Finite Group Databasebeta

← #393319

Subgroups — #393319

Order 512.

2640 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic
3C2×#1799C211yesmaximal, characteristic
4#13922C212yesmaximal, complemented
5#10918C214yesmaximal, complemented
6#10846C212yesmaximal, complemented
7#10745C214yesmaximal
8C24×Q8C2211yescharacteristic
9C22×C22⋊Q8—21no—
10C2×#518C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×#99C2212yes—
13#2409—24no—
14#2409C418yescomplemented
15C2×#395—21no—
16C2×#393—21no—
17C2×#391—21no—
18C2×#391—24no—
19C2×#386C2212yes—
20C2×#385C2212yes—
21C2×#380C2212yes—
22C2×#380C2211yescharacteristic
23#2275—24no—
24#2274—21no—
25#2273—21no—
26#2273C2212yes—
27C42×D4—24no—
28C4×#117—21no—
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4C2212yes—
31#1799—24no—
32C2×#342C2218yes—
33C2×#342C2212yes—
34#1446—24no—
35C22⋊#95C2218yes—
36C23×Q8—21no—
37C23×Q8—22no—
38C23×Q8—21no—
39C23×Q8—22no—
40C23×Q8—22no—
41C23×Q8—24no—
42C23×Q8—48no—
43C23×Q8C2×C414yescomplemented
44C23×D4C2311yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2312yes—
47C24×C4C2311yescharacteristic
48C2×C22⋊Q8—28no—
49C2×C22⋊Q8—48no—
50#518—24no—
51#518—28no—
52#518Q818yescomplemented
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—28no—
56C2×C4×D4—24no—
57C2×C4×D4C2×C414yes—
58C2×C4×D4D414yes—
59C2×#117—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2312yes—
69C22×C22⋊C4C2311yescharacteristic
70C22×C22⋊C4C2314yes—
71C22×C42—22no—
72C22×C42D411yescharacteristic
73C22×C42D414yes—
74C2×#99—24no—
75C2×#99—28no—
76#395—28no—
77#393—28no—
78#391—28no—
79#391—416no—
80#386—24no—
81#386C2×C418yes—
82#385—24no—
83#385C2×C418yescomplemented
84#380—22no—
85#380—24no—
86#380—24no—
87#380C2×C414yescomplemented
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—28no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—28no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—28no—
101#376—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—216no—
107C2×#95—22no—
108C2×#95D414yes—
109C43—22no—
110#342—24no—
111#342—216no—
112C25C2×D411yescharacteristic
113C25C2411yescharacteristic
114C22×Q8—28no—
115C22×Q8—216no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—28no—
119C22×Q8—28no—
120C22×Q8—48no—
121C22×Q8—44no—
122C22×Q8—416no—
123C22×Q8—48no—
124C22×Q8—416no—
125C22×Q8—416no—
126C22×Q8—432no—
127C22×Q8—41no—
128C22×Q8—42no—
129C22×Q8—41no—
130C22×Q8—44no—
131C22×Q8—44no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4C22×C412yes—
138C22×D4C2×D412yes—
139C22×D4C2×Q814yes—
140C22×D4#4114yes—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—48no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4C22⋊C414yes—
155C23×C4C22×C414yes—
156C23×C4C22×C412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D418yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4#4114yes—
167C22⋊Q8—432no—
168C4×D4—232no—
169C4×D4—28no—
170C4×D4—28no—
171C4×D4—28no—
172C4×D4—28no—
173C4×D4—232no—
174#117—28no—
175#117—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4C22⋊C414yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4C22⋊C414yes—
206C2×C22⋊C4C22×C414yes—
207C2×C22⋊C4C22×C412yes—
208C2×C22⋊C4C2×D412yes—
209C2×C22⋊C4C2×Q818yes—
210C2×C22⋊C4#4118yes—
211C2×C42—24no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—48no—
220C2×C42—84no—
221C2×C42C22⋊C414yes—
222#99—48no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—28no—
227#95—28no—
228#95—24no—
229#95—24no—
230#95—432no—
231#95—44no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24C2×C22⋊C412yes—
240C24C22⋊Q814yes—
241C24C23×C411yescharacteristic
242C24C22×D411yescharacteristic
243C24C22×D412yes—
244C24C22×D411yescharacteristic
245C24C22×Q812yes—
246C24C2×#4112yes—
247C24C2×#4112yes—
248C24C2×#4114yes—
249C2×Q8—216no—
250C2×Q8—232no—
251C2×Q8—416no—
252C2×Q8—48no—
253C2×Q8—48no—
254C2×Q8—432no—
255C2×Q8—416no—
256C2×Q8—416no—
257C2×Q8—416no—
258C2×Q8—432no—
259C2×Q8—432no—
260C2×Q8—416no—
261C2×Q8—44no—
262C2×Q8—48no—
263C2×Q8—44no—
264C2×Q8—48no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—28no—
269C2×D4—28no—
270C2×D4—28no—
271C2×D4—24no—
272C2×D4—28no—
