Finite Group Databasebeta

← #475513

Subgroups — #475513

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14682C211yesmaximal, characteristic
2#14682C211yesmaximal, characteristic
3#11372C212yesmaximal, complemented
4#11358C211yesmaximal, characteristic, complemented
5#11358C211yesmaximal, characteristic, complemented
6#11268C211yesmaximal, characteristic, complemented
7#11268C211yesmaximal, characteristic, complemented
8#11268C212yesmaximal, complemented
9#11200C212yesmaximal
10#11199C212yesmaximal
11#10712C211yesmaximal, characteristic
12C23×C42C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C2×#382C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#380C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×#380C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C4⋊#95C2211yescharacteristic
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2212yes—
35#1434—22no—
36#1434—22no—
37C22⋊#95—22no—
38C22⋊#95—22no—
39C24×C4D412yes—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4C2311yescharacteristic
44C22×C4⋊C4C2311yescharacteristic
45C22×C4⋊C4C2311yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—21no—
52C22×C42—22no—
53C22×C42—22no—
54C22×C42—24no—
55C22×C42—42no—
56C22×C42D412yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C22⋊C8—24no—
60C2×C22⋊C8—42no—
61C2×C22⋊C8—42no—
62#382—22no—
63#382—22no—
64#382—24no—
65#382—42no—
66#382—42no—
67#380—24no—
68#380—24no—
69#380—42no—
70#380—42no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—21no—
82#340—42no—
83#340—42no—
84#340—42no—
85#340—42no—
86#340—42no—
87#340—42no—
88#340—42no—
89#340—42no—
90#336—22no—
91#336—22no—
92#336—24no—
93C25C2×D411yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4#4114yes—
120C23×C4C2411yescharacteristic
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—
133C22×C8—22no—
134C22×C8—22no—
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—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
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—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—48no—
178C2×C42—48no—
179C2×C42—42no—
180C2×C42#4112yes—
181C2×C42#4112yes—
182C2×C42#4114yes—
183C22⋊C8—48no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—42no—
194#95—42no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24#12412yes—
205C24#13514yes—
206C24C22×D411yescharacteristic
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—48no—
236C22×C4—48no—
237C22×C4—48no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
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#12314yes—
283C22×C4#12314yes—
284C22×C4#12412yes—
285C22×C4C2×#4112yes—
286C22×C4C2×#4111yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4112yes—
289C22×C4H(5,𝔽2)11yescharacteristic
290C22×C4H(5,𝔽2)12yes—
291C22×C4H(5,𝔽2)11yescharacteristic
292C22×C4H(5,𝔽2)11yescharacteristic
293C22×C42−1+411yescharacteristic
294C22×C42−1+412yes—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
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—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
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—
325C22⋊C4—84no—
326C22⋊C4—88no—
327C22⋊C4—84no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—24no—
332C42—48no—
333C23—24no—
334C23—24no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—24no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23—22no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23—42no—
355C23#48814yes—
356C23C2×#12311yescharacteristic
357C23C2×#12311yescharacteristic
358C23C2×#12411yescharacteristic
359C23#53411yescharacteristic
360C23#53511yescharacteristic
361C23#53611yescharacteristic
362C23#53711yescharacteristic
363C23C2×#13512yes—
364C23#57512yes—
365C23#57512yes—
366C23#57512yes—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—432no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
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—81no—
426C2×C4—81no—
427C2×C4#53812yes—
428C2×C4#53912yes—
429C2×C4#54011yescharacteristic
430C2×C4#54011yescharacteristic
431C2×C4#54112yes—
432C2×C4#54212yes—
433C2×C4#54211yescharacteristic
434C2×C4#54211yescharacteristic
435C2×C4#55112yes—
436C2×C4#55112yes—
437C2×C4#55212yes—
438C2×C4#55212yes—
439C2×C4#55312yes—
440C2×C4#55312yes—
441C2×C4#56212yes—
442C2×C4#56212yes—
443C8—82no—
444C8—82no—
445C22—24no—
446C22—24no—
447C22—24no—
448C22—44no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—42no—
456C22#248311yescharacteristic
457C22#250311yescharacteristic
458C22#250411yescharacteristic
459C22#262411yescharacteristic
460C22#262511yescharacteristic
461C22#263911yescharacteristic
462C22#281611yescharacteristic
463C22#309112yes—
464C22#309112yes—
465C22#309312yes—
466C22#309312yes—
467C22C2×#48811yescharacteristic
468C22#313511yescharacteristic
469C22#314011yescharacteristic
470C22#314211yescharacteristic
471C22#319112yes—
472C22#319712yes—
473C22#321112yes—
474C22#321912yes—
475C22#323212yes—
476C22#324012yes—
477C22#325012yes—
478C22#325612yes—
479C4—42no—
480C4—44no—
481C4—44no—
482C4—162no—
483C4—161no—
484C4—161no—
485C4—162no—
486C4—161no—
487C4—161no—
488C4#274112yes—
489C4#274112yes—
490C4#274712yes—
491C4#274712yes—
492C2—44no—
493C2#1705811yescharacteristic
494C2#2760512yes—
495C2#2796411yescharacteristic
496C2#2796711yescharacteristic
497C2#2797711yescharacteristic
498C2#2797911yescharacteristic
499C2#2992412yes—
500C2#2994012yes—
501C2#3018611yescharacteristic
502C2#3019811yescharacteristic
503C2#3277812yes—
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. #11372C2semidirect
4. #11358C2semidirect
5. #11358C2semidirect
6. #11268C2semidirect
7. #11268C2semidirect
8. #11268C2semidirect

Maximal subgroups

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

subgroupindex
#146822
#113722
#113582
#112682
#112002
#111992
#107122