Finite Group Databasebeta

← #401692

Subgroups — #401692

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2C2×#2101C212yesmaximal, complemented
3#14682C211yesmaximal, characteristic
4#11293C212yesmaximal, complemented
5#11292C212yesmaximal, complemented
6#11208C212yesmaximal, complemented
7#10731C211yesmaximal, characteristic
8#10731C211yesmaximal, characteristic
9#10731C211yesmaximal, characteristic
10#10731C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2296—28no—
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#385—22no—
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4—22no—
26#2101—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39#1434—22no—
40C22⋊#95—22no—
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#117—22no—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—42no—
52C2×#117Q814yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42—21no—
66C22×C42—21no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—24no—
70C22×C42—42no—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80#389—24no—
81#389—24no—
82#389—24no—
83#389D418yes—
84#385—216no—
85#382—24no—
86#382—42no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91C4×C4⋊C4—216no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#336—24no—
109#336—42no—
110C25C2×D411yescharacteristic
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×Q812yes—
132C23×C4#4112yes—
133C23×C4#4112yes—
134C23×C4#4112yes—
135C23×C4#4112yes—
136C23×C4#4112yes—
137C23×C4C2411yescharacteristic
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—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152#117—216no—
153#117—216no—
154#117—48no—
155#117—48no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×Q814yes—
189C2×C4⋊C4#4114yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—48no—
214C2×C42—48no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42C2×D414yes—
219C22⋊C8—44no—
220C22⋊C8—44no—
221#95—28no—
222#95—28no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—42no—
227#95—42no—
228#95—44no—
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—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4#12014yes—
311C22×C4C22⋊Q812yes—
312C22×C4#12312yes—
313C22×C4#12412yes—
314C22×C4#12514yes—
315C22×C4#12514yes—
316C22×C4#12614yes—
317C22×C4#12614yes—
318C22×C4#13514yes—
319C22×C4C22×D412yes—
320C22×C4C22×Q811yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4H(5,𝔽2)12yes—
328C22×C42−1+412yes—
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—
341C2×C8—42no—
342C2×C8—42no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—416no—
349C4⋊C4—416no—
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—42no—
360C4⋊C4C22⋊Q818yes—
361C22⋊C4—216no—
362C22⋊C4—48no—
363C22⋊C4—84no—
364C22⋊C4—84no—
365C42—216no—
366C42—24no—
367C42—24no—
368C42—48no—
369C42—48no—
370C42—42no—
371C42—42no—
372C23—24no—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23#40014yes—
395C23C2×#12011yescharacteristic
396C23C2×C22⋊Q811yescharacteristic
397C23C2×#12311yescharacteristic
398C23#52511yescharacteristic
399C23C2×#12411yescharacteristic
400C23#53411yescharacteristic
401C23#53611yescharacteristic
402C23C2×#13512yes—
403C23#57512yes—
404C23#57512yes—
405C23#57512yes—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—432no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4#40014yes—
466C2×C4C2×C22⋊Q812yes—
467C2×C4#52512yes—
468C2×C4C2×#12511yescharacteristic
469C2×C4C2×#12511yescharacteristic
470C2×C4C2×#12611yescharacteristic
471C2×C4C2×#12611yescharacteristic
472C2×C4#53912yes—
473C2×C4#54412yes—
474C2×C4#54612yes—
475C2×C4#54612yes—
476C2×C4#54812yes—
477C2×C4#54812yes—
478C2×C4#55212yes—
479C2×C4#55212yes—
480C2×C4#55312yes—
481C2×C4#55312yes—
482C8—82no—
483C8—82no—
484C22—24no—
485C22—24no—
486C22—24no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22C2×#40011yescharacteristic
496C22#240511yescharacteristic
497C22#243611yescharacteristic
498C22#248411yescharacteristic
499C22#259111yescharacteristic
500C22#262711yescharacteristic
501C22#284811yescharacteristic
502C22#300212yes—
503C22#300512yes—
504C22#307612yes—
505C22#308012yes—
506C22#309112yes—
507C22#309312yes—
508C22#310212yes—
509C22#311112yes—
510C22#313111yescharacteristic
511C22#313511yescharacteristic
512C22#314011yescharacteristic
513C22#314011yescharacteristic
514C22#319112yes—
515C22#319712yes—
516C22#323312yes—
517C22#324012yes—
518C4—24no—
519C4—48no—
520C4—44no—
521C4—44no—
522C4—84no—
523C4—161no—
524C4—161no—
525C4—161no—
526C4—161no—
527C4#244212yes—
528C4#244212yes—
529C4#248112yes—
530C4#248112yes—
531C2—44no—
532C2#1482211yescharacteristic
533C2#2374411yescharacteristic
534C2#2374611yescharacteristic
535C2#2400812yes—
536C2#2590611yescharacteristic
537C2#2590911yescharacteristic
538C2#2761812yes—
539C2#2933311yescharacteristic
540C2#2933911yescharacteristic
541C2#2997412yes—
542C2#3296112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ 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. #11208C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#21012
#146822
#112932
#112922
#112082
#107312
#107122