Finite Group Databasebeta

← #409799

Subgroups — #409799

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1808C211yesmaximal, characteristic
2C2×#1752C211yesmaximal, characteristic
3#15132C212yesmaximal
4#11377C211yesmaximal, characteristic, complemented
5#11377C211yesmaximal, characteristic, complemented
6#11328C211yesmaximal, characteristic, complemented
7#11256C211yesmaximal, characteristic, complemented
8#11199C212yesmaximal
9#10881C212yesmaximal, complemented
10#10821C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C2×#397—22no—
19C2×#389C2212yes—
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1808—24no—
24#1752—24no—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1443—22no—
36#1442—22no—
37#1440—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C23×C8—24no—
43C23×C8—22no—
44C23×C8—22no—
45C23×C8—42no—
46C23×C8D414yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—42no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97—24no—
73C2×#97—42no—
74#397—48no—
75#389—24no—
76#389—42no—
77#380—24no—
78#380—24no—
79#380—24no—
80#380—42no—
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×#95—22no—
92C2×#95D411yescharacteristic
93C2×#95D411yescharacteristic
94#340—24no—
95#340—22no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—22no—
100#340—24no—
101#340—42no—
102#336—22no—
103#336—22no—
104C25C2×D411yescharacteristic
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4#4112yes—
124C23×C4C2411yescharacteristic
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
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—28no—
139C22×C8—28no—
140C22×C8—44no—
141C22×C8—48no—
142C22×C8—48no—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C22×C8#4112yes—
146C22×C8#4112yes—
147C22×C8#4114yes—
148C22×C8#4112yes—
149C22×C8#4112yes—
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—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4#4114yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—82no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195#107—44no—
196#107—44no—
197C22⋊C8—28no—
198#97—44no—
199#97—44no—
200#95—22no—
201#95—22no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—82no—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24#12112yes—
221C24AGL(1,ℤ/8)14yes—
222C24C22×D411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
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—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#12112yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4C2×#4111yescharacteristic
302C22×C4H(5,𝔽2)12yes—
303C22×C42−1+412yes—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—416no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8#12314yes—
324C2×C8#12314yes—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
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—88no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—88no—
345C22⋊C4—88no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—42no—
351C42—42no—
352C23—24no—
353C23—24no—
354C23—22no—
355C23—24no—
356C23—24no—
357C23—22no—
358C23—24no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—24no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23#46914yes—
375C23C2×#12111yescharacteristic
376C23#53211yescharacteristic
377C23#53211yescharacteristic
378C23#54612yes—
379C23#54712yes—
380C23C2×AGL(1,ℤ/8)12yes—
381C23#57512yes—
382C23#57611yescharacteristic
383C23#57811yescharacteristic
384C23#57811yescharacteristic
385C23#57811yescharacteristic
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
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—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×C4C2×#12311yescharacteristic
446C2×C4C2×#12311yescharacteristic
447C2×C4#52511yescharacteristic
448C2×C4#52911yescharacteristic
449C2×C4#52912yes—
450C2×C4#52912yes—
451C2×C4#52911yescharacteristic
452C2×C4#53211yescharacteristic
453C2×C4#53411yescharacteristic
454C2×C4#53611yescharacteristic
455C2×C4#53912yes—
456C2×C4#54212yes—
457C2×C4#55112yes—
458C2×C4#55112yes—
459C2×C4#55212yes—
460C2×C4#55212yes—
461C2×C4#55312yes—
462C2×C4#55612yes—
463C8—24no—
464C8—24no—
465C8—42no—
466C8—42no—
467C22—24no—
468C22—24no—
469C22—24no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—42no—
478C22#248611yescharacteristic
479C22#252112yes—
480C22#254311yescharacteristic
481C22#255312yes—
482C22#285411yescharacteristic
483C22C2×#46911yescharacteristic
484C22#305411yescharacteristic
485C22#305911yescharacteristic
486C22#306111yescharacteristic
487C22#317511yescharacteristic
488C22#317511yescharacteristic
489C22#317711yescharacteristic
490C22#317711yescharacteristic
491C22#318012yes—
492C22#318012yes—
493C22#330812yes—
494C22#331312yes—
495C22#331512yes—
496C22#331812yes—
497C22#334312yes—
498C22#334312yes—
499C22#334712yes—
500C22#334912yes—
501C4—42no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—84no—
507C4—162no—
508C4—162no—
509C4—162no—
510C4#239511yescharacteristic
511C4#248311yescharacteristic
512C4#249911yescharacteristic
513C4#253012yes—
514C4#257712yes—
515C4#263811yescharacteristic
516C2—44no—
517C2#1506111yescharacteristic
518C2#2766411yescharacteristic
519C2#2766711yescharacteristic
520C2#2807912yes—
521C2#2808112yes—
522C2#2856111yescharacteristic
523C2#2856411yescharacteristic
524C2#2884912yes—
525C2#2885612yes—
526C2#3313511yescharacteristic
527C2#3313911yescharacteristic
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
4. #11377C2semidirect
5. #11377C2semidirect
6. #11328C2semidirect
7. #11256C2semidirect
9. #10881C2semidirect
10. #10821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18082
C2×#17522
#151322
#113772
#113282
#112562
#111992
#108812
#108212
#107212