Finite Group Databasebeta

← #6481944

Subgroups — #6481944

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2C2×#2357C211yesmaximal, characteristic
3#33149C212yesmaximal, complemented
4#33147C212yesmaximal, complemented
5#29952C212yesmaximal
6#29950C212yesmaximal
7#26102C218yesmaximal, complemented
8#26036C218yesmaximal, complemented
9#21986C212yesmaximal, complemented
10#21955C211yesmaximal, characteristic, complemented
11#20738C212yesmaximal
12C2×#544—21no—
13C22×C4×D4C2211yescharacteristic
14C2×#432C2212yes—
15#2857C2212yes—
16#2857C2212yes—
17#2659—21no—
18#2659—21no—
19#2552C2214yescomplemented
20#2551C2214yescomplemented
21#2550—24no—
22#2357—24no—
23#2357C2218yes—
24#2316—21no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#388—21no—
28C2×#388—21no—
29#2285C2212yes—
30C2×#382C2211yescharacteristic
31C2×#381—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4—21no—
36#2094C2214yes—
37#2093C2214yes—
38#2067C2218yes—
39#2065C2218yes—
40#2023C2214yes—
41#2023C2214yes—
42#1988C2214yescomplemented
43#1987C2214yescomplemented
44#1949C2218yescomplemented
45#1948C2214yescomplemented
46#1947C2214yescomplemented
47#1940C2218yescomplemented
48#1921C2218yescomplemented
49#1919C2218yescomplemented
50#1853C2214yes—
51#1852C2214yes—
52#1847C2214yescomplemented
53#1836C2214yescomplemented
54#1824C2214yes—
55#1802C2214yes—
56#1802C2214yes—
57#1797C2212yes—
58#1797C2212yes—
59#1786C2214yes—
60#1762C2218yes—
61C23×D4C2311yescharacteristic
62C24×C4—21no—
63#544—48no—
64C2×C4⋊Q8—24no—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#125—24no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C4×D4—22no—
71C2×C4×D4—24no—
72C2×C4×D4Q818yescomplemented
73C2×C4×D4C2314yes—
74C2×#117—22no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#432—28no—
91C2×#107C2318yes—
92C2×#106C2318yes—
93C2×C4⋊C8C2314yes—
94C2×C4⋊C8C2314yes—
95C2×C4⋊C8C2314yes—
96C2×#102C2318yes—
97C2×#102C2318yes—
98C2×C22⋊C8C2314yes—
99C2×#97C2314yes—
100#398—24no—
101#398—24no—
102#395—24no—
103#395—24no—
104#391—22no—
105#391—22no—
106#390—22no—
107#390—22no—
108#390C2314yes—
109#390C2314yes—
110#389—24no—
111#389—24no—
112#389—22no—
113#389—22no—
114#389C2314yes—
115#389C2314yes—
116#388—28no—
117#388—28no—
118#384C2312yes—
119#384C2312yes—
120#384C2312yes—
121#384C2312yes—
122#382—22no—
123#382—22no—
124#382C2314yes—
125#381—28no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380C2314yes—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4D414yes—
134C4×C4⋊C4C2314yes—
135C4×C4⋊C4C2314yes—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C22⋊C4—28no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95D411yescharacteristic
144C2×#95D411yescharacteristic
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C25—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4C2×Q814yes—
159C22×D4C2412yes—
160C23×C4—21no—
161C23×C4—28no—
162C23×C4—24no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C22×C8C2418yes—
175C4⋊Q8—44no—
176C4⋊Q8—44no—
177C4⋊Q8—416no—
178#125—416no—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C4×D4—216no—
182C4×D4—28no—
183C4×D4C2×Q8116yes—
184#117—48no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×Q814yes—
208C2×C4⋊C4C2412yes—
209C2×C4⋊C4C2412yes—
210C2×C4⋊C4C2414yes—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2412yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—82no—
230C2×C42—82no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×Q814yes—
235C2×C42#4114yes—
236C2×C42C2412yes—
237C2×C42C2412yes—
238C2×C42C2412yes—
239#107—48no—
240#106—48no—
