Finite Group Databasebeta

← #225556

Subgroups — #225556

Order 512.

809 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22703C211yesmaximal, characteristic, complemented
2C4⋊C4×D8C211yesmaximal, characteristic
3#22670C211yesmaximal, characteristic
4D4×#106C211yesmaximal, characteristic, complemented
5C2×#1580C211yesmaximal, characteristic
6#9621C211yesmaximal, characteristic, complemented
7#9612C211yesmaximal, characteristic, complemented
8#8894C211yesmaximal, characteristic, complemented
9#8894C211yesmaximal, characteristic, complemented
10#8886C211yesmaximal, characteristic, complemented
11#8886C211yesmaximal, characteristic, complemented
12#8819C211yesmaximal, characteristic, complemented
13#8646C211yesmaximal, characteristic, complemented
14#8642C211yesmaximal, characteristic, complemented
15#8290C211yesmaximal, characteristic, complemented
16C2×#442—21no—
17C2×C4×D8—21no—
18C2×C4×D8C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C22×#106—21no—
22D4×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25D4×C4⋊C4C2211yescharacteristic
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1943—21no—
31#1942—21no—
32#1864—21no—
33#1864—21no—
34#1862—21no—
35#1862—21no—
36#1853—21no—
37#1853—21no—
38#1853—21no—
39#1853—21no—
40#1845C2211yescharacteristic
41#1844C2211yescharacteristic
42#1844C2211yescharacteristic
43#1844C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1807—21no—
47#1807—21no—
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53#1770C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C2×#340—21no—
56#1580C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×D8D411yescharacteristic
69C23×C8—21no—
70C23×C8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81#442—44no—
82C4×D8—44no—
83C4×D8C2×C414yes—
84C4×D8Q814yes—
85C8×D4—22no—
86C8×D4—22no—
87C8×D4C2×C414yescomplemented
88C8×D4C2×C414yescomplemented
89C2×#107—21no—
90C2×#107—21no—
91C2×#107D411yescharacteristic
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×#106—21no—
98C2×#106—21no—
99C2×#106—21no—
100C2×#106—44no—
101C2×#106—41no—
102C2×#106D411yescharacteristic
103C2×#106D411yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102D411yescharacteristic
113C2×#102D411yescharacteristic
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#385—41no—
126#385—41no—
127#385—41no—
128#385—41no—
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#380—21no—
134#380—21no—
135#380—21no—
136#380—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—41no—
150#340D411yescharacteristic
151#340D411yescharacteristic
152#336—21no—
153#336—21no—
154#334—22no—
155#334C2×C414yescomplemented
156D4⋊C8C2×C414yescomplemented
157D4⋊C8C2×C414yescomplemented
158#322C2×C414yescomplemented
159C22×D4C2×D411yescharacteristic
160C22×D4C2×D411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4—41no—
166C23×C4—41no—
167C2×D8C4⋊C414yes—
168C2×D8SD1614yes—
169C2×D8SD1614yes—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—24no—
174C22×C8—24no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8—81no—
180C22×C8—81no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—42no—
193C4×D4—81no—
194C4×D4—81no—
195C4×D4—81no—
196C4×D4—81no—
197C4×D4C22×C412yes—
198C4×D4C22×C412yes—
199C4×D4C2×Q812yes—
200C4×D4C2×Q812yes—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—81no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C2411yescharacteristic
240#107—42no—
241#107—81no—
242#107C4⋊C414yes—
243#106—22no—
244#106—41no—
245#106—42no—
246#106—42no—
247#106—42no—
248#106—42no—
249#106—81no—
250#106—81no—
251#106—81no—
252#106—81no—
253#106—82no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8C22×C412yes—
261C4⋊C8C22×C412yes—
262#102—42no—
263#102—42no—
264#102—42no—
265#102—42no—
266#102C4⋊C414yes—
267#102C4⋊C414yes—
268C22⋊C8—24no—
269C22⋊C8—24no—
270#97—42no—
271C4×C8—21no—
272C4×C8—42no—
273C4×C8C22×C412yes—
