Finite Group Databasebeta

← #472272

Subgroups — #472272

Order 512.

1134 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×#2036C212yesmaximal
2#14340C211yesmaximal, characteristic
3#14258C211yesmaximal, characteristic
4#11324C211yesmaximal, characteristic, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11256C212yesmaximal, complemented
7#11199C212yesmaximal
8#11190C212yesmaximal, complemented
9#11052C211yesmaximal, characteristic, complemented
10#11050C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2211yescharacteristic
24#2036—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1441—22no—
34#1441—22no—
35#1440—22no—
36#1439—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C23×Q8C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—42no—
46C2×C22⋊Q8—42no—
47C2×C22⋊Q8D414yes—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—42no—
51C2×#117—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D412yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#107—24no—
67C2×#107—42no—
68C2×#107—42no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—42no—
78#391—22no—
79#391—22no—
80#391—42no—
81#384—42no—
82#382—42no—
83#380—24no—
84#380—24no—
85#380—42no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95#340—24no—
96#340—24no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—22no—
101#336—22no—
102#336—42no—
103C25C2×D411yescharacteristic
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8C2×D412yes—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4#4112yes—
133C23×C4C2411yescharacteristic
134C22×C8—22no—
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#4114yes—
148C22⋊Q8—48no—
149C22⋊Q8—48no—
150#117—48no—
151#117—48no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—21no—
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—21no—
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—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C2×D412yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42#4112yes—
203C2×C42#4112yes—
204#107—48no—
205#103—48no—
206#103—48no—
207C22⋊C8—48no—
208#95—42no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—82no—
214#95—82no—
215#95—82no—
216#95—82no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24#12112yes—
228C24C22×D411yescharacteristic
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
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—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
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#12112yes—
317C22×C4#12314yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4H(5,𝔽2)12yes—
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)12yes—
326C22×C42−1+411yescharacteristic
327C22×C42−1+411yescharacteristic
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
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—
342C4⋊C4—24no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—42no—
346C4⋊C4—42no—
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—88no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C22⋊C4—88no—
362C22⋊C4—84no—
363C42—24no—
364C42—24no—
365C42—42no—
366C42—42no—
367C23—22no—
368C23—22no—
369C23—24no—
370C23—24no—
371C23—22no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—41no—
390C23C2×#12111yescharacteristic
391C23C2×#12311yescharacteristic
392C23#53211yescharacteristic
393C23#53411yescharacteristic
394C23#53411yescharacteristic
395C23#53511yescharacteristic
396C23#53611yescharacteristic
397C23#57612yes—
398C23#57712yes—
399C23#57812yes—
400C23#57812yes—
401Q8—44no—
402Q8—44no—
403Q8—82no—
404Q8—82no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4#53412yes—
467C2×C4#53812yes—
468C2×C4#53812yes—
469C2×C4#53812yes—
470C2×C4#53912yes—
471C2×C4#54212yes—
472C2×C4#54612yes—
473C2×C4#54711yescharacteristic
474C2×C4#54711yescharacteristic
475C2×C4#55211yescharacteristic
476C2×C4#55211yescharacteristic
477C2×C4#55312yes—
478C2×C4#55312yes—
479C2×C4#55512yes—
480C2×C4#56212yes—
481C2×C4#56212yes—
482C8—44no—
483C8—82no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22—44no—
488C22—42no—
489C22—81no—
490C22—81no—
491C22—82no—
492C22—82no—
493C22—81no—
494C22—81no—
495C22#248611yescharacteristic
496C22#250011yescharacteristic
497C22#254311yescharacteristic
498C22#261711yescharacteristic
499C22#262211yescharacteristic
500C22#266311yescharacteristic
501C22#281611yescharacteristic
502C22#304412yes—
503C22#304512yes—
504C22#309712yes—
505C22#309812yes—
506C22#317512yes—
507C22#317812yes—
508C22#319812yes—
509C22#320112yes—
510C22#320212yes—
511C22#320312yes—
512C22#322011yescharacteristic
513C22#322111yescharacteristic
514C22#322311yescharacteristic
515C22#322311yescharacteristic
516C22#324212yes—
517C22#324312yes—
518C4—42no—
519C4—44no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4—161no—
524C4—162no—
525C4—162no—
526C4—161no—
527C4#268612yes—
528C4#270812yes—
529C4#277912yes—
530C4#278112yes—
531C2—82no—
532C2#1681611yescharacteristic
533C2#2765612yes—
534C2#2789911yescharacteristic
535C2#2790211yescharacteristic
536C2#2859012yes—
537C2#2972411yescharacteristic
538C2#2972511yescharacteristic
539C2#2990112yes—
540C2#3037012yes—
541C2#3278311yescharacteristic
542C2#3278411yescharacteristic
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
4. #11324C2semidirect
5. #11268C2semidirect
6. #11256C2semidirect
8. #11190C2semidirect
9. #11052C2semidirect
10. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20362
#143402
#142582
#113242
#112682
#112562
#111992
#111902
#110522
#110502
#107132