Finite Group Databasebeta

← #393320

Subgroups — #393320

Order 512.

2036 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic
3C2×#1799C211yesmaximal, characteristic
4#13924C212yesmaximal, complemented
5#10918C214yesmaximal, complemented
6#10846C212yesmaximal, complemented
7#10745C214yesmaximal
8C2×#518C2212yes—
9C22×C4×D4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×#99C2212yes—
12#2487—24no—
13#2487C418yescomplemented
14C2×#400—21no—
15C2×#399—21no—
16C2×#389—21no—
17C2×#386C2212yes—
18C2×#385C2212yes—
19C2×#380—21no—
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22#2275—24no—
23#2274—21no—
24#2273—21no—
25#2273C2212yes—
26C42×D4—24no—
27C4×#117—21no—
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4C2212yes—
30#1799—24no—
31C2×#342C2218yes—
32C2×#342C2212yes—
33#1445—24no—
34C4⋊#95—24no—
35C22⋊#95C2218yes—
36C23×D4C2311yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4C2312yes—
39C24×C4C2311yescharacteristic
40#518—24no—
41#518—28no—
42#518Q818yes—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—28no—
46C2×C4×D4—24no—
47C2×C4×D4C2×C414yes—
48C2×C4×D4D414yes—
49C2×#117—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2×C414yescomplemented
60C22×C4⋊C4D411yescharacteristic
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4C2314yes—
65C22×C22⋊C4C2312yes—
66C22×C22⋊C4C2311yescharacteristic
67C22×C22⋊C4C2314yes—
68C22×C42—22no—
69C22×C42D411yescharacteristic
70C22×C42D414yes—
71C2×#99—24no—
72C2×#99—28no—
73#400—28no—
74#399—28no—
75#389—28no—
76#386—24no—
77#386C2×C418yes—
78#385—24no—
79#385C2×C418yescomplemented
80#380—28no—
81#380—22no—
82#380—24no—
83#380—24no—
84#380C2×C414yescomplemented
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—28no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—28no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—28no—
98#376—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—216no—
105C2×#95D414yes—
106C43—22no—
107#342—24no—
108#342—216no—
109C25C2×D411yescharacteristic
110C25C2411yescharacteristic
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—24no—
114C22×D4C22×C412yes—
115C22×D4C2×D412yes—
116C22×D4C2×Q814yes—
117C22×D4#4114yes—
118C23×C4—24no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4—84no—
131C23×C4C22⋊C414yes—
132C23×C4C22×C414yes—
133C23×C4C22×C412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D418yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4#4114yes—
144C4×D4—232no—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—28no—
148C4×D4—28no—
149C4×D4—232no—
150#117—28no—
151#117—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—84no—
172C2×C4⋊C4C22⋊C414yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—84no—
192C2×C22⋊C4C22⋊C414yes—
193C2×C22⋊C4C22×C414yes—
194C2×C22⋊C4C22×C412yes—
195C2×C22⋊C4C2×D412yes—
196C2×C22⋊C4C2×Q818yes—
197C2×C22⋊C4#4118yes—
198C2×C42—24no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—48no—
207C2×C42—84no—
208C2×C42C22⋊C414yes—
209#99—48no—
210#95—24no—
211#95—24no—
212#95—28no—
213#95—24no—
214#95—28no—
215#95—24no—
216#95—24no—
217#95—44no—
218#95—816no—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24C2×C22⋊C412yes—
227C24C22⋊Q814yes—
228C24C23×C411yescharacteristic
229C24C22×D411yescharacteristic
230C24C22×D412yes—
231C24C22×D411yescharacteristic
232C24C22×Q812yes—
233C24C2×#4112yes—
234C24C2×#4112yes—
235C24C2×#4114yes—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—28no—
240C2×D4—28no—
241C2×D4—28no—
242C2×D4—24no—
243C2×D4—28no—
244C2×D4C4×Q818yes—
245C2×D4#12414yes—
246C2×D4C4⋊Q814yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—28no—
