Finite Group Databasebeta

← #226124

Subgroups — #226124

Order 512.

732 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22807C211yesmaximal, characteristic
2#22766C211yesmaximal, characteristic
3#22665C211yesmaximal, characteristic, complemented
4#22657C211yesmaximal, characteristic, complemented
5C2×#1581C211yesmaximal, characteristic
6#9616C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8897C211yesmaximal, characteristic, complemented
9#8897C211yesmaximal, characteristic, complemented
10#8885C211yesmaximal, characteristic, complemented
11#8882C211yesmaximal, characteristic, complemented
12#8820C211yesmaximal, characteristic, complemented
13#8650C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic, complemented
15#8285C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#428—21no—
20C2×#427—21no—
21#2354C2211yescharacteristic
22#2354C2211yescharacteristic
23#2354C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1970—21no—
26#1948C2211yescharacteristic
27#1947C2211yescharacteristic
28#1947C2211yescharacteristic
29#1947C2211yescharacteristic
30#1934—21no—
31#1925—21no—
32#1925—21no—
33#1918—21no—
34#1879—21no—
35#1854—21no—
36#1854—21no—
37#1854—21no—
38#1852—21no—
39#1845C2211yescharacteristic
40#1845C2211yescharacteristic
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1809—21no—
46#1807—21no—
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51C4×#107—21no—
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102—21no—
56#1581C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×D8D411yescharacteristic
69C22×M16—21no—
70C23×C8—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C4×D8C2×C414yes—
79C4×D8Q814yes—
80#436—22no—
81#436C2×C414yes—
82C8×D4—22no—
83C8×D4C2×C414yescomplemented
84#428—44no—
85#427—44no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—41no—
94C2×#107D411yescharacteristic
95C2×#106—21no—
96C2×#106—21no—
97C2×#106—21no—
98C2×#106D411yescharacteristic
99C2×#106D411yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102D411yescharacteristic
109C2×#102D411yescharacteristic
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#390—21no—
118#390—21no—
119#390—21no—
120#389—21no—
121#388—41no—
122#388—41no—
123#388—41no—
124#385—41no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#380—21no—
130#380—21no—
131#380—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138#340—21no—
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—41no—
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#335—22no—
154#335C2×C414yescomplemented
155D4⋊C8C2×C414yes—
156D4⋊C8C2×C414yescomplemented
157#322C2×C414yescomplemented
158C22×D4C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—41no—
163C2×D8—41no—
164C2×D8—41no—
165C2×D8C4⋊C414yes—
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—42no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—24no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—42no—
186C4×D4—81no—
187C4×D4C22×C412yes—
188C4×D4C22×C412yes—
189C4×D4C2×Q812yes—
190C4×D4C2×Q812yes—
191#117—42no—
192#117—42no—
193#117—42no—
194#117—81no—
195#117—81no—
196#117—81no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
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—42no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—41no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42C2411yescharacteristic
233#107—42no—
234#107—42no—
235#107—42no—
236#107—42no—
237#107—42no—
238#107—81no—
239#107—81no—
240#107—82no—
241#107C4⋊C414yes—
242#106—22no—
243#106—42no—
244#106—42no—
245#106—81no—
246#106D814yescomplemented
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8C22×C412yes—
254C4⋊C8C22×C412yes—
255#102—42no—
256#102—81no—
257#102—81no—
258#102—81no—
259#102C4⋊C414yes—
260#102C4⋊C414yes—
261C22⋊C8—24no—
262C22⋊C8—24no—
263C4×C8—21no—
264C4×C8—42no—
265C4×C8—81no—
266C4×C8C22×C412yes—
267C4×C8C22×C412yes—
268C4×C8C22×C412yes—
269C4×C8C2×Q812yes—
270C4×C8#4112yes—
271#95—21no—
272#95—21no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277C24—21no—
278C24—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4C2×C4⋊C412yes—
290C2×D4C2×C4⋊C412yes—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—161no—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319D8—44no—
320M16—48no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—48no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8C2×C4⋊C412yes—
344C2×C8C2×C4⋊C412yes—
345C2×C8C4×D412yes—
346C2×C8C4×D412yes—
347C2×C8C2×D812yes—
348C2×C8#13512yes—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—82no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4C2×C4⋊C412yes—
379C4⋊C4C2×C4⋊C412yes—
380C4⋊C4C2×D812yes—
381C4⋊C4C2×D812yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—161no—
391C22⋊C4—161no—
392C22⋊C4—161no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—82no—
398C42—82no—
399C42—82no—
400C42C23×C411yescharacteristic
401C42C22×Q811yescharacteristic
402C42C2×#4111yescharacteristic
403C422−1+411yescharacteristic
404C23—22no—
405C23—22no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23D4211yescharacteristic
419D4—22no—
420D4—22no—
421D4—22no—
422D4—22no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—41no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—164no—
468C2×C4C22×C4⋊C411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4#52011yescharacteristic
471C2×C4#54711yescharacteristic
472C2×C4D4×Q811yescharacteristic
473C2×C4#55411yescharacteristic
474C2×C4C22×D811yescharacteristic
475C2×C4C2×#13511yescharacteristic
476C2×C4#57511yescharacteristic
477C2×C4#57511yescharacteristic
478C2×C4#57611yescharacteristic
479C2×C4#57611yescharacteristic
480C2×C4#57811yescharacteristic
481C2×C4#57811yescharacteristic
482C8—22no—
483C8—22no—
484C8—22no—
485C8—42no—
486C8—42no—
487C8—42no—
488C8—81no—
489C8—81no—
490C8C4×D814yes—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22D4×C4⋊C411yescharacteristic
506C22#326811yescharacteristic
507C22#326911yescharacteristic
508C22D4×D811yescharacteristic
509C22#328311yescharacteristic
510C22#328811yescharacteristic
511C22#328911yescharacteristic
512C4—21no—
513C4—21no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—81no—
519C4—81no—
520C4—161no—
521C4—161no—
522C4—161no—
523C4—161no—
524C4#291111yescharacteristic
525C4#291311yescharacteristic
526C4C2×C4×D811yescharacteristic
527C4#295111yescharacteristic
528C4#334711yescharacteristic
529C4#334911yescharacteristic
530C4Q8×D811yescharacteristic
531C4#338011yescharacteristic
532C2—82no—
533C2—82no—
534C2#1975911yescharacteristic
535C2#1976011yescharacteristic
536C2#1986311yescharacteristic
537C2#1986411yescharacteristic
538C2#2266011yescharacteristic
539C2C4⋊C4×D811yescharacteristic
540C2#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
3. #22665C2semidirect
4. #22657C2semidirect
6. #9616C2semidirect
7. #9614C2semidirect
8. #8897C2semidirect
9. #8897C2semidirect
10. #8885C2semidirect
11. #8882C2semidirect
12. #8820C2semidirect
13. #8650C2semidirect
14. #8644C2semidirect
15. #8285C2semidirect
56. #1581C4semidirect
83. C8×D4C2×C4semidirect
154. #335C2×C4semidirect
156. D4⋊C8C2×C4semidirect
157. #322C2×C4semidirect
246. #106D8semidirect

Maximal subgroups

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

subgroupindex
#228072
#227662
#226652
#226572
C2×#15812
#96162
#96142
#88972
#88852
#88822
#88202
#86502
#86442
#82852