273C2×D4C4×Q818yes—
274C2×D4#12414yes—
275C2×D4C4⋊Q814yes—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—28no—
280C22×C4—24no—
281C22×C4—216no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—21no—
286C22×C4—28no—
287C22×C4—21no—
288C22×C4—216no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—44no—
296C22×C4—48no—
297C22×C4—44no—
298C22×C4—48no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—48no—
302C22×C4—416no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—48no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—48no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—44no—
320C22×C4—48no—
321C22×C4—84no—
322C22×C4—82no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C4×D414yes—
331C22×C4C4×D414yes—
332C22×C4#12011yescharacteristic
333C22×C4#12012yes—
334C22×C4#12011yescharacteristic
335C22×C4#12114yes—
336C22×C4#12114yes—
337C22×C4C22⋊Q814yes—
338C22×C4C22⋊Q8116yescomplemented
339C22×C4#12312yes—
340C22×C4#12312yes—
341C4⋊C4—216no—
342C4⋊C4—28no—
343C4⋊C4—48no—
344C4⋊C4—416no—
345C4⋊C4—88no—
346C22⋊C4—216no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—28no—
350C22⋊C4—28no—
351C22⋊C4—28no—
352C22⋊C4—28no—
353C22⋊C4—28no—
354C22⋊C4—28no—
355C22⋊C4—416no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—48no—
359C22⋊C4—44no—
360C42—216no—
361C42—24no—
362C42—24no—
363C42—48no—
364C42—84no—
365C42—84no—
366C42—84no—
367C23—24no—
368C23—22no—
369C23—24no—
370C23—22no—
371C23—24no—
372C23—22no—
373C23—24no—
374C23—22no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—44no—
381C23—48no—
382C23—44no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—48no—
387C23—42no—
388C23—42no—
389C23—81no—
390C23—84no—
391C23—81no—
392C23#39114yescomplemented
393C23C22×C22⋊C411yescharacteristic
394C23C2×C4×D412yes—
395C23C2×C4×Q812yes—
396C23C4×#4112yes—
397C23C2×#12011yescharacteristic
398C23C2×#12112yes—
399C23C2×C22⋊Q812yes—
400C23C2×C22⋊Q814yes—
401C23C2×#12311yescharacteristic
402C23#52514yes—
403C23#52512yes—
404C23C2×#12411yescharacteristic
405C23#52914yes—
406C23C2×C4⋊Q811yescharacteristic
407C23#53212yes—
408C23#53314yes—
409Q8—416no—
410Q8—432no—
411Q8—416no—
412D4—28no—
413D4—28no—
414D4—216no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—216no—
427C2×C4—216no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—48no—
436C2×C4—416no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—48no—
443C2×C4—44no—
444C2×C4—48no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—416no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—84no—
452C2×C4—88no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—816no—
456C2×C4—88no—
457C2×C4—84no—
458C2×C4—816no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4#38612yes—
462C2×C4#38614yes—
463C2×C4#38612yes—
464C2×C4#39114yescomplemented
465C2×C4B(4,𝔽2)14yes—
466C22—28no—
467C22—28no—
468C22—24no—
469C22—24no—
470C22—21no—
471C22—22no—
472C22—22no—
473C22—21no—
474C22—21no—
475C22—22no—
476C22—21no—
477C22—22no—
478C22—24no—
479C22—42no—
480C22—42no—
481C22—41no—
482C22—44no—
483C22—41no—
484C22—42no—
485C22—44no—
486C22—44no—
487C22—48no—
488C22—42no—
489C22—42no—
490C22—82no—
491C22—84no—
492C22C2×#38611yescharacteristic
493C22C2×#39111yescharacteristic
494C22#229711yescharacteristic
495C22C4×C22⋊Q814yes—
496C22#284812yes—
497C22#285012yes—
498C22#285512yes—
499C22#285712yes—
500C22C2×B(4,𝔽2)12yes—
501C22#302812yes—
502C4—24no—
503C4—22no—
504C4—24no—
505C4—416no—
506C4—48no—
507C4—42no—
508C4—42no—
509C4—84no—
510C4—164no—
511C4#188914yes—
512C2—21no—
513C2—22no—
514C2—21no—
515C2—22no—
516C2—22no—
517C2—24no—
518C2—44no—
519C2#1470211yescharacteristic
520C2C2×#188911yescharacteristic
521C2#2096211yescharacteristic
522C2#3297514yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13922C2semidirect
5. #10918C2semidirect
6. #10846C2semidirect
14. #2409C4semidirect
43. C23×Q8C2×C4semidirect
52. #518Q8semidirect
83. #385C2×C4semidirect
87. #380C2×C4semidirect
338. C22×C4C22⋊Q8semidirect
392. C23#391semidirect
464. C2×C4#391semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C4×#5182
C2×#17992
#139222
#109182
#108462
#107452