254C22×C4—21no—
255C22×C4—216no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—28no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—48no—
270C22×C4—44no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—416no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4—84no—
289C22×C4—82no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4—88no—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C4×D414yes—
301C22×C4C4×D414yes—
302C22×C4#12011yescharacteristic
303C22×C4#12012yes—
304C22×C4#12011yescharacteristic
305C22×C4#12114yes—
306C22×C4#12114yes—
307C22×C4C22⋊Q814yes—
308C22×C4C22⋊Q8116yes—
309C22×C4#12312yes—
310C22×C4#12312yes—
311C4⋊C4—216no—
312C4⋊C4—28no—
313C4⋊C4—48no—
314C4⋊C4—44no—
315C4⋊C4—48no—
316C4⋊C4—44no—
317C4⋊C4—88no—
318C4⋊C4—88no—
319C22⋊C4—216no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—28no—
323C22⋊C4—28no—
324C22⋊C4—28no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—28no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—48no—
331C22⋊C4—44no—
332C22⋊C4—88no—
333C42—216no—
334C42—24no—
335C42—24no—
336C42—48no—
337C42—84no—
338C42—84no—
339C42—84no—
340C23—24no—
341C23—22no—
342C23—24no—
343C23—22no—
344C23—24no—
345C23—22no—
346C23—24no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—44no—
355C23—48no—
356C23—44no—
357C23—42no—
358C23—42no—
359C23—48no—
360C23—42no—
361C23—42no—
362C23—81no—
363C23—84no—
364C23—81no—
365C23#39114yes—
366C23C22×C22⋊C411yescharacteristic
367C23C2×C4×D412yes—
368C23C2×C4×Q812yes—
369C23C4×#4112yes—
370C23C2×#12011yescharacteristic
371C23C2×#12112yes—
372C23C2×C22⋊Q812yes—
373C23C2×C22⋊Q814yes—
374C23C2×#12311yescharacteristic
375C23#52514yes—
376C23#52512yes—
377C23C2×#12411yescharacteristic
378C23#52914yes—
379C23C2×C4⋊Q811yescharacteristic
380C23#53212yes—
381C23#53314yes—
382D4—28no—
383D4—28no—
384D4—216no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—28no—
393C2×C4—24no—
394C2×C4—28no—
395C2×C4—216no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—416no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—416no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—84no—
418C2×C4—88no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—816no—
422C2×C4—88no—
423C2×C4—88no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—816no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4#38612yes—
431C2×C4#38614yes—
432C2×C4#38612yes—
433C2×C4#39114yes—
434C2×C4#45814yes—
435C22—28no—
436C22—28no—
437C22—24no—
438C22—24no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—22no—
443C22—21no—
444C22—22no—
445C22—22no—
446C22—21no—
447C22—24no—
448C22—42no—
449C22—42no—
450C22—41no—
451C22—44no—
452C22—41no—
453C22—42no—
454C22—44no—
455C22—44no—
456C22—48no—
457C22—42no—
458C22—42no—
459C22—82no—
460C22—84no—
461C22C2×#38611yescharacteristic
462C22C2×#39111yescharacteristic
463C22#229711yescharacteristic
464C22C4×C22⋊Q814yes—
465C22#284812yes—
466C22#285012yes—
467C22#285512yes—
468C22#285712yes—
469C22C2×#45812yes—
470C22#302712yes—
471C4—24no—
472C4—22no—
473C4—24no—
474C4—416no—
475C4—42no—
476C4—42no—
477C4—84no—
478C4—84no—
479C4—164no—
480C4#189014yes—
481C2—21no—
482C2—22no—
483C2—21no—
484C2—22no—
485C2—22no—
486C2—24no—
487C2—44no—
488C2#1470211yescharacteristic
489C2C2×#189011yescharacteristic
490C2#2096111yescharacteristic
491C2#3297614yes—
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. #13924C2semidirect
5. #10918C2semidirect
6. #10846C2semidirect
13. #2487C4semidirect
59. C22×C4⋊C4C2×C4semidirect
79. #385C2×C4semidirect
84. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C4×#5182
C2×#17992
#139242
#109182
#108462
#107452