Finite Group Databasebeta

← #404882

Subgroups — #404882

Order 512.

1484 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1862C211yesmaximal, characteristic
4#15149C211yesmaximal, characteristic, complemented
5#11357C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11189C212yesmaximal
8#11066C212yesmaximal
9#10907C212yesmaximal, complemented
10#10884C212yesmaximal, complemented
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C22×#107C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C2×#398—21no—
19#2354—24no—
20#2354—24no—
21C2×#390C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×#388—22no—
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2099—24no—
30#1862—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1442—22no—
42#1441—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—22no—
53C2×C4×D4D414yes—
54C2×#117—28no—
55C2×#117—44no—
56C2×#117—44no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C42—22no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×#107—22no—
75C2×#107—24no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84#398—44no—
85#390—24no—
86#390—24no—
87#389—24no—
88#389—22no—
89#389—22no—
90#389—42no—
91#388—216no—
92#384—22no—
93#384D414yes—
94#380—28no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95D412yes—
107#340—24no—
108#340—24no—
109#340—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#336—24no—
114#336—22no—
115#336—22no—
116#336—42no—
117C25—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4C2×D412yes—
127C22×D4#4114yes—
128C23×C4—28no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—44no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8#4112yes—
160C4×D4—216no—
161C4×D4—28no—
162C4×D4—28no—
163#117—432no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4D814yes—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4#4114yes—
204C2×C4⋊C4#4112yes—
205C2×C4⋊C4#4114yes—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—28no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—44no—
224C2×C42—44no—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227#107—44no—
228#102—28no—
229#102—48no—
230#102—44no—
231#95—28no—
232#95—24no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—42no—
237#95—44no—
238#95—81no—
239C24—22no—
240C24—24no—
241C24—41no—
242C24—41no—
243C24—42no—
244C24—41no—
245C24—41no—
246C24—42no—
247C24—42no—
248C24—41no—
249C24—41no—
250C24C22×D411yescharacteristic
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4#12314yes—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—216no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—416no—
295C22×C4—416no—
296C22×C4—48no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—48no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4#12012yes—
325C22×C4#12012yes—
326C22×C4C2×D814yes—
327C22×C4C2×D812yes—
328C22×C4AGL(1,ℤ/8)12yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4112yes—
332C22×C4C2×#4112yes—
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C42−1+412yes—
338C22×C42−1+411yescharacteristic
339C2×C8—22no—
340C2×C8—24no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—44no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—48no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—84no—
377C4⋊C4—88no—
378C4⋊C4—84no—
379C4⋊C4#12314yes—
380C22⋊C4—216no—
381C22⋊C4—416no—
382C22⋊C4—88no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—416no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—42no—
391C23—24no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—42no—
397C23—42no—
398C23—44no—
399C23—44no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—81no—
405C23—81no—
406C23—82no—
407C23—81no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—81no—
413C23—82no—
414C23C2×#12011yescharacteristic
415C23#52512yes—
416C23#54611yescharacteristic
417C23#54711yescharacteristic
418C23#54812yes—
419C23C22×D811yescharacteristic
420C23C2×AGL(1,ℤ/8)11yescharacteristic
421C23#57512yes—
422C23#57712yes—
423C23#57711yescharacteristic
424C23#57811yescharacteristic
425D4—24no—
426D4—24no—
427D4—42no—
428D4—42no—
429D4—44no—
430D4—44no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—432no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4#44714yes—
487C2×C4C2×#12311yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4#52512yes—
491C2×C4#52912yes—
492C2×C4#52912yes—
493C2×C4#53211yescharacteristic
494C2×C4#53611yescharacteristic
495C2×C4#53711yescharacteristic
496C2×C4#54112yes—
497C2×C4#54212yes—
498C2×C4#54712yes—
499C2×C4#54811yescharacteristic
500C2×C4#55011yescharacteristic
501C2×C4#55112yes—
502C2×C4#55212yes—
503C2×C4#55212yes—
504C2×C4#55212yes—
505C8—42no—
506C8—82no—
507C8—81no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—84no—
512C22—82no—
513C22—84no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—84no—
518C22—84no—
519C22#240711yescharacteristic
520C22#243211yescharacteristic
521C22#247512yes—
522C22#247712yes—
523C22#284911yescharacteristic
524C22C2×#44711yescharacteristic
525C22#300511yescharacteristic
526C22#300911yescharacteristic
527C22#301311yescharacteristic
528C22#310012yes—
529C22#311412yes—
530C22#312012yes—
531C22#312512yes—
532C22#329411yescharacteristic
533C22#332112yes—
534C22#333011yescharacteristic
535C22#333411yescharacteristic
536C22#334812yes—
537C22#335411yescharacteristic
538C22#335812yes—
539C22#336812yes—
540C22#337612yes—
541C22#337712yes—
542C4—24no—
543C4—44no—
544C4—44no—
545C4—42no—
546C4—84no—
547C4—84no—
548C4—82no—
549C4—82no—
550C4—161no—
551C4#239511yescharacteristic
552C4#247512yes—
553C4#247912yes—
554C4#248411yescharacteristic
555C4#249811yescharacteristic
556C4#262511yescharacteristic
557C2—84no—
558C2#1490611yescharacteristic
559C2#2411611yescharacteristic
560C2#2413311yescharacteristic
561C2#2564511yescharacteristic
562C2#2565611yescharacteristic
563C2#2735812yes—
564C2#2737412yes—
565C2#2742212yes—
566C2#2744012yes—
567C2#3300211yescharacteristic
568C2#3301611yescharacteristic
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
2. C2×#2099C2semidirect
4. #15149C2semidirect
5. #11357C2semidirect
6. #11346C2semidirect
9. #10907C2semidirect
10. #10884C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#20992
C2×#18622
#151492
#113572
#113462
#111892
#110662
#109072
#108842
#107882