Finite Group Databasebeta

← #461706

Subgroups — #461706

Order 512.

1200 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1862C211yesmaximal, characteristic
2C2×#1802C211yesmaximal, characteristic
3#15197C211yesmaximal, characteristic, complemented
4#13920C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic, complemented
6#11322C211yesmaximal, characteristic, complemented
7#11228C212yesmaximal, complemented
8#11227C211yesmaximal, characteristic, complemented
9#11219C212yesmaximal, complemented
10#11189C212yesmaximal
11#11076C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C2×#390C2212yes—
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#380—21no—
24#1862—24no—
25#1802—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1442—22no—
35#1441—21no—
36#1441—21no—
37#1439—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C4×#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C2×C4×D4—24no—
49C2×C4×D4—22no—
50C2×C4×D4—42no—
51C2×C4×D4D414yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×#107—22no—
67C2×#107—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—42no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8—42no—
73C2×#102—22no—
74C2×#102—22no—
75C2×#102—24no—
76C2×#102—22no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102Q814yes—
81#390—24no—
82#390—42no—
83#389—24no—
84#389—22no—
85#389—22no—
86#389—42no—
87#384—22no—
88#384—22no—
89#380—44no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D411yescharacteristic
99C2×#95D412yes—
100C2×#95D411yescharacteristic
101#340—24no—
102#340—22no—
103#340—42no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—22no—
108#336—42no—
109C25—21no—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4C2×D412yes—
118C22×D4C2×Q812yes—
119C22×D4#4112yes—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—44no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—82no—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8C2×Q812yes—
152C4×D4—28no—
153C4×D4—44no—
154C4×D4—48no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
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—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4—81no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×Q812yes—
191C2×C4⋊C4#4114yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
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×C42C2×D412yes—
209C2×C42#4112yes—
210#107—44no—
211C4⋊C8—44no—
212C4⋊C8—48no—
213#102—28no—
214#102—44no—
215#102—48no—
216#95—22no—
217#95—22no—
218#95—24no—
219#95—44no—
220#95—42no—
221#95—42no—
222#95—82no—
223#95—81no—
224#95—82no—
225#95—82no—
226#95—82no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—42no—
231C24—42no—
232C24—41no—
233C24—41no—
234C24—42no—
235C24—41no—
236C24—42no—
237C24—41no—
238C24C22×D411yescharacteristic
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4C22⋊Q814yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—48no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—48no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—84no—
310C22×C4—88no—
311C22×C4—88no—
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4#13512yes—
315C22×C4AGL(1,ℤ/8)12yes—
316C22×C4#13714yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×Q811yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4112yes—
321C22×C4C2×#4111yescharacteristic
322C22×C4H(5,𝔽2)12yes—
323C22×C4H(5,𝔽2)12yes—
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C42−1+412yes—
326C22×C42−1+411yescharacteristic
327C2×C8—22no—
328C2×C8—24no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4—88no—
364C4⋊C4—84no—
365C4⋊C4#12314yes—
366C22⋊C4—48no—
367C22⋊C4—88no—
368C22⋊C4—88no—
369C42—22no—
370C42—22no—
371C42—24no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C23—24no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23—42no—
381C23—42no—
382C23—44no—
383C23—44no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—42no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23C2×#12111yescharacteristic
400C23#53712yes—
401C23#54612yes—
402C23#54611yescharacteristic
403C23D4×Q811yescharacteristic
404C23C2×#13511yescharacteristic
405C23C2×AGL(1,ℤ/8)11yescharacteristic
406C23C2×#13712yes—
407C23#57611yescharacteristic
408C23#57612yes—
409C23#57711yescharacteristic
410D4—24no—
411D4—24no—
412D4—42no—
413D4—44no—
414D4—44no—
415D4—42no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—816no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#46314yes—
472C2×C4C2×C22⋊Q811yescharacteristic
473C2×C4C2×#12311yescharacteristic
474C2×C4#53411yescharacteristic
475C2×C4#53511yescharacteristic
476C2×C4#53611yescharacteristic
477C2×C4#53711yescharacteristic
478C2×C4#53912yes—
479C2×C4#54112yes—
480C2×C4#54212yes—
481C2×C4#54212yes—
482C2×C4#54412yes—
483C2×C4#55112yes—
484C2×C4#55211yescharacteristic
485C2×C4#55312yes—
486C2×C4#55411yescharacteristic
487C2×C4#55412yes—
488C2×C4#56012yes—
489C2×C4#56212yes—
490C8—42no—
491C8—81no—
492C8—82no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—84no—
497C22—84no—
498C22—82no—
499C22—84no—
500C22—82no—
501C22—84no—
502C22—82no—
503C22—82no—
504C22#241211yescharacteristic
505C22#241711yescharacteristic
506C22#262011yescharacteristic
507C22#269112yes—
508C22#270412yes—
509C22C2×#46311yescharacteristic
510C22#303911yescharacteristic
511C22#304111yescharacteristic
512C22#304511yescharacteristic
513C22#324812yes—
514C22#325412yes—
515C22#325712yes—
516C22#325912yes—
517C22#329712yes—
518C22#330311yescharacteristic
519C22#331412yes—
520C22#331612yes—
521C22#331812yes—
522C22#332511yescharacteristic
523C22#333312yes—
524C22#338011yescharacteristic
525C22#338411yescharacteristic
526C22#338512yes—
527C4—24no—
528C4—44no—
529C4—44no—
530C4—42no—
531C4—82no—
532C4—82no—
533C4—162no—
534C4—162no—
535C4—161no—
536C4#248711yescharacteristic
537C4#262511yescharacteristic
538C4#262711yescharacteristic
539C4#274212yes—
540C4#274712yes—
541C4#281311yescharacteristic
542C2—84no—
543C2#1640011yescharacteristic
544C2#2453111yescharacteristic
545C2#2454611yescharacteristic
546C2#2481211yescharacteristic
547C2#2483311yescharacteristic
548C2#2982911yescharacteristic
549C2#2983411yescharacteristic
550C2#3149812yes—
551C2#3151012yes—
552C2#3187812yes—
553C2#3188212yes—
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
3. #15197C2semidirect
5. #11357C2semidirect
6. #11322C2semidirect
7. #11228C2semidirect
8. #11227C2semidirect
9. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18622
C2×#18022
#151972
#139202
#113572
#113222
#112282
#112272
#112192
#111892
#110762