Finite Group Databasebeta

← #461516

Subgroups — #461516

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1877C211yesmaximal, characteristic
2C2×#1802C211yesmaximal, characteristic
3#15185C211yesmaximal, characteristic, complemented
4#13916C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11322C211yesmaximal, characteristic, complemented
7#11316C212yesmaximal, complemented
8#11077C212yesmaximal, complemented
9#11077C211yesmaximal, characteristic, complemented
10#11076C212yesmaximal
11#11023C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#390C2212yes—
23C2×#384C2211yescharacteristic
24C2×#381—21no—
25#1877—24no—
26#1802—24no—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1445—21no—
35#1445—21no—
36#1442—22no—
37#1440—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C22×SD16—22no—
47C22×SD16—22no—
48C22×SD16—24no—
49C22×SD16—22no—
50C22×SD16—42no—
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C2×C4×D4—24no—
55C2×C4×D4—22no—
56C2×C4×D4—42no—
57C2×C4×D4D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8—42no—
73C2×C4⋊C8—42no—
74C2×#103—22no—
75C2×#103—42no—
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—24no—
85#391—22no—
86#391—22no—
87#391—42no—
88#390—24no—
89#390—42no—
90#384—22no—
91#384—22no—
92#381—44no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D411yescharacteristic
104C2×#95D412yes—
105C2×#95D411yescharacteristic
106#340—24no—
107#340—24no—
108#340—22no—
109#340—42no—
110#340—42no—
111#336—22no—
112#336—42no—
113C25—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8#4112yes—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4C2×D412yes—
133C22×D4#4114yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—44no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—82no—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C2×SD16—28no—
150C2×SD16—48no—
151C2×SD16—48no—
152C2×SD16—44no—
153C2×SD16—44no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8#4112yes—
171C4×D4—28no—
172C4×D4—44no—
173C4×D4—48no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
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—21no—
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—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42C2×D412yes—
217C2×C42#4112yes—
218C4⋊C8—44no—
219C4⋊C8—48no—
220#103—44no—
221#103—48no—
222#102—48no—
223#102—44no—
224#95—22no—
225#95—22no—
226#95—24no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—42no—
231#95—42no—
232#95—42no—
233#95—82no—
234#95—81no—
235#95—82no—
236#95—82no—
237C24—22no—
238C24—24no—
239C24—42no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24—41no—
246C24—42no—
247C24—41no—
248C24C22×D411yescharacteristic
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4#12314yes—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—48no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—48no—
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—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—84no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#13512yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4#13714yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4H(5,𝔽2)12yes—
349C22×C4H(5,𝔽2)11yescharacteristic
350C22×C4H(5,𝔽2)11yescharacteristic
351C22×C4H(5,𝔽2)12yes—
352C22×C42−1+412yes—
353SD16—416no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C22⋊C4—48no—
388C22⋊C4—88no—
389C22⋊C4—88no—
390C42—22no—
391C42—22no—
392C42—24no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—42no—
397C23—24no—
398C23—44no—
399C23—42no—
400C23—42no—
401C23—44no—
402C23—44no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—81no—
419C23—82no—
420C23C2×#12111yescharacteristic
421C23#53712yes—
422C23#54612yes—
423C23#54611yescharacteristic
424C23#55011yescharacteristic
425C23C2×#13511yescharacteristic
426C23C2×AGL(1,ℤ/8)11yescharacteristic
427C23C2×#13712yes—
428C23#57611yescharacteristic
429C23#57711yescharacteristic
430C23#57712yes—
431Q8—42no—
432Q8—42no—
433Q8—44no—
434Q8—81no—
435Q8—82no—
436Q8—81no—
437D4—24no—
438D4—24no—
439D4—42no—
440D4—44no—
441D4—44no—
442D4—42no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—816no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4#46314yes—
497C2×C4C2×#12311yescharacteristic
498C2×C4#52511yescharacteristic
499C2×C4#53411yescharacteristic
500C2×C4#53511yescharacteristic
501C2×C4#53511yescharacteristic
502C2×C4#53611yescharacteristic
503C2×C4#53912yes—
504C2×C4#53912yes—
505C2×C4#53912yes—
506C2×C4#54212yes—
507C2×C4#54212yes—
508C2×C4#55112yes—
509C2×C4#55112yes—
510C2×C4#55112yes—
511C2×C4#55311yescharacteristic
512C2×C4#55511yescharacteristic
513C2×C4#56012yes—
514C2×C4#56012yes—
515C8—42no—
516C8—81no—
517C8—82no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—82no—
522C22—84no—
523C22—82no—
524C22—84no—
525C22—82no—
526C22—82no—
527C22—84no—
528C22—84no—
529C22#241211yescharacteristic
530C22#241411yescharacteristic
531C22#262011yescharacteristic
532C22#269312yes—
533C22#270512yes—
534C22C2×#46311yescharacteristic
535C22#303911yescharacteristic
536C22#304211yescharacteristic
537C22#304511yescharacteristic
538C22#324812yes—
539C22#325412yes—
540C22#325712yes—
541C22#326312yes—
542C22#329612yes—
543C22#330511yescharacteristic
544C22#331012yes—
545C22#331412yes—
546C22#332012yes—
547C22#333112yes—
548C22#333211yescharacteristic
549C22#339411yescharacteristic
550C22#339911yescharacteristic
551C22#340012yes—
552C4—24no—
553C4—44no—
554C4—42no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4—162no—
559C4—161no—
560C4—162no—
561C4#248211yescharacteristic
562C4#262311yescharacteristic
563C4#263811yescharacteristic
564C4#273712yes—
565C4#274612yes—
566C4#281511yescharacteristic
567C2—84no—
568C2#1639511yescharacteristic
569C2#2452711yescharacteristic
570C2#2454811yescharacteristic
571C2#2463111yescharacteristic
572C2#2464611yescharacteristic
573C2#2982911yescharacteristic
574C2#2983811yescharacteristic
575C2#3155612yes—
576C2#3156612yes—
577C2#3190612yes—
578C2#3191812yes—
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. #15185C2semidirect
5. #11373C2semidirect
6. #11322C2semidirect
7. #11316C2semidirect
8. #11077C2semidirect
9. #11077C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18772
C2×#18022
#151852
#139162
#113732
#113222
#113162
#110772
#110762
#110232