Finite Group Databasebeta

← #401693

Subgroups — #401693

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#2101C212yesmaximal, complemented
3#14340C211yesmaximal, characteristic
4#11293C212yesmaximal, complemented
5#11292C212yesmaximal, complemented
6#11207C212yesmaximal, complemented
7#10731C212yesmaximal
8#10730C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14#2317—28no—
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#385—21no—
18C2×#385—21no—
19C2×#384—21no—
20C2×#384C2211yescharacteristic
21C2×#382—21no—
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24#2101—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1441—21no—
37#1441—21no—
38#1439—21no—
39#1439—21no—
40C4⋊#95C2212yes—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#117—22no—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117—24no—
50C2×#117—41no—
51C2×#117—41no—
52C2×#117Q814yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—41no—
61C22×C4⋊C4—41no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78#389—24no—
79#389—24no—
80#389—24no—
81#389D418yes—
82#385—28no—
83#385—28no—
84#384—28no—
85#384—41no—
86#384—41no—
87#382—28no—
88#382—24no—
89#380—24no—
90#380—24no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#336—24no—
107#336—41no—
108#336—41no—
109C25C2×D411yescharacteristic
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—82no—
124C23×C4—82no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×Q812yes—
131C23×C4#4112yes—
132C23×C4#4114yes—
133C23×C4#4114yes—
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149#117—216no—
150#117—216no—
151#117—48no—
152#117—48no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×Q814yes—
197C2×C4⋊C4#4114yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C2×D414yes—
221C22⋊C8—48no—
222#95—28no—
223#95—28no—
224#95—42no—
225#95—42no—
226#95—44no—
227#95—82no—
228#95—82no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24C22⋊Q812yes—
238C24#12312yes—
239C24#12412yes—
240C24C22×D411yescharacteristic
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—48no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—88no—
307C22×C4—88no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4#12012yes—
311C22×C4#12012yes—
312C22×C4C22⋊Q812yes—
313C22×C4#12312yes—
314C22×C4#12412yes—
315C22×C4#12518yes—
316C22×C4#12618yes—
317C22×C4C22×D412yes—
318C22×C4C22×Q811yescharacteristic
319C22×C4C2×#4112yes—
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4C2×#4112yes—
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C42−1+411yescharacteristic
326C22×C42−1+411yescharacteristic
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—88no—
360C4⋊C4—88no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4C22⋊Q818yes—
364C22⋊C4—216no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—88no—
368C42—216no—
369C42—24no—
370C42—24no—
371C42—42no—
372C42—42no—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—42no—
386C23—42no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23#40014yes—
396C23C2×#12011yescharacteristic
397C23C2×C22⋊Q811yescharacteristic
398C23C2×#12311yescharacteristic
399C23#52511yescharacteristic
400C23C2×#12411yescharacteristic
401C23#53411yescharacteristic
402C23#53611yescharacteristic
403C23#57612yes—
404C23#57712yes—
405C23#57712yes—
406C23#57812yes—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—816no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
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#40014yes—
467C2×C4C2×C22⋊Q812yes—
468C2×C4#52512yes—
469C2×C4C2×#12512yes—
470C2×C4C2×#12612yes—
471C2×C4#53911yescharacteristic
472C2×C4#53911yescharacteristic
473C2×C4#54411yescharacteristic
474C2×C4#54411yescharacteristic
475C2×C4#54612yes—
476C2×C4#54612yes—
477C2×C4#54812yes—
478C2×C4#54812yes—
479C2×C4#55212yes—
480C2×C4#55212yes—
481C2×C4#55312yes—
482C2×C4#55312yes—
483C8—82no—
484C8—82no—
485C22—28no—
486C22—24no—
487C22—42no—
488C22—42no—
489C22—44no—
490C22—42no—
491C22—42no—
492C22—44no—
493C22—42no—
494C22—42no—
495C22—44no—
496C22C2×#40011yescharacteristic
497C22#240511yescharacteristic
498C22#243611yescharacteristic
499C22#248411yescharacteristic
500C22#259111yescharacteristic
501C22#262711yescharacteristic
502C22#284811yescharacteristic
503C22#301212yes—
504C22#301312yes—
505C22#308312yes—
506C22#308412yes—
507C22#309612yes—
508C22#309712yes—
509C22#311912yes—
510C22#312012yes—
511C22#313612yes—
512C22#313712yes—
513C22#319911yescharacteristic
514C22#320011yescharacteristic
515C22#320011yescharacteristic
516C22#320211yescharacteristic
517C22#324312yes—
518C22#324412yes—
519C4—24no—
520C4—48no—
521C4—42no—
522C4—42no—
523C4—81no—
524C4—84no—
525C4—81no—
526C4—162no—
527C4—162no—
528C4#244212yes—
529C4#244212yes—
530C4#248112yes—
531C4#248112yes—
532C2—44no—
533C2#1482211yescharacteristic
534C2#2374812yes—
535C2#2401511yescharacteristic
536C2#2401611yescharacteristic
537C2#2591612yes—
538C2#2761912yes—
539C2#2934311yescharacteristic
540C2#2934411yescharacteristic
541C2#2997611yescharacteristic
542C2#2997711yescharacteristic
543C2#3296812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2101C2semidirect
4. #11293C2semidirect
5. #11292C2semidirect
6. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#21012
#143402
#112932
#112922
#112072
#107312
#107302
#107132