241C4⋊C8—28no—
242C4⋊C8—44no—
243C4⋊C8—44no—
244#102—48no—
245#102—48no—
246C22⋊C8—28no—
247#97—28no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95C2×D414yes—
257#95C2×D414yes—
258C24—24no—
259C24—22no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24C22×D411yescharacteristic
267C2×Q8—44no—
268C2×Q8—44no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—48no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—48no—
281C2×D4C22×Q814yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—216no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4—816no—
316C22×C4C22⋊Q814yes—
317C22×C4C22⋊Q814yes—
318C22×C4AGL(1,ℤ/8)14yes—
319C22×C4#13714yes—
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×D412yes—
327C22×C4C22×Q812yes—
328C22×C4C2×#4112yes—
329C22×C4C2511yescharacteristic
330C2×C8—28no—
331C2×C8—28no—
332C2×C8—28no—
333C4⋊C4—24no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—22no—
337C4⋊C4—28no—
338C4⋊C4—28no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—48no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4C22×Q814yes—
356C22⋊C4—216no—
357C22⋊C4—48no—
358C22⋊C4—48no—
359C22⋊C4—48no—
360C22⋊C4—84no—
361C22⋊C4—84no—
362C42—24no—
363C42—24no—
364C42—42no—
365C42—42no—
366C42—48no—
367C42—84no—
368C42—84no—
369C42C22⋊Q818yes—
370C42C22×Q814yes—
371C42C2×#4114yes—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—24no—
377C23—48no—
378C23—48no—
379C23—42no—
380C23—42no—
381C23—48no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—84no—
387C23—84no—
388C23—84no—
389C23—84no—
390C23C2×C22⋊Q812yes—
391C23C2×AGL(1,ℤ/8)12yes—
392C23C2×#13712yes—
393C23C23×D411yescharacteristic
394Q8—164no—
395Q8—164no—
396D4—28no—
397D4—48no—
398D4—44no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—28no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—816no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4C2×C22⋊Q812yes—
443C2×C4C2×C22⋊Q814yes—
444C2×C4C2×C22⋊Q814yes—
445C2×C4C2×AGL(1,ℤ/8)14yes—
446C2×C4C2×#13714yes—
447C2×C4C23×Q811yescharacteristic
448C2×C4C22×#4111yescharacteristic
449C2×C4#58512yes—
450C2×C4#58512yes—
451C8—48no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—44no—
459C22—44no—
460C22—44no—
461C22—816no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22#307914yes—
467C22C22×C22⋊Q811yescharacteristic
468C22#348511yescharacteristic
469C22#348711yescharacteristic
470C22C22×AGL(1,ℤ/8)11yescharacteristic
471C22C22×#13711yescharacteristic
472C22#358511yescharacteristic
473C22#358511yescharacteristic
474C4—24no—
475C4—24no—
476C4—24no—
477C4—24no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—44no—
482C4—84no—
483C4—84no—
484C4—82no—
485C4—82no—
486C4#345812yes—
487C4#349312yes—
488C2—24no—
489C2—84no—
490C2#3515111yescharacteristic
491C2C2×#307911yescharacteristic
492C2#6049011yescharacteristic
493C2#6170711yescharacteristic
494C2#6170811yescharacteristic
495C2#6189111yescharacteristic
496C2#6189211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35167C2semidirect
3. #33149C2semidirect
4. #33147C2semidirect
7. #26102C2semidirect
8. #26036C2semidirect
9. #21986C2semidirect
10. #21955C2semidirect
19. #2552C22semidirect
20. #2551C22semidirect
42. #1988C22semidirect
43. #1987C22semidirect
44. #1949C22semidirect
45. #1948C22semidirect
46. #1947C22semidirect
47. #1940C22semidirect
48. #1921C22semidirect
49. #1919C22semidirect
52. #1847C22semidirect
53. #1836C22semidirect
72. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#23572
#331492
#331472
#299522
#299502
#261022
#260362
#219862
#219552
#207382