Finite Group Databasebeta

← #228402

Subgroups — #228402

Order 512.

646 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22953C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic
3C4×#483C211yesmaximal, characteristic
4#9593C211yesmaximal, characteristic, complemented
5#9557C211yesmaximal, characteristic, complemented
6#9543C211yesmaximal, characteristic, complemented
7#9538C211yesmaximal, characteristic, complemented
8#9535C211yesmaximal, characteristic, complemented
9#9075C211yesmaximal, characteristic
10#8846C211yesmaximal, characteristic, complemented
11#8833C211yesmaximal, characteristic
12#8290C211yesmaximal, characteristic, complemented
13C8×#103C211yesmaximal, characteristic, complemented
14#8148C211yesmaximal, characteristic
15#7901C211yesmaximal, characteristic
16C2×#483C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×#431—21no—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C4×C22⋊Q8C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1984C2211yescharacteristic
24#1940—21no—
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1927C411yescharacteristic
29#1927C411yescharacteristic
30#1920C411yescharacteristic
31#1920C411yescharacteristic
32#1916—21no—
33#1916—21no—
34#1916—21no—
35#1872C411yescharacteristic
36#1853C411yescharacteristic
37#1851—21no—
38#1823C411yescharacteristic
39#1817—21no—
40#1797C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C4×#106C2211yescharacteristic
43#1770C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#103C2211yescharacteristic
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#483—24no—
75C8×Q8—22no—
76C8×Q8—41no—
77#431—22no—
78#431—41no—
79#431—44no—
80C2×#106C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84#423—22no—
85#423—41no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#103C2×C411yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8C2×C411yescharacteristic
103C2×C22⋊C8C2×C411yescharacteristic
104C2×C22⋊C8D411yescharacteristic
105C2×C22⋊C8D411yescharacteristic
106C2×#97—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#387—21no—
119#385—21no—
120#380C2×C411yescharacteristic
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C2×#95—21no—
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336C2×C411yescharacteristic
149#336C2×C411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152#334—22no—
153#334—41no—
154Q8⋊C8—22no—
155Q8⋊C8—41no—
156#322—22no—
157C82—41no—
158C22×Q8C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
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×C8—21no—
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22⋊Q8—22no—
182C22⋊Q8—41no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42C2411yescharacteristic
221#106—22no—
222#106—22no—
223#106—22no—
224#106—22no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8—81no—
238C4⋊C8#4112yes—
239#103—22no—
240#103—22no—
241#103—22no—
242#103M1614yescomplemented
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—41no—
246C22⋊C8—42no—
247C22⋊C8—42no—
248C22⋊C8SD1614yes—
249#97—81no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—42no—
262C4×C8—42no—
263C4×C8—42no—
264C4×C8#4112yes—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274C24C4×D411yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8C2×M1612yes—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—81no—
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C2×SD1612yes—
313C22×C4#13512yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317M16—84no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
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×C8C2×M1612yes—
358C2×C8C2×SD1612yes—
359C2×C8C2×SD1612yes—
360C4⋊C4—24no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4C2×M1612yes—
378C4⋊C4#13112yes—
379C4⋊C4#13112yes—
380C22⋊C4—42no—
381C22⋊C4—81no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—42no—
397C42—42no—
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C42H(5,𝔽2)11yescharacteristic
401C422−1+411yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23C4×SD1612yes—
409C23C2×C4×D411yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—41no—
413Q8—41no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—48no—
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—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—84no—
460C2×C4C4×SD1612yes—
461C2×C4C4×#4111yescharacteristic
462C2×C4#52011yescharacteristic
463C2×C4#54611yescharacteristic
464C2×C4#54711yescharacteristic
465C2×C4#55111yescharacteristic
466C2×C4#55211yescharacteristic
467C2×C4C22×M1611yescharacteristic
468C2×C4C2×#13111yescharacteristic
469C2×C4C2×#13111yescharacteristic
470C2×C4#56811yescharacteristic
471C2×C4C22×SD1611yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4#57711yescharacteristic
474C2×C4#57811yescharacteristic
475C8—22no—
476C8—22no—
477C8—22no—
478C8—42no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—42no—
485C8—81no—
486C8—81no—
487C8—81no—
488C8—81no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22#235911yescharacteristic
497C22#292911yescharacteristic
498C22D4×M1611yescharacteristic
499C22C2×C4×SD1611yescharacteristic
500C22#294711yescharacteristic
501C22#295511yescharacteristic
502C22#295711yescharacteristic
503C4—22no—
504C4—21no—
505C4—21no—
506C4—42no—
507C4—41no—
508C4—42no—
509C4—41no—
510C4—42no—
511C4—42no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4#296511yescharacteristic
517C4#296911yescharacteristic
518C4#299011yescharacteristic
519C4#299311yescharacteristic
520C4#329611yescharacteristic
521C4#329811yescharacteristic
522C4#333511yescharacteristic
523C4#333611yescharacteristic
524C2—42no—
525C2#1790211yescharacteristic
526C2#1791911yescharacteristic
527C2M16×SD1611yescharacteristic
528C2#1805311yescharacteristic
529C2#2290411yescharacteristic
530C2#2292011yescharacteristic
531C2#2294911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9593C2semidirect
5. #9557C2semidirect
6. #9543C2semidirect
7. #9538C2semidirect
8. #9535C2semidirect
10. #8846C2semidirect
12. #8290C2semidirect
13. C8×#103C2semidirect
242. #103M16semidirect

Maximal subgroups

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

subgroupindex
#229532
#225992
C4×#4832
#95932
#95572
#95432
#95382
#95352
#90752
#88462
#88332
#82902
C8×#1032
#81482
#79012