Finite Group Databasebeta

← #475515

Subgroups — #475515

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14340C211yesmaximal, characteristic
2#14340C211yesmaximal, characteristic
3#11372C212yesmaximal, complemented
4#11358C212yesmaximal, complemented
5#11267C212yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#11200C212yesmaximal
8#11199C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29#1441—21no—
30#1441—21no—
31#1441—21no—
32#1441—21no—
33#1439—21no—
34#1439—21no—
35#1439—21no—
36#1439—21no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C24×C4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×#117—24no—
43C2×#117—24no—
44C2×#117—41no—
45C2×#117—41no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8—41no—
67C2×C22⋊C8—41no—
68#384—41no—
69#384—41no—
70#384—41no—
71#384—41no—
72#382—24no—
73#382—24no—
74#380—24no—
75#380—24no—
76#380—42no—
77#380—42no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#336—24no—
97#336—24no—
98C25C2×D411yescharacteristic
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4#4112yes—
117C23×C4#4112yes—
118C23×C4C2411yescharacteristic
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133#117—48no—
134#117—48no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
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—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—82no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42#4114yes—
192C2×C42#4114yes—
193C22⋊C8—44no—
194C22⋊C8—44no—
195#95—44no—
196#95—42no—
197#95—42no—
198#95—42no—
199#95—42no—
200#95—44no—
201#95—82no—
202#95—82no—
203#95—82no—
204#95—82no—
205C24—21no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24#12412yes—
216C24C22×D411yescharacteristic
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
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—42no—
283C22×C4#12312yes—
284C22×C4#12312yes—
285C22×C4#12312yes—
286C22×C4#12312yes—
287C22×C4#12412yes—
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4112yes—
291C22×C4C2×#4112yes—
292C22×C4H(5,𝔽2)11yescharacteristic
293C22×C4H(5,𝔽2)11yescharacteristic
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C4H(5,𝔽2)12yes—
296C22×C42−1+411yescharacteristic
297C22×C42−1+412yes—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—84no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—88no—
337C22⋊C4—88no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C23—24no—
343C23—24no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—41no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23C2×#12311yescharacteristic
366C23C2×#12311yescharacteristic
367C23C2×#12411yescharacteristic
368C23#53411yescharacteristic
369C23#53511yescharacteristic
370C23#53611yescharacteristic
371C23#53711yescharacteristic
372C23#57612yes—
373C23#57712yes—
374C23#57712yes—
375C23#57812yes—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4#53811yescharacteristic
436C2×C4#53811yescharacteristic
437C2×C4#53911yescharacteristic
438C2×C4#53911yescharacteristic
439C2×C4#54012yes—
440C2×C4#54112yes—
441C2×C4#54212yes—
442C2×C4#54212yes—
443C2×C4#55112yes—
444C2×C4#55112yes—
445C2×C4#55212yes—
446C2×C4#55212yes—
447C2×C4#55312yes—
448C2×C4#55312yes—
449C2×C4#56212yes—
450C2×C4#56212yes—
451C8—82no—
452C8—82no—
453C22—44no—
454C22—44no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—82no—
463C22—82no—
464C22#248311yescharacteristic
465C22#250311yescharacteristic
466C22#250411yescharacteristic
467C22#262411yescharacteristic
468C22#262511yescharacteristic
469C22#263911yescharacteristic
470C22#281611yescharacteristic
471C22#309812yes—
472C22#309812yes—
473C22#309912yes—
474C22#309912yes—
475C22#313612yes—
476C22#313712yes—
477C22#319911yescharacteristic
478C22#320011yescharacteristic
479C22#320011yescharacteristic
480C22#320211yescharacteristic
481C22#322412yes—
482C22#322512yes—
483C22#324112yes—
484C22#324212yes—
485C22#325712yes—
486C22#325812yes—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—162no—
493C4—162no—
494C4—162no—
495C4—162no—
496C4#274112yes—
497C4#274112yes—
498C4#274712yes—
499C4#274712yes—
500C2—82no—
501C2#1705811yescharacteristic
502C2#2760811yescharacteristic
503C2#2760911yescharacteristic
504C2#2797012yes—
505C2#2798412yes—
506C2#2993512yes—
507C2#2994612yes—
508C2#3019211yescharacteristic
509C2#3019311yescharacteristic
510C2#3277911yescharacteristic
511C2#3278011yescharacteristic
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
3. #11372C2semidirect
4. #11358C2semidirect
5. #11267C2semidirect
6. #11267C2semidirect

Maximal subgroups

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

subgroupindex
#143402
#113722
#113582
#112672
#112002
#111992
#107132