Finite Group Databasebeta

← #7231171

Subgroups — #7231171

Order 512.

1463 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#36597C211yesmaximal, characteristic
3#33069C212yesmaximal, complemented
4#32819C212yesmaximal, complemented
5#32694C212yesmaximal, complemented
6#32036C212yesmaximal
7#31504C214yesmaximal, complemented
8#30657C214yesmaximal, complemented
9#27056C214yesmaximal, complemented
10#26357C214yesmaximal, complemented
11#25660C212yesmaximal, complemented
12#24108C212yesmaximal, complemented
13#23353C211yesmaximal, characteristic, complemented
14C2×#559C2211yescharacteristic
15C2×#556C2211yescharacteristic
16C2×#489C2212yes—
17C2×#483C2212yes—
18C2×#474C2212yes—
19C2×#428C2212yes—
20#2859C2212yes—
21#2858—21no—
22#2858C2211yescharacteristic
23#2844—21no—
24#2809—21no—
25#2720—24no—
26#2683—21no—
27#2567C2212yes—
28#2554—24no—
29#2546C2211yescharacteristic
30#2545—21no—
31#2544—21no—
32#2541C2211yescharacteristic
33#2497C2214yes—
34#2487C2212yes—
35#2440C2214yes—
36#2426C2214yes—
37#2379—21no—
38#2363C2214yes—
39C4×#124—21no—
40C4×#124C2211yescharacteristic
41C42×Q8C2211yescharacteristic
42#2094C2214yes—
43#2088C2214yes—
44#2071C2214yes—
45#2065C2214yes—
46#2065C2214yes—
47#2058C2214yes—
48#2058C2214yes—
49#2042C2214yes—
50#2041C2214yes—
51#2035C2214yes—
52#2033C2214yes—
53#2033C2214yes—
54#1985C2212yes—
55#1983C2212yes—
56#1947C2212yes—
57#1947C2212yes—
58#1943C2214yes—
59#1924C2214yes—
60#1921C2214yes—
61#1921C2214yes—
62#1918C2214yes—
63#1895C2214yes—
64#1894C2212yes—
65#1874C2214yes—
66#1870C2212yes—
67#1863C2214yes—
68#1863C2214yes—
69#1862C2214yes—
70#1854C2214yes—
71#1851C2214yes—
72#1831C2212yes—
73#1825C2212yes—
74#1803C2212yes—
75#1803C2212yes—
76#1762C2212yes—
77C22×M16C2312yes—
78#559—42no—
79#559—42no—
80#556—42no—
81#556—42no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127—21no—
86C2×#127C2311yescharacteristic
87C2×#126—22no—
88C2×#125—22no—
89C2×#124—21no—
90C2×#124—21no—
91C2×#124—21no—
92C2×#124—21no—
93C2×#124D411yescharacteristic
94C2×#124C2312yes—
95C2×#123—22no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8C2312yes—
98C2×#121C2314yes—
99C2×C4×Q8—21no—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2311yescharacteristic
103C2×C4×D4—21no—
104C2×C4×D4C2312yes—
105C2×C4×D4C2311yescharacteristic
106C2×#117C2312yes—
107C22×C4⋊C4C2312yes—
108#489—44no—
109#483—44no—
110#474—44no—
111#428—44no—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2314yes—
127C2×#97C2312yes—
128#400—22no—
129#400—24no—
130#399—22no—
131#398—22no—
132#398—24no—
133#396—24no—
134#396—24no—
135#395—22no—
136#395—22no—
137#395—24no—
138#394—24no—
139#392—22no—
140#391—22no—
141#391—21no—
142#391D411yescharacteristic
143#390—21no—
144#390—24no—
145#390—24no—
146#390—22no—
147#390D411yescharacteristic
148#390C2314yes—
149#389—22no—
150#389—22no—
151#389C2314yes—
152#389C2314yes—
153#389C2314yes—
154#389C2314yes—
155#388—24no—
156#388—24no—
157#388—24no—
158#388D414yes—
159#387—24no—
160#386—22no—
161#386—24no—
162#385—24no—
163#385—22no—
164#384—22no—
165#384C2312yes—
166#384C2312yes—
167#382—22no—
168#382D411yescharacteristic
169#380C2312yes—
170#380C2314yes—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2311yescharacteristic
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—
189C22×Q8—21no—
190C22×Q8—22no—
191C22×Q8C2×D411yescharacteristic
192C22×Q8C2411yescharacteristic
193C22×D4—24no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4C2×D411yescharacteristic
197C22×D4C2412yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4C2412yes—
201C2×M16—44no—
202C2×M16—42no—
203C22×C8C2414yes—
204C22×C8C2414yes—
205C4⋊Q8—41no—
206C4⋊Q8—41no—
207C4⋊Q8—42no—
208C4⋊Q8—42no—
209C4⋊Q8—82no—
210#127—41no—
211#127—41no—
212#127—82no—
213#126—48no—
214#125—84no—
215#124—24no—
216#124—24no—
217#124—42no—
218#124—44no—
219#124—44no—
220#124—44no—
221#124—82no—
222#123—48no—
