Finite Group Databasebeta

← #449026

Subgroups — #449026

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2497C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1844C211yesmaximal, characteristic, complemented
4#15150C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11220C212yesmaximal, complemented
7#11095C212yesmaximal
8#11095C212yesmaximal
9#10921C211yesmaximal, characteristic
10#10908C212yesmaximal, complemented
11#10880C211yesmaximal, characteristic, complemented
12C22×#106C2212yes—
13C22×#106C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16#2497—24no—
17#2497—24no—
18C2×#400—21no—
19C2×#397—21no—
20C2×#397—21no—
21C2×#396—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388—21no—
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#2099—24no—
34#1844—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⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—42no—
75C2×#106—42no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C4×C8—24no—
80C2×C4×C8—22no—
81#400—28no—
82#397—44no—
83#397—44no—
84#396—28no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389D414yes—
89#388—28no—
90#387—28no—
91#384—22no—
92#384—22no—
93#384—22no—
94#384D414yes—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—42no—
100#380—42no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
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—24no—
187C2×C4⋊C4—24no—
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—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4SD1614yes—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—82no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226#106—48no—
227#106—48no—
228C22⋊C8—48no—
229C4×C8—28no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—22no—
236#95—22no—
237#95—44no—
238#95—44no—
239#95—42no—
240#95—44no—
241#95—42no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24C2×SD1612yes—
253C24C22×D411yescharacteristic
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
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#12012yes—
323C22×C4#12012yes—
324C22×C4C2×SD1612yes—
325C22×C4C2×SD1612yes—
326C22×C4#13512yes—
327C22×C4AGL(1,ℤ/8)12yes—
328C22×C4#13712yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4112yes—
332C22×C4C2×#4112yes—
333C22×C4C2×#4112yes—
334C22×C4H(5,𝔽2)12yes—
335C22×C42−1+412yes—
336C22×C42−1+412yes—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8#12618yes—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—48no—
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—84no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C22⋊C4—28no—
376C22⋊C4—44no—
377C22⋊C4—84no—
378C22⋊C4—84no—
379C22⋊C4—88no—
380C42—22no—
381C42—22no—
382C42—24no—
383C42—41no—
384C42—41no—
385C42—42no—
386C42—84no—
387C23—24no—
388C23—24no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—42no—
408C23—41no—
409C23—41no—
410C23C2×#12011yescharacteristic
411C23#53211yescharacteristic
412C23#53511yescharacteristic
413C23#54712yes—
414C23#54712yes—
415C23C22×SD1611yescharacteristic
416C23C2×#13511yescharacteristic
417C23C2×AGL(1,ℤ/8)11yescharacteristic
418C23C2×#13711yescharacteristic
419C23#57712yes—
420C23#57712yes—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
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—41no—
455C2×C4—42no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
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#44814yes—
483C2×C4#44914yes—
484C2×C4#52511yescharacteristic
485C2×C4C2×#12612yes—
486C2×C4#53211yescharacteristic
487C2×C4#53611yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#53912yes—
490C2×C4#54112yes—
491C2×C4#54212yes—
492C2×C4#54612yes—
493C2×C4#54712yes—
494C2×C4#54812yes—
495C2×C4#55012yes—
496C2×C4#55212yes—
497C2×C4#55312yes—
498C2×C4#55612yes—
499C2×C4#56312yes—
500C8—28no—
501C8—82no—
502C22—44no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—42no—
507C22—81no—
508C22—81no—
509C22—82no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22#240611yescharacteristic
514C22#243111yescharacteristic
515C22#243211yescharacteristic
516C22#249612yes—
517C22#271812yes—
518C22C2×#44811yescharacteristic
519C22C2×#44911yescharacteristic
520C22#300911yescharacteristic
521C22#300911yescharacteristic
522C22#316711yescharacteristic
523C22#317412yes—
524C22#318211yescharacteristic
525C22#320811yescharacteristic
526C22#321011yescharacteristic
527C22#322112yes—
528C22#333512yes—
529C22#333512yes—
530C22#333712yes—
531C22#334012yes—
532C22#334412yes—
533C22#334512yes—
534C22#335412yes—
535C22#335612yes—
536C4—24no—
537C4—24no—
538C4—42no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—81no—
543C4—81no—
544C4—82no—
545C4—162no—
546C4—162no—
547C4#248112yes—
548C4#251212yes—
549C4#271612yes—
550C4#272112yes—
551C2—82no—
552C2#1592811yescharacteristic
553C2#2409111yescharacteristic
554C2#2409711yescharacteristic
555C2#2559811yescharacteristic
556C2#2560611yescharacteristic
557C2#2570011yescharacteristic
558C2#2570211yescharacteristic
559C2#2783512yes—
560C2#2785312yes—
561C2#3228212yes—
562C2#3228512yes—
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×#2099C2semidirect
3. C2×#1844C2semidirect
5. #11371C2semidirect
6. #11220C2semidirect
10. #10908C2semidirect
11. #10880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24972
C2×#20992
C2×#18442
#151502
#113712
#112202
#110952
#109212
#109082
#108802