Finite Group Databasebeta

← #461450

Subgroups — #461450

Order 512.

1138 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×#2096C211yesmaximal, characteristic, complemented
2C2×#1818C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#14258C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11268C211yesmaximal, characteristic, complemented
7#11257C212yesmaximal, complemented
8#11190C212yesmaximal, complemented
9#11095C212yesmaximal
10#11052C211yesmaximal, characteristic, complemented
11#10970C212yesmaximal
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#106C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2211yescharacteristic
24#2096—24no—
25#1818—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34#1441—22no—
35#1441—22no—
36#1440—22no—
37#1439—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×Q8C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—42no—
49C2×#117—24no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4D412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#106—24no—
68C2×#106—42no—
69C2×#106—42no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—42no—
79#391—22no—
80#391—22no—
81#391—42no—
82#384—22no—
83#384—22no—
84#382—42no—
85#380—42no—
86#380—42no—
87#380—42no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—24no—
103#336—22no—
104#336D414yes—
105C25C2×D411yescharacteristic
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8#4112yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4#4114yes—
135C23×C4C2411yescharacteristic
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—
149C22×C8C2×D414yes—
150C22⋊Q8—48no—
151C22⋊Q8—48no—
152#117—48no—
153#117—48no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
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—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4#4112yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42C2×D412yes—
205C2×C42#4112yes—
206#106—48no—
207#103—48no—
208#103—48no—
209C22⋊C8—48no—
210#95—44no—
211#95—42no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—82no—
216#95—82no—
217#95—82no—
218#95—82no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24#12312yes—
230C24C22×D411yescharacteristic
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—42no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
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#12114yes—
319C22×C4#12312yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4H(5,𝔽2)12yes—
326C22×C4H(5,𝔽2)12yes—
327C22×C4H(5,𝔽2)12yes—
328C22×C42−1+411yescharacteristic
329C22×C42−1+411yescharacteristic
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
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—
344C4⋊C4—24no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—42no—
348C4⋊C4—42no—
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—88no—
361C22⋊C4—48no—
362C22⋊C4—48no—
363C22⋊C4—84no—
364C22⋊C4—88no—
365C42—24no—
366C42—24no—
367C42—42no—
368C42—42no—
369C23—24no—
370C23—24no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23C2×#12111yescharacteristic
393C23C2×#12311yescharacteristic
394C23#52511yescharacteristic
395C23#53411yescharacteristic
396C23#53511yescharacteristic
397C23#53511yescharacteristic
398C23#53611yescharacteristic
399C23#57712yes—
400C23#57712yes—
401C23#57712yes—
402C23#57812yes—
403Q8—44no—
404Q8—44no—
405Q8—82no—
406Q8—82no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4#53512yes—
469C2×C4#53812yes—
470C2×C4#53912yes—
471C2×C4#53912yes—
472C2×C4#54012yes—
473C2×C4#54012yes—
474C2×C4#54612yes—
475C2×C4#54711yescharacteristic
476C2×C4#54811yescharacteristic
477C2×C4#55112yes—
478C2×C4#55112yes—
479C2×C4#55112yes—
480C2×C4#55211yescharacteristic
481C2×C4#55611yescharacteristic
482C2×C4#56212yes—
483C2×C4#56212yes—
484C8—44no—
485C8—82no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—44no—
490C22—44no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—82no—
496C22—82no—
497C22#241211yescharacteristic
498C22#241511yescharacteristic
499C22#248211yescharacteristic
500C22#261911yescharacteristic
501C22#262311yescharacteristic
502C22#263511yescharacteristic
503C22#281611yescharacteristic
504C22#305712yes—
505C22#306312yes—
506C22#309712yes—
507C22#309912yes—
508C22#312112yes—
509C22#312512yes—
510C22#320112yes—
511C22#320412yes—
512C22#322111yescharacteristic
513C22#322111yescharacteristic
514C22#322211yescharacteristic
515C22#322212yes—
516C22#322311yescharacteristic
517C22#322512yes—
518C22#324212yes—
519C22#324412yes—
520C4—42no—
521C4—44no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4—161no—
526C4—162no—
527C4—161no—
528C4—162no—
529C4#269612yes—
530C4#270612yes—
531C4#274012yes—
532C4#274312yes—
533C2—82no—
534C2#1639211yescharacteristic
535C2#2456612yes—
536C2#2471812yes—
537C2#2759611yescharacteristic
538C2#2759711yescharacteristic
539C2#2981111yescharacteristic
540C2#2981711yescharacteristic
541C2#2991912yes—
542C2#3012312yes—
543C2#3278411yescharacteristic
544C2#3278511yescharacteristic
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
1. C2×#2096C2semidirect
5. #11371C2semidirect
6. #11268C2semidirect
7. #11257C2semidirect
8. #11190C2semidirect
10. #11052C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20962
C2×#18182
#143402
#142582
#113712
#112682
#112572
#111902
#110952
#110522
#109702