Finite Group Databasebeta

← #6479525

Subgroups — #6479525

Order 512.

2144 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#33095C214yesmaximal, complemented
4#29907C214yesmaximal, complemented
5#27857C218yesmaximal, complemented
6#27613C214yesmaximal, complemented
7#27585C214yesmaximal, complemented
8#21938C212yesmaximal, complemented
9#21758C212yesmaximal
10C2×#1817C211yesmaximal, characteristic, complemented
11C22×#123—21no—
12C22×#121C2211yescharacteristic
13C22×#117—21no—
14C23×C42C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16#2854—21no—
17#2854—21no—
18#2854—22no—
19#2854C2212yes—
20#2854C2214yes—
21#2854C2212yes—
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#388—22no—
25C2×#386—21no—
26#2285—22no—
27#2285—22no—
28#2285C2214yes—
29#2285C2214yescomplemented
30C2×#384C2212yescomplemented
31C2×#382—21no—
32C2×#380—21no—
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2212yescomplemented
37#2100C2218yescomplemented
38#2094C2218yescomplemented
39#2084C2218yescomplemented
40#2071C2218yescomplemented
41#1926C22116yescomplemented
42#1919C22116yescomplemented
43#1893C2218yes—
44#1874C2218yes—
45#1853C2218yes—
46#1853C2218yescomplemented
47#1823C2214yes—
48#1817C2214yes—
49#1817C2212yescomplemented
50#1817C2212yescomplemented
51#1807C2218yescomplemented
52#1786C2214yes—
53C2×#336C2214yescomplemented
54C23×D4—21no—
55C23×D4C2311yescharacteristic
56C24×C4D411yescharacteristic
57C24×C4D411yescharacteristic
58C24×C4C2311yescharacteristic
59C23×C8C2314yes—
60C2×#123—22no—
61C2×#123—24no—
62C2×#123—22no—
63C2×#123—48no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—28no—
67C2×#121C2312yes—
68C2×#121C2312yes—
69C2×#117—24no—
70C2×#117—24no—
71C2×#117—48no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C42—28no—
84C22×C42—22no—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D412yes—
88C22×C42D412yes—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2312yes—
93C2×#106C23116yes—
94C2×#102C23116yes—
95C2×C22⋊C8—28no—
96C2×C22⋊C8C2314yes—
97C2×C22⋊C8C2314yes—
98#390—22no—
99#390—22no—
100#390C2314yes—
101#390C2314yes—
102#389—22no—
103#389—22no—
104#389C2314yes—
105#389C2314yes—
106#388—24no—
107#388—24no—
108#388—24no—
109#388—24no—
110#386—22no—
111#386—24no—
112#386—22no—
113#384—22no—
114#384—22no—
115#384C2314yes—
116#384C2314yes—
117#382—24no—
118#382—24no—
119#380—21no—
120#380—21no—
121#380—24no—
122#380—24no—
123#380—22no—
124#380—22no—
125#380C2312yes—
126#380C2314yes—
127#380C2314yes—
128#380C2312yes—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4C2318yes—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—22no—
137#340C23116yes—
138#340C23116yes—
139#336C2318yes—
140#336C2318yes—
141C25—21no—
142C25C2×D411yescharacteristic
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—24no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4C2412yes—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4SD1618yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4118yes—
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8—28no—
177C22×C8—48no—
178C22×C8C2418yes—
179#123—48no—
180#123—416no—
181#123—48no—
182#121—28no—
183#121—28no—
184#121—44no—
185#121—44no—
186#117—416no—
187#117—416no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4C2412yes—
204C2×C4⋊C4C2418yes—
205C2×C4⋊C4C2414yes—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—48no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—48no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—48no—
221C2×C42—24no—
222C2×C42—28no—
223C2×C42—24no—
224C2×C42—28no—
225C2×C42—24no—
226C2×C42—28no—
227C2×C42—24no—
228C2×C42—28no—
229C2×C42—28no—
230C2×C42—48no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2412yes—
238C2×C42C2412yes—
