Finite Group Databasebeta

← #305440

Subgroups — #305440

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042C211yesmaximal, characteristic, complemented
2#26020C211yesmaximal, characteristic
3#22522C211yesmaximal, characteristic
4#22502C211yesmaximal, characteristic
5#10684C211yesmaximal, characteristic, complemented
6#10597C211yesmaximal, characteristic, complemented
7#10593C211yesmaximal, characteristic, complemented
8#10349C211yesmaximal, characteristic, complemented
9#8885C211yesmaximal, characteristic
10#8884C211yesmaximal, characteristic, complemented
11#8849C211yesmaximal, characteristic
12#8848C211yesmaximal, characteristic, complemented
13#8657C211yesmaximal, characteristic, complemented
14#8650C211yesmaximal, characteristic
15#8450C211yesmaximal, characteristic
16C2×#554C2211yescharacteristic
17C2×#475—21no—
18C2×#446C2211yescharacteristic
19C2×#436C2211yescharacteristic
20#2551C2211yescharacteristic
21#2354C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23#2088—21no—
24#2067—21no—
25#2058—21no—
26#2023—21no—
27#1992—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1989—21no—
31#1988—21no—
32#1987C2211yescharacteristic
33#1987C2211yescharacteristic
34#1956C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1918—21no—
40#1894—21no—
41#1873—21no—
42#1865—21no—
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1834C2211yescharacteristic
46#1823—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1802—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1762—21no—
53C4×#103C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1736—21no—
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68#554—22no—
69#554—22no—
70#554—41no—
71#554Q814yescomplemented
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×#125—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—41no—
78C2×C4×Q8—21no—
79C2×C4×Q8C2311yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4C2311yescharacteristic
82C2×#117—21no—
83C22×C4⋊C4—21no—
84#475—44no—
85#446—22no—
86#446—41no—
87#436—22no—
88#436—41no—
89C2×#107—21no—
90C2×#107—41no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—41no—
104C2×#103D411yescharacteristic
105C2×#103D411yescharacteristic
106C2×#102—21no—
107C2×#102—21no—
108C2×#102D411yescharacteristic
109C2×#102D411yescharacteristic
110C2×C22⋊C8—21no—
111C2×#97—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#395—41no—
116#390—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—41no—
121#388—41no—
122#384—21no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#380—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
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#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#335—41no—
154#334—41no—
155#333—41no—
156Q8⋊C8—22no—
157D4⋊C8—22no—
158C22×Q8—21no—
159C22×Q8—41no—
160C22×Q8C2×D411yescharacteristic
161C22×D4C2×D411yescharacteristic
162C23×C4—21no—
163C23×C4—41no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16—81no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8—81no—
182C4⋊Q8—81no—
183C4⋊Q8C2×Q812yes—
184#125—24no—
185#125—42no—
186C22⋊Q8—24no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C22⋊Q8—42no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—42no—
193C4×Q8—81no—
194C4×Q8C2×Q812yes—
195C4×Q8#4112yes—
196C4×D4—24no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—42no—
200C4×D4C2×Q812yes—
201C4×D4#4112yes—
202#117—42no—
203#117—81no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42C2411yescharacteristic
240#107—42no—
241#107—82no—
242#106—42no—
243#106—42no—
244#106—81no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—81no—
256#103—22no—
257#103—22no—
258#103—42no—
259#103—81no—
260#103—82no—
261#102—22no—
262#102—41no—
263#102—42no—
264#102—81no—
265C22⋊C8—42no—
266#97—42no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273#95—21no—
274#95—21no—
275#95—41no—
276#95—41no—
277#95—41no—
278C24—21no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—82no—
287C2×Q8—82no—
288C2×Q8—81no—
289C2×Q8—81no—
290C2×Q8—161no—
291C2×Q8C22⋊Q812yes—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4C22⋊Q812yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—44no—
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—161no—
317C22×C4—161no—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4H(5,𝔽2)11yescharacteristic
321M16—84no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8#13712yes—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—44no—
362C4⋊C4—44no—
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—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4—81no—
382C4⋊C4—82no—
383C4⋊C4—161no—
384C4⋊C4C22⋊Q812yes—
385C4⋊C4C22⋊Q812yes—
386C4⋊C4AGL(1,ℤ/8)12yes—
387C4⋊C4#13712yes—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—44no—
391C22⋊C4—82no—
392C22⋊C4—161no—
393C42—22no—
394C42—22no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—82no—
400C42—82no—
401C42C22×Q811yescharacteristic
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C42H(5,𝔽2)11yescharacteristic
405C23—22no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23D4211yescharacteristic
413Q8—22no—
414Q8—22no—
415Q8—41no—
416Q8—41no—
417Q8—82no—
418Q8—161no—
419Q8—164no—
420D4—22no—
421D4—22no—
422D4—41no—
423D4—41no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
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—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
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—164no—
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4C2×C22⋊Q811yescharacteristic
466C2×C4#54311yescharacteristic
467C2×C4#54611yescharacteristic
468C2×C4#54611yescharacteristic
469C2×C4#55111yescharacteristic
470C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
471C2×C4C2×#13711yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C2×C4#57511yescharacteristic
474C2×C4#57611yescharacteristic
475C2×C4#57711yescharacteristic
476C2×C4#57811yescharacteristic
477C2×C4#57811yescharacteristic
478C8—22no—
479C8—42no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—81no—
484C8—81no—
485C8—81no—
486C8—81no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22#243711yescharacteristic
495C22#326411yescharacteristic
496C22#326611yescharacteristic
497C22#327611yescharacteristic
498C22#327711yescharacteristic
499C22#328811yescharacteristic
500C22#329111yescharacteristic
501C4—21no—
502C4—21no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—81no—
510C4—81no—
511C4—161no—
512C4—161no—
513C4—161no—
514C4#307811yescharacteristic
515C4#307911yescharacteristic
516C4#308211yescharacteristic
517C4#308311yescharacteristic
518C4#331411yescharacteristic
519C4#332111yescharacteristic
520C4#332811yescharacteristic
521C4#333011yescharacteristic
522C2—82no—
523C2#2003811yescharacteristic
524C2#2017311yescharacteristic
525C2#2018311yescharacteristic
526C2#2031811yescharacteristic
527C2#2597011yescharacteristic
528C2#2597111yescharacteristic
529C2#2598311yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×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. #26042C2semidirect
5. #10684C2semidirect
6. #10597C2semidirect
7. #10593C2semidirect
8. #10349C2semidirect
10. #8884C2semidirect
12. #8848C2semidirect
13. #8657C2semidirect
71. #554Q8semidirect

Maximal subgroups

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

subgroupindex
#260422
#260202
#225222
#225022
#106842
#105972
#105932
#103492
#88852
#88842
#88492
#88482
#86572
#86502
#84502