Finite Group Databasebeta

← #7195468

Subgroups — #7195468

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic
2#59821C211yesmaximal, characteristic
3#32016C214yesmaximal, complemented
4#31343C214yesmaximal, complemented
5#29272C211yesmaximal, characteristic, complemented
6#29255C211yesmaximal, characteristic, complemented
7#28606C212yesmaximal, complemented
8#26303C212yesmaximal, complemented
9#26236C212yesmaximal, complemented
10#26148C214yesmaximal, complemented
11#24107C214yesmaximal, complemented
12#23835C211yesmaximal, characteristic, complemented
13#22663C212yesmaximal, complemented
14#22264C211yesmaximal, characteristic
15#22218C211yesmaximal, characteristic, complemented
16C2×#559—21no—
17C2×#556C2212yes—
18C2×#487C2211yescharacteristic
19C2×#484C2214yes—
20C2×#436C2212yes—
21C2×#428C2214yes—
22#2727—22no—
23#2724—22no—
24#2582—21no—
25#2548—21no—
26#2547C2212yes—
27#2547C2212yes—
28#2542C2212yes—
29#2538C2212yes—
30#2512—22no—
31#2486C2214yes—
32#2448—22no—
33#2447—22no—
34#2447—22no—
35#2415C2214yes—
36#2354C2212yes—
37Q8×C4⋊C4—21no—
38D4×C4⋊C4C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2099C2212yes—
46#2096C2212yes—
47#2083C2212yes—
48#2083C2212yes—
49#2077C2214yes—
50#2076C2214yes—
51#2057C2212yes—
52#2057C2212yes—
53#2057C2212yes—
54#2036C2214yes—
55#2036C2214yes—
56#2036C2214yes—
57#2036C2214yes—
58#2024C2212yes—
59#1990C2211yescharacteristic
60#1988C2212yes—
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1954C2212yes—
64#1948C2212yes—
65#1947C2212yes—
66#1927C2214yes—
67#1918C2214yes—
68#1894C2214yes—
69#1894C2214yes—
70#1872C2214yes—
71#1870C2214yes—
72#1870C2214yes—
73#1865C2214yes—
74#1865C2214yes—
75#1854C2214yes—
76#1854C2214yes—
77#1853C2214yes—
78#1844C2211yescharacteristic
79#1831C2211yescharacteristic
80#1825C2214yes—
81#1809C2214yes—
82C4×#107C2212yes—
83C4×#107C2212yes—
84#1744C2211yescharacteristic
85C22×M16C2314yes—
86#559—48no—
87#556—48no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127D411yescharacteristic
93C2×#126—24no—
94C2×#124—22no—
95C2×#124—22no—
96C2×#124—22no—
97C2×#123—24no—
98C2×C22⋊Q8C2314yes—
99C2×C22⋊Q8C2314yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117C2314yes—
111C22×C4⋊C4C2314yes—
112#487—24no—
113#484—48no—
114#436—28no—
115#428—48no—
116C2×#107C2312yes—
117C2×#107C2314yes—
118C2×#107C2312yes—
119C2×#107C2312yes—
120C2×#107C2312yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2312yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#399—24no—
134#398—24no—
135#397—24no—
136#397—24no—
137#397—24no—
138#397—24no—
139#396—24no—
140#394—24no—
141#391—22no—
142#391—22no—
143#391—21no—
144#391—21no—
145#391—21no—
146#391C2312yes—
147#390—22no—
148#390—22no—
149#390—22no—
150#390D412yes—
151#390D412yes—
152#390D412yes—
153#389—22no—
154#389—22no—
155#389—24no—
156#389—22no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—24no—
166#388—24no—
167#385—24no—
168#385—24no—
169#384—22no—
170#384D411yescharacteristic
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#381—24no—
176#380C2314yes—
177#380C2314yes—
178#380C2314yes—
179#380C2314yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C43C2311yescharacteristic
190#340C2312yes—
191#340C2312yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2312yes—
198#340C2312yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2312yes—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×Q8C2412yes—
207C22×D4C2×D412yes—
