Finite Group Databasebeta

← #461658

Subgroups — #461658

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2073C212yesmaximal
2C2×#1864C211yesmaximal, characteristic
3C2×#1802C211yesmaximal, characteristic
4#15242C211yesmaximal, characteristic, complemented
5#13920C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic, complemented
7#11255C211yesmaximal, characteristic, complemented
8#11228C211yesmaximal, characteristic, complemented
9#11220C212yesmaximal, complemented
10#11208C212yesmaximal, complemented
11#11189C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C2×#390C2212yes—
19C2×#389C2211yescharacteristic
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23C2×#380—21no—
24#2073—28no—
25#1864—24no—
26#1802—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1445—21no—
37#1445—21no—
38#1441—22no—
39#1439—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C22⋊#95—21no—
43C4×#95C2211yescharacteristic
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×D4—24no—
50C2×C4×D4—22no—
51C2×C4×D4—42no—
52C2×C4×D4D414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#106—22no—
68C2×#106—24no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—42no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—22no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102D414yes—
79#390—24no—
80#390—42no—
81#389—24no—
82#389—22no—
83#389—22no—
84#389—42no—
85#384—22no—
86#384—22no—
87#382—24no—
88#382—42no—
89#380—44no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D411yescharacteristic
99C2×#95D411yescharacteristic
100C2×#95D412yes—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—22no—
105#340—42no—
106#340—42no—
107#336—24no—
108#336—22no—
109#336—42no—
110C25—21no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4C2×D412yes—
119C22×D4C2×D412yes—
120C22×D4#4112yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—44no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—82no—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8C2×D412yes—
153C4×D4—28no—
154C4×D4—44no—
155C4×D4—48no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
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—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—81no—
188C2×C4⋊C4—81no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4#4114yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42C2×D412yes—
210C2×C42#4112yes—
211#106—44no—
212C4⋊C8—44no—
213#102—28no—
214#102—44no—
215#102—48no—
216#95—22no—
217#95—22no—
218#95—42no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—42no—
223#95—42no—
224#95—81no—
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—24no—
244C2×D4—22no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—22no—
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×D4#12114yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
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—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—48no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—48no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—84no—
309C22×C4—88no—
310C22×C4—88no—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#13512yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4112yes—
319C22×C4C2×#4111yescharacteristic
320C22×C4H(5,𝔽2)12yes—
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C42−1+412yes—
323C22×C42−1+412yes—
324C22×C42−1+411yescharacteristic
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C4⋊C4—88no—
362C4⋊C4—84no—
363C4⋊C4#12314yes—
364C22⋊C4—48no—
365C22⋊C4—88no—
366C22⋊C4—88no—
367C42—22no—
368C42—22no—
369C42—24no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—42no—
374C23—24no—
375C23—42no—
376C23—42no—
377C23—44no—
378C23—42no—
379C23—42no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23C2×#12111yescharacteristic
398C23#53612yes—
399C23D4211yescharacteristic
400C23#54612yes—
401C23#54811yescharacteristic
402C23C2×#13511yescharacteristic
403C23C2×AGL(1,ℤ/8)11yescharacteristic
404C23#57512yes—
405C23#57611yescharacteristic
406C23#57711yescharacteristic
407C23#57712yes—
408D4—24no—
409D4—24no—
410D4—42no—
411D4—44no—
412D4—44no—
413D4—42no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—816no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—81no—
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—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4#46314yes—
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×#12311yescharacteristic
470C2×C4#53411yescharacteristic
471C2×C4#53611yescharacteristic
472C2×C4#53611yescharacteristic
473C2×C4#53712yes—
474C2×C4#53711yescharacteristic
475C2×C4#53912yes—
476C2×C4#54112yes—
477C2×C4#54112yes—
478C2×C4#54212yes—
479C2×C4#54711yescharacteristic
480C2×C4#54712yes—
481C2×C4#55112yes—
482C2×C4#55212yes—
483C2×C4#55311yescharacteristic
484C2×C4#56212yes—
485C2×C4#56312yes—
486C8—42no—
487C8—82no—
488C8—81no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—82no—
493C22—84no—
494C22—82no—
495C22—84no—
496C22—82no—
497C22—82no—
498C22—84no—
499C22—84no—
500C22#241211yescharacteristic
501C22#241611yescharacteristic
502C22#262111yescharacteristic
503C22#267212yes—
504C22#269012yes—
505C22C2×#46311yescharacteristic
506C22#304011yescharacteristic
507C22#304211yescharacteristic
508C22#304511yescharacteristic
509C22#323312yes—
510C22#323512yes—
511C22#324112yes—
512C22#324412yes—
513C22#326712yes—
514C22#327411yescharacteristic
515C22#328411yescharacteristic
516C22#329612yes—
517C22#330912yes—
518C22#331012yes—
519C22#332612yes—
520C22#336111yescharacteristic
521C22#337311yescharacteristic
522C22#337612yes—
523C4—24no—
524C4—44no—
525C4—42no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—162no—
530C4—162no—
531C4—161no—
532C4#248611yescharacteristic
533C4#261811yescharacteristic
534C4#262511yescharacteristic
535C4#272212yes—
536C4#274712yes—
537C4#281411yescharacteristic
538C2—84no—
539C2#1639911yescharacteristic
540C2#2452711yescharacteristic
541C2#2454711yescharacteristic
542C2#2473111yescharacteristic
543C2#2474511yescharacteristic
544C2#2986511yescharacteristic
545C2#2987411yescharacteristic
546C2#3063812yes—
547C2#3064812yes—
548C2#3146212yes—
549C2#3148212yes—
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
4. #15242C2semidirect
6. #11373C2semidirect
7. #11255C2semidirect
8. #11228C2semidirect
9. #11220C2semidirect
10. #11208C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20732
C2×#18642
C2×#18022
#152422
#139202
#113732
#112552
#112282
#112202
#112082
#111892