Finite Group Databasebeta

← #461688

Subgroups — #461688

Order 512.

1156 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2087C211yesmaximal, characteristic, complemented
2C2×#1850C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C2×#1807C211yesmaximal, characteristic
5#15132C212yesmaximal
6#11377C211yesmaximal, characteristic, complemented
7#11256C211yesmaximal, characteristic, complemented
8#11232C211yesmaximal, characteristic, complemented
9#11231C212yesmaximal, complemented
10#11220C212yesmaximal, complemented
11#11199C212yesmaximal
12C24×C8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#397—22no—
20C2×#389C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#2087—24no—
25#1850—24no—
26#1808—24no—
27#1807—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1443—22no—
38#1442—22no—
39#1440—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45C23×C8—22no—
46C23×C8—24no—
47C23×C8—42no—
48C23×C8D414yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4D411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—42no—
69C2×#106—24no—
70C2×#106—24no—
71C2×#106—42no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—24no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—42no—
79#397—48no—
80#389—24no—
81#389—42no—
82#380—24no—
83#380—24no—
84#380—24no—
85#380—42no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D411yescharacteristic
98C2×#95D411yescharacteristic
99#340—22no—
100#340—24no—
101#340—24no—
102#340—42no—
103#340—42no—
104#340—42no—
105#340D414yes—
106#336—22no—
107#336—22no—
108C25C2×D411yescharacteristic
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4#4112yes—
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—28no—
135C22×C8—28no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—44no—
147C22×C8—48no—
148C22×C8—48no—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8#4112yes—
153C22×C8#4114yes—
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—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
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—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×D414yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—82no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—21no—
202#107—44no—
203#107—44no—
204#106—48no—
205C4⋊C8—44no—
206C4⋊C8—44no—
207C22⋊C8—44no—
208#95—22no—
209#95—22no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—82no—
219C24—22no—
220C24—21no—
221C24—22no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—22no—
226C24—21no—
227C24—21no—
228C24#12112yes—
229C24AGL(1,ℤ/8)14yes—
230C24C22×D411yescharacteristic
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
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—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#12112yes—
302C22×C4#13512yes—
303C22×C4#13512yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4C2×#4111yescharacteristic
309C22×C4H(5,𝔽2)12yes—
310C22×C4H(5,𝔽2)11yescharacteristic
311C22×C42−1+412yes—
312C22×C42−1+412yes—
313C22×C42−1+411yescharacteristic
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—416no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8#12114yes—
334C2×C8#12314yes—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
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—42no—
351C4⋊C4—88no—
352C22⋊C4—28no—
353C22⋊C4—44no—
354C22⋊C4—88no—
355C22⋊C4—88no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—42no—
362C42—41no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—22no—
367C23—22no—
368C23—24no—
369C23—24no—
370C23—22no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23#46914yes—
386C23C2×#12111yescharacteristic
387C23#53712yes—
388C23D4211yescharacteristic
389C23#54711yescharacteristic
390C23#54712yes—
391C23C2×#13511yescharacteristic
392C23C2×#13511yescharacteristic
393C23C2×AGL(1,ℤ/8)12yes—
394C23#57712yes—
395C23#57811yescharacteristic
396C23#57811yescharacteristic
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
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×C4C2×#12111yescharacteristic
459C2×C4C2×#12311yescharacteristic
460C2×C4#53411yescharacteristic
461C2×C4#53611yescharacteristic
462C2×C4#53612yes—
463C2×C4#53611yescharacteristic
464C2×C4#53711yescharacteristic
465C2×C4#53912yes—
466C2×C4#54112yes—
467C2×C4#54112yes—
468C2×C4#54212yes—
469C2×C4#54612yes—
470C2×C4#54811yescharacteristic
471C2×C4#55112yes—
472C2×C4#55212yes—
473C2×C4#55311yescharacteristic
474C2×C4#56212yes—
475C2×C4#56312yes—
476C8—24no—
477C8—24no—
478C8—81no—
479C8—81no—
480C22—24no—
481C22—24no—
482C22—24no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—44no—
487C22—44no—
488C22—42no—
489C22—44no—
490C22—44no—
491C22#241611yescharacteristic
492C22#248611yescharacteristic
493C22#261811yescharacteristic
494C22#267212yes—
495C22#272212yes—
496C22#305011yescharacteristic
497C22C2×#46911yescharacteristic
498C22#305911yescharacteristic
499C22#306011yescharacteristic
500C22#324712yes—
501C22#324712yes—
502C22#325812yes—
503C22#325812yes—
504C22#326512yes—
505C22#328311yescharacteristic
506C22#328511yescharacteristic
507C22#333611yescharacteristic
508C22#333712yes—
509C22#334012yes—
510C22#334211yescharacteristic
511C22#334312yes—
512C22#334312yes—
513C22#334512yes—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—81no—
518C4—84no—
519C4—81no—
520C4—162no—
521C4—162no—
522C4—162no—
523C4#241211yescharacteristic
524C4#262111yescharacteristic
525C4#262511yescharacteristic
526C4#269012yes—
527C4#274712yes—
528C4#281411yescharacteristic
529C2—44no—
530C2#1639911yescharacteristic
531C2#2477911yescharacteristic
532C2#2478211yescharacteristic
533C2#2765911yescharacteristic
534C2#2766711yescharacteristic
535C2#2977211yescharacteristic
536C2#2977711yescharacteristic
537C2#3066712yes—
538C2#3067612yes—
539C2#3241212yes—
540C2#3241412yes—
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×#2087C2semidirect
6. #11377C2semidirect
7. #11256C2semidirect
8. #11232C2semidirect
9. #11231C2semidirect
10. #11220C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20872
C2×#18502
C2×#18082
C2×#18072
#151322
#113772
#112562
#112322
#112312
#112202
#111992