Finite Group Databasebeta

← #461626

Subgroups — #461626

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1877C211yesmaximal, characteristic
2C2×#1802C211yesmaximal, characteristic
3#15199C211yesmaximal, characteristic, complemented
4#13920C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic, complemented
6#11255C211yesmaximal, characteristic, complemented
7#11207C212yesmaximal, complemented
8#11077C211yesmaximal, characteristic, complemented
9#11067C212yesmaximal, complemented
10#11049C212yesmaximal
11#11023C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2212yes—
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2212yes—
25C2×#380—21no—
26#1877—24no—
27#1802—24no—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1444—21no—
37#1444—22no—
38#1444—21no—
39#1439—22no—
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C22×SD16—22no—
47C22×SD16—22no—
48C22×SD16—42no—
49C22×SD16—42no—
50C22×SD16Q814yes—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4×D4—24no—
55C2×C4×D4—22no—
56C2×C4×D4—42no—
57C2×C4×D4D414yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8—42no—
74C2×#103—22no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#102—24no—
80C2×#102—22no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102—42no—
84#391—22no—
85#391—22no—
86#391—42no—
87#391—42no—
88#390—24no—
89#390—42no—
90#389—24no—
91#389—42no—
92#384—22no—
93#384—22no—
94#384—42no—
95#384—42no—
96#380—44no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95D411yescharacteristic
105C2×#95D412yes—
106C2×#95D411yescharacteristic
107#340—24no—
108#340—22no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—42no—
114#336—42no—
115C25—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8C2×Q812yes—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4C2×D412yes—
134C22×D4C2×Q812yes—
135C22×D4#4112yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—44no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—82no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C2×SD16—28no—
152C2×SD16—48no—
153C2×SD16—44no—
154C2×SD16—44no—
155C2×SD16—48no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8C2×Q812yes—
173C4×D4—28no—
174C4×D4—44no—
175C4×D4—48no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—81no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4#4114yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42C2×D412yes—
227C2×C42#4112yes—
228C4⋊C8—44no—
229#103—48no—
230#103—44no—
231#102—48no—
232#102—44no—
233#95—24no—
234#95—22no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—81no—
242#95SD1614yescomplemented
243C24—22no—
244C24—22no—
245C24—22no—
246C24—42no—
247C24—42no—
248C24—41no—
249C24—42no—
250C24—41no—
251C24—41no—
252C24—42no—
253C24—41no—
254C24C22×D411yescharacteristic
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—24no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4C22⋊Q814yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—48no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—48no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—84no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4#12112yes—
344C22×C4#12112yes—
345C22×C4C2×SD1614yes—
346C22×C4C2×SD1612yes—
347C22×C4#13714yes—
348C22×C4#13712yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q811yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C4H(5,𝔽2)12yes—
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C42−1+411yescharacteristic
359SD16—416no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—88no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C4⋊C4#12314yes—
396C22⋊C4—48no—
397C22⋊C4—88no—
398C22⋊C4—88no—
399C42—22no—
400C42—22no—
401C42—24no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—42no—
406C23—24no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—44no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23C2×#12111yescharacteristic
430C23#53512yes—
431C23#54611yescharacteristic
432C23#54612yes—
433C23D4×Q811yescharacteristic
434C23C22×SD1611yescharacteristic
435C23C2×#13711yescharacteristic
436C23C2×#13712yes—
437C23#57611yescharacteristic
438C23#57612yes—
439C23#57711yescharacteristic
440Q8—42no—
441Q8—42no—
442Q8—44no—
443Q8—81no—
444Q8—82no—
445Q8—81no—
446D4—24no—
447D4—24no—
448D4—42no—
449D4—44no—
450D4—44no—
451D4—42no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—816no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4#46114yes—
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×#12311yescharacteristic
508C2×C4#53411yescharacteristic
509C2×C4#53511yescharacteristic
510C2×C4#53511yescharacteristic
511C2×C4#53711yescharacteristic
512C2×C4#53812yes—
513C2×C4#54012yes—
514C2×C4#54012yes—
515C2×C4#54212yes—
516C2×C4#54312yes—
517C2×C4#55112yes—
518C2×C4#55211yescharacteristic
519C2×C4#55312yes—
520C2×C4#55412yes—
521C2×C4#55411yescharacteristic
522C2×C4#56012yes—
523C2×C4#56212yes—
524C8—42no—
525C8—81no—
526C8—82no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—84no—
531C22—84no—
532C22—84no—
533C22—82no—
534C22—82no—
535C22—84no—
536C22—82no—
537C22—82no—
538C22#241211yescharacteristic
539C22#241711yescharacteristic
540C22#261911yescharacteristic
541C22#269212yes—
542C22#270312yes—
543C22C2×#46111yescharacteristic
544C22#303911yescharacteristic
545C22#304111yescharacteristic
546C22#304511yescharacteristic
547C22#320612yes—
548C22#321812yes—
549C22#322012yes—
550C22#322412yes—
551C22#329512yes—
552C22#329511yescharacteristic
553C22#331412yes—
554C22#331412yes—
555C22#331612yes—
556C22#333111yescharacteristic
557C22#333312yes—
558C22Q8×SD1611yescharacteristic
559C22#338512yes—
560C22#338511yescharacteristic
561C4—24no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—42no—
566C4—82no—
567C4—162no—
568C4—161no—
569C4—162no—
570C4#248711yescharacteristic
571C4#262411yescharacteristic
572C4#262611yescharacteristic
573C4#274112yes—
574C4#274812yes—
575C4#281311yescharacteristic
576C2—84no—
577C2#1639811yescharacteristic
578C2#2453011yescharacteristic
579C2#2454611yescharacteristic
580C2#2481511yescharacteristic
581C2#2483311yescharacteristic
582C2#2979811yescharacteristic
583C2#2980211yescharacteristic
584C2#3152212yes—
585C2#3153412yes—
586C2#3183812yes—
587C2#3185812yes—
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
3. #15199C2semidirect
5. #11357C2semidirect
6. #11255C2semidirect
7. #11207C2semidirect
8. #11077C2semidirect
9. #11067C2semidirect
242. #95SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#18772
C2×#18022
#151992
#139202
#113572
#112552
#112072
#110772
#110672
#110492
#110232