Finite Group Databasebeta

← #410099

Subgroups — #410099

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2C2×#1918C212yesmaximal
3C4×#389C211yesmaximal, characteristic
4#11328C212yesmaximal
5#11324C211yesmaximal, characteristic
6#11292C212yesmaximal
7#11267C211yesmaximal, characteristic
8#11207C211yesmaximal, characteristic
9#11207C211yesmaximal, characteristic
10#10884C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×#125—21no—
13C22×#107C2212yes—
14#2859—28no—
15C2×#389—21no—
16C2×#389—21no—
17C2×#389C2212yes—
18C2×#389C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382—21no—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C43C2211yescharacteristic
28#1918—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40#1436—21no—
41#1436—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4×#95—21no—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#125—28no—
48C2×#125—44no—
49C2×#125—44no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42—21no—
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—24no—
75C2×#107—24no—
76#389—28no—
77#389—28no—
78#389—24no—
79#389—24no—
80#389—22no—
81#389—22no—
82#389—24no—
83#389—22no—
84#389D418yes—
85#389Q814yes—
86#384—24no—
87#384—24no—
88#382—28no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—44no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C43—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#336—24no—
112#336—24no—
113#336—22no—
114#336—22no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8#4114yes—
142#125—432no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
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—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×Q812yes—
192C2×C4⋊C4C2×Q812yes—
193C2×C4⋊C4#4112yes—
194C2×C4⋊C4#4112yes—
195C2×C4⋊C4#4112yes—
196C2×C4⋊C4#4112yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42D814yes—
224C2×C42C2×D414yes—
225C2×C42C2×Q812yes—
226C2×C42#4112yes—
227#107—216no—
228#95—24no—
229#95—28no—
230#95—24no—
231#95—28no—
232#95—24no—
233#95—42no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238C24C22×D411yescharacteristic
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4#12012yes—
312C22×C4#12012yes—
313C22×C4C22⋊Q812yes—
314C22×C4C22⋊Q812yes—
315C22×C4C2×D812yes—
316C22×C4C2×D812yes—
317C22×C4C2×D812yes—
318C22×C4#13512yes—
319C22×C4AGL(1,ℤ/8)12yes—
320C22×C4#13712yes—
321C22×C4#13712yes—
322C22×C4#13712yes—
323C22×C4C22×D412yes—
324C22×C4C22×Q811yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C42−1+412yes—
332C22×C42−1+412yes—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—416no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—416no—
371C4⋊C4—48no—
372C4⋊C4#12514yescomplemented
373C42—28no—
374C42—24no—
375C42—24no—
376C42—24no—
377C42—24no—
378C42—28no—
379C42—44no—
380C42—44no—
381C42—48no—
382C42—42no—
383C42—42no—
384C42—84no—
385C23C2×#12011yescharacteristic
386C23C2×C22⋊Q811yescharacteristic
387C23#52511yescharacteristic
388C23#52511yescharacteristic
389C23#53211yescharacteristic
390C23#53611yescharacteristic
391C23#53611yescharacteristic
392C23C22×D811yescharacteristic
393C23C2×#13511yescharacteristic
394C23C2×AGL(1,ℤ/8)11yescharacteristic
395C23C2×#13711yescharacteristic
396C23C2×#13711yescharacteristic
397C23C2×#13711yescharacteristic
398C23#57511yescharacteristic
399C23#57511yescharacteristic
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
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—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
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—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4#44714yes—
482C2×C4#47414yes—
483C2×C4#47914yes—
484C2×C4#52512yes—
485C2×C4C2×#12511yescharacteristic
486C2×C4#52911yescharacteristic
487C2×C4#52911yescharacteristic
488C2×C4#53212yes—
489C2×C4#53311yescharacteristic
490C2×C4#54112yes—
491C2×C4#54112yes—
492C2×C4#54712yes—
493C2×C4#54712yes—
494C2×C4#54812yes—
495C2×C4D4×Q812yes—
496C2×C4#55212yes—
497C2×C4#55212yes—
498C2×C4#55212yes—
499C2×C4#55712yes—
500C8—44no—
501C8—82no—
502C22#238811yescharacteristic
503C22#240611yescharacteristic
504C22#240911yescharacteristic
505C22#256511yescharacteristic
506C22#261111yescharacteristic
507C22#263411yescharacteristic
508C22#285111yescharacteristic
509C22C2×#44711yescharacteristic
510C22#300211yescharacteristic
511C22#300411yescharacteristic
512C22#300711yescharacteristic
513C22C2×#47411yescharacteristic
514C22C2×#47911yescharacteristic
515C22#307911yescharacteristic
516C22#308011yescharacteristic
517C22#310011yescharacteristic
518C22#310011yescharacteristic
519C22#310611yescharacteristic
520C22#310711yescharacteristic
521C22#311211yescharacteristic
522C22#311311yescharacteristic
523C22#311611yescharacteristic
524C22#311611yescharacteristic
525C22#316911yescharacteristic
526C22#317111yescharacteristic
527C22#318111yescharacteristic
528C22#318511yescharacteristic
529C22#322611yescharacteristic
530C22#322611yescharacteristic
531C22#323111yescharacteristic
532C22#323211yescharacteristic
533C22#323411yescharacteristic
534C22#323611yescharacteristic
535C22#323711yescharacteristic
536C22#323911yescharacteristic
537C4—24no—
538C4—24no—
539C4—24no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—44no—
544C4—82no—
545C4—84no—
546C4—82no—
547C4—82no—
548C4—84no—
549C4#246212yes—
550C4#247512yes—
551C4#250612yes—
552C4#251912yes—
553C2#1507111yescharacteristic
554C2#2356711yescharacteristic
555C2#2357711yescharacteristic
556C2#2406011yescharacteristic
557C2#2406511yescharacteristic
558C2#2429811yescharacteristic
559C2#2430711yescharacteristic
560C2#2917811yescharacteristic
561C2#2918311yescharacteristic
562C2#2957211yescharacteristic
563C2#2957911yescharacteristic
564C2#3009111yescharacteristic
565C2#3009811yescharacteristic
566C2#3306611yescharacteristic
567C2#3307311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
372. C4⋊C4#125semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#19182
C4×#3892
#113282
#113242
#112922
#112672
#112072
#108842
#107952