Finite Group Databasebeta

← #475590

Subgroups — #475590

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 540 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×#1846C211yesmaximal, characteristic, complemented
5#15223C212yesmaximal
6#11358C212yesmaximal, complemented
7#11328C211yesmaximal, characteristic, complemented
8#11328C211yesmaximal, characteristic, complemented
9#11096C211yesmaximal, characteristic
10#11096C211yesmaximal, characteristic
11#10820C211yesmaximal, characteristic, complemented
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#382C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#2088—28no—
26#2077—28no—
27#1923—24no—
28#1846—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1445—22no—
39#1445—22no—
40#1439—22no—
41#1439—22no—
42C4⋊#95C2212yes—
43C24×C4C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4D411yescharacteristic
57C22×C22⋊C4D411yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—42no—
63C2×#107—42no—
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76C2×C4×C8D414yes—
77#389—24no—
78#389—24no—
79#389—42no—
80#389—42no—
81#382—24no—
82#382—24no—
83#380—24no—
84#380—24no—
85#380—42no—
86#380—42no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D411yescharacteristic
98C2×#95D411yescharacteristic
99#340—22no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340D418yes—
106#336—22no—
107#336—22no—
108C25C2×D411yescharacteristic
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
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×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8C2×D412yes—
145C22×C8C2×D414yes—
146C22×C8C2×D412yes—
147C22×C8#4112yes—
148C22×C8#4112yes—
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—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4C2×D414yes—
178C2×C4⋊C4C2×D414yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
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—22no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42C2×D412yes—
199C2×C42#4112yes—
200#107—44no—
201#107—44no—
202#106—48no—
203C4⋊C8—44no—
204C22⋊C8—44no—
205C22⋊C8—44no—
206C4×C8—28no—
207#95—22no—
208#95—22no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—82no—
216#95—82no—
217C24—21no—
218C24—22no—
219C24—22no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—21no—
225C24—22no—
226C24—21no—
227C24#13512yes—
228C24C22×D411yescharacteristic
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
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—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
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#12112yes—
300C22×C4#12112yes—
301C22×C4#13512yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D412yes—
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4H(5,𝔽2)12yes—
308C22×C42−1+412yes—
309C22×C42−1+411yescharacteristic
310C22×C42−1+411yescharacteristic
311C22×C42−1+412yes—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8#12314yes—
331C2×C8#12714yes—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—88no—
349C4⋊C4—88no—
350C22⋊C4—28no—
351C22⋊C4—44no—
352C22⋊C4—88no—
353C22⋊C4—88no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—41no—
359C42—41no—
360C42—42no—
361C23—22no—
362C23—22no—
363C23—24no—
364C23—24no—
365C23—22no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23C2×#12111yescharacteristic
385C23#53612yes—
386C23D4212yes—
387C23#54711yescharacteristic
388C23#54711yescharacteristic
389C23C2×#13511yescharacteristic
390C23C2×#13511yescharacteristic
391C23#57512yes—
392C23#57611yescharacteristic
393C23#57712yes—
394C23#57811yescharacteristic
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
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—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4C2×#12311yescharacteristic
457C2×C4C2×#12711yescharacteristic
458C2×C4#53511yescharacteristic
459C2×C4#53611yescharacteristic
460C2×C4#53611yescharacteristic
461C2×C4#53611yescharacteristic
462C2×C4#53612yes—
463C2×C4#53612yes—
464C2×C4#53912yes—
465C2×C4#53912yes—
466C2×C4#54112yes—
467C2×C4#54712yes—
468C2×C4#54712yes—
469C2×C4#55012yes—
470C2×C4#55211yescharacteristic
471C2×C4#55211yescharacteristic
472C2×C4#56312yes—
473C2×C4#56312yes—
474C8—24no—
475C8—24no—
476C8—81no—
477C8—81no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—44no—
482C22—42no—
483C22—81no—
484C22—81no—
485C22—82no—
486C22—82no—
487C22—81no—
488C22—81no—
489C22#248611yescharacteristic
490C22#254111yescharacteristic
491C22#262111yescharacteristic
492C22#268012yes—
493C22#268012yes—
494C22#305011yescharacteristic
495C22#305411yescharacteristic
496C22#306011yescharacteristic
497C22#306111yescharacteristic
498C22#323212yes—
499C22#323312yes—
500C22#324112yes—
501C22#324212yes—
502C22#327312yes—
503C22#327712yes—
504C22#328212yes—
505C22#328212yes—
506C22#333611yescharacteristic
507C22#333911yescharacteristic
508C22#334111yescharacteristic
509C22#334211yescharacteristic
510C22#334812yes—
511C22#335012yes—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—84no—
517C4—84no—
518C4—81no—
519C4—162no—
520C4—162no—
521C4#250111yescharacteristic
522C4#262311yescharacteristic
523C4#265811yescharacteristic
524C4#272112yes—
525C4#272112yes—
526C4#282111yescharacteristic
527C2—82no—
528C2#1706211yescharacteristic
529C2#2766011yescharacteristic
530C2#2766511yescharacteristic
531C2#2848311yescharacteristic
532C2#2849111yescharacteristic
533C2#2987911yescharacteristic
534C2#2988211yescharacteristic
535C2#3108712yes—
536C2#3108812yes—
537C2#3109012yes—
538C2#3109312yes—
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×#1846C2semidirect
6. #11358C2semidirect
7. #11328C2semidirect
8. #11328C2semidirect
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×#18462
#152232
#113582
#113282
#110962
#108202