Finite Group Databasebeta

← #7231154

Subgroups — #7231154

Order 512.

1505 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#36496C211yesmaximal, characteristic
3#33073C212yesmaximal, complemented
4#32818C212yesmaximal, complemented
5#32695C212yesmaximal, complemented
6#32028C212yesmaximal
7#31500C214yesmaximal, complemented
8#30659C214yesmaximal, complemented
9#27049C214yesmaximal, complemented
10#26336C214yesmaximal, complemented
11#25660C212yesmaximal, complemented
12#24098C212yesmaximal, complemented
13#23352C211yesmaximal, characteristic, complemented
14C2×#559C2211yescharacteristic
15C2×#554C2211yescharacteristic
16C2×#489C2212yes—
17C2×#484C2212yes—
18C2×#474C2212yes—
19C22×#107C2212yes—
20#2861—21no—
21#2859C2212yes—
22#2858C2211yescharacteristic
23#2844—21no—
24#2809—21no—
25#2720—24no—
26#2683—21no—
27#2567C2212yes—
28#2567C2211yescharacteristic
29#2566—21no—
30#2544—21no—
31#2541C2211yescharacteristic
32#2497C2214yes—
33#2487C2212yes—
34#2461—24no—
35#2440C2214yes—
36#2426C2214yes—
37#2379—21no—
38#2361C2214yes—
39C4×#127—21no—
40C4×#124C2211yescharacteristic
41C42×Q8C2211yescharacteristic
42#2093C2214yes—
43#2086C2214yes—
44#2071C2214yes—
45#2067C2214yes—
46#2067C2214yes—
47#2057C2214yes—
48#2057C2214yes—
49#2042C2214yes—
50#2041C2214yes—
51#2036C2214yes—
52#2034C2214yes—
53#2034C2214yes—
54#1985C2212yes—
55#1983C2212yes—
56#1948C2212yes—
57#1947C2212yes—
58#1944C2214yes—
59#1924C2214yes—
60#1920C2214yes—
61#1920C2214yes—
62#1918C2214yes—
63#1893C2214yes—
64#1892C2212yes—
65#1874C2214yes—
66#1869C2212yes—
67#1865C2214yes—
68#1863C2214yes—
69#1862C2214yes—
70#1852C2214yes—
71#1850C2214yes—
72#1831C2212yes—
73#1824C2212yes—
74#1803C2212yes—
75#1803C2212yes—
76#1762C2212yes—
77C23×C8C2312yes—
78#559—42no—
79#559—42no—
80#554—42no—
81#554—42no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2312yes—
84C2×#127—21no—
85C2×#127C2311yescharacteristic
86C2×#125—21no—
87C2×#125—22no—
88C2×#124—21no—
89C2×#124—22no—
90C2×#124D411yescharacteristic
91C2×#124C2312yes—
92C2×#123—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2312yes—
96C2×#121C2314yes—
97C2×C4×Q8C2311yescharacteristic
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2311yescharacteristic
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4—24no—
103C2×C4×D4C2312yes—
104C2×C4×D4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107#489—44no—
108#484—44no—
109#474—44no—
110C2×#107—44no—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2314yes—
126C2×#97C2312yes—
127#400—22no—
128#400—24no—
129#399—22no—
130#398—24no—
131#396—24no—
132#396—24no—
133#395—22no—
134#395—22no—
135#395—22no—
136#395—24no—
137#394—24no—
138#392—22no—
139#391—21no—
140#391—22no—
141#391D411yescharacteristic
142#390—21no—
143#390—24no—
144#390—21no—
145#390—21no—
146#390D411yescharacteristic
147#390C2314yes—
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#389C2314yes—
154#388—24no—
155#388—24no—
156#388—24no—
157#388D414yes—
158#387—24no—
159#386—22no—
160#386—24no—
161#385—24no—
162#385—22no—
163#385—24no—
164#384—21no—
165#384—22no—
166#384C2312yes—
167#384C2312yes—
168#382—22no—
169#382D411yescharacteristic
170#380C2312yes—
171#380C2314yes—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
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×C4—22no—
201C23×C4C2412yes—
202C22×C8—44no—
203C22×C8—42no—
204C22×C8C2414yes—
205C22×C8C2414yes—
206C4⋊Q8—41no—
207C4⋊Q8—41no—
208C4⋊Q8—42no—
209C4⋊Q8—42no—
210#127—41no—
211#127—41no—
212#127—44no—
213#125—44no—
214#125—84no—
215#124—24no—
216#124—24no—
217#124—42no—
218#124—44no—
219#124—84no—
