Finite Group Databasebeta

← #403689

Subgroups — #403689

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic
3C2×#2058C212yesmaximal
4C2×#1948C211yesmaximal, characteristic
5#15196C211yesmaximal, characteristic
6#11227C211yesmaximal, characteristic
7#11220C212yesmaximal
8#11202C211yesmaximal, characteristic
9#10908C212yesmaximal
10#10885C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#106C2212yes—
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17#2365—24no—
18#2365—24no—
19C2×#391—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#387—21no—
24C2×#387—21no—
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2084—24no—
31#2058—28no—
32#1948—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1444—21no—
44#1441—21no—
45#1441—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C23×Q8—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8D414yes—
57C2×C4×Q8—28no—
58C2×C4×Q8—44no—
59C2×C4×Q8—44no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—22no—
88#391—28no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—24no—
93#389—24no—
94#389D414yes—
95#387—28no—
96#387—28no—
97#384—22no—
98#384—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D414yes—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340Q818yescomplemented
125#336—24no—
126#336—24no—
127#336—22no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8C2×Q814yes—
161C22×C8#4114yes—
162C4⋊Q8—28no—
163C4⋊Q8—28no—
164C4⋊Q8—28no—
165C4⋊Q8—28no—
166C4×Q8—432no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4SD1614yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×Q814yes—
218C2×C4⋊C4C2×Q814yes—
219C2×C4⋊C4#4112yes—
220C2×C4⋊C4#4112yes—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42#4112yes—
244#106—48no—
245#106—48no—
246C4⋊C8—28no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254#95—42no—
255#95—44no—
256#95—42no—
257#95—82no—
258#95—82no—
259C24C22×D411yescharacteristic
260C2×Q8—216no—
261C2×Q8—48no—
262C2×Q8—48no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—48no—
266C2×Q8—48no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—41no—
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—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4#12012yes—
329C22×C4#12012yes—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4C2×SD1612yes—
333C22×C4C2×SD1612yes—
334C22×C4C2×SD1612yes—
335C22×C4AGL(1,ℤ/8)12yes—
336C22×C4#13712yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×Q812yes—
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C42−1+412yes—
346C22×C42−1+412yes—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—216no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—44no—
375C4⋊C4—48no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—48no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C4⋊C4#12114yes—
389C4⋊C4#12114yes—
390C4⋊C4C4⋊Q814yes—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—24no—
398C42—48no—
399C42—48no—
400C42—41no—
401C42—41no—
402C42—42no—
403C42—48no—
404C23C2×#12011yescharacteristic
405C23C2×#12111yescharacteristic
406C23#52511yescharacteristic
407C23#54712yes—
408C23D4×Q812yes—
409C23C22×SD1611yescharacteristic
410C23C2×AGL(1,ℤ/8)11yescharacteristic
411C23C2×#13711yescharacteristic
412C23#57511yescharacteristic
413C23#57712yes—
414C23#57712yes—
415Q8—432no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#45014yes—
485C2×C4#46114yes—
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×#12111yescharacteristic
488C2×C4#52511yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4#52511yescharacteristic
491C2×C4C2×C4⋊Q811yescharacteristic
492C2×C4#53312yes—
493C2×C4#53312yes—
494C2×C4#53611yescharacteristic
495C2×C4#53611yescharacteristic
496C2×C4#54112yes—
497C2×C4#54112yes—
498C2×C4#54712yes—
499C2×C4#54812yes—
500C2×C4D4×Q812yes—
501C2×C4D4×Q812yes—
502C2×C4#55212yes—
503C2×C4#55712yes—
504C8—44no—
505C8—44no—
506C22#240611yescharacteristic
507C22#242912yes—
508C22#243411yescharacteristic
509C22#246212yes—
510C22#284711yescharacteristic
511C22C2×#45011yescharacteristic
512C22#300711yescharacteristic
513C22#301011yescharacteristic
514C22#301011yescharacteristic
515C22C2×#46111yescharacteristic
516C22#303611yescharacteristic
517C22#304312yes—
518C22#310111yescharacteristic
519C22#311011yescharacteristic
520C22#312612yes—
521C22#333512yes—
522C22#334512yes—
523C22#335012yes—
524C22#335612yes—
525C22Q8×SD1612yes—
526C22#338412yes—
527C22#338712yes—
528C22#338812yes—
529C4—24no—
530C4—24no—
531C4—42no—
532C4—42no—
533C4—44no—
534C4—44no—
535C4—42no—
536C4—44no—
537C4—82no—
538C4—81no—
539C4—84no—
540C4—162no—
541C4#239911yescharacteristic
542C4#242912yes—
543C4#246212yes—
544C4#253611yescharacteristic
545C4#254611yescharacteristic
546C4#263411yescharacteristic
547C2#1488011yescharacteristic
548C2#2408911yescharacteristic
549C2#2410811yescharacteristic
550C2#2545312yes—
551C2#2546112yes—
552C2#2583411yescharacteristic
553C2#2584811yescharacteristic
554C2#2685812yes—
555C2#2687512yes—
556C2#3290511yescharacteristic
557C2#3291211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
124. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#20842
C2×#20582
C2×#19482
#151962
#112272
#112202
#112022
#109082
#108852
#108062