Finite Group Databasebeta

← #416805

Subgroups — #416805

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1835C211yesmaximal, characteristic, complemented
2C2×#1750C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#14340C211yesmaximal, characteristic
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11200C212yesmaximal
8#11199C212yesmaximal
9#10843C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#1835—24no—
24#1750—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1441—22no—
35#1441—22no—
36#1439—22no—
37#1439—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C22⋊C8—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×#97—22no—
69C2×#97—22no—
70C2×#97—22no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8D414yes—
74#384—22no—
75#384—22no—
76#384—22no—
77#384—22no—
78#382—22no—
79#382—22no—
80#382—22no—
81#382—22no—
82#380—24no—
83#380—24no—
84#380—42no—
85#380—42no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
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—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4#4114yes—
124C23×C4C2411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
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×C8C2×D414yes—
138C22×C8#4114yes—
139#117—44no—
140#117—44no—
141#117—44no—
142#117—44no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
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—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
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×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C2×D412yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200C2×C42#4112yes—
201C2×C42#4112yes—
202C22⋊C8—216no—
203#97—44no—
204C4×C8—28no—
205#95—42no—
206#95—42no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—82no—
211#95—82no—
212#95—82no—
213#95—82no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24#12412yes—
225C24C22×D411yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
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#12114yes—
294C22×C4#12314yes—
295C22×C4#12314yes—
296C22×C4#12412yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4H(5,𝔽2)12yes—
306C22×C42−1+412yes—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C4⋊C4—48no—
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—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—88no—
339C4⋊C4—88no—
340C22⋊C4—48no—
341C22⋊C4—88no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42#12314yes—
349C42#12414yes—
350C42#12614yes—
351C42#12614yes—
352C23—22no—
353C23—24no—
354C23—24no—
355C23—22no—
356C23—22no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23C2×#12111yescharacteristic
376C23C2×#12311yescharacteristic
377C23C2×#12311yescharacteristic
378C23C2×#12411yescharacteristic
379C23#53211yescharacteristic
380C23#53511yescharacteristic
381C23#53611yescharacteristic
382C23#57612yes—
383C23#57712yes—
384C23#57712yes—
385C23#57812yes—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4#39718yes—
445C2×C4C2×#12311yescharacteristic
446C2×C4#52512yes—
447C2×C4C2×#12411yescharacteristic
448C2×C4C2×#12611yescharacteristic
449C2×C4C2×#12611yescharacteristic
450C2×C4#52912yes—
451C2×C4#53712yes—
452C2×C4#54212yes—
453C2×C4#55212yes—
454C2×C4#55212yes—
455C2×C4#55212yes—
456C2×C4#55312yes—
457C2×C4#55312yes—
458C2×C4#55312yes—
459C2×C4#55512yes—
460C2×C4#55612yes—
461C8—44no—
462C8—44no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—82no—
469C22—81no—
470C22—81no—
471C22—82no—
472C22—81no—
473C22—81no—
474C22C2×#39711yescharacteristic
475C22#249911yescharacteristic
476C22#250011yescharacteristic
477C22#250311yescharacteristic
478C22#250411yescharacteristic
479C22#262011yescharacteristic
480C22#285011yescharacteristic
481C22#306012yes—
482C22#306112yes—
483C22#309612yes—
484C22#309612yes—
485C22#309712yes—
486C22#309712yes—
487C22#313612yes—
488C22#313712yes—
489C22#317812yes—
490C22#317912yes—
491C22#322011yescharacteristic
492C22#322111yescharacteristic
493C22#322211yescharacteristic
494C22#322311yescharacteristic
495C22#324312yes—
496C22#324412yes—
497C4—28no—
498C4—42no—
499C4—82no—
500C4—162no—
501C4—162no—
502C4—161no—
503C4—161no—
504C4—161no—
505C4—161no—
506C4#248512yes—
507C4#249412yes—
508C4#249512yes—
509C4#256212yes—
510C2—82no—
511C2#1523311yescharacteristic
512C2#2364112yes—
513C2#2788712yes—
514C2#2790211yescharacteristic
515C2#2790311yescharacteristic
516C2#2797112yes—
517C2#2798511yescharacteristic
518C2#2798611yescharacteristic
519C2#2984911yescharacteristic
520C2#2985011yescharacteristic
521C2#3304712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ 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. #10843C2semidirect
10. #10843C2semidirect
11. #10841C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18352
C2×#17502
#143402
#113162
#113152
#112002
#111992
#108432
#108412