Finite Group Databasebeta

← #400827

Subgroups — #400827

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1947C211yesmaximal, characteristic, complemented
2C2×#1925C211yesmaximal, characteristic
3C2×#1925C212yesmaximal
4#15136C211yesmaximal, characteristic
5#13821C211yesmaximal, characteristic, complemented
6#11373C211yesmaximal, characteristic, complemented
7#11255C211yesmaximal, characteristic, complemented
8#11219C212yesmaximal, complemented
9#11076C211yesmaximal, characteristic
10#11066C212yesmaximal
11#10907C212yesmaximal, complemented
12C22×#107C2212yes—
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C2×#397—21no—
19C2×#397—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1947—24no—
28#1925—28no—
29#1925—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38#1439—22no—
39#1436—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42#1434—21no—
43C4×#95C2211yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C42—21no—
61C22×C42—22no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×#107—24no—
65C2×#107—22no—
66C2×#107—24no—
67C2×#107—24no—
68C2×#107—24no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—22no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—22no—
74C2×#102—24no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—24no—
78C2×#102D414yes—
79#397—44no—
80#397—44no—
81#390—24no—
82#390—22no—
83#390—22no—
84#390—42no—
85#389—24no—
86#389—24no—
87#384—22no—
88#384—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—42no—
92C4×C4⋊C4—44no—
93C4×C4⋊C4D414yes—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95D411yescharacteristic
102C2×#95D412yes—
103C2×#95D411yescharacteristic
104#340—24no—
105#340—22no—
106#340—24no—
107#340—24no—
108#340—42no—
109#336—24no—
110#336—24no—
111#336—22no—
112C25—21no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4C2×D412yes—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8C2×D412yes—
149C22×C8#4114yes—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—81no—
178C2×C4⋊C4—81no—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4#4114yes—
183C2×C4⋊C4#4112yes—
184C2×C4⋊C4#4114yes—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—44no—
208C2×C42—44no—
209C2×C42—82no—
210C2×C42C2×D412yes—
211C2×C42#4112yes—
212#107—48no—
213#107—44no—
214C4⋊C8—28no—
215#102—216no—
216#102—28no—
217#102—44no—
218#95—28no—
219#95—24no—
220#95—22no—
221#95—42no—
222#95—44no—
223#95—44no—
224#95—42no—
225#95—44no—
226#95—81no—
227#95D814yes—
228C24—22no—
229C24—41no—
230C24—42no—
231C24—42no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24—42no—
236C24—41no—
237C24—41no—
238C24—42no—
239C24C22×D411yescharacteristic
240C2×D4—22no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4#12112yes—
309C22×C4#12112yes—
310C22×C4C2×D814yes—
311C22×C4C2×D812yes—
312C22×C4AGL(1,ℤ/8)12yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4112yes—
318C22×C4C2×#4112yes—
319C22×C4C2×#4111yescharacteristic
320C22×C4H(5,𝔽2)11yescharacteristic
321C22×C42−1+412yes—
322C22×C42−1+411yescharacteristic
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—416no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4#12114yes—
360C4⋊C4#12414yes—
361C22⋊C4—88no—
362C22⋊C4—84no—
363C22⋊C4—88no—
364C22⋊C4—84no—
365C42—22no—
366C42—22no—
367C42—24no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—48no—
373C42—88no—
374C23—42no—
375C23—44no—
376C23—44no—
377C23—42no—
378C23—42no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—81no—
392C23—82no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23C2×#12111yescharacteristic
398C23#52512yes—
399C23D4211yescharacteristic
400C23#54712yes—
401C23#54811yescharacteristic
402C23C22×D811yescharacteristic
403C23C2×AGL(1,ℤ/8)11yescharacteristic
404C23#57512yes—
405C23#57611yescharacteristic
406C23#57711yescharacteristic
407C23#57812yes—
408D4—42no—
409D4—44no—
410D4—42no—
411D4—81no—
412D4—82no—
413D4—81no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4#45914yes—
469C2×C4C2×#12111yescharacteristic
470C2×C4#52511yescharacteristic
471C2×C4#52512yes—
472C2×C4C2×#12411yescharacteristic
473C2×C4#52912yes—
474C2×C4#52912yes—
475C2×C4#53211yescharacteristic
476C2×C4#53611yescharacteristic
477C2×C4#53711yescharacteristic
478C2×C4#53912yes—
479C2×C4#54112yes—
480C2×C4#54711yescharacteristic
481C2×C4#54712yes—
482C2×C4#55212yes—
483C2×C4#55311yescharacteristic
484C2×C4#55612yes—
485C2×C4#55612yes—
486C2×C4#55912yes—
487C8—44no—
488C8—42no—
489C8—81no—
490C22—82no—
491C22—82no—
492C22—84no—
493C22—84no—
494C22—82no—
495C22—161no—
496C22—161no—
497C22—161no—
498C22—162no—
499C22—161no—
500C22—162no—
501C22#241311yescharacteristic
502C22#241611yescharacteristic
503C22#245512yes—
504C22#246912yes—
505C22#285011yescharacteristic
506C22C2×#45911yescharacteristic
507C22#304011yescharacteristic
508C22#304411yescharacteristic
509C22#304511yescharacteristic
510C22#310012yes—
511C22#311512yes—
512C22#312112yes—
513C22#312412yes—
514C22#326912yes—
515C22#327211yescharacteristic
516C22D4×D811yescharacteristic
517C22#333412yes—
518C22#334312yes—
519C22#335012yes—
520C22#335512yes—
521C22#335811yescharacteristic
522C22#336611yescharacteristic
523C22#337712yes—
524C4—24no—
525C4—44no—
526C4—42no—
527C4—44no—
528C4—44no—
529C4—84no—
530C4—82no—
531C4—161no—
532C4—162no—
533C4#239211yescharacteristic
534C4#247812yes—
535C4#254711yescharacteristic
536C4#259211yescharacteristic
537C4#259612yes—
538C4#266411yescharacteristic
539C2—162no—
540C2#1480811yescharacteristic
541C2#2457611yescharacteristic
542C2#2459711yescharacteristic
543C2#2472611yescharacteristic
544C2#2474311yescharacteristic
545C2#2654612yes—
546C2#2656912yes—
547C2#2712612yes—
548C2#2715412yes—
549C2#3302411yescharacteristic
550C2#3303511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1947C2semidirect
5. #13821C2semidirect
6. #11373C2semidirect
7. #11255C2semidirect
8. #11219C2semidirect
11. #10907C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19472
C2×#19252
#151362
#138212
#113732
#112552
#112192
#110762
#110662
#109072