Finite Group Databasebeta

← #7228752

Subgroups — #7228752

Order 512.

1519 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#36457C211yesmaximal, characteristic
3#33073C212yesmaximal, complemented
4#32818C212yesmaximal, complemented
5#32629C212yesmaximal, complemented
6#32268C212yesmaximal, complemented
7#32222C214yesmaximal, complemented
8#30773C214yesmaximal, complemented
9#27529C214yesmaximal, complemented
10#25975C214yesmaximal, complemented
11#25841C212yesmaximal, complemented
12#24046C212yesmaximal
13#23347C211yesmaximal, characteristic, complemented
14C2×Q82C2211yescharacteristic
15C2×#550C2211yescharacteristic
16C2×#500C2212yes—
17C2×#480C2212yes—
18C2×#477C2212yes—
19C2×#417C2212yes—
20#2859C2212yes—
21#2858—21no—
22#2858C2211yescharacteristic
23#2844—21no—
24#2809—21no—
25#2720—24no—
26#2683—21no—
27#2567C2212yes—
28#2563—21no—
29#2563C2211yescharacteristic
30#2548—21no—
31#2548C2211yescharacteristic
32#2497C2214yes—
33#2491—24no—
34#2491C2214yes—
35#2487C2212yes—
36#2440C2214yes—
37#2379—21no—
38#2341C2214yes—
39C4×#124—21no—
40C4×#124C2211yescharacteristic
41C42×Q8C2211yescharacteristic
42#2093C2214yes—
43#2077C2214yes—
44#2075C2214yes—
45#2071C2214yes—
46#2067C2214yes—
47#2067C2214yes—
48#2065C2214yes—
49#2057C2214yes—
50#2057C2214yes—
51#2041C2214yes—
52#2033C2214yes—
53#2023C2214yes—
54#1987C2212yes—
55#1984C2212yes—
56#1948C2212yes—
57#1947C2212yes—
58#1934C2214yes—
59#1926C2214yes—
60#1923C2214yes—
61#1920C2214yes—
62#1918C2214yes—
63#1895C2212yes—
64#1870C2212yes—
65#1865C2214yes—
66#1863C2214yes—
67#1863C2214yes—
68#1852C2214yes—
69#1851C2214yes—
70#1831C2212yes—
71#1809C2214yes—
72#1803C2212yes—
73#1803C2212yes—
74#1788C2214yes—
75#1788C2212yes—
76C4×#103C2212yes—
77C22×#41C2312yes—
78C22×M16C2312yes—
79Q82—42no—
80Q82—42no—
81#550—42no—
82#550—42no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#127—21no—
89C2×#127C2311yescharacteristic
90C2×#125—22no—
91C2×#124—21no—
92C2×#124—24no—
93C2×#124—22no—
94C2×#124D411yescharacteristic
95C2×C22⋊Q8—24no—
96C2×#121—22no—
97C2×#121C2312yes—
98C2×C4×Q8—21no—
99C2×C4×Q8C2311yescharacteristic
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2311yescharacteristic
103C2×C4×D4—21no—
104C2×C4×D4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106#500—44no—
107#480—44no—
108#477—44no—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116#417—44no—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2314yes—
125C2×C4×C8C2312yes—
126#400—22no—
127#400—24no—
128#399—22no—
129#398—24no—
130#396—24no—
131#395—22no—
132#395—22no—
133#395—24no—
134#394—22no—
135#394—24no—
136#393—22no—
137#393—24no—
138#391—22no—
139#391D411yescharacteristic
140#390—21no—
141#390—21no—
142#390—24no—
143#390—24no—
144#390—21no—
145#390—21no—
146#390D411yescharacteristic
147#390C2314yes—
148#390C2314yes—
149#389—22no—
150#389—22no—
151#389C2314yes—
152#389C2314yes—
153#389C2314yes—
154#388—24no—
155#388—24no—
156#388D414yes—
157#387—24no—
158#386—22no—
159#385—24no—
160#385—22no—
161#384—22no—
162#384C2312yes—
163#384C2312yes—
164#382—21no—
165#382—22no—
166#382D411yescharacteristic
167#380—24no—
168#380C2314yes—
169#380C2314yes—
170#380C2312yes—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2311yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—24no—
177C4×C22⋊C4C2314yes—
178C43—21no—
179C43C2311yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C2×#41—24no—
190C2×#41—28no—
191C2×#41—42no—
192C2×#41—44no—
193C22×Q8—22no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8C2×D411yescharacteristic
197C22×Q8C2411yescharacteristic
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4C2×D411yescharacteristic
203C22×D4C2412yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4C2412yes—
207C2×M16—44no—
208C2×M16—42no—
209C22×C8C2414yes—
210C22×C8C2414yes—
211C4⋊Q8—24no—
212C4⋊Q8—42no—
213C4⋊Q8—41no—
