Finite Group Databasebeta

← #395859

Subgroups — #395859

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#2085C212yesmaximal, complemented
3C2×#1854C211yesmaximal, characteristic
4#13951C211yesmaximal, characteristic
5#13881C211yesmaximal, characteristic
6#11325C212yesmaximal, complemented
7#11256C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428—28no—
16C2×#428—24no—
17C22×#107C2212yes—
18C22×#107C2211yescharacteristic
19C22×#106C2212yes—
20C22×#106C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#389C2212yes—
23C2×#382C2212yes—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#2085—28no—
28#1854—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1442—21no—
38#1442—21no—
39#1439—22no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43#1434—21no—
44C22⋊#95—22no—
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—42no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2312yes—
77#428—232no—
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—22no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—22no—
86C2×#106—42no—
87C2×#106D418yes—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8Q814yes—
91#389—24no—
92#389—42no—
93#382—24no—
94#382—42no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340—24no—
113#340—24no—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—42no—
118#336—24no—
119#336—24no—
120C25C2×D411yescharacteristic
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—44no—
132C23×C4—42no—
133C23×C4—44no—
134C23×C4—81no—
135C23×C4—81no—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×Q812yes—
141C23×C4C2×Q812yes—
142C23×C4#4112yes—
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
146C2×M16—216no—
147C2×M16—28no—
148C2×M16—28no—
149C2×M16—28no—
150C2×M16—216no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8C2×D414yes—
164C22×C8C2×Q812yes—
165C22×C8C2×Q812yes—
166C22×C8C2×Q814yes—
167C22×C8#4114yes—
168#117—28no—
169#117—44no—
170#117—48no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
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—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D412yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—81no—
216C2×C22⋊C4—81no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—44no—
225#107—216no—
226#107—44no—
227#106—216no—
228#106—44no—
229C22⋊C8—28no—
230#95—22no—
231#95—22no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—42no—
237#95—42no—
238#95—82no—
239#95—82no—
240#95—82no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24#12112yes—
250C24#12312yes—
251C24C4⋊Q812yes—
252C24C22×D411yescharacteristic
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
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—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—48no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
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—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—88no—
319C22×C4—88no—
320C22×C4—84no—
321C22×C4—84no—
322C22×C4#12112yes—
323C22×C4C22⋊Q814yes—
324C22×C4C22⋊Q814yes—
325C22×C4#12312yes—
326C22×C4#12514yes—
327C22×C4#12514yes—
328C22×C4C4⋊Q812yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D412yes—
331C22×C4C22×Q811yescharacteristic
332C22×C4C22×Q811yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4112yes—
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+412yes—
339M16—232no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8C22⋊Q818yes—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—88no—
378C4⋊C4#12014yes—
379C22⋊C4—48no—
380C22⋊C4—48no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—48no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C23—24no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—22no—
395C23—22no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23#39814yes—
411C23C2×#12111yescharacteristic
412C23C2×#12311yescharacteristic
413C23C2×C4⋊Q811yescharacteristic
414C23D4212yes—
415C23#54712yes—
416C23#57512yes—
417C23#57512yes—
418C23#57611yescharacteristic
419C23#57711yescharacteristic
420C23#57711yescharacteristic
421C23#57811yescharacteristic
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—24no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—816no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4#39814yes—
482C2×C4C2×#12011yescharacteristic
483C2×C4C2×C22⋊Q811yescharacteristic
484C2×C4C2×C22⋊Q811yescharacteristic
485C2×C4C2×C22⋊Q812yes—
486C2×C4#52511yescharacteristic
487C2×C4#52512yes—
488C2×C4C2×#12511yescharacteristic
489C2×C4C2×#12511yescharacteristic
490C2×C4#53411yescharacteristic
491C2×C4#53611yescharacteristic
492C2×C4#53912yes—
493C2×C4#54412yes—
494C2×C4#54612yes—
495C2×C4#54812yes—
496C2×C4D4×Q812yes—
497C2×C4#55212yes—
498C2×C4#55312yes—
499C2×C4#55412yes—
500C8—28no—
501C8—42no—
502C8—42no—
503C22—24no—
504C22—24no—
505C22—24no—
506C22—42no—
507C22—42no—
508C22—44no—
509C22—44no—
510C22—42no—
511C22—42no—
512C22—44no—
513C22—44no—
514C22C2×#39811yescharacteristic
515C22#241011yescharacteristic
516C22#241311yescharacteristic
517C22#244012yes—
518C22#244312yes—
519C22#304011yescharacteristic
520C22#304011yescharacteristic
521C22#304511yescharacteristic
522C22#304611yescharacteristic
523C22#309312yes—
524C22#309611yescharacteristic
525C22#309711yescharacteristic
526C22#316311yescharacteristic
527C22#316411yescharacteristic
528C22#316612yes—
529C22#326812yes—
530C22#326912yes—
531C22#327512yes—
532C22#327512yes—
533C22#334812yes—
534C22#335012yes—
535C22#335212yes—
536C22#335512yes—
537C4—24no—
538C4—44no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—84no—
543C4—161no—
544C4—162no—
545C4—161no—
546C4#240911yescharacteristic
547C4#244212yes—
548C4#247312yes—
549C4#257911yescharacteristic
550C4#262711yescharacteristic
551C4#284811yescharacteristic
552C2—44no—
553C2#1472811yescharacteristic
554C2#2365911yescharacteristic
555C2#2366211yescharacteristic
556C2#2437111yescharacteristic
557C2#2437711yescharacteristic
558C2#2459511yescharacteristic
559C2#2459711yescharacteristic
560C2#2607312yes—
561C2#2607612yes—
562C2#2615212yes—
563C2#2615412yes—
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×#2085C2semidirect
6. #11325C2semidirect
7. #11256C2semidirect
8. #11220C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#20852
C2×#18542
#139512
#138812
#113252
#112562
#112202
#107302
#107222
#107152