Finite Group Databasebeta

← #440939

Subgroups — #440939

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2025C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic, complemented
4#15218C211yesmaximal, characteristic, complemented
5#11384C212yesmaximal, complemented
6#11371C211yesmaximal, characteristic, complemented
7#11207C212yesmaximal, complemented
8#11066C212yesmaximal
9#10927C211yesmaximal, characteristic
10#10875C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C22⋊Q8—22no—
15C22×C4×D4C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#97C2211yescharacteristic
19#2542—24no—
20#2542—24no—
21C2×#399—22no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#2025—24no—
29#1992—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1444—22no—
40#1441—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—42no—
52C2×#125—24no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—42no—
56C2×C22⋊Q8—216no—
57C2×C22⋊Q8—48no—
58C2×C22⋊Q8—48no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—42no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102Q818yescomplemented
86C2×#97—22no—
87C2×#97—42no—
88#399—216no—
89#390—22no—
90#390—22no—
91#390—22no—
92#390D414yes—
93#389—24no—
94#389—42no—
95#384—22no—
96#384—22no—
97#384—22no—
98#384D414yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—24no—
114#336—42no—
115C25—21no—
116C22×Q8—24no—
117C22×Q8—44no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—44no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4C2×D412yes—
128C22×D4C2×Q814yes—
129C23×C4—24no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—42no—
136C23×C4—44no—
137C23×C4—42no—
138C23×C4—44no—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
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×C8C2×Q814yes—
158C4⋊Q8—48no—
159C4⋊Q8—48no—
160#125—28no—
161#125—48no—
162#125—44no—
163C22⋊Q8—464no—
164C4×D4—48no—
165C4×D4—48no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
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—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×Q814yes—
207C2×C4⋊C4#4114yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42#4114yes—
229#102—48no—
230#102—48no—
231#97—44no—
232#95—24no—
233#95—44no—
234#95—44no—
235#95—82no—
236#95—82no—
237C24—22no—
238C24—24no—
239C24—41no—
240C24—41no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24C22×D411yescharacteristic
247C2×Q8—416no—
248C2×Q8—48no—
249C2×Q8—84no—
250C2×Q8—88no—
251C2×Q8—84no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4C4⋊Q814yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—48no—
293C22×C4—416no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
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—84no—
319C22×C4—88no—
320C22×C4—84no—
321C22×C4#12014yes—
322C22×C4#12412yes—
323C22×C4#12412yes—
324C22×C4AGL(1,ℤ/8)14yes—
325C22×C4AGL(1,ℤ/8)14yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×Q812yes—
329C22×C4C2×#4112yes—
330C22×C4C2×#4112yes—
331C22×C4H(5,𝔽2)12yes—
332C22×C4H(5,𝔽2)12yes—
333C22×C42−1+412yes—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—216no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—416no—
357C4⋊C4—416no—
358C4⋊C4—48no—
359C4⋊C4—416no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—88no—
368C22⋊C4—48no—
369C22⋊C4—416no—
370C22⋊C4—416no—
371C42—22no—
372C42—22no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—41no—
377C42—41no—
378C42—42no—
379C23—24no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—42no—
385C23—42no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—82no—
391C23—82no—
392C23—81no—
393C23—82no—
394C23—81no—
395C23—82no—
396C23—81no—
397C23—81no—
398C23—82no—
399C23—82no—
400C23C2×#12011yescharacteristic
401C23C2×#12411yescharacteristic
402C23#53511yescharacteristic
403C23#54612yes—
404C23D4×Q812yes—
405C23C2×AGL(1,ℤ/8)12yes—
406C23C2×AGL(1,ℤ/8)12yes—
407C23#57712yes—
408C23#57812yes—
409Q8—816no—
410D4—28no—
411D4—44no—
412D4—44no—
413D4—44no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—816no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
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#49014yes—
468C2×C4#52511yescharacteristic
469C2×C4C2×C4⋊Q811yescharacteristic
470C2×C4#53312yes—
471C2×C4#53511yescharacteristic
472C2×C4#53611yescharacteristic
473C2×C4#54212yes—
474C2×C4#54212yes—
475C2×C4#54312yes—
476C2×C4#54612yes—
477C2×C4#54612yes—
478C2×C4#54612yes—
479C2×C4D4×Q812yes—
480C2×C4#55412yes—
481C2×C4#55412yes—
482C2×C4#55612yes—
483C2×C4#56212yes—
484C8—44no—
485C8—82no—
486C22—44no—
487C22—48no—
488C22—82no—
489C22—82no—
490C22—84no—
491C22—84no—
492C22—84no—
493C22—84no—
494C22—84no—
495C22#240411yescharacteristic
496C22#243411yescharacteristic
497C22#243511yescharacteristic
498C22#249212yes—
499C22#269212yes—
500C22#300312yes—
501C22#301312yes—
502C22C2×#49011yescharacteristic
503C22#313712yes—
504C22#313811yescharacteristic
505C22#321712yes—
506C22#322111yescharacteristic
507C22#322311yescharacteristic
508C22#330612yes—
509C22#330812yes—
510C22#333012yes—
511C22#333212yes—
512C22#338412yes—
513C22#338412yes—
514C22#338612yes—
515C22#338612yes—
516C4—24no—
517C4—42no—
518C4—44no—
519C4—44no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4—84no—
524C4—84no—
525C4#246512yes—
526C4#254912yes—
527C4#269212yes—
528C4#270012yes—
529C2—84no—
530C2#1567111yescharacteristic
531C2#2397212yes—
532C2#2584412yes—
533C2#2586011yescharacteristic
534C2#2586811yescharacteristic
535C2#2782212yes—
536C2#2783012yes—
537C2#3153812yes—
538C2#3154012yes—
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
3. C2×#1992C2semidirect
4. #15218C2semidirect
5. #11384C2semidirect
6. #11371C2semidirect
7. #11207C2semidirect
10. #10875C2semidirect
85. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#20252
C2×#19922
#152182
#113842
#113712
#112072
#110662
#109272
#108752
#107882