214C4⋊Q8—41no—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4⋊Q8—44no—
218C4⋊Q8—48no—
219#127—41no—
220#127—41no—
221#127—44no—
222#125—84no—
223#124—24no—
224#124—44no—
225#124—84no—
226#124—88no—
227C22⋊Q8—88no—
228#121—24no—
229#121—42no—
230#121—48no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—42no—
238C4×Q8—42no—
239C4×Q8—44no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4—44no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2411yescharacteristic
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×D412yes—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2412yes—
289C2×C42C2412yes—
290#107—28no—
291#107—44no—
292#107—44no—
293#106—44no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296C4⋊C8—42no—
297#103—28no—
298#103—28no—
299#102—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303C22⋊C8—44no—
304C22⋊C8—42no—
305C4×C8—42no—
306#95—24no—
307#95—22no—
308#95—21no—
309#95—22no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95C2×D414yes—
316#95C2×D414yes—
317C24—22no—
318C24C22×D412yes—
319#41—48no—
320#41—84no—
321#41—88no—
322C2×Q8—21no—
323C2×Q8—22no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—44no—
327C2×Q8—42no—
328C2×Q8—84no—
329C2×Q8—82no—
330C2×Q8—84no—
331C2×Q8—82no—
332C2×Q8#13714yes—
333C2×Q8H(5,𝔽2)12yes—
334C2×Q82−1+414yes—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—48no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—44no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—82no—
350C2×D4—84no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—44no—
362C22×C4—48no—
363C22×C4—82no—
364C22×C4—84no—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C2511yescharacteristic
373M16—84no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—84no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—84no—
405C4⋊C4—82no—
406C4⋊C4H(5,𝔽2)12yes—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—84no—
414C22⋊C4—84no—
415C22⋊C4—82no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C42—24no—
419C42—22no—
420C42—24no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—21no—
425C42—22no—
426C42—22no—
427C42—41no—
428C42—44no—
429C42—44no—
430C42—42no—
431C42—44no—
432C42—82no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—22no—
437C23—24no—
438C23—42no—
439C23—42no—
440C23—44no—
441C23—42no—
442C23—44no—
443C23C23×D411yescharacteristic
444Q8—22no—
445Q8—24no—
446Q8—44no—
447Q8—164no—
448Q8—162no—
449Q8—164no—
450Q8#56314yes—
451D4—42no—
452D4—42no—
453D4—44no—
454D4—82no—
455D4—88no—
456D4—84no—
457D4—164no—
458D4—162no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—88no—
493C2×C4—164no—
494C2×C4C2×#13712yes—
495C2×C4C2×#13714yes—
496C2×C4#57614yes—
497C2×C4C2×H(5,𝔽2)11yescharacteristic
498C2×C4C2×H(5,𝔽2)11yescharacteristic
499C2×C4C2×2−1+412yes—
500C2×C4#58512yes—
501C8—44no—
502C8—44no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—42no—
508C22—82no—
509C22—82no—
510C22—84no—
511C22—84no—
512C22—82no—
513C22#348411yescharacteristic
514C22#348912yes—
515C22C22×#13711yescharacteristic
516C22C2×#57612yes—
517C22#358611yescharacteristic
518C4—24no—
519C4—22no—
520C4—24no—
521C4—24no—
522C4—42no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—84no—
529C4—84no—
530C4—82no—
531C4C2×#56311yescharacteristic
532C4#354511yescharacteristic
533C4#356112yes—
534C2—82no—
535C2—162no—
536C2#4613611yescharacteristic
537C2#6157311yescharacteristic
538C2#6166111yescharacteristic
539C2#6214912yes—
540C2#6217012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #46152C2semidirect
3. #33073C2semidirect
4. #32818C2semidirect
5. #32629C2semidirect
6. #32268C2semidirect
7. #32222C2semidirect
8. #30773C2semidirect
9. #27529C2semidirect
10. #25975C2semidirect
11. #25841C2semidirect
13. #23347C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#364572
#330732
#328182
#326292
#322682
#322222
#307732
#275292
#259752
#258412
#240462
#233472