Finite Group Databasebeta

← #473838

Subgroups — #473838

Order 512.

1228 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14702C211yesmaximal, characteristic
2#14682C211yesmaximal, characteristic
3#11315C212yesmaximal, complemented
4#11268C211yesmaximal, characteristic, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11200C212yesmaximal
7#11190C212yesmaximal, complemented
8#11052C211yesmaximal, characteristic, complemented
9#11052C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C22⋊C8C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32#1436—22no—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36#1434—22no—
37C22⋊#95—22no—
38C22⋊#95—22no—
39C23×Q8C2311yescharacteristic
40C24×C4D412yes—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—42no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C42—22no—
58C22×C42—22no—
59C22×C42—42no—
60C22×C42—42no—
61C22×C42D412yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#103—24no—
65C2×#103—42no—
66C2×#103—42no—
67C2×#103—42no—
68C2×#103—42no—
69C2×#103—42no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—42no—
73#391—22no—
74#391—22no—
75#391—42no—
76#382—42no—
77#382—42no—
78#380—24no—
79#380—42no—
80#380—42no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91#340—24no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#336—22no—
98#336—22no—
99#336—42no—
100C25C2×D411yescharacteristic
101C22×Q8—21no—
102C22×Q8—22no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8#4112yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—24no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4#4114yes—
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—
152C22⋊Q8—48no—
153C22⋊Q8—48no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
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—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4#4112yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—44no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—48no—
197C2×C42—48no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42#4112yes—
201C2×C42#4112yes—
202#103—48no—
203#103—48no—
204C22⋊C8—48no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—42no—
213#95—44no—
214#95—44no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24#12412yes—
225C24#13714yes—
226C24C22×D411yescharacteristic
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
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—22no—
268C22×C4—22no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
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—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4#12314yes—
321C22×C4#12412yes—
322C22×C4#13514yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4H(5,𝔽2)12yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C42−1+411yescharacteristic
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—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—24no—
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—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C22⋊C4—48no—
363C22⋊C4—84no—
364C22⋊C4—88no—
365C42—24no—
366C42—24no—
367C42—48no—
368C42—42no—
369C42—42no—
370C23—22no—
371C23—24no—
372C23—24no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—41no—
390C23—42no—
391C23—41no—
392C23#49114yes—
393C23C2×#12311yescharacteristic
394C23#52511yescharacteristic
395C23C2×#12411yescharacteristic
396C23#53411yescharacteristic
397C23#53411yescharacteristic
398C23#53511yescharacteristic
399C23#53711yescharacteristic
400C23C2×#13512yes—
401C23C2×#13712yes—
402C23#57512yes—
403C23#57512yes—
404Q8—44no—
405Q8—44no—
406Q8—82no—
407Q8—82no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—432no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—81no—
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—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4#53812yes—
471C2×C4#53812yes—
472C2×C4#53812yes—
473C2×C4#53812yes—
474C2×C4#53912yes—
475C2×C4#54012yes—
476C2×C4#55112yes—
477C2×C4#55112yes—
478C2×C4#55211yescharacteristic
479C2×C4#55312yes—
480C2×C4#55311yescharacteristic
481C2×C4#55312yes—
482C2×C4#55511yescharacteristic
483C2×C4#55611yescharacteristic
484C2×C4#56012yes—
485C2×C4#56212yes—
486C8—82no—
487C8—82no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—42no—
492C22—44no—
493C22—42no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—42no—
498C22—42no—
499C22#248511yescharacteristic
500C22#250411yescharacteristic
501C22#256111yescharacteristic
502C22#262211yescharacteristic
503C22#263511yescharacteristic
504C22#263911yescharacteristic
505C22#281311yescharacteristic
506C22#309112yes—
507C22#309512yes—
508C22#310212yes—
509C22#311712yes—
510C22#313111yescharacteristic
511C22C2×#49111yescharacteristic
512C22#314011yescharacteristic
513C22#314211yescharacteristic
514C22#319012yes—
515C22#319112yes—
516C22#319612yes—
517C22#319712yes—
518C22#321212yes—
519C22#321512yes—
520C22#324812yes—
521C22#325612yes—
522C4—44no—
523C4—44no—
524C4—42no—
525C4—82no—
526C4—82no—
527C4—161no—
528C4—162no—
529C4—162no—
530C4—161no—
531C4#275012yes—
532C4#275612yes—
533C4#277812yes—
534C4#278612yes—
535C2—44no—
536C2#1688411yescharacteristic
537C2#2763512yes—
538C2#2797611yescharacteristic
539C2#2798311yescharacteristic
540C2#2910811yescharacteristic
541C2#2911211yescharacteristic
542C2#2989712yes—
543C2#3011412yes—
544C2#3019011yescharacteristic
545C2#3019811yescharacteristic
546C2#3272912yes—
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
3. #11315C2semidirect
4. #11268C2semidirect
5. #11268C2semidirect
7. #11190C2semidirect
8. #11052C2semidirect
9. #11052C2semidirect

Maximal subgroups

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

subgroupindex
#147022
#146822
#113152
#112682
#112002
#111902
#110522
#109702
#107122