Finite Group Databasebeta

← #411965

Subgroups — #411965

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2030C212yesmaximal
3#14685C212yesmaximal
4C4×#381C211yesmaximal, characteristic
5#11390C212yesmaximal
6#11287C212yesmaximal
7#11070C212yesmaximal
8#10854C212yesmaximal
9#10846C211yesmaximal, characteristic
10C2×#543C2212yes—
11C23×C42C2211yescharacteristic
12#2857—22no—
13#2659—22no—
14#2549—24no—
15#2440—24no—
16#2315C2212yes—
17C2×#381C2214yes—
18C2×#381C2211yescharacteristic
19C2×#381C2211yescharacteristic
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#2030—28no—
25C2×#342C2212yes—
26C2×#342C2214yes—
27C2×#342C2212yes—
28C2×#342C2214yes—
29C4⋊#95—24no—
30#1434—24no—
31#1434—22no—
32C22⋊#95C2211yescharacteristic
33C22⋊#95C2214yes—
34C22⋊#95C2211yescharacteristic
35C4×#95—22no—
36C24×C4C2311yescharacteristic
37C24×C4C2311yescharacteristic
38C24×C4C2311yescharacteristic
39#543—24no—
40#543—24no—
41#543—44no—
42C2×C4⋊Q8—22no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—28no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4D414yes—
47C22×C22⋊C4C2314yes—
48C22×C22⋊C4C2314yes—
49C22×C22⋊C4C2311yescharacteristic
50C22×C22⋊C4C2311yescharacteristic
51C22×C22⋊C4C2311yescharacteristic
52C22×C22⋊C4C2311yescharacteristic
53C22×C42—21no—
54C22×C42—22no—
55C22×C42—21no—
56C22×C42—22no—
57C22×C42—24no—
58C22×C42—22no—
59C22×C42—21no—
60C22×C42D412yes—
61C22×C42Q814yescomplemented
62#398—28no—
63#395—28no—
64#391—24no—
65#389—24no—
66#389—24no—
67#385—28no—
68#384—22no—
69#381—22no—
70#381—22no—
71#381—28no—
72#381—44no—
73#381D414yes—
74#381Q814yescomplemented
75#380—24no—
76#380—22no—
77#380—22no—
78#380—22no—
79#380—22no—
80#380—24no—
81#380—24no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—24no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—28no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—28no—
93C2×#95D412yes—
94C2×#95D414yes—
95#342—28no—
96#342—24no—
97#342—24no—
98#342—44no—
99C25C2411yescharacteristic
100C22×Q8—24no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D418yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×Q812yes—
127C23×C4C2×Q812yes—
128C23×C4C2×Q812yes—
129C23×C4C2×Q812yes—
130C23×C4#4114yes—
131C23×C4#4112yes—
132C4⋊Q8—84no—
133C22⋊Q8—416no—
134C22⋊Q8—416no—
135C22⋊Q8—48no—
136C22⋊Q8—48no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—48no—
144C2×C4⋊C4—48no—
145C2×C4⋊C4—44no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4C2×D412yes—
169C2×C22⋊C4C2×D412yes—
170C2×C22⋊C4C2×Q812yes—
171C2×C22⋊C4C2×Q812yes—
172C2×C22⋊C4#4112yes—
173C2×C22⋊C4#4112yes—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42—28no—
177C2×C42—24no—
178C2×C42—28no—
179C2×C42—22no—
180C2×C42—41no—
181C2×C42—42no—
182C2×C42—41no—
183C2×C42—42no—
184C2×C42—44no—
185C2×C42—42no—
186C2×C42—44no—
187C2×C42—44no—
188C2×C42—44no—
189C2×C42—42no—
190#95—28no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—28no—
195#95—28no—
196#95—416no—
197#95—44no—
198#95—44no—
199#95—416no—
200#95—42no—
201#95—44no—
202C24—42no—
203C24—42no—
204C24C22×D411yescharacteristic
205C24C22×D412yes—
206C24C22×D411yescharacteristic
207C24C22×Q811yescharacteristic
208C24C22×Q811yescharacteristic
209C24C2×#4111yescharacteristic
210C24C2×#4111yescharacteristic
211C24C2×#4112yes—
