Finite Group Databasebeta

← #374465

Subgroups — #374465

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic
3C2×#1991C212yesmaximal, complemented
4C2×C4×C22⋊C8C211yesmaximal, characteristic
5#11200C212yesmaximal
6#11199C212yesmaximal
7#10885C212yesmaximal, complemented
8#10884C212yesmaximal, complemented
9#10842C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13C22×C4×C8C2212yes—
14#2851—24no—
15#2851C418yescomplemented
16C2×#389C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#387—21no—
19C2×#387—21no—
20C2×#385—21no—
21C2×#385—21no—
22#2283—28no—
23C2×#383—21no—
24C2×#382C2211yescharacteristic
25C2×#382C2212yes—
26C2×#381—21no—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C22⋊C4—21no—
32C2×#376—21no—
33#1991—28no—
34C4×C22⋊C8—24no—
35C2×#340C2214yes—
36C2×#340C2214yes—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#336C2212yes—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2314yes—
44C2×#125—22no—
45C2×#125—24no—
46C2×#125—24no—
47C2×#125C2×C414yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2314yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C42—24no—
61C22×C42—24no—
62C22×C42—22no—
63C22×C42C2×C414yes—
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D414yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C22⋊C8—24no—
70C2×C22⋊C8—28no—
71C2×C4×C8—24no—
72C2×C4×C8—24no—
73C2×C4×C8—24no—
74#389—22no—
75#389—22no—
76#389C2×C414yes—
77#389C2×C414yes—
78#387—28no—
79#387—28no—
80#385—28no—
81#385—28no—
82#383—28no—
83#382—22no—
84#382—24no—
85#382C2×C414yes—
86#382D418yes—
87#381—28no—
88#380—22no—
89#380—22no—
90#380—28no—
91#380C2×C414yes—
92#380C2×C414yes—
93C4×C4⋊C4—28no—
94C4×C22⋊C4—28no—
95#376—28no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106#340—28no—
107#340—28no—
108#340—28no—
109#340—28no—
110#336—24no—
111C25C2×D411yescharacteristic
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—22no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—24no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C22⋊C414yes—
130C23×C4C22⋊C414yes—
131C23×C4C22×C412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—24no—
143C22×C8—24no—
144C22×C8—24no—
145C22×C8#4118yes—
146#125—216no—
147#125—48no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C2×D418yes—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—216no—
192C2×C42—216no—
193C2×C42—28no—
194C2×C42—44no—
195C2×C42—44no—
196C2×C42C22⋊C414yes—
197C2×C42C22⋊C414yes—
198C2×C42C22×C412yes—
199C2×C42C22×C412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D414yes—
202C2×C42C2×D412yes—
203C2×C42#4114yes—
204C22⋊C8—216no—
205C4×C8—48no—
206#95—28no—
207#95—28no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—28no—
215#95—28no—
216C24—21no—
217C24—21no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24C2×C22⋊C412yes—
224C24#13514yes—
225C24C22×D411yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—28no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—28no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4#12012yes—
292C22×C4#12012yes—
293C22×C4#12312yes—
294C22×C4#12312yes—
295C22×C4#13514yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D412yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4112yes—
300C22×C4C2×#4114yes—
301C22×C42−1+414yes—
302C2×C8—28no—
303C2×C8—28no—
304C2×C8—28no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—28no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—416no—
327C22⋊C4—48no—
328C42—24no—
329C42—24no—
330C42—28no—
331C42—28no—
332C42—416no—
333C42C4×D418yes—
334C42#12118yes—
335C23—22no—
336C23—24no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—28no—
341C23—28no—
342C23—24no—
343C23—22no—
344C23—22no—
345C23—42no—
346C23—44no—
347C23—42no—
348C23—44no—
349C23—42no—
350C23—42no—
351C23#41614yes—
352C23C22×C22⋊C411yescharacteristic
353C23C2×#12011yescharacteristic
354C23C2×#12311yescharacteristic
355C23#53212yes—
356C23#53612yes—
357C23C2×#13512yes—
358C23C2×#13512yes—
359C23#57512yes—
360C23#57512yes—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—416no—
380C2×C4—416no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#38614yes—
413C2×C4#38614yes—
414C2×C4#41614yes—
415C2×C4C2×C4×D412yes—
416C2×C4C4×#4112yes—
417C2×C4#51912yes—
418C2×C4C2×#12112yes—
419C2×C4#52512yes—
420C2×C4#53612yes—
421C2×C4#54714yes—
422C2×C4#54814yes—
423C2×C4#55214yes—
424C2×C4#55614yes—
425C8—48no—
426C22—24no—
427C22—24no—
428C22—24no—
429C22—42no—
430C22—48no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—48no—
435C22C2×#38611yescharacteristic
436C22#229711yescharacteristic
437C22#231011yescharacteristic
438C22#243212yes—
439C22#249912yes—
440C22C2×#41611yescharacteristic
441C22C2×#41611yescharacteristic
442C22#289611yescharacteristic
443C22#289611yescharacteristic
444C22#300212yes—
445C22#300212yes—
446C22#309112yes—
447C22#309112yes—
448C22#316814yes—
449C22#318314yes—
450C22#323214yes—
451C22#323314yes—
452C4—24no—
453C4—24no—
454C4—24no—
455C4—42no—
456C4—42no—
457C4—88no—
458C4—82no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4#236214yes—
463C4#241514yes—
464C2—44no—
465C2#1413811yescharacteristic
466C2#2090811yescharacteristic
467C2#2090811yescharacteristic
468C2#2140211yescharacteristic
469C2#2141111yescharacteristic
470C2#2183611yescharacteristic
471C2#2183711yescharacteristic
472C2#2567514yes—
473C2#2787914yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#1991C2semidirect
7. #10885C2semidirect
8. #10884C2semidirect
9. #10842C2semidirect
15. #2851C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#22832
C2×#19912
C2×C4×C22⋊C82
#112002
#111992
#108852
#108842
#108422