Finite Group Databasebeta

← #403688

Subgroups — #403688

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3C2×#2057C212yesmaximal
4C2×#1947C211yesmaximal, characteristic
5#15196C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic
7#11219C212yesmaximal
8#11202C211yesmaximal, characteristic
9#10907C212yesmaximal
10#10885C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16#2369—24no—
17#2369—24no—
18C2×#391—21no—
19C2×#391—21no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2083—24no—
31#2057—28no—
32#1947—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—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85#391—28no—
86#391—28no—
87#389—28no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—24no—
92#389—24no—
93#389D414yes—
94#384—22no—
95#384—22no—
96C4×C4⋊C4—28no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340Q818yescomplemented
123#336—24no—
124#336—24no—
125#336—22no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—82no—
130C22×Q8—82no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
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×C8C2×Q814yes—
157C22×C8#4114yes—
158C4⋊Q8—28no—
159C4⋊Q8—28no—
160C4⋊Q8—28no—
161C4⋊Q8—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4Q1614yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×Q814yes—
211C2×C4⋊C4C2×Q814yes—
212C2×C4⋊C4#4112yes—
213C2×C4⋊C4#4112yes—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—82no—
230C2×C42—82no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42#4112yes—
235#107—48no—
236#107—48no—
237C4⋊C8—28no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—22no—
244#95—22no—
245#95—42no—
246#95—44no—
247#95—42no—
248#95—82no—
249#95—82no—
250C24C22×D411yescharacteristic
251C2×Q8—216no—
252C2×Q8—88no—
253C2×Q8—88no—
254C2×Q8—84no—
255C2×Q8—84no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—41no—
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—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4#12012yes—
314C22×C4#12012yes—
315C22×C4#12112yes—
316C22×C4#12112yes—
317C22×C4C2×Q1612yes—
318C22×C4C2×Q1612yes—
319C22×C4C2×Q1612yes—
320C22×C4#13712yes—
321C22×C4#13712yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×Q812yes—
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4112yes—
330C22×C42−1+412yes—
331C22×C42−1+412yes—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—216no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—44no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—84no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—88no—
372C4⋊C4—84no—
373C4⋊C4#12114yes—
374C4⋊C4#12114yes—
375C4⋊C4C4⋊Q814yes—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—24no—
383C42—48no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—84no—
388C42—84no—
389C23C2×#12011yescharacteristic
390C23C2×#12111yescharacteristic
391C23#52511yescharacteristic
392C23#54712yes—
393C23D4×Q812yes—
394C23C22×Q1611yescharacteristic
395C23C2×#13711yescharacteristic
396C23C2×#13711yescharacteristic
397C23#57511yescharacteristic
398C23#57612yes—
399C23#57612yes—
400Q8—816no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
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—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4#45114yes—
466C2×C4#46214yes—
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4#52511yescharacteristic
470C2×C4#52511yescharacteristic
471C2×C4#52511yescharacteristic
472C2×C4C2×C4⋊Q811yescharacteristic
473C2×C4#53312yes—
474C2×C4#53312yes—
475C2×C4#53611yescharacteristic
476C2×C4#53611yescharacteristic
477C2×C4#54112yes—
478C2×C4#54112yes—
479C2×C4#54712yes—
480C2×C4#54812yes—
481C2×C4D4×Q812yes—
482C2×C4D4×Q812yes—
483C2×C4#55212yes—
484C2×C4#55712yes—
485C8—44no—
486C8—44no—
487C22#240611yescharacteristic
488C22#242912yes—
489C22#243411yescharacteristic
490C22#246212yes—
491C22#284711yescharacteristic
492C22C2×#45111yescharacteristic
493C22#300711yescharacteristic
494C22#300811yescharacteristic
495C22#300811yescharacteristic
496C22C2×#46211yescharacteristic
497C22#303711yescharacteristic
498C22#304112yes—
499C22#310411yescharacteristic
500C22#311311yescharacteristic
501C22#312212yes—
502C22#333812yes—
503C22#334612yes—
504C22#334712yes—
505C22#335712yes—
506C22Q8×Q1612yes—
507C22#338512yes—
508C22#338512yes—
509C22#338912yes—
510C4—24no—
511C4—24no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—44no—
516C4—44no—
517C4—44no—
518C4—81no—
519C4—82no—
520C4—84no—
521C4—162no—
522C4#239911yescharacteristic
523C4#242912yes—
524C4#246212yes—
525C4#253611yescharacteristic
526C4#254611yescharacteristic
527C4#263411yescharacteristic
528C2#1488011yescharacteristic
529C2#2409811yescharacteristic
530C2#2410511yescharacteristic
531C2#2545512yes—
532C2#2546212yes—
533C2#2584111yescharacteristic
534C2#2585011yescharacteristic
535C2#2686212yes—
536C2#2687912yes—
537C2#3290611yescharacteristic
538C2#3291111yescharacteristic
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
122. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#20832
C2×#20572
C2×#19472
#151962
#112282
#112192
#112022
#109072
#108852
#108072