Finite Group Databasebeta

← #374603

Subgroups — #374603

Order 512.

2524 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#13922C212yesmaximal, complemented
5#11222C214yesmaximal, complemented
6#10854C212yesmaximal, 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—
17#2349—24no—
18D4×C4⋊C4—24no—
19#2316—21no—
20#2315—21no—
21#2315C2212yes—
22C2×#391—21no—
23C2×#391—24no—
24C2×#386C2212yes—
25C2×#385C2212yes—
26#2285—21no—
27C2×#380—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C4×#99—24no—
32C2×#342C2212yes—
33C2×#342C2218yes—
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—28no—
51#518—44no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—24no—
55C2×C4×D4—28no—
56C2×C4×D4C2×C414yes—
57C2×C4×D4D414yes—
58C2×#117—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D414yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2312yes—
69C22×C22⋊C4C2314yes—
70C22×C22⋊C4C2311yescharacteristic
71C22×C42—22no—
72C22×C42D411yescharacteristic
73C2×#99—24no—
74C2×#99—24no—
75C2×#99Q818yescomplemented
76#395—28no—
77#393—28no—
78#391—28no—
79#391—416no—
80#389—28no—
81#386—24no—
82#386C2×C418yes—
83#385—28no—
84#385—28no—
85#385—24no—
86#385C2×C418yescomplemented
87#384—22no—
88#382—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380C2×C414yescomplemented
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C22⋊C4—24no—
102C4×C22⋊C4—24no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—216no—
108C2×#95—22no—
109C2×#95D414yes—
110#342—24no—
111#342—48no—
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—22no—
137C22×D4—22no—
138C22×D4C22×C412yes—
139C22×D4C2×D412yes—
140C22×D4C2×Q814yes—
141C22×D4#4114yes—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—48no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4C22⋊C414yes—
156C23×C4C22×C414yes—
157C23×C4C22×C412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D418yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4#4114yes—
168C22⋊Q8—432no—
169C4×D4—28no—
170C4×D4—28no—
171C4×D4—28no—
172C4×D4—28no—
173C4×D4—432no—
174#117—44no—
175#117—44no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—84no—
189C2×C4⋊C4C22⋊C414yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—48no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4C22⋊C414yes—
212C2×C22⋊C4C22×C414yes—
213C2×C22⋊C4C22×C412yes—
214C2×C22⋊C4C2×D412yes—
215C2×C22⋊C4C2×Q818yes—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—48no—
221C2×C42—44no—
222C2×C42C22⋊C414yes—
223#99—216no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—28no—
228#95—28no—
229#95—24no—
230#95—24no—
231#95—432no—
232#95—44no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24C2×C22⋊C412yes—
242C24#12412yes—
243C24C4⋊Q812yes—
244C24C23×C411yescharacteristic
245C24C22×D411yescharacteristic
246C24C22×D412yes—
247C24C22×D411yescharacteristic
248C24C22×Q812yes—
249C24C2×#4112yes—
250C24C2×#4112yes—
251C24H(5,𝔽2)12yes—
252C242−1+412yes—
253C2×Q8—216no—
254C2×Q8—232no—
255C2×Q8—416no—
256C2×Q8—48no—
257C2×Q8—432no—
258C2×Q8—48no—
259C2×Q8—416no—
260C2×Q8—416no—
261C2×Q8—416no—
262C2×Q8—432no—
263C2×Q8—432no—
264C2×Q8—416no—
265C2×Q8—44no—
266C2×Q8—48no—
267C2×Q8—44no—
268C2×Q8—48no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—28no—
274C2×D4—28no—
275C2×D4—28no—
276C2×D4—28no—
277C2×D4C4×Q818yes—
278C2×D4C22⋊Q818yes—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—28no—
284C22×C4—24no—
285C22×C4—216no—
286C22×C4—28no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—216no—
299C22×C4—44no—
