C2211yescharacteristic
53#1770C2211yescharacteristic
54#1762—21no—
55C4×#102C2211yescharacteristic
56#1665C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70C22×D8—21no—
71C23×C8—21no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×#127C2311yescharacteristic
74C2×C4×Q8—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4C2311yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82#492—22no—
83#492C2×C414yescomplemented
84#474—22no—
85#474C2×C414yescomplemented
86#459—22no—
87#459C2×C414yes—
88#442—44no—
89#440—44no—
90C8×D4—22no—
91C8×D4C2×C414yescomplemented
92C2×#107—21no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#107—21no—
96C2×#107—21no—
97C2×#107—44no—
98C2×#107D411yescharacteristic
99C2×#107D411yescharacteristic
100C2×#106—21no—
101C2×#106—21no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—41no—
115C2×#102D411yescharacteristic
116C2×#102D411yescharacteristic
117C2×C22⋊C8—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×#97—21no—
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391—21no—
125#390—21no—
126#390—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#387—41no—
131#386—41no—
132#385—41no—
133#385—41no—
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#380—21no—
137#380—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—41no—
149#340D411yescharacteristic
150#340D411yescharacteristic
151#340D411yescharacteristic
152#340D411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#334C2×C414yescomplemented
157#332C2×C414yescomplemented
158#331C2×C414yescomplemented
159D4⋊C8—22no—
160#322—22no—
161C22×Q8—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C2×D411yescharacteristic
166C23×C4—21no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—41no—
170C2×SD16—24no—
171C2×SD16—24no—
172C2×SD16—42no—
173C2×SD16—42no—
174C2×SD16—81no—
175C2×SD16—81no—
176C2×D8—24no—
177C2×D8—42no—
178C2×D8—42no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—24no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8C22×C412yes—
195#127—21no—
196#127—21no—
197#127—21no—
198#127C22×C412yes—
199C4×Q8—42no—
200C4×Q8—81no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—81no—
208C4×D4—81no—
209C4×D4—81no—
210C4×D4—81no—
211C4×D4C22×C412yes—
212C4×D4#4112yes—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—81no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—41no—
245C2×C42C2411yescharacteristic
246#107—22no—
247#107—22no—
248#107—24no—
249#107—42no—
250#107—42no—
251#107—42no—
252#107—81no—
253#106—42no—
254#106—81no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8C22×C412yes—
261C4⋊C8C22×C412yes—
262C4⋊C8C22×C412yes—
263#103—42no—
264#103—81no—
265#102—24no—
266#102—22no—
267#102—22no—
268#102—24no—
269#102—42no—
270#102—42no—
271#102—81no—
272#102—81no—
273#102—82no—
274C22⋊C8—24no—
275#97—42no—
276#97—42no—
277#97—81no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8C22×C412yes—
283C4×C8#4112yes—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—41no—
288#95—41no—
289#95—41no—
290#95—41no—
291C24—21no—
292C24—41no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4C4×D412yes—
319C2×D4AGL(1,ℤ/8)12yes—
320C2×D4#13712yes—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—161no—
344C22×C4—161no—
345C22×C4—161no—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349SD16—48no—
350SD16—84no—
351D8—48no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—48no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8C4×D412yes—
377C2×C8C4×D412yes—
378C2×C8AGL(1,ℤ/8)12yes—
379C2×C8AGL(1,ℤ/8)12yes—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—81no—
406C4⋊C4—81no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4C4×D412yes—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—161no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—82no—
425C42C23×C411yescharacteristic
426C42C2×#4111yescharacteristic
427C42C2×#4111yescharacteristic
428C422−1+411yescharacteristic
429C23—22no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23D4211yescharacteristic
444Q8—42no—
445Q8—42no—
446Q8—81no—
447Q8—81no—
448D4—22no—
449D4—22no—
450D4—22no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—82no—
456D4—81no—
457D4—81no—
458D4—81no—
459D4—81no—
460D4—161no—
461D4#44014yescomplemented
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
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—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—164no—
499C2×C4—164no—
500C2×C4—164no—
501C2×C4C2×C4×D411yescharacteristic
502C2×C4C2×C4×D411yescharacteristic
503C2×C4#52011yescharacteristic
504C2×C4#54711yescharacteristic
505C2×C4#54811yescharacteristic
506C2×C4#55311yescharacteristic
507C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
508C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
509C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
510C2×C4C2×#13711yescharacteristic
511C2×C4#57611yescharacteristic
512C2×C4#57611yescharacteristic
513C2×C4#57711yescharacteristic
514C2×C4#57811yescharacteristic
515C8—22no—
516C8—41no—
517C8—41no—
518C8—42no—
519C8—42no—
520C8—42no—
521C8—81no—
522C8—81no—
523C8#44214yescomplemented
524C22—24no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—161no—
535C22—161no—
536C22—161no—
537C22—161no—
538C22#236111yescharacteristic
539C22#326411yescharacteristic
540C22#327011yescharacteristic
541C22#327211yescharacteristic
542C22#327411yescharacteristic
543C22#328811yescharacteristic
544C22#328911yescharacteristic
545C4—21no—
546C4—21no—
547C4—42no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C4—82no—
554C4—81no—
555C4—161no—
556C4—161no—
557C4C2×#44011yescharacteristic
558C4C2×#44211yescharacteristic
559C4#294611yescharacteristic
560C4#295111yescharacteristic
561C4#334311yescharacteristic
562C4#334611yescharacteristic
563C4#336611yescharacteristic
564C4#337311yescharacteristic
565C2—44no—
566C2—162no—
567C2#2016011yescharacteristic
568C2#2016311yescharacteristic
569C2#2020811yescharacteristic
570C2#2021111yescharacteristic
571C2#2300111yescharacteristic
572C2#2303211yescharacteristic
573C2#2304911yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
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
1. #23190C2semidirect
3. #22968C2semidirect
4. D4×#107C2semidirect
6. #9842C2semidirect
7. #9621C2semidirect
8. #9574C2semidirect
9. #9571C2semidirect
10. #8981C2semidirect
12. #8926C2semidirect
13. #8921C2semidirect
14. #8287C2semidirect
15. #8202C2semidirect
56. #1665C4semidirect
83. #492C2×C4semidirect
85. #474C2×C4semidirect
91. C8×D4C2×C4semidirect
156. #334C2×C4semidirect
157. #332C2×C4semidirect
158. #331C2×C4semidirect
461. D4#440semidirect
523. C8#442semidirect

Maximal subgroups

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

subgroupindex
#231902
#230012
#229682
D4×#1072
C2×#16652
#98422
#96212
#95742
#95712
#89812
#89772
#89262
#89212
#82872
#82022