Finite Group Databasebeta

← #416803

Subgroups — #416803

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1835C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊C8C211yesmaximal, characteristic
3#14682C211yesmaximal, characteristic
4#14682C211yesmaximal, characteristic
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11200C212yesmaximal
8#11199C212yesmaximal
9#10842C211yesmaximal, characteristic, complemented
10#10842C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×C4×C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2212yes—
23#1835—24no—
24C4×C22⋊C8—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37#1434—22no—
38#1434—22no—
39C22⋊#95—22no—
40C22⋊#95—22no—
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4C2311yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4—21no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—21no—
54C22×C42—22no—
55C22×C42—22no—
56C22×C42—22no—
57C22×C42—22no—
58C22×C42—22no—
59C22×C42—22no—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—24no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8D414yes—
72#382—22no—
73#382—22no—
74#382—22no—
75#382—22no—
76#382—22no—
77#382—22no—
78#382—22no—
79#382—22no—
80#380—24no—
81#380—24no—
82#380—42no—
83#380—42no—
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×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—42no—
100#340—42no—
101#340—42no—
102#340—42no—
103#336—24no—
104C25C2×D411yescharacteristic
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4#4114yes—
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8C2×D414yes—
145C22×C8#4114yes—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
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—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
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×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—44no—
191C2×C42—44no—
192C2×C42—44no—
193C2×C42—44no—
194C2×C42C2×D412yes—
195C2×C42#4112yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200C22⋊C8—216no—
201C4×C8—28no—
202C4×C8—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—44no—
213#95—44no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24#12412yes—
224C24#13514yes—
225C24C22×D411yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—216no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—48no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—48no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
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#12114yes—
300C22×C4#12314yes—
301C22×C4#12314yes—
302C22×C4#12412yes—
303C22×C4#13514yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4C2×#4112yes—
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4H(5,𝔽2)12yes—
313C22×C42−1+412yes—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—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—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C22⋊C4—84no—
345C22⋊C4—88no—
346C22⋊C4—84no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—48no—
352C42#12314yes—
353C42#12414yes—
354C42#12614yes—
355C42#12614yes—
356C23—24no—
357C23—24no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—22no—
366C23—22no—
367C23—24no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23#48814yes—
379C23C2×#12111yescharacteristic
380C23C2×#12311yescharacteristic
381C23C2×#12311yescharacteristic
382C23C2×#12411yescharacteristic
383C23#53211yescharacteristic
384C23#53511yescharacteristic
385C23#53611yescharacteristic
386C23C2×#13512yes—
387C23C2×#13512yes—
388C23#57512yes—
389C23#57512yes—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—432no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
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—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4#39718yes—
450C2×C4C2×#12311yescharacteristic
451C2×C4#52512yes—
452C2×C4C2×#12411yescharacteristic
453C2×C4C2×#12611yescharacteristic
454C2×C4C2×#12611yescharacteristic
455C2×C4#52912yes—
456C2×C4#53712yes—
457C2×C4#54212yes—
458C2×C4#55212yes—
459C2×C4#55212yes—
460C2×C4#55212yes—
461C2×C4#55312yes—
462C2×C4#55312yes—
463C2×C4#55312yes—
464C2×C4#55512yes—
465C2×C4#55612yes—
466C8—44no—
467C8—44no—
468C22—24no—
469C22—24no—
470C22—24no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—42no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22C2×#39711yescharacteristic
480C22#249911yescharacteristic
481C22#250011yescharacteristic
482C22#250311yescharacteristic
483C22#250411yescharacteristic
484C22#262011yescharacteristic
485C22#285011yescharacteristic
486C22#305012yes—
487C22#305412yes—
488C22#309112yes—
489C22#309112yes—
490C22#309112yes—
491C22#309112yes—
492C22C2×#48811yescharacteristic
493C22#313111yescharacteristic
494C22#314011yescharacteristic
495C22#314211yescharacteristic
496C22#316812yes—
497C22#318312yes—
498C22#321112yes—
499C22#321212yes—
500C22#323212yes—
501C22#323312yes—
502C4—28no—
503C4—44no—
504C4—42no—
505C4—161no—
506C4—161no—
507C4—161no—
508C4—161no—
509C4—162no—
510C4—162no—
511C4#248512yes—
512C4#249412yes—
513C4#249512yes—
514C4#256212yes—
515C2—44no—
516C2#1523311yescharacteristic
517C2#2363812yes—
518C2#2787912yes—
519C2#2789312yes—
520C2#2796311yescharacteristic
521C2#2796411yescharacteristic
522C2#2797611yescharacteristic
523C2#2797711yescharacteristic
524C2#2984612yes—
525C2#3304411yescharacteristic
526C2#3305011yescharacteristic
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
1. C2×#1835C2semidirect
5. #11316C2semidirect
6. #11315C2semidirect
9. #10842C2semidirect
10. #10842C2semidirect
11. #10842C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18352
C2×C4×C22⋊C82
#146822
#113162
#113152
#112002
#111992
#108422