Finite Group Databasebeta

← #416742

Subgroups — #416742

Order 512.

1522 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
1#35002C211yesmaximal, characteristic
2#14130C211yesmaximal, characteristic
3#13881C212yesmaximal
4#11390C214yesmaximal
5#11222C214yesmaximal
6#10854C212yesmaximal
7C4×#342C211yesmaximal, characteristic
8C2×#543C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10#2857—21no—
11#2857—21no—
12#2659—21no—
13#2659—21no—
14#2549—24no—
15#2440—22no—
16#2440—22no—
17#2315C2212yes—
18C2×#389—21no—
19C2×#385C2212yes—
20C2×#382—22no—
21C2×#380C2211yescharacteristic
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×#342C2212yes—
26C2×#342C2212yes—
27C2×#342C2218yes—
28#1444—21no—
29#1442—21no—
30#1442—24no—
31C4⋊#95—21no—
32C4⋊#95—22no—
33C22⋊#95C2212yes—
34C22⋊#95C2218yes—
35C24×C4C2312yes—
36C24×C4C2311yescharacteristic
37#543—28no—
38#543—44no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—24no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—44no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D412yes—
54C22×C22⋊C4C2314yes—
55C22×C22⋊C4C2314yes—
56C22×C22⋊C4C2312yes—
57C22×C22⋊C4C2311yescharacteristic
58C22×C22⋊C4C2311yescharacteristic
59C22×C42—22no—
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62#398—24no—
63#398—24no—
64#395—24no—
65#395—24no—
66#391—22no—
67#391—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—44no—
73#385—24no—
74#385—24no—
75#385—28no—
76#384—21no—
77#384—21no—
78#382—48no—
79#380—22no—
80#380—22no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380—22no—
85#380—22no—
86#380Q814yescomplemented
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4Q818yescomplemented
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—28no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—28no—
97C2×#95—22no—
98C2×#95D411yescharacteristic
99C2×#95D411yescharacteristic
100C2×#95D414yes—
101#342—24no—
102#342—24no—
103#342—48no—
104C25C2411yescharacteristic
105C22×Q8—22no—
106C22×Q8—22no—
107C23×C4—22no—
108C23×C4—24no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—44no—
116C23×C4—82no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D414yes—
120C23×C4C2×D414yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×Q814yes—
128C23×C4C2×Q812yes—
129C23×C4#4112yes—
130C4⋊Q8—82no—
131C4⋊Q8—82no—
132C22⋊Q8—416no—
133C22⋊Q8—416no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137C22⋊Q8—44no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—82no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4C2×Q818yes—
184C2×C22⋊C4C2×Q814yes—
185C2×C22⋊C4#4114yes—
186C2×C22⋊C4#4114yes—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—28no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—44no—
193C2×C42—44no—
194C2×C42—42no—
195C2×C42—42no—
196#95—24no—
197#95—24no—
198#95—22no—
199#95—22no—
200#95—24no—
201#95—24no—
202#95—28no—
203#95—24no—
204#95—416no—
205#95—44no—
206#95—44no—
207#95—82no—
208#95—88no—
209C24—41no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24C22×D411yescharacteristic
214C24C22×D412yes—
215C24C22×Q812yes—
216C24C22×Q811yescharacteristic
217C24C2×#4111yescharacteristic
218C24C2×#4112yes—
219C24C2×#4112yes—
220C24H(5,𝔽2)12yes—
221C242−1+412yes—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—82no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8—82no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—28no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—28no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—28no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—48no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—48no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—88no—
289C22×C4—88no—
290C22×C4#12011yescharacteristic
291C22×C4#12012yes—
292C22×C4#12011yescharacteristic
293C22×C4C22⋊Q814yes—
294C22×C4C22⋊Q812yes—
295C22×C4C22⋊Q812yes—
296C22×C4C22⋊Q814yes—
297C22×C4#12412yes—
298C22×C4#12412yes—
299C22×C4#12412yes—
300C22×C4#12412yes—
301C22×C4C4⋊Q812yes—
302C22×C4C4⋊Q814yes—
303C22×C4C4⋊Q812yes—
304C22×C4C4⋊Q814yes—
305C4⋊C4—48no—
306C4⋊C4—44no—
307C4⋊C4—48no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—88no—
322C22⋊C4—28no—
323C22⋊C4—28no—
324C22⋊C4—28no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—48no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C42—416no—
334C42—42no—
335C42—42no—
336C42—81no—
337C42—81no—
338C23—21no—
339C23—22no—
340C23—21no—
341C23—22no—
342C23—21no—
343C23—22no—
344C23—21no—
345C23—22no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23C2×#12011yescharacteristic
361C23C2×C22⋊Q811yescharacteristic
362C23C2×C22⋊Q812yes—
363C23C2×#12411yescharacteristic
364C23C2×#12411yescharacteristic
365C23C2×C4⋊Q811yescharacteristic
366C23C2×C4⋊Q812yes—
367C23#53314yes—
368C23#53511yescharacteristic
369C23#53611yescharacteristic
370C23#54414yes—
371C23#54614yes—
372C23#54814yes—
373C23#55412yes—
374C23#55512yes—
375C23#55612yes—
376C23#55712yes—
377Q8—164no—
378Q8—164no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—28no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—28no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—416no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—84no—
430C2×C4—82no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—84no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—88no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—88no—
441C2×C4—88no—
442C2×C4#39512yes—
443C2×C4#39514yes—
444C2×C4#39512yes—
445C2×C4B(4,𝔽2)12yes—
446C2×C4#45812yes—
447C22—21no—
448C22—22no—
449C22—21no—
450C22—22no—
451C22—21no—
452C22—22no—
453C22—21no—
454C22—22no—
455C22—21no—
456C22—21no—
457C22—24no—
458C22—21no—
459C22—21no—
460C22—24no—
461C22—21no—
462C22—21no—
463C22—24no—
464C22—21no—
465C22—21no—
466C22—24no—
467C22—44no—
468C22—44no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22C2×#39511yescharacteristic
474C22#243511yescharacteristic
475C22#243611yescharacteristic
476C22#243914yes—
477C22#255014yes—
478C22#256311yescharacteristic
479C22#256811yescharacteristic
480C22#266011yescharacteristic
481C22#286211yescharacteristic
482C22C2×B(4,𝔽2)11yescharacteristic
483C22C2×#45811yescharacteristic
484C22#302711yescharacteristic
485C22#302811yescharacteristic
486C4—22no—
487C4—24no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—84no—
493C4—88no—
494C4—84no—
495C4—84no—
496C4—84no—
497C4#206214yes—
498C2—21no—
499C2—24no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—24no—
504C2#1523011yescharacteristic
505C2C2×#206211yescharacteristic
506C2#2356611yescharacteristic
507C2#2589911yescharacteristic
508C2#2590011yescharacteristic
509C2#2594711yescharacteristic
510C2#2594811yescharacteristic
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
86. #380Q8semidirect
90. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#141302
#138812
#113902
#112222
#108542
C4×#3422