Finite Group Databasebeta

← #475522

Subgroups — #475522

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C212yesmaximal
2C2×#2077C212yesmaximal
3#14686C211yesmaximal, characteristic
4#14682C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11373C211yesmaximal, characteristic, complemented
7#11372C212yesmaximal, complemented
8#11325C212yesmaximal, complemented
9#11324C211yesmaximal, characteristic, complemented
10#11323C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic
12C23×C42C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#380—22no—
21C2×#380C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2211yescharacteristic
24#2077—28no—
25#2076—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C4⋊#95—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39#1434—22no—
40C22⋊#95—22no—
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—22no—
56C22×C42—24no—
57C22×C42—22no—
58C22×C42—22no—
59C22×C42—42no—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#107—24no—
64C2×#107—42no—
65C2×#107—42no—
66C2×#106—24no—
67C2×#106—42no—
68C2×#106—42no—
69C2×C22⋊C8—42no—
70C2×C22⋊C8—42no—
71C2×C22⋊C8D418yes—
72#384—22no—
73#384—22no—
74#384—42no—
75#382—24no—
76#382—42no—
77#380—24no—
78#380—24no—
79#380—24no—
80#380—24no—
81#380—48no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#336—24no—
97#336—22no—
98#336—22no—
99C25C2×D411yescharacteristic
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D414yes—
125C23×C4C2×D412yes—
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8C2×D414yes—
140C22×C8C2×D414yes—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
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—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—48no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—44no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—48no—
189C2×C42—48no—
190C2×C42—42no—
191C2×C42#4114yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194#107—48no—
195#106—48no—
196C22⋊C8—48no—
197#95—48no—
198#95—44no—
199#95—44no—
200#95—42no—
201#95—42no—
202#95—44no—
203#95—44no—
204#95—44no—
205C24—21no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24#12712yes—
215C24#13714yes—
216C24C22×D411yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—216no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—48no—
241C22×C4—48no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—48no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4#12314yes—
287C22×C4#12314yes—
288C22×C4#12712yes—
289C22×C4#13514yes—
290C22×C4AGL(1,ℤ/8)14yes—
291C22×C4C22×D412yes—
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4112yes—
295C22×C4H(5,𝔽2)11yescharacteristic
296C22×C4H(5,𝔽2)12yes—
297C22×C42−1+411yescharacteristic
298C22×C42−1+412yes—
299C22×C42−1+411yescharacteristic
300C22×C42−1+411yescharacteristic
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—88no—
332C4⋊C4—88no—
333C22⋊C4—88no—
334C22⋊C4—84no—
335C22⋊C4—84no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—48no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—22no—
352C23—22no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—42no—
363C23#49714yes—
364C23C2×#12311yescharacteristic
365C23C2×#12311yescharacteristic
366C23C2×#12711yescharacteristic
367C23#53411yescharacteristic
368C23#53611yescharacteristic
369C23#53611yescharacteristic
370C23#53711yescharacteristic
371C23C2×#13512yes—
372C23C2×AGL(1,ℤ/8)12yes—
373C23C2×#13712yes—
374C23#57512yes—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—432no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—84no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4—84no—
420C2×C4—82no—
421C2×C4—84no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4#48218yes—
431C2×C4#53412yes—
432C2×C4#53712yes—
433C2×C4#54111yescharacteristic
434C2×C4#54111yescharacteristic
435C2×C4#54112yes—
436C2×C4#54211yescharacteristic
437C2×C4#54211yescharacteristic
438C2×C4#54212yes—
439C2×C4#54712yes—
440C2×C4#54712yes—
441C2×C4#54712yes—
442C2×C4#54712yes—
443C2×C4#55112yes—
444C2×C4#55112yes—
445C2×C4#56312yes—
446C2×C4#56312yes—
447C8—44no—
448C8—44no—
449C22—24no—
450C22—24no—
451C22—24no—
452C22—42no—
453C22—42no—
454C22—44no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22#248311yescharacteristic
461C22#250111yescharacteristic
462C22#250111yescharacteristic
463C22#262511yescharacteristic
464C22#262511yescharacteristic
465C22#265611yescharacteristic
466C22#281711yescharacteristic
467C22C2×#48212yes—
468C22#309112yes—
469C22#309212yes—
470C22#309512yes—
471C22#314711yescharacteristic
472C22C2×#49711yescharacteristic
473C22#315111yescharacteristic
474C22#315311yescharacteristic
475C22#319012yes—
476C22#319412yes—
477C22#322812yes—
478C22#323312yes—
479C22#323612yes—
480C22#323912yes—
481C22#324812yes—
482C22#325112yes—
483C4—28no—
484C4—42no—
485C4—44no—
486C4—162no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4—162no—
491C4—161no—
492C4#271712yes—
493C4#271712yes—
494C4#272212yes—
495C4#272212yes—
496C2—44no—
497C2#1705911yescharacteristic
498C2#2760212yes—
499C2#2790711yescharacteristic
500C2#2791111yescharacteristic
501C2#2791311yescharacteristic
502C2#2791911yescharacteristic
503C2#2994112yes—
504C2#2994312yes—
505C2#3024411yescharacteristic
506C2#3024511yescharacteristic
507C2#3278912yes—
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
5. #11373C2semidirect
6. #11373C2semidirect
7. #11372C2semidirect
8. #11325C2semidirect
9. #11324C2semidirect
10. #11323C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×#20772
#146862
#146822
#113732
#113722
#113252
#113242
#113232
#107122