212C24C2×#4111yescharacteristic
213C24H(5,𝔽2)12yes—
214C242−1+412yes—
215C2×Q8—48no—
216C2×Q8—84no—
217C2×Q8—84no—
218C22×C4—24no—
219C22×C4—28no—
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—28no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—48no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4#12011yescharacteristic
288C22×C4#12012yes—
289C22×C4#12011yescharacteristic
290C22×C4#12114yes—
291C22×C4C22⋊Q812yes—
292C22×C4C22⋊Q814yes—
293C22×C4C22⋊Q814yes—
294C22×C4C22⋊Q812yes—
295C22×C4C22⋊Q814yes—
296C22×C4#12412yes—
297C22×C4#12414yes—
298C22×C4#12412yes—
299C22×C4#12414yes—
300C22×C4C4⋊Q814yescomplemented
301C4⋊C4—48no—
302C4⋊C4—416no—
303C4⋊C4—48no—
304C4⋊C4—84no—
305C4⋊C4—88no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C42—44no—
325C42—48no—
326C42—44no—
327C42—44no—
328C42—82no—
329C42—84no—
330C23—22no—
331C23—24no—
332C23—22no—
333C23—24no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—42no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—44no—
345C23—44no—
346C23C2×#12011yescharacteristic
347C23C2×#12111yescharacteristic
348C23C2×C22⋊Q811yescharacteristic
349C23C2×C22⋊Q811yescharacteristic
350C23C2×C22⋊Q812yes—
351C23C2×#12411yescharacteristic
352C23C2×#12412yes—
353C23C2×C4⋊Q811yescharacteristic
354C23#53211yescharacteristic
355C23#53311yescharacteristic
356C23#53311yescharacteristic
357C23#53511yescharacteristic
358C23#53611yescharacteristic
359C23#54212yes—
360C23#54412yes—
361C23D4212yes—
362C23#54612yes—
363C23#54712yes—
364C23#54812yes—
365C23D4×Q812yes—
366C23#55412yes—
367C23#55512yes—
368C23#55612yes—
369Q8—168no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—28no—
374C2×C4—24no—
375C2×C4—28no—
376C2×C4—24no—
377C2×C4—28no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—28no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—88no—
423C2×C4—88no—
424C2×C4—84no—
425C2×C4—88no—
426C2×C4—84no—
427C2×C4—88no—
428C2×C4—84no—
429C2×C4#39312yes—
430C2×C4#39314yes—
431C2×C4#39312yes—
432C2×C4#45814yes—
433C22—22no—
434C22—22no—
435C22—24no—
436C22—24no—
437C22—21no—
438C22—22no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—22no—
443C22—21no—
444C22—22no—
445C22—22no—
446C22—24no—
447C22—22no—
448C22—24no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22C2×#39311yescharacteristic
460C22#240811yescharacteristic
461C22#240911yescharacteristic
462C22#243912yes—
463C22#244012yes—
464C22#249112yes—
465C22#249212yes—
466C22#254711yescharacteristic
467C22#256311yescharacteristic
468C22#262811yescharacteristic
469C22#285711yescharacteristic
470C22C2×#45812yes—
471C22#302512yes—
472C4—22no—
473C4—24no—
474C4—44no—
475C4—44no—
476C4—44no—
477C4—44no—
478C4—82no—
479C4—82no—
480C4—88no—
481C4—164no—
482C4#203014yescomplemented
483C2—21no—
484C2—21no—
485C2—22no—
486C2—21no—
487C2—21no—
488C2—22no—
489C2—22no—
490C2—22no—
491C2#1512511yescharacteristic
492C2C2×#203011yescharacteristic
493C2#2346711yescharacteristic
494C2#2426112yes—
495C2#2435112yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ 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
61. C22×C42Q8semidirect
74. #381Q8semidirect
300. C22×C4C4⋊Q8semidirect
482. C4#2030semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#20302
#146852
C4×#3812
#113902
#112872
#110702
#108542
#108462