208C22×D4C2×D412yes—
209C22×D4C2×D412yes—
210C23×C4—24no—
211C23×C4C2414yes—
212C2×M16—28no—
213C2×M16—44no—
214C2×M16—44no—
215C22×C8C2414yes—
216C22×C8C2412yes—
217C22×C8C2412yes—
218C4⋊Q8—24no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—42no—
224#127—24no—
225#126—416no—
226#124—48no—
227#124—48no—
228#124—48no—
229#123—416no—
230C22⋊Q8—28no—
231C22⋊Q8—44no—
232C22⋊Q8—44no—
233C22⋊Q8—44no—
234C4×Q8—44no—
235C4×Q8—48no—
236C4×Q8—44no—
237C4×Q8—44no—
238C4×Q8—44no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—44no—
242C4×D4—44no—
243C4×D4—48no—
244C4×D4—48no—
245#117—28no—
246#117—44no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4C2×D414yes—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301#107—24no—
302#107—28no—
303#107—24no—
304#107—42no—
305#107—42no—
306#106—44no—
307#106—44no—
308C4⋊C8—24no—
309#103—28no—
310#103—44no—
311#103—44no—
312#103—44no—
313C22⋊C8—28no—
314C22⋊C8—44no—
315C22⋊C8—44no—
316C4×C8—22no—
317C4×C8#4114yes—
318#95—24no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—24no—
330C24C22×D414yes—
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—48no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D414yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2511yescharacteristic
378M16—416no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8C2×#4114yes—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—48no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—81no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4H(5,𝔽2)14yes—
423C22⋊C4—28no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—41no—
437C42—41no—
438C42—48no—
439C42—44no—
440C42—44no—
441C42—44no—
442C42—44no—
443C42AGL(1,ℤ/8)14yes—
444C42C2×#4112yes—
445C422−1+412yes—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23C23×D411yescharacteristic
454Q8—48no—
455Q8—44no—
456Q8—44no—
457Q8—84no—
458Q8—84no—
459D4—48no—
460D4—48no—
461D4—84no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—168no—
506C2×C4#54712yes—
507C2×C4#54712yes—
508C2×C4C2×AGL(1,ℤ/8)12yes—
509C2×C4C2×AGL(1,ℤ/8)12yes—
510C2×C4C2×AGL(1,ℤ/8)12yes—
511C2×C4#57812yes—
512C2×C4C22×#4111yescharacteristic
513C2×C4C2×H(5,𝔽2)12yes—
514C2×C4C2×2−1+411yescharacteristic
515C2×C4#58512yes—
516C8—24no—
517C8—44no—
518C8—42no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22C2×#54711yescharacteristic
527C22#348411yescharacteristic
528C22#348511yescharacteristic
529C22C22×AGL(1,ℤ/8)11yescharacteristic
530C22C2×#57811yescharacteristic
531C22#358611yescharacteristic
532C22#358611yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—44no—
537C4—41no—
538C4—44no—
539C4—41no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4#335314yes—
548C4#350612yes—
549C4#355412yes—
550C2—84no—
551C2#4535411yescharacteristic
552C2C2×#335311yescharacteristic
553C2#6129211yescharacteristic
554C2#6158511yescharacteristic
555C2#6168311yescharacteristic
556C2#6171711yescharacteristic
557C2#6180611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32016C2semidirect
4. #31343C2semidirect
5. #29272C2semidirect
6. #29255C2semidirect
7. #28606C2semidirect
8. #26303C2semidirect
9. #26236C2semidirect
10. #26148C2semidirect
11. #24107C2semidirect
12. #23835C2semidirect
13. #22663C2semidirect
15. #22218C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#598212
#320162
#313432
#292722
#292552
#286062
#263032
#262362
#261482
#241072
#238352
#226632
#222642
#222182