Finite Group Databasebeta

← #404888

Subgroups — #404888

Order 512.

1474 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×#2354C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1864C211yesmaximal, characteristic
4#15194C211yesmaximal, characteristic, complemented
5#11357C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11189C212yesmaximal
8#11066C212yesmaximal
9#10908C212yesmaximal, complemented
10#10885C212yesmaximal, complemented
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C22×#106C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18#2354—24no—
19#2354—24no—
20C2×#390C2212yes—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#388—22no—
24C2×#384C2211yescharacteristic
25C2×#380—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#2099—24no—
29#1864—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1445—21no—
41#1444—21no—
42#1441—22no—
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—22no—
60C22×C4⋊C4—21no—
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×#106—22no—
75C2×#106—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#390—24no—
85#390—24no—
86#389—24no—
87#389—22no—
88#389—22no—
89#389—42no—
90#388—216no—
91#384—22no—
92#384D414yes—
93#380—28no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—24no—
110#340—24no—
111#340—24no—
112#336—24no—
113#336—22no—
114#336—22no—
115#336—42no—
116C25—21no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4C2×D412yes—
126C22×D4#4114yes—
127C23×C4—28no—
128C23×C4—21no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—44no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—21no—
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—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8#4112yes—
159C4×D4—216no—
160C4×D4—28no—
161C4×D4—28no—
162#117—432no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4SD1614yes—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4#4114yes—
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#106—44no—
226#102—28no—
227#102—48no—
228#102—44no—
229#95—28no—
230#95—24no—
231#95—44no—
232#95—42no—
233#95—42no—
234#95—81no—
235#95—82no—
236#95—82no—
237C24—22no—
238C24—24no—
239C24—41no—
240C24—41no—
241C24—42no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24—42no—
246C24—41no—
247C24—41no—
248C24C22×D411yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4#12314yes—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—216no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—416no—
293C22×C4—416no—
294C22×C4—48no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—48no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
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—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4#12012yes—
321C22×C4#12012yes—
322C22×C4C2×SD1614yes—
323C22×C4C2×SD1612yes—
324C22×C4AGL(1,ℤ/8)12yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4112yes—
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4112yes—
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C42−1+412yes—
334C22×C42−1+411yescharacteristic
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—28no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—48no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—88no—
375C4⋊C4#12314yes—
376C22⋊C4—216no—
377C22⋊C4—416no—
378C22⋊C4—88no—
379C42—22no—
380C42—22no—
381C42—24no—
382C42—416no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C23—24no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—42no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—81no—
401C23—81no—
402C23—82no—
403C23—81no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—81no—
409C23—82no—
410C23C2×#12011yescharacteristic
411C23#52512yes—
412C23#54611yescharacteristic
413C23#54711yescharacteristic
414C23#54812yes—
415C23C22×SD1611yescharacteristic
416C23C2×AGL(1,ℤ/8)11yescharacteristic
417C23#57512yes—
418C23#57612yes—
419C23#57711yescharacteristic
420C23#57811yescharacteristic
421D4—24no—
422D4—24no—
423D4—42no—
424D4—42no—
425D4—44no—
426D4—44no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—432no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
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—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4#45014yes—
481C2×C4C2×#12311yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4#52511yescharacteristic
484C2×C4#52512yes—
485C2×C4#52912yes—
486C2×C4#52912yes—
487C2×C4#53211yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#53711yescharacteristic
490C2×C4#54112yes—
491C2×C4#54212yes—
492C2×C4#54712yes—
493C2×C4#54811yescharacteristic
494C2×C4#55011yescharacteristic
495C2×C4#55112yes—
496C2×C4#55212yes—
497C2×C4#55212yes—
498C2×C4#55212yes—
499C8—42no—
500C8—82no—
501C8—81no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—84no—
506C22—84no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—84no—
512C22—84no—
513C22#240711yescharacteristic
514C22#243211yescharacteristic
515C22#247512yes—
516C22#247712yes—
517C22#284911yescharacteristic
518C22C2×#45011yescharacteristic
519C22#300511yescharacteristic
520C22#300811yescharacteristic
521C22#301311yescharacteristic
522C22#310112yes—
523C22#311412yes—
524C22#311912yes—
525C22#312412yes—
526C22#329811yescharacteristic
527C22#331912yes—
528C22#333211yescharacteristic
529C22#333511yescharacteristic
530C22#334712yes—
531C22#335311yescharacteristic
532C22#336212yes—
533C22#336612yes—
534C22#337612yes—
535C22#337712yes—
536C4—24no—
537C4—44no—
538C4—44no—
539C4—42no—
540C4—84no—
541C4—84no—
542C4—82no—
543C4—82no—
544C4—161no—
545C4#239511yescharacteristic
546C4#247512yes—
547C4#247912yes—
548C4#248411yescharacteristic
549C4#249811yescharacteristic
550C4#262511yescharacteristic
551C2—84no—
552C2#1490611yescharacteristic
553C2#2412011yescharacteristic
554C2#2412911yescharacteristic
555C2#2564911yescharacteristic
556C2#2565611yescharacteristic
557C2#2736012yes—
558C2#2737612yes—
559C2#2742412yes—
560C2#2743812yes—
561C2#3300711yescharacteristic
562C2#3301211yescharacteristic
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. #15194C2semidirect
5. #11357C2semidirect
6. #11346C2semidirect
9. #10908C2semidirect
10. #10885C2semidirect

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×#18642
#151942
#113572
#113462
#111892
#110662
#109082
#108852
#107882