Finite Group Databasebeta

← #475591

Subgroups — #475591

Order 512.

1168 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2088C212yesmaximal, complemented
2C2×#2077C212yesmaximal
3C2×#1923C211yesmaximal, characteristic
4C2×#1844C211yesmaximal, characteristic, complemented
5#15135C211yesmaximal, characteristic
6#15135C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11357C211yesmaximal, characteristic, complemented
9#11327C212yesmaximal, complemented
10#11095C212yesmaximal
11#10820C211yesmaximal, characteristic, complemented
12C22×#106C2212yes—
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#399—21no—
19C2×#399—21no—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2088—28no—
28#2077—28no—
29#1923—24no—
30#1844—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1445—21no—
40#1445—21no—
41#1444—21no—
42#1444—21no—
43#1439—21no—
44#1439—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#106—24no—
69C2×#106—24no—
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—42no—
73C2×#106—42no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—24no—
80C2×C4×C8—24no—
81C2×C4×C8D414yes—
82#399—44no—
83#399—44no—
84#389—24no—
85#389—24no—
86#389—42no—
87#389—42no—
88#384—22no—
89#384—22no—
90#384—22no—
91#384—22no—
92#380—24no—
93#380—24no—
94#380—42no—
95#380—42no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340D418yes—
114#336—22no—
115#336—22no—
116C25C2×D411yescharacteristic
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8C2×D412yes—
154C22×C8C2×D414yes—
155C22×C8C2×D412yes—
156C22×C8#4114yes—
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—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×D414yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4SD1614yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42C2×D412yes—
216C2×C42#4112yes—
217#106—48no—
218#106—48no—
219C4⋊C8—44no—
220C22⋊C8—44no—
221C22⋊C8—44no—
222C4×C8—28no—
223#95—22no—
224#95—22no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—42no—
230#95—42no—
231#95—42no—
232#95—42no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—22no—
237C24—22no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—22no—
243C24C2×SD1612yes—
244C24C22×D411yescharacteristic
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
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—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
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—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4#12112yes—
310C22×C4#12112yes—
311C22×C4C2×SD1612yes—
312C22×C4C2×SD1612yes—
313C22×C4#13512yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4#13712yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D412yes—
318C22×C4C2×#4112yes—
319C22×C4C2×#4111yescharacteristic
320C22×C4H(5,𝔽2)12yes—
321C22×C42−1+411yescharacteristic
322C22×C42−1+411yescharacteristic
323C22×C42−1+412yes—
324C22×C42−1+411yescharacteristic
325C22×C42−1+411yescharacteristic
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8#12314yes—
345C2×C8#12714yes—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—88no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—88no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C22⋊C4—28no—
369C22⋊C4—44no—
370C22⋊C4—88no—
371C22⋊C4—88no—
372C42—22no—
373C42—22no—
374C42—24no—
375C42—41no—
376C42—41no—
377C42—42no—
378C23—24no—
379C23—22no—
380C23—22no—
381C23—24no—
382C23—22no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—41no—
390C23—41no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23C2×#12111yescharacteristic
402C23#53611yescharacteristic
403C23#53611yescharacteristic
404C23D4212yes—
405C23#54712yes—
406C23C22×SD1611yescharacteristic
407C23C2×#13511yescharacteristic
408C23C2×AGL(1,ℤ/8)11yescharacteristic
409C23C2×#13711yescharacteristic
410C23#57712yes—
411C23#57712yes—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
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—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4#46514yes—
470C2×C4#46814yes—
471C2×C4C2×#12311yescharacteristic
472C2×C4C2×#12711yescharacteristic
473C2×C4#53511yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53611yescharacteristic
476C2×C4#53611yescharacteristic
477C2×C4#53612yes—
478C2×C4#53612yes—
479C2×C4#53912yes—
480C2×C4#53912yes—
481C2×C4#54111yescharacteristic
482C2×C4#54111yescharacteristic
483C2×C4#54712yes—
484C2×C4#54712yes—
485C2×C4#55012yes—
486C2×C4#55212yes—
487C2×C4#56312yes—
488C2×C4#56312yes—
489C8—24no—
490C8—24no—
491C8—81no—
492C8—81no—
493C22—42no—
494C22—42no—
495C22—44no—
496C22—44no—
497C22—42no—
498C22—81no—
499C22—81no—
500C22—82no—
501C22—82no—
502C22—81no—
503C22—81no—
504C22#248611yescharacteristic
505C22#254111yescharacteristic
506C22#262111yescharacteristic
507C22#268012yes—
508C22#268012yes—
509C22C2×#46511yescharacteristic
510C22C2×#46811yescharacteristic
511C22#305711yescharacteristic
512C22#305711yescharacteristic
513C22#322711yescharacteristic
514C22#322811yescharacteristic
515C22#322911yescharacteristic
516C22#323111yescharacteristic
517C22#324412yes—
518C22#324412yes—
519C22#327212yes—
520C22#327812yes—
521C22D4×SD1612yes—
522C22#328412yes—
523C22#333512yes—
524C22#333712yes—
525C22#334412yes—
526C22#334512yes—
527C4—24no—
528C4—24no—
529C4—42no—
530C4—81no—
531C4—84no—
532C4—84no—
533C4—81no—
534C4—162no—
535C4—162no—
536C4#250111yescharacteristic
537C4#262311yescharacteristic
538C4#265811yescharacteristic
539C4#272112yes—
540C4#272112yes—
541C4#282111yescharacteristic
542C2—82no—
543C2#1706211yescharacteristic
544C2#2765811yescharacteristic
545C2#2766311yescharacteristic
546C2#2848211yescharacteristic
547C2#2849011yescharacteristic
548C2#2987711yescharacteristic
549C2#2988111yescharacteristic
550C2#3108612yes—
551C2#3108912yes—
552C2#3109112yes—
553C2#3109212yes—
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
1. C2×#2088C2semidirect
4. C2×#1844C2semidirect
7. #11357C2semidirect
8. #11357C2semidirect
9. #11327C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20882
C2×#20772
C2×#19232
C2×#18442
#151352
#113572
#113272
#110952
#108202