Finite Group Databasebeta

← #404926

Subgroups — #404926

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1862C211yesmaximal, characteristic
4#15156C211yesmaximal, characteristic, complemented
5#11373C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11189C212yesmaximal
8#11123C212yesmaximal
9#10907C212yesmaximal, complemented
10#10907C212yesmaximal, complemented
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C2×#400—21no—
20#2354—24no—
21#2354—24no—
22C2×#390C2212yes—
23C2×#389C2211yescharacteristic
24C2×#388—22no—
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2083—24no—
30#1862—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1443—22no—
41#1441—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
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⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—22no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—22no—
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—42no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—22no—
81C2×#102—22no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—42no—
86#400—44no—
87#390—24no—
88#390—24no—
89#389—22no—
90#389—22no—
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—22no—
106C2×#95—21no—
107C2×#95D412yes—
108#340—24no—
109#340—24no—
110#340—22no—
111#340—24no—
112#340—42no—
113#336—24no—
114#336—24no—
115#336—22no—
116#336—22no—
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—21no—
134C23×C4—22no—
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—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8#4114yes—
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—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4D814yes—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4#4112yes—
203C2×C4⋊C4#4114yes—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—28no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—44no—
222C2×C42—44no—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225#107—48no—
226#107—44no—
227#102—28no—
228#102—48no—
229#102—44no—
230#95—28no—
231#95—24no—
232#95—44no—
233#95—42no—
234#95—44no—
235#95—42no—
236#95—44no—
237#95—81no—
238#95—82no—
239C24—22no—
240C24—24no—
241C24—42no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—42no—
247C24—41no—
248C24—41no—
249C24—42no—
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—41no—
266C2×D4—41no—
267C2×D4—42no—
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—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—416no—
294C22×C4—416no—
295C22×C4—48no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—48no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
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—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—48no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4#12314yes—
379C22⋊C4—216no—
380C22⋊C4—416no—
381C22⋊C4—88no—
382C42—22no—
383C42—22no—
384C42—24no—
385C42—416no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C23—24no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—44no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—81no—
408C23—81no—
409C23—82no—
410C23—81no—
411C23—81no—
412C23—82no—
413C23C2×#12011yescharacteristic
414C23#52512yes—
415C23#54712yes—
416C23#54811yescharacteristic
417C23#55011yescharacteristic
418C23C22×D811yescharacteristic
419C23C2×AGL(1,ℤ/8)11yescharacteristic
420C23#57512yes—
421C23#57611yescharacteristic
422C23#57711yescharacteristic
423C23#57812yes—
424D4—24no—
425D4—24no—
426D4—42no—
427D4—42no—
428D4—44no—
429D4—44no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—432no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
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—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
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—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4#44714yes—
486C2×C4C2×#12311yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4#52511yescharacteristic
489C2×C4#52512yes—
490C2×C4#52912yes—
491C2×C4#52912yes—
492C2×C4#53211yescharacteristic
493C2×C4#53611yescharacteristic
494C2×C4#53711yescharacteristic
495C2×C4#54112yes—
496C2×C4#54212yes—
497C2×C4#54611yescharacteristic
498C2×C4#54711yescharacteristic
499C2×C4#54812yes—
500C2×C4#55112yes—
501C2×C4#55212yes—
502C2×C4#55212yes—
503C2×C4#55212yes—
504C8—42no—
505C8—44no—
506C8—81no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—84no—
511C22—82no—
512C22—82no—
513C22—84no—
514C22—82no—
515C22—82no—
516C22—84no—
517C22—84no—
518C22#240711yescharacteristic
519C22#243211yescharacteristic
520C22#247512yes—
521C22#247912yes—
522C22#284911yescharacteristic
523C22C2×#44711yescharacteristic
524C22#300511yescharacteristic
525C22#301111yescharacteristic
526C22#301211yescharacteristic
527C22#310012yes—
528C22#311412yes—
529C22#312012yes—
530C22#312412yes—
531C22#333412yes—
532C22#334712yes—
533C22#334812yes—
534C22#335312yes—
535C22#335811yescharacteristic
536C22#336611yescharacteristic
537C22#337712yes—
538C22#339111yescharacteristic
539C22#339711yescharacteristic
540C22#340212yes—
541C4—24no—
542C4—44no—
543C4—44no—
544C4—42no—
545C4—82no—
546C4—82no—
547C4—84no—
548C4—162no—
549C4—161no—
550C4#239511yescharacteristic
551C4#247512yes—
552C4#247712yes—
553C4#248411yescharacteristic
554C4#249811yescharacteristic
555C4#262511yescharacteristic
556C2—84no—
557C2#1490611yescharacteristic
558C2#2418411yescharacteristic
559C2#2420111yescharacteristic
560C2#2570311yescharacteristic
561C2#2571011yescharacteristic
562C2#2735812yes—
563C2#2737612yes—
564C2#2748612yes—
565C2#2750212yes—
566C2#3300211yescharacteristic
567C2#3301711yescharacteristic
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×#2083C2semidirect
4. #15156C2semidirect
5. #11373C2semidirect
6. #11346C2semidirect
9. #10907C2semidirect
10. #10907C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#20832
C2×#18622
#151562
#113732
#113462
#111892
#111232
#109072
#107882