Finite Group Databasebeta

← #470641

Subgroups — #470641

Order 512.

1254 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2094C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic, complemented
3C2×#1968C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic, complemented
5#15199C211yesmaximal, characteristic, complemented
6#15136C211yesmaximal, characteristic
7#11323C212yesmaximal, complemented
8#11122C212yesmaximal, complemented
9#11049C212yesmaximal
10#11023C212yesmaximal
11#10797C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2212yes—
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#397—22no—
21C2×#391C2211yescharacteristic
22C2×#391C2212yes—
23C2×#390C2211yescharacteristic
24C2×#390C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2212yes—
28#2094—24no—
29#2072—24no—
30#1968—24no—
31#1834—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1444—22no—
39#1444—22no—
40#1439—22no—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C22×SD16—22no—
46C22×SD16—22no—
47C22×SD16—22no—
48C22×SD16—24no—
49C22×SD16—42no—
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—24no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#106—24no—
68C2×#106—22no—
69C2×#106—42no—
70C2×#106—42no—
71C2×#106—42no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—22no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#102—24no—
78C2×#102—22no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102—42no—
82C2×C4×C8—24no—
83C2×C4×C8D414yes—
84#397—48no—
85#391—22no—
86#391—22no—
87#391—42no—
88#391—42no—
89#390—24no—
90#390—22no—
91#390—22no—
92#390—42no—
93#384—22no—
94#384—22no—
95#384—22no—
96#384—22no—
97#384—42no—
98#384—42no—
99C2×#95—24no—
100C2×#95—24no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95D412yes—
104C2×#95D412yes—
105#340—24no—
106#340—24no—
107#340—24no—
108#336—24no—
109#336—42no—
110#336D414yes—
111C25—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8#4112yes—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4#4112yes—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C2×D414yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C2×SD16—28no—
142C2×SD16—48no—
143C2×SD16—48no—
144C2×SD16—44no—
145C2×SD16—44no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8C2×D414yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8#4114yes—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—84no—
193C2×C4⋊C4—84no—
194C2×C22⋊C4—48no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—48no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—84no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211#106—44no—
212#106—48no—
213#103—48no—
214#103—44no—
215#102—44no—
216#102—48no—
217C4×C8—28no—
218#95—24no—
219#95—48no—
220#95—42no—
221#95—42no—
222#95—48no—
223#95SD1618yescomplemented
224C24—22no—
225C24—41no—
226C24—42no—
227C24—42no—
228C24—41no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24C22×D411yescharacteristic
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—22no—
237C2×Q8—24no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4#12712yes—
304C22×C4#12712yes—
305C22×C4C2×SD1618yes—
306C22×C4C2×SD1614yes—
307C22×C4#13714yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)12yes—
314C22×C4H(5,𝔽2)12yes—
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C42−1+412yes—
317C22×C42−1+411yescharacteristic
318SD16—416no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—24no—
326C2×C8—22no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8#12314yes—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C4⋊C4—816no—
354C4⋊C4—84no—
355C22⋊C4—816no—
356C22⋊C4—84no—
357C22⋊C4—84no—
358C42—22no—
359C42—22no—
360C42—24no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—42no—
369C23—84no—
370C23—84no—
371C23—82no—
372C23—82no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—84no—
379C23—84no—
380C23—82no—
381C23—81no—
382C23C2×#12711yescharacteristic
383C23#53512yes—
384C23#54611yescharacteristic
385C23#54712yes—
386C23#54811yescharacteristic
387C23C22×SD1612yes—
388C23C2×#13712yes—
389C23#57612yes—
390C23#57712yes—
391Q8—42no—
392Q8—44no—
393Q8—42no—
394Q8—81no—
395Q8—81no—
396Q8—82no—
397D4—42no—
398D4—44no—
399D4—42no—
400D4—81no—
401D4—81no—
402D4—82no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—84no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—82no—
439C2×C4#49214yes—
440C2×C4#49714yes—
441C2×C4C2×#12311yescharacteristic
442C2×C4#52511yescharacteristic
443C2×C4#53511yescharacteristic
444C2×C4#53511yescharacteristic
445C2×C4#53512yes—
446C2×C4#53611yescharacteristic
447C2×C4#53711yescharacteristic
448C2×C4#53912yes—
449C2×C4#53912yes—
450C2×C4#54012yes—
451C2×C4#54212yes—
452C2×C4#54611yescharacteristic
453C2×C4#54712yes—
454C2×C4#54811yescharacteristic
455C2×C4#55112yes—
456C2×C4#55112yes—
457C2×C4#56212yes—
458C2×C4#56312yes—
459C8—24no—
460C8—42no—
461C8—44no—
462C22—82no—
463C22—82no—
464C22—88no—
465C22—82no—
466C22—161no—
467C22—161no—
468C22—161no—
469C22—161no—
470C22—164no—
471C22#250111yescharacteristic
472C22#254011yescharacteristic
473C22#265711yescharacteristic
474C22#270612yes—
475C22#272312yes—
476C22C2×#49211yescharacteristic
477C22C2×#49711yescharacteristic
478C22#315411yescharacteristic
479C22#315511yescharacteristic
480C22#320614yes—
481C22#322414yes—
482C22#329512yes—
483C22#333112yes—
484C22#333514yes—
485C22#334614yes—
486C22#336212yes—
487C22#336912yes—
488C4—24no—
489C4—24no—
490C4—48no—
491C4—82no—
492C4—161no—
493C4—162no—
494C4—161no—
495C4—162no—
496C4#248211yescharacteristic
497C4#262411yescharacteristic
498C4#263311yescharacteristic
499C4#270612yes—
500C4#272312yes—
501C4#282011yescharacteristic
502C2—162no—
503C2#1669211yescharacteristic
504C2#2792411yescharacteristic
505C2#2793811yescharacteristic
506C2#2840411yescharacteristic
507C2#2841911yescharacteristic
508C2#3026311yescharacteristic
509C2#3027411yescharacteristic
510C2#3193414yes—
511C2#3243014yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2072C2semidirect
4. C2×#1834C2semidirect
5. #15199C2semidirect
7. #11323C2semidirect
8. #11122C2semidirect
11. #10797C2semidirect
223. #95SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#20942
C2×#20722
C2×#19682
C2×#18342
#151992
#151362
#113232
#111222
#110492
#110232
#107972