Finite Group Databasebeta

← #374604

Subgroups — #374604

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#13924C212yesmaximal, complemented
5#11222C214yesmaximal, complemented
6#10854C212yesmaximal, complemented
7#10745C214yesmaximal
8C2×#518C2212yes—
9C22×C4×D4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×#99C2212yes—
12#2487—24no—
13#2487C418yescomplemented
14C2×#400—21no—
15C2×#399—21no—
16#2349—24no—
17D4×C4⋊C4—24no—
18#2316—21no—
19#2315—21no—
20#2315C2212yes—
21C2×#389—21no—
22C2×#386C2212yes—
23C2×#385C2212yes—
24#2285—21no—
25C2×#380—21no—
26C2×#380—21no—
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C4×#99—24no—
31C2×#342C2212yes—
32C2×#342C2218yes—
33#1445—24no—
34C4⋊#95—24no—
35C22⋊#95C2218yes—
36C23×D4C2311yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4C2312yes—
39C24×C4C2311yescharacteristic
40#518—28no—
41#518—44no—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4—24no—
45C2×C4×D4—28no—
46C2×C4×D4C2×C414yes—
47C2×C4×D4D414yes—
48C2×#117—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C414yescomplemented
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D414yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4C2314yes—
65C22×C22⋊C4C2312yes—
66C22×C22⋊C4C2314yes—
67C22×C22⋊C4C2311yescharacteristic
68C22×C42—22no—
69C22×C42D411yescharacteristic
70C2×#99—24no—
71C2×#99—24no—
72C2×#99Q818yes—
73#400—28no—
74#399—28no—
75#389—28no—
76#389—28no—
77#386—24no—
78#386C2×C418yes—
79#385—28no—
80#385—28no—
81#385—24no—
82#385C2×C418yescomplemented
83#384—22no—
84#382—22no—
85#380—22no—
86#380—22no—
87#380—28no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380C2×C414yescomplemented
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—24no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—216no—
106C2×#95D414yes—
107#342—24no—
108#342—48no—
109C25C2×D411yescharacteristic
110C25C2411yescharacteristic
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4C22×C412yes—
116C22×D4C2×D412yes—
117C22×D4C2×Q814yes—
118C22×D4#4114yes—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4—84no—
132C23×C4C22⋊C414yes—
133C23×C4C22×C414yes—
134C23×C4C22×C412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D418yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4#4114yes—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—28no—
148C4×D4—28no—
149C4×D4—432no—
150#117—44no—
151#117—44no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—84no—
175C2×C4⋊C4—84no—
176C2×C4⋊C4C22⋊C414yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—84no—
198C2×C22⋊C4C22⋊C414yes—
199C2×C22⋊C4C22×C414yes—
200C2×C22⋊C4C22×C412yes—
201C2×C22⋊C4C2×D412yes—
202C2×C22⋊C4C2×Q818yes—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—48no—
208C2×C42—44no—
209C2×C42C22⋊C414yes—
210#99—216no—
211#95—24no—
212#95—24no—
213#95—28no—
214#95—24no—
215#95—28no—
216#95—24no—
217#95—24no—
218#95—44no—
219#95—816no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24C2×C22⋊C412yes—
229C24#12412yes—
230C24C4⋊Q812yes—
231C24C23×C411yescharacteristic
232C24C22×D411yescharacteristic
233C24C22×D412yes—
234C24C22×D411yescharacteristic
235C24C22×Q812yes—
236C24C2×#4112yes—
237C24C2×#4112yes—
238C24H(5,𝔽2)12yes—
239C242−1+412yes—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—28no—
245C2×D4—28no—
246C2×D4—28no—
247C2×D4—28no—
248C2×D4C4×Q818yes—
249C2×D4C22⋊Q818yes—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—216no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—48no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—48no—
278C22×C4—416no—
279C22×C4—48no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—84no—
295C22×C4—82no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—88no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C4×D414yes—
307C22×C4C4×D414yes—
308C22×C4#12011yescharacteristic
309C22×C4#12012yes—
310C22×C4#12011yescharacteristic
311C22×C4#12114yes—
312C22×C4#12114yes—
313C22×C4#12312yes—
314C22×C4#12312yes—
315C22×C4#12412yes—
316C22×C4C4⋊Q812yes—
317C4⋊C4—28no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—48no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C4⋊C4—88no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—28no—
334C22⋊C4—48no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—48no—
340C22⋊C4—44no—
341C22⋊C4—88no—
342C42—24no—
343C42—24no—
344C42—48no—
345C42—88no—
346C23—24no—
347C23—22no—
348C23—24no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—42no—
367C23—42no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—81no—
373C23—82no—
374C23—81no—
375C23—82no—
376C23#39114yes—
377C23C22×C22⋊C411yescharacteristic
378C23C2×C4×D412yes—
379C23C2×C4×Q812yes—
380C23#52012yes—
381C23C2×#12011yescharacteristic
382C23C2×#12112yes—
383C23C2×C22⋊Q812yes—
384C23C2×#12311yescharacteristic
385C23C2×#12411yescharacteristic
386C23C2×C4⋊Q811yescharacteristic
387C23#53511yescharacteristic
388C23#53611yescharacteristic
389C23#53712yes—
390C23#54614yes—
391C23D4×Q814yes—
392C23#55214yes—
393C23#55414yes—
394D4—216no—
395D4—216no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—28no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—28no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—48no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—416no—
424C2×C4—416no—
425C2×C4—44no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—48no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—84no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—84no—
435C2×C4—88no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—88no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—816no—
443C2×C4—168no—
444C2×C4#38612yes—
445C2×C4#38614yes—
446C2×C4#38612yes—
447C2×C4#39114yes—
448C2×C4#45814yes—
449C22—28no—
450C22—22no—
451C22—22no—
452C22—21no—
453C22—22no—
454C22—21no—
455C22—22no—
456C22—21no—
457C22—22no—
458C22—22no—
459C22—21no—
460C22—28no—
461C22—22no—
462C22—22no—
463C22—24no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—44no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22C2×#38611yescharacteristic
479C22C2×#39111yescharacteristic
480C22#231111yescharacteristic
481C22#236014yes—
482C22#241714yes—
483C22#243411yescharacteristic
484C22#243511yescharacteristic
485C22#250211yescharacteristic
486C22#250311yescharacteristic
487C22C2×#45812yes—
488C22#302712yes—
489C4—24no—
490C4—22no—
491C4—24no—
492C4—42no—
493C4—42no—
494C4—84no—
495C4—88no—
496C4—88no—
497C4—84no—
498C4#189014yes—
499C2—21no—
500C2—24no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—24no—
505C2—44no—
506C2#1413911yescharacteristic
507C2C2×#189011yescharacteristic
508C2#2096111yescharacteristic
509C2#2585312yes—
510C2#2590012yes—
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. #11222C2semidirect
6. #10854C2semidirect
13. #2487C4semidirect
58. C22×C4⋊C4C2×C4semidirect
82. #385C2×C4semidirect
95. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#341812
C2×C4×#992
#139242
#112222
#108542
#107452