223C22⋊Q8—24no—
224C22⋊Q8—42no—
225C22⋊Q8—88no—
226#121—28no—
227#121—44no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—24no—
231C4×Q8—41no—
232C4×Q8—41no—
233C4×Q8—42no—
234C4×Q8—44no—
235C4×D4—24no—
236C4×D4—42no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—44no—
240#117—24no—
241#117—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2411yescharacteristic
265C2×C4⋊C4C2412yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D414yes—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D412yes—
287C2×C42C2×D411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2412yes—
290C2×C42C2412yes—
291#107—28no—
292#107—28no—
293#106—44no—
294#106—44no—
295C4⋊C8—42no—
296C4⋊C8—42no—
297C4⋊C8—42no—
298#103—28no—
299#103—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303#102—44no—
304C22⋊C8—44no—
305C22⋊C8—42no—
306#97—42no—
307#95—24no—
308#95—22no—
309#95—21no—
310#95—22no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95C2×D414yes—
317#95C2×D414yes—
318C24—22no—
319C24C22×D412yes—
320C2×Q8—22no—
321C2×Q8—21no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—84no—
330C2×Q8—82no—
331C2×Q8H(5,𝔽2)12yes—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—48no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—82no—
345C2×D4—84no—
346C2×D4—84no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—44no—
358C22×C4—84no—
359C22×C4—82no—
360C22×C4—84no—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C2511yescharacteristic
369M16—48no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C4⋊C4—24no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—42no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—84no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—84no—
399C4⋊C4—82no—
400C4⋊C4—84no—
401C4⋊C4H(5,𝔽2)12yes—
402C4⋊C42−1+414yes—
403C22⋊C4—28no—
404C22⋊C4—42no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—48no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—48no—
413C22⋊C4—84no—
414C22⋊C4—82no—
415C22⋊C4—84no—
416C42—24no—
417C42—22no—
418C42—22no—
419C42—21no—
420C42—21no—
421C42—22no—
422C42—22no—
423C42—41no—
424C42—42no—
425C42—44no—
426C42—44no—
427C42—42no—
428C42—82no—
429C42—82no—
430C42#13714yes—
431C23—24no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—42no—
437C23—44no—
438C23—42no—
439C23—44no—
440C23—42no—
441C23C23×D411yescharacteristic
442Q8—24no—
443Q8—41no—
444Q8—41no—
445Q8—82no—
446Q8—84no—
447Q8—164no—
448D4—42no—
449D4—42no—
450D4—44no—
451D4—82no—
452D4—84no—
453D4—168no—
454D4—162no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—164no—
491C2×C4—164no—
492C2×C4C2×#13714yes—
493C2×C4C2×#13712yes—
494C2×C4C2×H(5,𝔽2)11yescharacteristic
495C2×C4C2×H(5,𝔽2)11yescharacteristic
496C2×C4C2×2−1+412yes—
497C2×C4#58512yes—
498C8—44no—
499C8—44no—
500C22—44no—
501C22—42no—
502C22—42no—
503C22—44no—
504C22—42no—
505C22—84no—
506C22—82no—
507C22—82no—
508C22—84no—
509C22—82no—
510C22#348411yescharacteristic
511C22#348912yes—
512C22C22×#13711yescharacteristic
513C22#358511yescharacteristic
514C22#358512yes—
515C4—22no—
516C4—24no—
517C4—42no—
518C4—44no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—41no—
524C4—42no—
525C4—84no—
526C4—82no—
527C4—84no—
528C4#352911yescharacteristic
529C4#355011yescharacteristic
530C4#355912yes—
531C2—82no—
532C2—162no—
533C2#4622811yescharacteristic
534C2#6158811yescharacteristic
535C2#6167311yescharacteristic
536C2#6211212yes—
537C2#6223512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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. #33069C2semidirect
4. #32819C2semidirect
5. #32694C2semidirect
7. #31504C2semidirect
8. #30657C2semidirect
9. #27056C2semidirect
10. #26357C2semidirect
11. #25660C2semidirect
12. #24108C2semidirect
13. #23353C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#365972
#330692
#328192
#326942
#320362
#315042
#306572
#270562
#263572
#256602
#241082
#233532