Finite Group Databasebeta

← #475178

Subgroups — #475178

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14348C211yesmaximal, characteristic, complemented
2#14260C211yesmaximal, characteristic
3#11323C211yesmaximal, characteristic, complemented
4#11292C212yesmaximal, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11257C212yesmaximal, complemented
7#11189C212yesmaximal
8#11121C212yesmaximal
9#11051C211yesmaximal, characteristic
10#11049C211yesmaximal, characteristic
11#10796C211yesmaximal, characteristic, complemented
12C22×#106C2212yes—
13C22×#102C2212yes—
14C22×#102C2212yes—
15C2×#390C2211yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#376C2211yescharacteristic
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31#1442—22no—
32#1442—22no—
33#1440—22no—
34#1439—22no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C23×D4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C22⋊C4—22no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#106—42no—
57C2×#106—42no—
58C2×#106—42no—
59C2×#102—24no—
60C2×#102—24no—
61C2×#102—42no—
62C2×#102—42no—
63C2×#102—42no—
64C2×#102—42no—
65#390—22no—
66#390—22no—
67#390—22no—
68#390—22no—
69#390—42no—
70#389—42no—
71#384—42no—
72#382—42no—
73#376—22no—
74#376—22no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95D412yes—
84C2×#95D412yes—
85#340—24no—
86#340—24no—
87#340—42no—
88#340—42no—
89#340—42no—
90#336—24no—
91#336—22no—
92#336—22no—
93#336—42no—
94C25—21no—
95C22×D4—21no—
96C22×D4—21no—
97C22×D4—22no—
98C22×D4—21no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—21no—
102C22×D4—22no—
103C22×D4—21no—
104C22×D4#4112yes—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2411yescharacteristic
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—
128C22×C8#4114yes—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
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—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4#4112yes—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—82no—
163C2×C22⋊C4—82no—
164C2×C22⋊C4—82no—
165C2×C22⋊C4—82no—
166C2×C22⋊C4—82no—
167C2×C22⋊C4—82no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42#4112yes—
181C2×C42#4112yes—
182#106—48no—
183#102—48no—
184#102—48no—
185#95—24no—
186#95—24no—
187#95—44no—
188#95—42no—
189#95—82no—
190#95—82no—
191#95—82no—
192#95—82no—
193#95—82no—
194#95—82no—
195C24—22no—
196C24—41no—
197C24—41no—
198C24—41no—
199C24—41no—
200C24—41no—
201C24—41no—
202C24—42no—
203C24—42no—
204C24—42no—
205C24—42no—
206C24C22×D411yescharacteristic
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—42no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—41no—
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×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4H(5,𝔽2)12yes—
282C22×C4H(5,𝔽2)11yescharacteristic
283C22×C4H(5,𝔽2)12yes—
284C22×C42−1+412yes—
285C22×C42−1+411yescharacteristic
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
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—
300C4⋊C4—24no—
301C4⋊C4—42no—
302C4⋊C4—42no—
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—88no—
315C4⋊C4—84no—
316C22⋊C4—88no—
317C22⋊C4—88no—
318C22⋊C4—84no—
319C22⋊C4—84no—
320C42—24no—
321C42—24no—
322C42—41no—
323C42—41no—
324C42—42no—
325C42—41no—
326C42—42no—
327C42—41no—
328C23—44no—
329C23—42no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23—81no—
334C23—82no—
335C23—82no—
336C23—81no—
337C23—82no—
338C23—81no—
339C23—82no—
340C23—81no—
341C23—82no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—82no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23—82no—
350C23—81no—
351C23C2×#12311yescharacteristic
352C23#52511yescharacteristic
353C23#53211yescharacteristic
354C23#53511yescharacteristic
355C23#53511yescharacteristic
356C23#53611yescharacteristic
357C23#53711yescharacteristic
358C23#57612yes—
359C23#57712yes—
360C23#57712yes—
361C23#57712yes—
362D4—44no—
363D4—44no—
364D4—82no—
365D4—82no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4#53912yes—
420C2×C4#53912yes—
421C2×C4#53912yes—
422C2×C4#53912yes—
423C2×C4#53912yes—
424C2×C4#54112yes—
425C2×C4#55112yes—
426C2×C4#55211yescharacteristic
427C2×C4#55311yescharacteristic
428C2×C4#55312yes—
429C2×C4#55512yes—
430C2×C4#55512yes—
431C2×C4#55511yescharacteristic
432C2×C4#55611yescharacteristic
433C2×C4H(3,𝔽4)12yes—
434C2×C4#56212yes—
435C8—44no—
436C8—82no—
437C22—84no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—84no—
442C22—161no—
443C22—161no—
444C22—162no—
445C22—162no—
446C22—161no—
447C22—161no—
448C22#248511yescharacteristic
449C22#250011yescharacteristic
450C22#261011yescharacteristic
451C22#262311yescharacteristic
452C22#263311yescharacteristic
453C22#266411yescharacteristic
454C22#281911yescharacteristic
455C22#309612yes—
456C22#309912yes—
457C22#311812yes—
458C22#312512yes—
459C22#317312yes—
460C22#317912yes—
461C22#322011yescharacteristic
462C22#322111yescharacteristic
463C22#322112yes—
464C22#322211yescharacteristic
465C22#322211yescharacteristic
466C22#322412yes—
467C22#324112yes—
468C22#324412yes—
469C22#325712yes—
470C22#326012yes—
471C4—44no—
472C4—82no—
473C4—82no—
474C4—81no—
475C4—81no—
476C4—162no—
477C4—161no—
478C4—162no—
479C4—161no—
480C4#274412yes—
481C4#278212yes—
482C4#278312yes—
483C4#278512yes—
484C2—162no—
485C2#1702911yescharacteristic
486C2#2764112yes—
487C2#2789811yescharacteristic
488C2#2790311yescharacteristic
489C2#2957012yes—
490C2#2991612yes—
491C2#3008411yescharacteristic
492C2#3009011yescharacteristic
493C2#3039212yes—
494C2#3281211yescharacteristic
495C2#3281511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #14348C2semidirect
3. #11323C2semidirect
4. #11292C2semidirect
5. #11268C2semidirect
6. #11257C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
#143482
#142602
#113232
#112922
#112682
#112572
#111892
#111212
#110512
#110492
#107962