Finite Group Databasebeta

← #410212

Subgroups — #410212

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#382C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5C2×#1823C211yesmaximal, characteristic
6#14697C211yesmaximal, characteristic
7#11268C214yesmaximal, complemented
8#11208C214yesmaximal, complemented
9#10712C211yesmaximal, characteristic
10C23×C4⋊C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C2×#389C2212yes—
16C2×#385—21no—
17C2×#382—22no—
18C2×#382—24no—
19C2×#382—24no—
20C2×#382C2214yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C22×#95—21no—
24C22×#95—21no—
25#1824—24no—
26#1824—24no—
27#1823—24no—
28#1823—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2214yes—
32C2×#340C2214yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2214yes—
36#1436—22no—
37C4⋊#95C2212yes—
38C22⋊#95—21no—
39C24×C4—21no—
40C24×C4—21no—
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4D414yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—28no—
71C22×C42D412yes—
72C22×C42C2312yes—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79#389—28no—
80#385—44no—
81#382—216no—
82#382—28no—
83#382—28no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380D414yes—
88C2×#95—28no—
89C2×#95—28no—
90C2×#95—21no—
91C2×#95—24no—
92C2×#95—24no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97#340—28no—
98#340—24no—
99#340—24no—
100#340—28no—
101#340—24no—
102#340—24no—
103#336—28no—
104C25C2×D411yescharacteristic
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4#4112yes—
141C23×C4#4112yes—
142C23×C4#4112yes—
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—24no—
149C22×C8—24no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—24no—
153C22×C8—24no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
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—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C42—216no—
200C2×C42—216no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42#4118yes—
208C22⋊C8—44no—
209C22⋊C8—44no—
210#95—216no—
211#95—216no—
212#95—24no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—44no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24#12012yes—
222C24#12112yes—
223C24#12412yes—
224C24#13512yes—
225C24#13512yes—
226C24AGL(1,ℤ/8)12yes—
227C24#13712yes—
228C24C22×D411yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—28no—
236C22×C4—28no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—48no—
257C22×C4—48no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—48no—
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—44no—
281C22×C4—44no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—88no—
297C22×C4—88no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4#12012yes—
303C22×C4#12114yes—
304C22×C4#12114yes—
305C22×C4#12112yes—
306C22×C4#12314yes—
307C22×C4#12314yes—
308C22×C4#12314yes—
309C22×C4#12314yes—
310C22×C4#12412yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4114yes—
318C22×C42−1+414yes—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—44no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—88no—
346C22⋊C4—48no—
347C22⋊C4—84no—
348C42—216no—
349C42—28no—
350C42—28no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—22no—
356C23—22no—
357C23—24no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23#39414yes—
368C23#44914yes—
369C23#45214yes—
370C23#46314yes—
371C23#46414yes—
372C23#48814yes—
373C23#48914yes—
374C23C2×#12011yescharacteristic
375C23C2×#12111yescharacteristic
376C23C2×#12312yes—
377C23C2×#12411yescharacteristic
378C23#53612yes—
379C23C2×#13511yescharacteristic
380C23C2×#13511yescharacteristic
381C23C2×AGL(1,ℤ/8)11yescharacteristic
382C23C2×#13711yescharacteristic
383C23#57512yes—
384C23#57512yes—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—416no—
400C2×C4—416no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—816no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—84no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—84no—
429C2×C4—816no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4#39414yes—
441C2×C4C2×#12111yescharacteristic
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4C2×#12311yescharacteristic
445C2×C4#52512yes—
446C2×C4#52512yes—
447C2×C4#53612yes—
448C2×C4#53612yes—
449C2×C4#54714yes—
450C2×C4#55214yes—
451C2×C4#55214yes—
452C2×C4#55214yes—
453C8—82no—
454C8—82no—
455C22—28no—
456C22—28no—
457C22—28no—
458C22—28no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22#204218yes—
466C22C2×#39411yescharacteristic
467C22#248612yes—
468C22#250312yes—
469C22#261411yescharacteristic
470C22#284911yescharacteristic
471C22C2×#44911yescharacteristic
472C22C2×#45211yescharacteristic
473C22#300511yescharacteristic
474C22#300511yescharacteristic
475C22C2×#46311yescharacteristic
476C22C2×#46411yescharacteristic
477C22#304012yes—
478C22#309112yes—
479C22#309112yes—
480C22#309412yes—
481C22#309512yes—
482C22C2×#48811yescharacteristic
483C22C2×#48911yescharacteristic
484C22#314012yes—
485C22#323212yes—
486C22#323212yes—
487C22#323412yes—
488C22#323612yes—
489C4—24no—
490C4—44no—
491C4—44no—
492C4—84no—
493C4—84no—
494C4—84no—
495C4—161no—
496C4—161no—
497C4#241312yes—
498C4#241312yes—
499C4#248412yes—
500C4#248412yes—
501C2—28no—
502C2#1507311yescharacteristic
503C2C2×#204211yescharacteristic
504C2#2350611yescharacteristic
505C2#2765112yes—
506C2#2765312yes—
507C2#2796412yes—
508C2#2796912yes—
509C2#2964011yescharacteristic
510C2#2964311yescharacteristic
511C2#3298011yescharacteristic
512C2#3298411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
7. #11268C2semidirect
8. #11208C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3822
C2×#18242
C2×#18232
#146972
#112682
#112082
#107122