Finite Group Databasebeta

← #458589

Subgroups — #458589

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2088C212yesmaximal, complemented
2C2×#1853C212yesmaximal
3#15135C211yesmaximal, characteristic
4#14130C211yesmaximal, characteristic
5#11327C212yesmaximal, complemented
6#11323C212yesmaximal, complemented
7#11316C212yesmaximal, complemented
8#11199C212yesmaximal
9#11042C211yesmaximal, characteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×#106C2212yes—
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C2×#399—21no—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#385—21no—
19C2×#384C2211yescharacteristic
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23#2088—28no—
24#1853—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1445—21no—
34#1444—21no—
35#1444—21no—
36#1442—21no—
37#1439—21no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C22⋊#95—21no—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—41no—
52C22×C4⋊C4—41no—
53C22×C4⋊C4—42no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4D411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—24no—
67C2×#106—42no—
68C2×#106—42no—
69C2×#106—42no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—41no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75#399—44no—
76#389—24no—
77#389—24no—
78#389—42no—
79#389—42no—
80#385—44no—
81#384—41no—
82#384—41no—
83#380—22no—
84#380—24no—
85#380—42no—
86#380Q814yes—
87C4×C22⋊C4—41no—
88C4×C22⋊C4—41no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D411yescharacteristic
98C2×#95D411yescharacteristic
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#340D418yes—
105#336—24no—
106#336—41no—
107C25C2×D411yescharacteristic
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—44no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—82no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×Q812yes—
131C23×C4#4112yes—
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8C2×D414yes—
146C22×C8#4114yes—
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—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2×Q814yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4SD1614yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207#106—48no—
208#106—48no—
209C4⋊C8—44no—
210C22⋊C8—216no—
211#95—24no—
212#95—41no—
213#95—41no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—42no—
218#95—42no—
219#95—44no—
220#95—82no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—22no—
229C24C22⋊Q812yes—
230C24C2×SD1612yes—
231C24AGL(1,ℤ/8)12yes—
232C24C22×D411yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—48no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
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—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—88no—
302C22×C4—88no—
303C22×C4—84no—
304C22×C4—84no—
305C22×C4C22⋊Q812yes—
306C22×C4C2×SD1612yes—
307C22×C4C2×SD1612yes—
308C22×C4#13712yes—
309C22×C4C22×D412yes—
310C22×C4C22×Q811yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4112yes—
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C4H(5,𝔽2)12yes—
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C42−1+411yescharacteristic
317C22×C42−1+412yes—
318C22×C42−1+411yescharacteristic
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—48no—
340C4⋊C4—42no—
341C4⋊C4—42no—
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—88no—
351C4⋊C4—84no—
352C4⋊C4—88no—
353C4⋊C4—84no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—48no—
357C22⋊C4—88no—
358C42—42no—
359C42—42no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—24no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23#47614yes—
387C23C2×C22⋊Q811yescharacteristic
388C23#53411yescharacteristic
389C23#53611yescharacteristic
390C23D4212yes—
391C23#54712yes—
392C23C22×SD1611yescharacteristic
393C23C2×AGL(1,ℤ/8)11yescharacteristic
394C23C2×#13711yescharacteristic
395C23#57511yescharacteristic
396C23#57712yes—
397C23#57712yes—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—816no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4#52512yes—
459C2×C4#53512yes—
460C2×C4#53612yes—
461C2×C4#53912yes—
462C2×C4#53911yescharacteristic
463C2×C4#53911yescharacteristic
464C2×C4#53912yes—
465C2×C4#53912yes—
466C2×C4#54411yescharacteristic
467C2×C4#54411yescharacteristic
468C2×C4D4211yescharacteristic
469C2×C4#54712yes—
470C2×C4D4×Q812yes—
471C2×C4#55412yes—
472C2×C4#55911yescharacteristic
473C2×C4#56011yescharacteristic
474C2×C4#56311yescharacteristic
475C2×C4#56312yes—
476C8—44no—
477C8—44no—
478C22—24no—
479C22—28no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—44no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22#244912yes—
490C22#262711yescharacteristic
491C22#266911yescharacteristic
492C22#268012yes—
493C22#272411yescharacteristic
494C22C2×#47611yescharacteristic
495C22#307811yescharacteristic
496C22#308412yes—
497C22#318711yescharacteristic
498C22#319311yescharacteristic
499C22#320011yescharacteristic
500C22#320011yescharacteristic
501C22#322911yescharacteristic
502C22#323411yescharacteristic
503C22#324412yes—
504C22#326512yes—
505C22#326712yes—
506C22#327812yes—
507C22D4×SD1612yes—
508C22#333512yes—
509C22#334412yes—
510C22#334512yes—
511C22#335112yes—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—84no—
517C4—81no—
518C4—84no—
519C4—84no—
520C4—162no—
521C4#245511yescharacteristic
522C4#263312yes—
523C4#267711yescharacteristic
524C4#272012yes—
525C4#280711yescharacteristic
526C4#283411yescharacteristic
527C2—44no—
528C2#1632111yescharacteristic
529C2#2633812yes—
530C2#2636012yes—
531C2#2996811yescharacteristic
532C2#2997711yescharacteristic
533C2#3050011yescharacteristic
534C2#3051611yescharacteristic
535C2#3108912yes—
536C2#3109412yes—
537C2#3245911yescharacteristic
538C2#3246811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2088C2semidirect
5. #11327C2semidirect
6. #11323C2semidirect
7. #11316C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20882
C2×#18532
#151352
#141302
#113272
#113232
#113162
#111992
#110422