Finite Group Databasebeta

← #398654

Subgroups — #398654

Order 512.

1526 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1862C211yesmaximal, characteristic
2C2×#1804C212yesmaximal
3C2×#1802C211yesmaximal, characteristic
4#15242C211yesmaximal, characteristic, complemented
5D4×#95C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic, complemented
7#11322C211yesmaximal, characteristic, complemented
8#11268C212yesmaximal, complemented
9#11232C212yesmaximal, complemented
10#11227C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×#107C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23C22×#95—21no—
24#1862—24no—
25#1804—28no—
26#1802—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35#1445—22no—
36#1441—21no—
37#1441—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C22⋊#95—22no—
43C4×#95C2211yescharacteristic
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4—22no—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—22no—
54C2×C4×D4—24no—
55C2×C4×D4—42no—
56C2×C4×D4—42no—
57C2×C4×D4—42no—
58C2×C4×D4D414yes—
59C2×C4×D4Q818yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72C2×#107—22no—
73C2×#107—42no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—42no—
78C2×C4⋊C8—42no—
79C2×#102—22no—
80C2×#102—22no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—22no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102D414yes—
87#389—22no—
88#389—22no—
89#384—22no—
90#384—22no—
91#382—24no—
92#382—42no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—44no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105C2×#95D412yes—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—22no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—24no—
114C25—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4C2×D412yes—
121C22×D4C2×D412yes—
122C22×D4C2×Q814yes—
123C22×D4#4112yes—
124C22×D4#4114yes—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—24no—
129C23×C4—28no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—82no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8C2×D412yes—
164C4×D4—216no—
165C4×D4—28no—
166C4×D4—44no—
167C4×D4—48no—
168C4×D4—48no—
169C4×D4—48no—
170C4×D4—48no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×Q814yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42C2×D412yes—
216C2×C42C2×Q814yes—
217C2×C42#4112yes—
218C2×C42#4114yes—
219#107—44no—
220C4⋊C8—48no—
221C4⋊C8—44no—
222#102—28no—
223#102—48no—
224#102—44no—
225#95—22no—
226#95—22no—
227#95—42no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—82no—
234#95—82no—
235#95—82no—
236#95—81no—
237C24—22no—
238C24—24no—
239C24—22no—
240C24—24no—
241C24—22no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24—42no—
246C24—41no—
247C24—41no—
248C24C22×D411yescharacteristic
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4#12114yes—
263C2×D4C22⋊Q818yes—
264C2×D4#12314yes—
265C2×D4#12518yes—
266C2×D4C4⋊Q814yes—
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—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—48no—
289C22×C4—416no—
290C22×C4—416no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—48no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—84no—
321C22×C4—88no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4C22⋊Q814yes—
331C22×C4C22⋊Q814yes—
332C22×C4#13512yes—
333C22×C4AGL(1,ℤ/8)12yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×Q812yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4112yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)12yes—
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+411yescharacteristic
344C2×C8—22no—
345C2×C8—24no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C22⋊C4—48no—
383C22⋊C4—416no—
384C42—24no—
385C42—22no—
386C42—22no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—28no—
395C23—28no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—42no—
401C23—42no—
402C23—44no—
403C23—44no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23C2×#12111yescharacteristic
415C23C2×C22⋊Q812yes—
416C23D4211yescharacteristic
417C23#54612yes—
418C23#54811yescharacteristic
419C23C2×#13511yescharacteristic
420C23C2×AGL(1,ℤ/8)11yescharacteristic
421C23#57512yes—
422C23#57612yes—
423C23#57711yescharacteristic
424C23#57811yescharacteristic
425D4—28no—
426D4—24no—
427D4—28no—
428D4—24no—
429D4—24no—
430D4#39818yes—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—816no—
464C2×C4—832no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4#46314yes—
488C2×C4C2×#12111yescharacteristic
489C2×C4C2×C22⋊Q812yes—
490C2×C4C2×#12311yescharacteristic
491C2×C4C2×#12512yes—
492C2×C4C2×C4⋊Q811yescharacteristic
493C2×C4#53211yescharacteristic
494C2×C4#53312yes—
495C2×C4#53411yescharacteristic
496C2×C4#53711yescharacteristic
497C2×C4#54312yes—
498C2×C4#54412yes—
499C2×C4#54612yes—
500C2×C4#54711yescharacteristic
501C2×C4#55112yes—
502C2×C4#55112yes—
503C2×C4#55212yes—
504C2×C4#55311yescharacteristic
505C2×C4#55312yes—
506C8—42no—
507C8—81no—
508C8—82no—
509C22—28no—
510C22—48no—
511C22—48no—
512C22—48no—
513C22—48no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22#241211yescharacteristic
521C22#241611yescharacteristic
522C22#243712yes—
523C22#244212yes—
524C22#285711yescharacteristic
525C22C2×#46311yescharacteristic
526C22#304011yescharacteristic
527C22#304111yescharacteristic
528C22#304611yescharacteristic
529C22#307612yes—
530C22#307712yes—
531C22#308112yes—
532C22#308312yes—
533C22#326812yes—
534C22#327211yescharacteristic
535C22#328311yescharacteristic
536C22#329712yes—
537C22#331512yes—
538C22#331812yes—
539C22#332312yes—
540C22#336011yescharacteristic
541C22#336711yescharacteristic
542C22#337012yes—
543C4—24no—
544C4—44no—
545C4—44no—
546C4—42no—
547C4—84no—
548C4—82no—
549C4—82no—
550C4—162no—
551C4—161no—
552C4C2×#39811yescharacteristic
553C4#243812yes—
554C4#249811yescharacteristic
555C4#254311yescharacteristic
556C4#256912yes—
557C4#265911yescharacteristic
558C2—48no—
559C2#1476611yescharacteristic
560C2#2453111yescharacteristic
561C2#2454511yescharacteristic
562C2#2472711yescharacteristic
563C2#2474911yescharacteristic
564C2#2595612yes—
565C2#2597612yes—
566C2#2611712yes—
567C2#2611912yes—
568C2#3320111yescharacteristic
569C2#3320311yescharacteristic
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. #11371C2semidirect
7. #11322C2semidirect
8. #11268C2semidirect
9. #11232C2semidirect
10. #11227C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18622
C2×#18042
C2×#18022
#152422
D4×#952
#113712
#113222
#112682
#112322
#112272
#107882