Finite Group Databasebeta

← #470792

Subgroups — #470792

Order 512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C211yesmaximal, characteristic
2C2×#1834C211yesmaximal, characteristic, complemented
3#15150C211yesmaximal, characteristic
4#15135C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11228C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#11200C212yesmaximal
10#11095C212yesmaximal
11#10861C211yesmaximal, characteristic, complemented
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#399—21no—
18C2×#397—21no—
19C2×#397—21no—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1923—24no—
27#1834—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1445—21no—
38#1445—21no—
39#1445—21no—
40#1444—21no—
41#1439—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C24×C4C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D411yescharacteristic
62C22×C22⋊C4D411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#106—24no—
66C2×#106—24no—
67C2×#106—42no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—42no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—42no—
74C2×C4×C8—24no—
75C2×C4×C8D414yes—
76#399—44no—
77#397—44no—
78#397—44no—
79#389—24no—
80#389—42no—
81#384—22no—
82#384—22no—
83#384—22no—
84#384—22no—
85#380—24no—
86#380—42no—
87#380—42no—
88#380—42no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95D411yescharacteristic
101C2×#95D411yescharacteristic
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340—42no—
107#340—42no—
108#340—42no—
109#336—22no—
110#336—22no—
111C25C2×D411yescharacteristic
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8#4114yes—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4#4114yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4SD1614yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42C2×D412yes—
209C2×C42#4112yes—
210#106—48no—
211C4⋊C8—44no—
212C22⋊C8—44no—
213C22⋊C8—44no—
214C4×C8—28no—
215#95—22no—
216#95—22no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—42no—
222#95—44no—
223#95—42no—
224#95—42no—
225#95—42no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24C2×SD1612yes—
237C24C22×D411yescharacteristic
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4C2×SD1612yes—
305C22×C4C2×SD1612yes—
306C22×C4#13512yes—
307C22×C4AGL(1,ℤ/8)12yes—
308C22×C4AGL(1,ℤ/8)12yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)12yes—
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C42−1+412yes—
317C22×C42−1+411yescharacteristic
318C22×C42−1+411yescharacteristic
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8#12314yes—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—84no—
356C4⋊C4—88no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C22⋊C4—28no—
361C22⋊C4—44no—
362C22⋊C4—88no—
363C22⋊C4—88no—
364C42—22no—
365C42—22no—
366C42—24no—
367C42—41no—
368C42—41no—
369C42—42no—
370C23—24no—
371C23—22no—
372C23—22no—
373C23—24no—
374C23—22no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—42no—
393C23C2×#12111yescharacteristic
394C23#53511yescharacteristic
395C23#53611yescharacteristic
396C23#54612yes—
397C23#54712yes—
398C23C22×SD1611yescharacteristic
399C23C2×#13511yescharacteristic
400C23C2×AGL(1,ℤ/8)11yescharacteristic
401C23C2×AGL(1,ℤ/8)11yescharacteristic
402C23#57712yes—
403C23#57812yes—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4#46514yes—
462C2×C4#46914yes—
463C2×C4C2×#12311yescharacteristic
464C2×C4#53211yescharacteristic
465C2×C4#53511yescharacteristic
466C2×C4#53611yescharacteristic
467C2×C4#53711yescharacteristic
468C2×C4#53711yescharacteristic
469C2×C4#53912yes—
470C2×C4#54012yes—
471C2×C4#54112yes—
472C2×C4#54212yes—
473C2×C4#54211yescharacteristic
474C2×C4#54211yescharacteristic
475C2×C4#55112yes—
476C2×C4#55212yes—
477C2×C4#55212yes—
478C2×C4#55912yes—
479C2×C4#56212yes—
480C2×C4#56312yes—
481C8—24no—
482C8—42no—
483C8—81no—
484C8—81no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—82no—
495C22—82no—
496C22#248611yescharacteristic
497C22#254411yescharacteristic
498C22#262011yescharacteristic
499C22#269812yes—
500C22#270612yes—
501C22C2×#46511yescharacteristic
502C22C2×#46911yescharacteristic
503C22#305711yescharacteristic
504C22#305911yescharacteristic
505C22#320711yescharacteristic
506C22#320811yescharacteristic
507C22#322512yes—
508C22#322711yescharacteristic
509C22#322811yescharacteristic
510C22#324212yes—
511C22#329812yes—
512C22#330212yes—
513C22#332412yes—
514C22#333212yes—
515C22#333512yes—
516C22#333712yes—
517C22#334312yes—
518C22#334312yes—
519C4—24no—
520C4—24no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—84no—
525C4—162no—
526C4—162no—
527C4—162no—
528C4#249811yescharacteristic
529C4#262411yescharacteristic
530C4#266411yescharacteristic
531C4#275412yes—
532C4#277312yes—
533C4#282011yescharacteristic
534C2—82no—
535C2#1670111yescharacteristic
536C2#2765811yescharacteristic
537C2#2766711yescharacteristic
538C2#2860011yescharacteristic
539C2#2860711yescharacteristic
540C2#2984311yescharacteristic
541C2#2984811yescharacteristic
542C2#3168812yes—
543C2#3169312yes—
544C2#3193212yes—
545C2#3193712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1834C2semidirect
5. #11373C2semidirect
6. #11357C2semidirect
7. #11228C2semidirect
8. #11220C2semidirect
11. #10861C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×#18342
#151502
#151352
#113732
#113572
#112282
#112202
#112002
#110952
#108612