Finite Group Databasebeta

← #449033

Subgroups — #449033

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#1847C211yesmaximal, characteristic, complemented
4#15150C211yesmaximal, characteristic
5#11372C211yesmaximal, characteristic, complemented
6#11219C212yesmaximal, complemented
7#11096C212yesmaximal
8#11095C212yesmaximal
9#10922C211yesmaximal, characteristic
10#10908C212yesmaximal, complemented
11#10881C211yesmaximal, characteristic, complemented
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×C22⋊C8C2212yes—
15C22×#97C2211yescharacteristic
16#2494—24no—
17#2494—24no—
18C2×#401—21no—
19C2×#397—21no—
20C2×#397—21no—
21C2×#397—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388—21no—
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#2100—24no—
34#1847—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—24no—
70C2×#107—42no—
71C2×#106—24no—
72C2×#106—24no—
73C2×#106—42no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×#97—24no—
78C2×#97—22no—
79#401—28no—
80#397—28no—
81#397—44no—
82#397—44no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389D414yes—
87#388—28no—
88#387—28no—
89#384—22no—
90#384—22no—
91#382—22no—
92#382D414yes—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—42no—
98#380—42no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—24no—
124#336—22no—
125C25C2×D411yescharacteristic
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8#4114yes—
158C22×C8#4114yes—
159C22×C8#4114yes—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
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—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—82no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—82no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223#107—48no—
224#106—48no—
225C22⋊C8—48no—
226#97—28no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—22no—
235#95—22no—
236#95—44no—
237#95—42no—
238#95—42no—
239#95—44no—
240#95—44no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—22no—
245C24—22no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—22no—
250C24—22no—
251C24AGL(1,ℤ/8)12yes—
252C24C22×D411yescharacteristic
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4#12012yes—
328C22×C4#12012yes—
329C22×C4AGL(1,ℤ/8)12yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4112yes—
333C22×C4C2×#4112yes—
334C22×C4C2×#4112yes—
335C22×C4H(5,𝔽2)12yes—
336C22×C42−1+412yes—
337C22×C42−1+412yes—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8#12618yes—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—48no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—48no—
377C22⋊C4—84no—
378C22⋊C4—88no—
379C42—22no—
380C42—22no—
381C42—24no—
382C42—41no—
383C42—41no—
384C42—42no—
385C42—84no—
386C23—24no—
387C23—24no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23—41no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23C2×#12011yescharacteristic
410C23#53211yescharacteristic
411C23#53511yescharacteristic
412C23#54712yes—
413C23#54712yes—
414C23C2×AGL(1,ℤ/8)11yescharacteristic
415C23C2×AGL(1,ℤ/8)11yescharacteristic
416C23#57511yescharacteristic
417C23#57511yescharacteristic
418C23#57712yes—
419C23#57812yes—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
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—41no—
456C2×C4—42no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#52511yescharacteristic
486C2×C4C2×#12612yes—
487C2×C4#53211yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#53611yescharacteristic
490C2×C4#53912yes—
491C2×C4#54112yes—
492C2×C4#54212yes—
493C2×C4#54612yes—
494C2×C4#54712yes—
495C2×C4#54812yes—
496C2×C4#55012yes—
497C2×C4#55212yes—
498C2×C4#55312yes—
499C2×C4#55612yes—
500C2×C4#56312yes—
501C8—28no—
502C8—82no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—42no—
507C22—44no—
508C22—81no—
509C22—81no—
510C22—82no—
511C22—81no—
512C22—81no—
513C22—82no—
514C22#240611yescharacteristic
515C22#243111yescharacteristic
516C22#243211yescharacteristic
517C22#249612yes—
518C22#271812yes—
519C22#300611yescharacteristic
520C22#300611yescharacteristic
521C22#300911yescharacteristic
522C22#301111yescharacteristic
523C22#317111yescharacteristic
524C22#317812yes—
525C22#318011yescharacteristic
526C22#321711yescharacteristic
527C22#321911yescharacteristic
528C22#322512yes—
529C22#334312yes—
530C22#334512yes—
531C22#334512yes—
532C22#334912yes—
533C22#334912yes—
534C22#335012yes—
535C22#335112yes—
536C22#335312yes—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—81no—
544C4—82no—
545C4—81no—
546C4—162no—
547C4—162no—
548C4#248112yes—
549C4#251212yes—
550C4#271612yes—
551C4#272112yes—
552C2—82no—
553C2#1592811yescharacteristic
554C2#2410111yescharacteristic
555C2#2410711yescharacteristic
556C2#2560511yescharacteristic
557C2#2561311yescharacteristic
558C2#2571411yescharacteristic
559C2#2571611yescharacteristic
560C2#2784212yes—
561C2#2784712yes—
562C2#3230312yes—
563C2#3230412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2100C2semidirect
3. C2×#1847C2semidirect
5. #11372C2semidirect
6. #11219C2semidirect
10. #10908C2semidirect
11. #10881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
C2×#21002
C2×#18472
#151502
#113722
#112192
#110962
#110952
#109222
#109082
#108812