Finite Group Databasebeta

← #395856

Subgroups — #395856

Order 512.

1548 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#2083C212yesmaximal, complemented
3C2×#1854C211yesmaximal, characteristic
4#13951C211yesmaximal, characteristic
5#13880C211yesmaximal, characteristic
6#11323C212yesmaximal, complemented
7#11257C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428—28no—
16C2×#428—24no—
17C22×#107C2211yescharacteristic
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×#106C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#389C2212yes—
23C2×#385—22no—
24C2×#384C2212yes—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#2083—28no—
29#1854—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1444—21no—
39#1444—21no—
40#1439—22no—
41C4⋊#95—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—21no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—24no—
50C22×M16—24no—
51C22×M16Q814yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—42no—
58C2×#117D414yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D412yes—
77C22×C42—21no—
78C22×C42C2312yes—
79#428—232no—
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—22no—
84C2×#107D418yes—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—22no—
89C2×#106—42no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8Q814yes—
93#389—24no—
94#389—42no—
95#385—48no—
96#384—24no—
97#384—42no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95D411yescharacteristic
112C2×#95D411yescharacteristic
113#340—24no—
114#340—24no—
115#340—24no—
116#340—22no—
117#340—22no—
118#340—42no—
119#336—24no—
120#336—24no—
121C25C2×D411yescharacteristic
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—44no—
135C23×C4—81no—
136C23×C4—81no—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×Q812yes—
142C23×C4C2×Q812yes—
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4#4112yes—
146C23×C4C2411yescharacteristic
147C2×M16—216no—
148C2×M16—28no—
149C2×M16—28no—
150C2×M16—28no—
151C2×M16—216no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C2×D414yes—
165C22×C8C2×Q812yes—
166C22×C8C2×Q812yes—
167C22×C8C2×Q814yes—
168C22×C8#4114yes—
169#117—28no—
170#117—44no—
171#117—48no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
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—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D412yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—81no—
223C2×C22⋊C4—81no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—44no—
232#107—216no—
233#107—44no—
234#106—216no—
235#106—44no—
236C22⋊C8—28no—
237#95—22no—
238#95—22no—
239#95—44no—
240#95—44no—
241#95—42no—
242#95—42no—
243#95—44no—
244#95—44no—
245#95—82no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24#12112yes—
255C24#12312yes—
256C24C4⋊Q812yes—
257C24C22×D411yescharacteristic
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—48no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—88no—
322C22×C4—88no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4#12112yes—
326C22×C4C22⋊Q814yes—
327C22×C4C22⋊Q814yes—
328C22×C4#12312yes—
329C22×C4#12514yes—
330C22×C4#12514yes—
331C22×C4C4⋊Q812yes—
332C22×C4AGL(1,ℤ/8)14yes—
333C22×C4#13714yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C22×Q811yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4H(5,𝔽2)12yes—
343C22×C42−1+412yes—
344M16—232no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8C22⋊Q818yes—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—88no—
383C4⋊C4—88no—
384C4⋊C4—88no—
385C4⋊C4#12014yes—
386C22⋊C4—216no—
387C22⋊C4—48no—
388C22⋊C4—84no—
389C22⋊C4—84no—
390C42—48no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—24no—
401C23—22no—
402C23—22no—
403C23—24no—
404C23—24no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23#39814yes—
418C23C2×#12111yescharacteristic
419C23C2×#12311yescharacteristic
420C23C2×C4⋊Q811yescharacteristic
421C23D4212yes—
422C23#54712yes—
423C23C2×AGL(1,ℤ/8)12yes—
424C23C2×#13712yes—
425C23#57611yescharacteristic
426C23#57711yescharacteristic
427C23#57711yescharacteristic
428C23#57811yescharacteristic
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
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—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—816no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4#39814yes—
489C2×C4C2×#12011yescharacteristic
490C2×C4C2×C22⋊Q811yescharacteristic
491C2×C4C2×C22⋊Q811yescharacteristic
492C2×C4C2×C22⋊Q812yes—
493C2×C4#52511yescharacteristic
494C2×C4#52512yes—
495C2×C4C2×#12511yescharacteristic
496C2×C4C2×#12511yescharacteristic
497C2×C4#53411yescharacteristic
498C2×C4#53611yescharacteristic
499C2×C4#53912yes—
500C2×C4#54412yes—
501C2×C4#54612yes—
502C2×C4#54812yes—
503C2×C4D4×Q812yes—
504C2×C4#55212yes—
505C2×C4#55312yes—
506C2×C4#55412yes—
507C8—28no—
508C8—42no—
509C8—42no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—44no—
516C22—44no—
517C22—42no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22C2×#39811yescharacteristic
522C22#241011yescharacteristic
523C22#241311yescharacteristic
524C22#244012yes—
525C22#244312yes—
526C22#303811yescharacteristic
527C22#303911yescharacteristic
528C22#304511yescharacteristic
529C22#304611yescharacteristic
530C22#309212yes—
531C22#309611yescharacteristic
532C22#309711yescharacteristic
533C22#316311yescharacteristic
534C22#316411yescharacteristic
535C22#316512yes—
536C22#326412yes—
537C22#327012yes—
538C22#327212yes—
539C22#327812yes—
540C22#334412yes—
541C22#334512yes—
542C22#335312yes—
543C22#335712yes—
544C4—24no—
545C4—44no—
546C4—44no—
547C4—44no—
548C4—84no—
549C4—82no—
550C4—161no—
551C4—162no—
552C4—161no—
553C4#240911yescharacteristic
554C4#244212yes—
555C4#247312yes—
556C4#257911yescharacteristic
557C4#262711yescharacteristic
558C4#284811yescharacteristic
559C2—44no—
560C2#1472811yescharacteristic
561C2#2365811yescharacteristic
562C2#2366211yescharacteristic
563C2#2437211yescharacteristic
564C2#2437411yescharacteristic
565C2#2459211yescharacteristic
566C2#2459811yescharacteristic
567C2#2607412yes—
568C2#2607812yes—
569C2#2614212yes—
570C2#2614812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2083C2semidirect
6. #11323C2semidirect
7. #11257C2semidirect
8. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#20832
C2×#18542
#139512
#138802
#113232
#112572
#112192
#107302
#107222
#107152