274C4×C8C22×C412yes—
275C4×C8C22×C412yes—
276C4×C8C2×Q812yes—
277C4×C8#4112yes—
278#95—21no—
279#95—21no—
280#95—41no—
281#95—41no—
282#95—41no—
283#95—41no—
284C24—21no—
285C24—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4C2×C4⋊C412yes—
293C2×D4C2×C4⋊C412yes—
294C2×D4C2×SD1612yes—
295C2×D4C2×SD1612yes—
296C2×D4C2×SD1612yes—
297C2×D4C2×SD1612yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—161no—
329C22×C4—161no—
330C22×C4—161no—
331C22×C4—161no—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4H(5,𝔽2)11yescharacteristic
335D8#106116yescomplemented
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—48no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—84no—
359C2×C8C2×C4⋊C412yes—
360C2×C8C2×C4⋊C412yes—
361C2×C8C4×D412yes—
362C2×C8C4×D412yes—
363C2×C8C2×SD1612yes—
364C2×C8C2×SD1612yes—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—81no—
393C4⋊C4—81no—
394C4⋊C4—81no—
395C4⋊C4—81no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4C2×C4⋊C412yes—
399C4⋊C4C2×C4⋊C412yes—
400C4⋊C4AGL(1,ℤ/8)12yes—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42C23×C411yescharacteristic
414C42C22×Q811yescharacteristic
415C42C2×#4111yescharacteristic
416C422−1+411yescharacteristic
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23D4211yescharacteristic
432D4—22no—
433D4—22no—
434D4—22no—
435D4—22no—
436D4—22no—
437D4—22no—
438D4C2×#10614yes—
439D4C2×#10614yescomplemented
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—164no—
481C2×C4—164no—
482C2×C4—164no—
483C2×C4—164no—
484C2×C4C22×C4⋊C411yescharacteristic
485C2×C4C2×C4×D411yescharacteristic
486C2×C4#52011yescharacteristic
487C2×C4#54711yescharacteristic
488C2×C4D4×Q811yescharacteristic
489C2×C4#55411yescharacteristic
490C2×C4C22×SD1611yescharacteristic
491C2×C4C22×SD1611yescharacteristic
492C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
493C2×C4#57511yescharacteristic
494C2×C4#57611yescharacteristic
495C2×C4#57711yescharacteristic
496C2×C4#57711yescharacteristic
497C2×C4#57711yescharacteristic
498C8—22no—
499C8—22no—
500C8—22no—
501C8—42no—
502C8—42no—
503C8—42no—
504C8—81no—
505C8—81no—
506C8C2×#10614yescomplemented
507C22—24no—
508C22—24no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22D4×C4⋊C411yescharacteristic
522C22#326411yescharacteristic
523C22#326711yescharacteristic
524C22D4×SD1611yescharacteristic
525C22D4×SD1611yescharacteristic
526C22#328811yescharacteristic
527C22#328911yescharacteristic
528C4—21no—
529C4—21no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—161no—
537C4—161no—
538C4—161no—
539C4—161no—
540C4C22×#10611yescharacteristic
541C4#291211yescharacteristic
542C4C4×AGL(1,ℤ/8)11yescharacteristic
543C4#294611yescharacteristic
544C4#333511yescharacteristic
545C4#333511yescharacteristic
546C4#338411yescharacteristic
547C4#338911yescharacteristic
548C2—44no—
549C2—44no—
550C2#1976311yescharacteristic
551C2#1976611yescharacteristic
552C2#1980311yescharacteristic
553C2#1980611yescharacteristic
554C2D4×#10611yescharacteristic
555C2#2267611yescharacteristic
556C2#2269311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
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. #22703C2semidirect
4. D4×#106C2semidirect
6. #9621C2semidirect
7. #9612C2semidirect
8. #8894C2semidirect
9. #8894C2semidirect
10. #8886C2semidirect
11. #8886C2semidirect
12. #8819C2semidirect
13. #8646C2semidirect
14. #8642C2semidirect
15. #8290C2semidirect
56. #1580C4semidirect
87. C8×D4C2×C4semidirect
88. C8×D4C2×C4semidirect
155. #334C2×C4semidirect
156. D4⋊C8C2×C4semidirect
157. D4⋊C8C2×C4semidirect
158. #322C2×C4semidirect
335. D8#106semidirect
439. D4C2×#106semidirect
506. C8C2×#106semidirect

Maximal subgroups

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

subgroupindex
#227032
C4⋊C4×D82
#226702
D4×#1062
C2×#15802
#96212
#96122
#88942
#88862
#88192
#86462
#86422
#82902