Finite Group Databasebeta

← #475268

Subgroups — #475268

Order 512.

1072 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×#1939C211yesmaximal, characteristic, complemented
2C2×#1934C211yesmaximal, characteristic
3#15156C211yesmaximal, characteristic, complemented
4#15148C211yesmaximal, characteristic
5#11293C212yesmaximal, complemented
6#11228C212yesmaximal, complemented
7#11227C211yesmaximal, characteristic, complemented
8#11189C212yesmaximal
9#11076C212yesmaximal
10#11076C211yesmaximal, characteristic
11#10820C211yesmaximal, characteristic, complemented
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#102C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#400—21no—
19C2×#399—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1939—24no—
26#1934—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1443—22no—
36#1441—21no—
37#1439—22no—
38#1439—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—24no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—42no—
65C2×C4⋊C8—42no—
66C2×C4⋊C8—42no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—22no—
70C2×#102—24no—
71C2×#102—22no—
72C2×#102—42no—
73C2×#102—42no—
74C2×#102—42no—
75C2×C4×C8—42no—
76C2×C4×C8D414yes—
77#400—44no—
78#399—44no—
79#390—24no—
80#390—22no—
81#390—22no—
82#390—42no—
83#389—24no—
84#389—22no—
85#389—22no—
86#389—42no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95D412yes—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—24no—
108C25—21no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4#4112yes—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
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—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4#4112yes—
182C2×C4⋊C4#4114yes—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42C2×D412yes—
203C2×C42#4112yes—
204C4⋊C8—44no—
205C4⋊C8—48no—
206#102—28no—
207#102—44no—
208#102—48no—
209C4×C8—44no—
210#95—24no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—42no—
215#95—44no—
216#95—42no—
217#95—42no—
218#95—81no—
219#95—81no—
220#95—82no—
221C24—22no—
222C24—41no—
223C24—42no—
224C24—42no—
225C24—41no—
226C24—42no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24—42no—
231C24—41no—
232C24C22×D411yescharacteristic
233C2×D4—22no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
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—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4#12112yes—
302C22×C4#12112yes—
303C22×C4#13514yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4AGL(1,ℤ/8)12yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4112yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4H(5,𝔽2)12yes—
312C22×C4H(5,𝔽2)12yes—
313C22×C42−1+411yescharacteristic
314C22×C42−1+412yes—
315C22×C42−1+411yescharacteristic
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—28no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—88no—
354C4⋊C4—84no—
355C22⋊C4—88no—
356C22⋊C4—84no—
357C22⋊C4—88no—
358C22⋊C4—84no—
359C42—22no—
360C42—22no—
361C42—24no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C23—44no—
367C23—42no—
368C23—42no—
369C23—44no—
370C23—42no—
371C23—82no—
372C23—82no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—82no—
389C23C2×#12111yescharacteristic
390C23#53712yes—
391C23#54612yes—
392C23#54811yescharacteristic
393C23#54811yescharacteristic
394C23C2×#13512yes—
395C23C2×AGL(1,ℤ/8)11yescharacteristic
396C23C2×AGL(1,ℤ/8)11yescharacteristic
397C23#57611yescharacteristic
398C23#57712yes—
399C23#57811yescharacteristic
400D4—42no—
401D4—42no—
402D4—44no—
403D4—81no—
404D4—81no—
405D4—82no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4#46914yes—
461C2×C4#52511yescharacteristic
462C2×C4#53211yescharacteristic
463C2×C4#53411yescharacteristic
464C2×C4#53511yescharacteristic
465C2×C4#53611yescharacteristic
466C2×C4#53611yescharacteristic
467C2×C4#53812yes—
468C2×C4#53912yes—
469C2×C4#53912yes—
470C2×C4#54012yes—
471C2×C4#54212yes—
472C2×C4#55212yes—
473C2×C4#55212yes—
474C2×C4#55211yescharacteristic
475C2×C4#55211yescharacteristic
476C2×C4#55912yes—
477C2×C4#56312yes—
478C2×C4#56312yes—
479C8—42no—
480C8—42no—
481C8—82no—
482C22—82no—
483C22—82no—
484C22—84no—
485C22—84no—
486C22—82no—
487C22—161no—
488C22—161no—
489C22—161no—
490C22—161no—
491C22—162no—
492C22—162no—
493C22#253611yescharacteristic
494C22#254411yescharacteristic
495C22#261811yescharacteristic
496C22#270212yes—
497C22#270712yes—
498C22#305011yescharacteristic
499C22C2×#46911yescharacteristic
500C22#305711yescharacteristic
501C22#306111yescharacteristic
502C22#324712yes—
503C22#324712yes—
504C22#325712yes—
505C22#325812yes—
506C22#329712yes—
507C22#330012yes—
508C22#332412yes—
509C22#333212yes—
510C22#336012yes—
511C22#336512yes—
512C22#336611yescharacteristic
513C22#336611yescharacteristic
514C22#336711yescharacteristic
515C22#336711yescharacteristic
516C4—24no—
517C4—42no—
518C4—44no—
519C4—82no—
520C4—82no—
521C4—84no—
522C4—162no—
523C4—161no—
524C4—161no—
525C4#261111yescharacteristic
526C4#263511yescharacteristic
527C4#266311yescharacteristic
528C4#277112yes—
529C4#277312yes—
530C4#282111yescharacteristic
531C2—162no—
532C2#1703911yescharacteristic
533C2#2818911yescharacteristic
534C2#2819411yescharacteristic
535C2#2860111yescharacteristic
536C2#2860711yescharacteristic
537C2#2977211yescharacteristic
538C2#2978011yescharacteristic
539C2#3181112yes—
540C2#3181412yes—
541C2#3197212yes—
542C2#3197712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1939C2semidirect
3. #15156C2semidirect
5. #11293C2semidirect
6. #11228C2semidirect
7. #11227C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19392
C2×#19342
#151562
#151482
#112932
#112282
#112272
#111892
#110762
#108202