Finite Group Databasebeta

← #475518

Subgroups — #475518

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14340C211yesmaximal, characteristic
2#14340C211yesmaximal, characteristic
3#11373C211yesmaximal, characteristic, complemented
4#11371C211yesmaximal, characteristic, complemented
5#11357C212yesmaximal, complemented
6#11268C212yesmaximal, complemented
7#11268C211yesmaximal, characteristic, complemented
8#11268C211yesmaximal, characteristic, complemented
9#11200C212yesmaximal
10#11199C212yesmaximal
11#10713C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C2×#384C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#380C2212yes—
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32#1441—22no—
33#1441—22no—
34#1439—22no—
35#1439—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×#117—24no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—42no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D412yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C22⋊C8—24no—
62C2×C22⋊C8—42no—
63C2×C22⋊C8—42no—
64#384—24no—
65#384—22no—
66#384—22no—
67#382—42no—
68#382—42no—
69#380—24no—
70#380—24no—
71#380—42no—
72#380—42no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
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—24no—
91#336—22no—
92#336—22no—
93C25C2×D411yescharacteristic
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4#4114yes—
113C23×C4C2411yescharacteristic
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
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—
128#117—48no—
129#117—48no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—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—21no—
142C2×C4⋊C4—21no—
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—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—82no—
174C2×C22⋊C4—82no—
175C2×C22⋊C4—82no—
176C2×C22⋊C4—82no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42#4114yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C22⋊C8—48no—
186#95—42no—
187#95—42no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—82no—
192#95—82no—
193#95—82no—
194#95—82no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24#12412yes—
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—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
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#12314yes—
276C22×C4#12314yes—
277C22×C4#12412yes—
278C22×C4C2×#4112yes—
279C22×C4C2×#4112yes—
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4H(5,𝔽2)11yescharacteristic
283C22×C4H(5,𝔽2)12yes—
284C22×C4H(5,𝔽2)12yes—
285C22×C42−1+411yescharacteristic
286C22×C42−1+411yescharacteristic
287C22×C42−1+411yescharacteristic
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—48no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—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—88no—
320C4⋊C4—88no—
321C22⋊C4—48no—
322C22⋊C4—88no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C23—24no—
330C23—24no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23C2×#12311yescharacteristic
353C23C2×#12311yescharacteristic
354C23C2×#12411yescharacteristic
355C23#53411yescharacteristic
356C23#53511yescharacteristic
357C23#53611yescharacteristic
358C23#53711yescharacteristic
359C23#57612yes—
360C23#57712yes—
361C23#57712yes—
362C23#57812yes—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4#53812yes—
423C2×C4#53912yes—
424C2×C4#54012yes—
425C2×C4#54111yescharacteristic
426C2×C4#54111yescharacteristic
427C2×C4#54211yescharacteristic
428C2×C4#54211yescharacteristic
429C2×C4#54212yes—
430C2×C4#55112yes—
431C2×C4#55112yes—
432C2×C4#55212yes—
433C2×C4#55212yes—
434C2×C4#55312yes—
435C2×C4#55312yes—
436C2×C4#56212yes—
437C2×C4#56212yes—
438C8—82no—
439C8—82no—
440C22—44no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—44no—
445C22—81no—
446C22—81no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22—82no—
451C22#248311yescharacteristic
452C22#250311yescharacteristic
453C22#250411yescharacteristic
454C22#262411yescharacteristic
455C22#262511yescharacteristic
456C22#263911yescharacteristic
457C22#281611yescharacteristic
458C22#309612yes—
459C22#309612yes—
460C22#309712yes—
461C22#309712yes—
462C22#313612yes—
463C22#313712yes—
464C22#320312yes—
465C22#320412yes—
466C22#322011yescharacteristic
467C22#322111yescharacteristic
468C22#322211yescharacteristic
469C22#322311yescharacteristic
470C22#324312yes—
471C22#324412yes—
472C22#325712yes—
473C22#325812yes—
474C4—42no—
475C4—44no—
476C4—44no—
477C4—161no—
478C4—161no—
479C4—162no—
480C4—161no—
481C4—161no—
482C4—162no—
483C4#274112yes—
484C4#274112yes—
485C4#274712yes—
486C4#274712yes—
487C2—82no—
488C2#1705811yescharacteristic
489C2#2760712yes—
490C2#2797112yes—
491C2#2798511yescharacteristic
492C2#2798611yescharacteristic
493C2#2993311yescharacteristic
494C2#2993411yescharacteristic
495C2#2994512yes—
496C2#3019112yes—
497C2#3278311yescharacteristic
498C2#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
3. #11373C2semidirect
4. #11371C2semidirect
5. #11357C2semidirect
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
#143402
#113732
#113712
#113572
#112682
#112002
#111992
#107132