220#123—48no—
221C22⋊Q8—24no—
222C22⋊Q8—42no—
223C22⋊Q8—48no—
224C22⋊Q8—88no—
225#121—28no—
226#121—44no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—24no—
230C4×Q8—41no—
231C4×Q8—41no—
232C4×Q8—42no—
233C4×D4—24no—
234C4×D4—42no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—88no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2411yescharacteristic
266C2×C4⋊C4C2412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D412yes—
279C2×C42—21no—
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—28no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296C4⋊C8—42no—
297#103—28no—
298#103—44no—
299#102—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303C22⋊C8—44no—
304C22⋊C8—42no—
305#97—42no—
306#95—22no—
307#95—21no—
308#95—22no—
309#95—24no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95C2×D414yes—
316#95C2×D414yes—
317C24—22no—
318C24C22×D412yes—
319C2×Q8—22no—
320C2×Q8—21no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—84no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×Q8H(5,𝔽2)12yes—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—48no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—48no—
341C2×D4—42no—
342C2×D4—82no—
343C2×D4—84no—
344C2×D4#13714yes—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—84no—
358C22×C4—82no—
359C22×C4—84no—
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
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—48no—
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—42no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—84no—
401C4⋊C4—82no—
402C4⋊C4H(5,𝔽2)12yes—
403C4⋊C42−1+414yes—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—48no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—48no—
413C22⋊C4—84no—
414C22⋊C4—82no—
415C22⋊C4—84no—
416C42—24no—
417C42—22no—
418C42—24no—
419C42—22no—
420C42—22no—
421C42—21no—
422C42—21no—
423C42—22no—
424C42—22no—
425C42—41no—
426C42—44no—
427C42—42no—
428C42—44no—
429C42—82no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—42no—
438C23—44no—
439C23C2×#13714yes—
440C23C23×D411yescharacteristic
441Q8—24no—
442Q8—41no—
443Q8—41no—
444Q8—82no—
445Q8—162no—
446Q8—164no—
447D4—44no—
448D4—42no—
449D4—42no—
450D4—44no—
451D4—84no—
452D4—816no—
453D4—162no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—164no—
489C2×C4—164no—
490C2×C4—164no—
491C2×C4C2×#13712yes—
492C2×C4#57614yes—
493C2×C4C2×H(5,𝔽2)11yescharacteristic
494C2×C4C2×H(5,𝔽2)11yescharacteristic
495C2×C4C2×2−1+412yes—
496C2×C4#58512yes—
497C8—44no—
498C8—44no—
499C22—24no—
500C22—28no—
501C22—42no—
502C22—44no—
503C22—42no—
504C22—82no—
505C22—82no—
506C22—84no—
507C22—84no—
508C22—82no—
509C22#348411yescharacteristic
510C22#348912yes—
511C22C22×#13711yescharacteristic
512C22C2×#57612yes—
513C22#358611yescharacteristic
514C4—22no—
515C4—24no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—44no—
521C4—42no—
522C4—41no—
523C4—42no—
524C4—84no—
525C4—82no—
526C4—84no—
527C4#352911yescharacteristic
528C4#355011yescharacteristic
529C4#355912yes—
530C2—44no—
531C2—162no—
532C2#4622811yescharacteristic
533C2#6157311yescharacteristic
534C2#6166111yescharacteristic
535C2#6212712yes—
536C2#6217012yes—
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. #33073C2semidirect
4. #32818C2semidirect
5. #32695C2semidirect
7. #31500C2semidirect
8. #30659C2semidirect
9. #27049C2semidirect
10. #26336C2semidirect
11. #25660C2semidirect
12. #24098C2semidirect
13. #23352C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#364962
#330732
#328182
#326952
#320282
#315002
#306592
#270492
#263362
#256602
#240982
#233522