Finite Group Databasebeta

← #122018

Subgroups — #122018

Order 512.

1274 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2646C211yesmaximal, characteristic
2#10730C212yesmaximal
3C2×#1294C212yesmaximal
4C2×#1280C211yesmaximal, characteristic
5#7557C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8#2646—24no—
9#2646—22no—
10#2646C414yescomplemented
11C2×#400—22no—
12C2×#398—22no—
13C2×#385—22no—
14C2×#380C2212yes—
15C22×#95—22no—
16C22×#95C2211yescharacteristic
17C22×#95C2211yescharacteristic
18C2×#340—22no—
19C2×#340—22no—
20#1294—24no—
21#1280—22no—
22C24×C4—21no—
23C24×C4—21no—
24C24×C4D412yes—
25C24×C4C2311yescharacteristic
26C23×C8—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—44no—
33C22×C4⋊C4D412yes—
34C22×C4⋊C4D412yes—
35C22×C4⋊C4Q812yes—
36C22×C22⋊C4—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8Q814yes—
43#400—28no—
44#400—28no—
45#398—28no—
46#398—28no—
47#385—28no—
48#385—28no—
49#380—22no—
50#380—22no—
51#380C2×C414yes—
52#380D414yes—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—21no—
61C2×#95—44no—
62C2×#95—44no—
63C2×#95—44no—
64C2×#95C2×C412yes—
65C2×#95D412yes—
66C2×#95Q812yes—
67#340—44no—
68#340—44no—
69C25C2×D411yescharacteristic
70C23×C4—24no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—41no—
91C23×C4—41no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4C22⋊C414yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×Q812yes—
110C23×C4C2×Q811yescharacteristic
111C23×C4C2×Q811yescharacteristic
112C23×C4#4112yes—
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C22×C8—24no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—42no—
121C22×C8—42no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—48no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—82no—
140C2×C4⋊C4—82no—
141C2×C4⋊C4—82no—
142C2×C4⋊C4—82no—
143C2×C4⋊C4C22⋊C414yes—
144C2×C4⋊C4C22⋊C414yes—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C22⋊C8—24no—
152C22⋊C8—24no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155#95—28no—
156#95—28no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—416no—
163#95—42no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24—22no—
169C24—22no—
170C24C2×C22⋊C411yescharacteristic
171C24#12011yescharacteristic
172C24#12011yescharacteristic
173C24C22⋊Q811yescharacteristic
174C24C22⋊Q811yescharacteristic
175C24#12411yescharacteristic
176C24C4⋊Q811yescharacteristic
177C24#13512yes—
178C24#13512yes—
179C24AGL(1,ℤ/8)12yes—
180C24#13712yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4C2×C22⋊C412yes—
281C22×C4C4×Q811yescharacteristic
282C22×C4C4×Q811yescharacteristic
283C22×C4C4×Q812yes—
284C22×C4#12012yes—
285C22×C4C22⋊Q812yes—
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4C22⋊Q812yes—
289C22×C4C22⋊Q812yes—
290C22×C4#12412yes—
291C22×C4#12412yes—
292C22×C4#12411yescharacteristic
293C22×C4C4⋊Q812yes—
294C22×C4C4⋊Q812yes—
295C22×C4C4⋊Q811yescharacteristic
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—82no—
300C2×C8—82no—
301C2×C8—82no—
302C2×C8—82no—
303C4⋊C4—28no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—88no—
312C4⋊C4—88no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—22no—
352C23—22no—
353C23#37911yescharacteristic
354C23#39111yescharacteristic
355C23#39111yescharacteristic
356C23#39311yescharacteristic
357C23#39311yescharacteristic
358C23#39511yescharacteristic
359C23#39511yescharacteristic
360C23#41612yes—
361C23#41712yes—
362C23#44912yes—
363C23#44912yes—
364C23#45212yes—
365C23#45212yes—
366C23#45311yescharacteristic
367C23#45411yescharacteristic
368C23#45511yescharacteristic
369C23#45611yescharacteristic
370C23B(4,𝔽2)12yes—
371C23#45812yes—
372C23#47812yes—
373C23#47812yes—
374C23#47912yes—
375C23#47912yes—
376C23#48811yescharacteristic
377C23#48811yescharacteristic
378C23#49011yescharacteristic
379C23#49111yescharacteristic
380C23#49912yes—
381C23#50112yes—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—48no—
405C2×C4—44no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—84no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—82no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4#39112yes—
450C2×C4#39112yes—
451C2×C4#39312yes—
452C2×C4#39512yes—
453C8—84no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—22no—
467C22—21no—
468C22—21no—
469C22—22no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22#144611yescharacteristic
491C22#159712yes—
492C22#160012yes—
493C22#160312yes—
494C22#160612yes—
495C22#160912yes—
496C22#161212yes—
497C22#161512yes—
498C22#161812yes—
499C22#178812yes—
500C22#179012yes—
501C22#179211yescharacteristic
502C22#179311yescharacteristic
503C22#198912yes—
504C22#199111yescharacteristic
505C22#199211yescharacteristic
506C22#202511yescharacteristic
507C22#202511yescharacteristic
508C22#202611yescharacteristic
509C22#202611yescharacteristic
510C22#202711yescharacteristic
511C22#202811yescharacteristic
512C22#202911yescharacteristic
513C22#203011yescharacteristic
514C22#205912yes—
515C22#205912yes—
516C22#206011yescharacteristic
517C22#206111yescharacteristic
518C22#206212yes—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—84no—
524C4—84no—
525C4—84no—
526C4—84no—
527C4—84no—
528C2—21no—
529C2—22no—
530C2—22no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2#840712yes—
535C2#840812yes—
536C2#1034411yescharacteristic
537C2#1034511yescharacteristic
538C2#1034611yescharacteristic
539C2#1034711yescharacteristic
540C2#1057112yes—
541C2#1057212yes—
542C2#1138511yescharacteristic
543C2#1139011yescharacteristic
544C2#1139211yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2646C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
#107302
C2×#12942
C2×#12802
#75572