Finite Group Databasebeta

← #467272

Subgroups — #467272

Order 512.

1148 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1948C211yesmaximal, characteristic, complemented
2C2×#1926C211yesmaximal, characteristic
3#15155C211yesmaximal, characteristic
4#13933C211yesmaximal, characteristic, complemented
5#11357C211yesmaximal, characteristic, complemented
6#11323C212yesmaximal, complemented
7#11255C211yesmaximal, characteristic, complemented
8#11207C212yesmaximal, complemented
9#11121C212yesmaximal
10#11076C211yesmaximal, characteristic
11#11066C212yesmaximal
12C22×#106C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C2×#399—21no—
19C2×#399—22no—
20C2×#399—21no—
21C2×#390C2211yescharacteristic
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#384C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1948—24no—
28#1926—24no—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1442—22no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C42—21no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×#106—22no—
65C2×#106—24no—
66C2×#106—42no—
67C2×#106—42no—
68C2×#106—42no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—22no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—24no—
74C2×#102—22no—
75C2×#102—42no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102Q814yes—
79#399—48no—
80#399—44no—
81#399—44no—
82#390—22no—
83#390—22no—
84#390—42no—
85#390—42no—
86#389—24no—
87#389—42no—
88#384—22no—
89#384—22no—
90#384—42no—
91#384—42no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4D414yes—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95D411yescharacteristic
101C2×#95D412yes—
102C2×#95D411yescharacteristic
103#340—22no—
104#340—24no—
105#340—42no—
106#340—42no—
107#340—42no—
108#336—24no—
109#336—22no—
110#336—42no—
111C25—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4C2×Q812yes—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8C2×Q812yes—
148C22×C8#4114yes—
149C2×C4⋊C4—21no—
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—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×Q812yes—
188C2×C4⋊C4#4112yes—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—82no—
210C2×C42C2×D412yes—
211C2×C42#4112yes—
212#106—44no—
213#106—48no—
214C4⋊C8—28no—
215#102—28no—
216#102—44no—
217#102—48no—
218#95—24no—
219#95—24no—
220#95—22no—
221#95—44no—
222#95—42no—
223#95—82no—
224#95—82no—
225#95—81no—
226#95SD1614yes—
227C24—22no—
228C24—41no—
229C24—42no—
230C24—41no—
231C24—42no—
232C24—41no—
233C24—42no—
234C24—41no—
235C24—41no—
236C24—42no—
237C24—41no—
238C24C22×D411yescharacteristic
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
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—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4C2×SD1614yes—
307C22×C4C2×SD1612yes—
308C22×C4AGL(1,ℤ/8)14yes—
309C22×C4AGL(1,ℤ/8)12yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×Q811yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4H(5,𝔽2)12yes—
316C22×C4H(5,𝔽2)12yes—
317C22×C4H(5,𝔽2)11yescharacteristic
318C22×C42−1+412yes—
319C22×C42−1+411yescharacteristic
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—88no—
355C4⋊C4—88no—
356C4⋊C4—88no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4C22⋊Q814yes—
360C22⋊C4—88no—
361C22⋊C4—88no—
362C22⋊C4—84no—
363C22⋊C4—84no—
364C42—22no—
365C42—22no—
366C42—24no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—84no—
372C23—42no—
373C23—44no—
374C23—42no—
375C23—42no—
376C23—44no—
377C23—81no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—81no—
383C23—81no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—82no—
395C23C2×#12111yescharacteristic
396C23#53512yes—
397C23#54611yescharacteristic
398C23#54712yes—
399C23D4×Q811yescharacteristic
400C23C22×SD1611yescharacteristic
401C23C2×AGL(1,ℤ/8)11yescharacteristic
402C23C2×AGL(1,ℤ/8)12yes—
403C23#57711yescharacteristic
404C23#57712yes—
405C23#57811yescharacteristic
406D4—42no—
407D4—42no—
408D4—44no—
409D4—81no—
410D4—81no—
411D4—82no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4#46014yes—
465C2×C4C2×C22⋊Q811yescharacteristic
466C2×C4#52511yescharacteristic
467C2×C4#53511yescharacteristic
468C2×C4#53511yescharacteristic
469C2×C4#53611yescharacteristic
470C2×C4#53711yescharacteristic
471C2×C4#53812yes—
472C2×C4#53912yes—
473C2×C4#54012yes—
474C2×C4#54212yes—
475C2×C4#54412yes—
476C2×C4#55212yes—
477C2×C4#55211yescharacteristic
478C2×C4#55412yes—
479C2×C4#55411yescharacteristic
480C2×C4#55912yes—
481C2×C4#56212yes—
482C2×C4#56312yes—
483C8—42no—
484C8—44no—
485C8—81no—
486C22—84no—
487C22—84no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—161no—
492C22—161no—
493C22—162no—
494C22—162no—
495C22—161no—
496C22—161no—
497C22#241311yescharacteristic
498C22#241711yescharacteristic
499C22#261911yescharacteristic
500C22#271012yes—
501C22#272012yes—
502C22C2×#46011yescharacteristic
503C22#303811yescharacteristic
504C22#304211yescharacteristic
505C22#304611yescharacteristic
506C22#320812yes—
507C22#321712yes—
508C22#322112yes—
509C22#322512yes—
510C22#329811yescharacteristic
511C22#330612yes—
512C22#333211yescharacteristic
513C22#333512yes—
514C22#334512yes—
515C22#334512yes—
516C22#335312yes—
517C22Q8×SD1611yescharacteristic
518C22#338412yes—
519C22#338611yescharacteristic
520C4—24no—
521C4—42no—
522C4—44no—
523C4—44no—
524C4—44no—
525C4—82no—
526C4—162no—
527C4—162no—
528C4—161no—
529C4#258011yescharacteristic
530C4#262811yescharacteristic
531C4#263611yescharacteristic
532C4#276812yes—
533C4#277312yes—
534C4#282011yescharacteristic
535C2—162no—
536C2#1656411yescharacteristic
537C2#2457811yescharacteristic
538C2#2459211yescharacteristic
539C2#2481311yescharacteristic
540C2#2482911yescharacteristic
541C2#2979611yescharacteristic
542C2#2980411yescharacteristic
543C2#3205012yes—
544C2#3207212yes—
545C2#3234412yes—
546C2#3236212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1948C2semidirect
4. #13933C2semidirect
5. #11357C2semidirect
6. #11323C2semidirect
7. #11255C2semidirect
8. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19482
C2×#19262
#151552
#139332
#113572
#113232
#112552
#112072
#111212
#110762
#110662