Finite Group Databasebeta

← #410417

Subgroups — #410417

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C211yesmaximal, characteristic
2C2×C4×#107C211yesmaximal, characteristic, complemented
3#13994C212yesmaximal
4#11328C211yesmaximal, characteristic, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11228C212yesmaximal, complemented
7#11200C212yesmaximal
8#11096C211yesmaximal, characteristic
9#11096C211yesmaximal, characteristic
10#10820C211yesmaximal, characteristic, complemented
11#10806C212yesmaximal, complemented
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#388—22no—
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1923—24no—
25C4×#107—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1442—22no—
37#1440—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40#1434—22no—
41C24×C4C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D411yescharacteristic
54C22×C22⋊C4D411yescharacteristic
55C22×C42—22no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—24no—
61C2×#107—24no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—42no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—42no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8D414yes—
72#389—24no—
73#389—42no—
74#388—48no—
75#380—24no—
76#380—24no—
77#380—24no—
78#380—42no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—24no—
81C4×C22⋊C4—22no—
82C4×C22⋊C4—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D411yescharacteristic
92C2×#95D411yescharacteristic
93#340—24no—
94#340—22no—
95#340—22no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—42no—
100#340—42no—
101#336—22no—
102#336—22no—
103C25C2×D411yescharacteristic
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8#4112yes—
143C22×C8#4112yes—
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—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—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4#4114yes—
169C2×C4⋊C4#4114yes—
170C2×C4⋊C4#4114yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—82no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—44no—
191C2×C42—44no—
192C2×C42—44no—
193C2×C42C2×D412yes—
194C2×C42#4112yes—
195#107—28no—
196#107—28no—
197C4⋊C8—44no—
198C22⋊C8—44no—
199C22⋊C8—44no—
200C4×C8—28no—
201#95—28no—
202#95—22no—
203#95—22no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—82no—
211C24—21no—
212C24—22no—
213C24—22no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24#13512yes—
222C24C22×D411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
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#12112yes—
293C22×C4#12112yes—
294C22×C4#13512yes—
295C22×C4AGL(1,ℤ/8)14yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4112yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4112yes—
303C22×C4H(5,𝔽2)12yes—
304C22×C42−1+411yescharacteristic
305C22×C42−1+411yescharacteristic
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8#12314yes—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—88no—
343C22⋊C4—28no—
344C22⋊C4—416no—
345C22⋊C4—44no—
346C22⋊C4—88no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—41no—
352C42—41no—
353C42—42no—
354C42—88no—
355C23—22no—
356C23—22no—
357C23—24no—
358C23—24no—
359C23—22no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23C2×#12111yescharacteristic
379C23#52512yes—
380C23#54612yes—
381C23#54711yescharacteristic
382C23#54711yescharacteristic
383C23C2×#13511yescharacteristic
384C23C2×#13511yescharacteristic
385C23C2×AGL(1,ℤ/8)12yes—
386C23#57611yescharacteristic
387C23#57811yescharacteristic
388C23#57812yes—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
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#46914yes—
450C2×C4C2×#12311yescharacteristic
451C2×C4#52511yescharacteristic
452C2×C4#52511yescharacteristic
453C2×C4#52912yes—
454C2×C4#52912yes—
455C2×C4#52912yes—
456C2×C4#53211yescharacteristic
457C2×C4#53511yescharacteristic
458C2×C4#53611yescharacteristic
459C2×C4#54212yes—
460C2×C4#54212yes—
461C2×C4#55212yes—
462C2×C4#55211yescharacteristic
463C2×C4#55211yescharacteristic
464C2×C4#55312yes—
465C2×C4#55312yes—
466C2×C4#55512yes—
467C2×C4#55612yes—
468C8—24no—
469C8—42no—
470C8—81no—
471C8—81no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—44no—
476C22—42no—
477C22—81no—
478C22—81no—
479C22—82no—
480C22—82no—
481C22—81no—
482C22—81no—
483C22#239211yescharacteristic
484C22#246812yes—
485C22#247412yes—
486C22#248611yescharacteristic
487C22#253511yescharacteristic
488C22#305011yescharacteristic
489C22C2×#46911yescharacteristic
490C22#305911yescharacteristic
491C22#306111yescharacteristic
492C22#310512yes—
493C22#310612yes—
494C22#312412yes—
495C22#312712yes—
496C22#330012yes—
497C22#330012yes—
498C22#332512yes—
499C22#333012yes—
500C22#333611yescharacteristic
501C22#333911yescharacteristic
502C22#334111yescharacteristic
503C22#334211yescharacteristic
504C22#334312yes—
505C22#334312yes—
506C4—24no—
507C4—42no—
508C4—42no—
509C4—84no—
510C4—84no—
511C4—81no—
512C4—81no—
513C4—84no—
514C4—162no—
515C4#250011yescharacteristic
516C4#252512yes—
517C4#252912yes—
518C4#260411yescharacteristic
519C4#263211yescharacteristic
520C4#285311yescharacteristic
521C2—82no—
522C2#1507811yescharacteristic
523C2#2369711yescharacteristic
524C2#2370511yescharacteristic
525C2#2709512yes—
526C2#2710312yes—
527C2#2733112yes—
528C2#2733912yes—
529C2#2766011yescharacteristic
530C2#2766711yescharacteristic
531C2#2814511yescharacteristic
532C2#2814911yescharacteristic
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
2. C2×C4×#107C2semidirect
4. #11328C2semidirect
5. #11256C2semidirect
6. #11228C2semidirect
10. #10820C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×C4×#1072
#139942
#113282
#112562
#112282
#112002
#110962
#108202
#108062