300C22×C4—48no—
301C22×C4—44no—
302C22×C4—48no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—48no—
306C22×C4—416no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—48no—
317C22×C4—44no—
318C22×C4—48no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—84no—
328C22×C4—82no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C4×D414yes—
337C22×C4C4×D414yes—
338C22×C4#12011yescharacteristic
339C22×C4#12012yes—
340C22×C4#12011yescharacteristic
341C22×C4#12114yes—
342C22×C4#12114yes—
343C22×C4#12312yes—
344C22×C4#12312yes—
345C22×C4#12412yes—
346C22×C4C4⋊Q812yes—
347C4⋊C4—28no—
348C4⋊C4—48no—
349C4⋊C4—416no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—28no—
358C22⋊C4—28no—
359C22⋊C4—28no—
360C22⋊C4—28no—
361C22⋊C4—416no—
362C22⋊C4—48no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—48no—
368C22⋊C4—44no—
369C42—24no—
370C42—24no—
371C42—48no—
372C42—88no—
373C23—24no—
374C23—22no—
375C23—24no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—44no—
389C23—44no—
390C23—42no—
391C23—44no—
392C23—44no—
393C23—42no—
394C23—42no—
395C23—44no—
396C23—44no—
397C23—42no—
398C23—42no—
399C23—81no—
400C23—82no—
401C23—81no—
402C23—82no—
403C23#39114yescomplemented
404C23C22×C22⋊C411yescharacteristic
405C23C2×C4×D412yes—
406C23C2×C4×Q812yes—
407C23#52012yes—
408C23C2×#12011yescharacteristic
409C23C2×#12112yes—
410C23C2×C22⋊Q812yes—
411C23C2×#12311yescharacteristic
412C23C2×#12411yescharacteristic
413C23C2×C4⋊Q811yescharacteristic
414C23#53511yescharacteristic
415C23#53611yescharacteristic
416C23#53712yes—
417C23#54614yes—
418C23D4×Q814yes—
419C23#55214yes—
420C23#55414yes—
421Q8—416no—
422Q8—432no—
423Q8—416no—
424D4—216no—
425D4—216no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—216no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—48no—
448C2×C4—416no—
449C2×C4—48no—
450C2×C4—42no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—416no—
460C2×C4—44no—
461C2×C4—48no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—48no—
465C2×C4—84no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—84no—
469C2×C4—88no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4—816no—
474C2×C4—168no—
475C2×C4#38612yes—
476C2×C4#38614yes—
477C2×C4#38612yes—
478C2×C4#39114yescomplemented
479C2×C4B(4,𝔽2)14yes—
480C22—28no—
481C22—22no—
482C22—22no—
483C22—21no—
484C22—22no—
485C22—22no—
486C22—21no—
487C22—21no—
488C22—22no—
489C22—21no—
490C22—22no—
491C22—28no—
492C22—22no—
493C22—22no—
494C22—24no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22C2×#38611yescharacteristic
510C22C2×#39111yescharacteristic
511C22#231111yescharacteristic
512C22#236014yes—
513C22#241714yes—
514C22#243411yescharacteristic
515C22#243511yescharacteristic
516C22#250211yescharacteristic
517C22#250311yescharacteristic
518C22C2×B(4,𝔽2)12yes—
519C22#302812yes—
520C4—24no—
521C4—22no—
522C4—24no—
523C4—48no—
524C4—42no—
525C4—42no—
526C4—84no—
527C4—88no—
528C4—88no—
529C4#188914yes—
530C2—21no—
531C2—24no—
532C2—21no—
533C2—21no—
534C2—21no—
535C2—24no—
536C2—44no—
537C2#1413911yescharacteristic
538C2C2×#188911yescharacteristic
539C2#2096211yescharacteristic
540C2#2585212yes—
541C2#2589912yes—
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. #11222C2semidirect
6. #10854C2semidirect
14. #2409C4semidirect
43. C23×Q8C2×C4semidirect
75. C2×#99Q8semidirect
86. #385C2×C4semidirect
98. #380C2×C4semidirect
403. C23#391semidirect
478. 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
#341812
C2×C4×#992
#139222
#112222
#108542
#107452