239#106—416no—
240#102—416no—
241C22⋊C8—28no—
242C22⋊C8—28no—
243#95—24no—
244#95—24no—
245#95—28no—
246#95—22no—
247#95—22no—
248#95—24no—
249#95—24no—
250C24—22no—
251C24—42no—
252C24—42no—
253C24—44no—
254C24—44no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24#12314yes—
259C24C2×SD1614yes—
260C24AGL(1,ℤ/8)14yes—
261C24C22×D411yescharacteristic
262C24C22×D411yescharacteristic
263C24C22×D411yescharacteristic
264C2×D4—24no—
265C2×D4—28no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—88no—
275C2×D4—816no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—232no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—416no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—416no—
306C22×C4—44no—
307C22×C4—48no—
308C22×C4—44no—
309C22×C4—48no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—48no—
314C22×C4—42no—
315C22×C4#12314yes—
316C22×C4#12314yes—
317C22×C4#12314yes—
318C22×C4C2×SD1618yes—
319C22×C4C2×SD1614yes—
320C22×C4C2×SD1618yes—
321C22×C4C2×SD1614yes—
322C22×C4C22×D412yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C2×#4114yes—
330C22×C4C2×#4114yes—
331C22×C4C2×#4114yes—
332C22×C4C2511yescharacteristic
333C2×C8—28no—
334C2×C8—216no—
335C2×C8—88no—
336C4⋊C4—24no—
337C4⋊C4—216no—
338C4⋊C4—28no—
339C4⋊C4—48no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—48no—
344C4⋊C4—42no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—24no—
366C42—416no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—44no—
374C23—44no—
375C23—48no—
376C23—48no—
377C23—42no—
378C23—44no—
379C23—42no—
380C23—42no—
381C23—44no—
382C23—84no—
383C23—88no—
384C23—88no—
385C23—84no—
386C23—84no—
387C23#481116yes—
388C23C2×#12312yes—
389C23C2×#12312yes—
390C23C2×#12312yes—
391C23C22×SD1612yes—
392C23C22×SD1612yes—
393C23C22×SD1612yes—
394C23C2×AGL(1,ℤ/8)12yes—
395C23C2×AGL(1,ℤ/8)12yes—
396C23C2×AGL(1,ℤ/8)12yes—
397C23C23×D411yescharacteristic
398D4—48no—
399D4—84no—
400D4—88no—
401D4—88no—
402C2×C4—28no—
403C2×C4—216no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—216no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—88no—
438C2×C4—816no—
439C2×C4—84no—
440C2×C4C2×#12312yes—
441C2×C4C2×#12312yes—
442C2×C4C2×#12312yes—
443C2×C4C22×SD1614yes—
444C2×C4C22×SD1614yes—
445C2×C4#57514yes—
446C2×C4C22×#4112yes—
447C2×C4#58514yes—
448C8—48no—
449C22—24no—
450C22—24no—
451C22—24no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—24no—
456C22—24no—
457C22—24no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—416no—
462C22—84no—
463C22—82no—
464C22—84no—
465C22—82no—
466C22—82no—
467C22—168no—
468C22C2×#48114yes—
469C22C2×#48114yes—
470C22C2×#48114yes—
471C22C22×#12311yescharacteristic
472C22#344912yes—
473C22C23×SD1611yescharacteristic
474C22C22×AGL(1,ℤ/8)11yescharacteristic
475C22C2×#57512yes—
476C4—22no—
477C4—22no—
478C4—22no—
479C4—24no—
480C4—24no—
481C4—24no—
482C4—42no—
483C4—84no—
484C4—88no—
485C4—82no—
486C4#349114yes—
487C2—24no—
488C2—24no—
489C2—24no—
490C2—162no—
491C2#3509711yescharacteristic
492C2C22×#48111yescharacteristic
493C2#6051111yescharacteristic
494C2#6094112yes—
495C2#6095812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
3. #33095C2semidirect
4. #29907C2semidirect
5. #27857C2semidirect
6. #27613C2semidirect
7. #27585C2semidirect
8. #21938C2semidirect
10. C2×#1817C2semidirect
29. #2285C22semidirect
30. C2×#384C22semidirect
36. C2×C4×C4⋊C4C22semidirect
37. #2100C22semidirect
38. #2094C22semidirect
39. #2084C22semidirect
40. #2071C22semidirect
41. #1926C22semidirect
42. #1919C22semidirect
46. #1853C22semidirect
49. #1817C22semidirect
50. #1817C22semidirect
51. #1807C22semidirect
53. C2×#336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#22852
#330952
#299072
#278572
#276132
#275852
#219382
#217582
C2×#18172