Finite Group Databasebeta

← #6532002

Subgroups — #6532002

Order 512.

1808 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic, complemented
3#27614C212yesmaximal, complemented
4#27611C212yesmaximal, complemented
5#27585C212yesmaximal
6#27582C212yesmaximal
7#25068C218yesmaximal, complemented
8#25002C218yesmaximal, complemented
9#21986C212yesmaximal, complemented
10#21822C211yesmaximal, characteristic, complemented
11#20738C212yesmaximal
12C2×#519—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#432C2212yes—
16#2580—24no—
17#2580C2218yes—
18C2×#399—21no—
19C2×#393—21no—
20#2351—24no—
21Q8×C4⋊C4C2214yescomplemented
22Q8×C4⋊C4C2214yescomplemented
23#2316—21no—
24#2316—21no—
25#2316—21no—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#385—21no—
31C2×#385—21no—
32#2285C2212yes—
33#2285C2212yes—
34#2285C2212yes—
35C2×#382C2211yescharacteristic
36#2100C2214yescomplemented
37#2099C2214yescomplemented
38#2094C2214yes—
39#2093C2214yes—
40#2084C2214yescomplemented
41#2083C2214yescomplemented
42#2071C2214yes—
43#2071C2214yes—
44#1985C2218yescomplemented
45#1949C2218yescomplemented
46#1926C2218yes—
47#1925C2218yes—
48#1921C2218yescomplemented
49#1920C2218yescomplemented
50#1853C2214yes—
51#1852C2214yes—
52#1847C2214yescomplemented
53#1836C2214yescomplemented
54#1824C2214yes—
55#1817C2212yes—
56#1817C2212yes—
57#1802C2214yes—
58#1802C2214yes—
59#1786C2214yes—
60#1762C2218yes—
61C23×Q8D411yescharacteristic
62C23×D4C2311yescharacteristic
63C24×C4—21no—
64#519—48no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—28no—
68C2×C4×Q8D414yes—
69C2×C4×Q8C2314yes—
70C2×C4×D4—22no—
71C2×C4×D4—28no—
72C2×C4×D4C2314yes—
73C2×#117—22no—
74C2×#117—22no—
75C2×#117—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—28no—
93C2×#107C2318yes—
94C2×#106C2318yes—
95C2×C4⋊C8C2314yes—
96C2×#102C2318yes—
97C2×#102C2318yes—
98C2×C22⋊C8C2314yes—
99C2×#97C2314yes—
100#399—28no—
101#393—28no—
102#390—22no—
103#390—22no—
104#390C2314yes—
105#390C2314yes—
106#389—24no—
107#389—24no—
108#389—24no—
109#389—24no—
110#389—24no—
111#389—24no—
112#389—22no—
113#389—22no—
114#389C2314yes—
115#389C2314yes—
116#385—28no—
117#385—28no—
118#384C2312yes—
119#384C2312yes—
120#384C2312yes—
121#384C2312yes—
122#384C2312yes—
123#384C2312yes—
124#382—22no—
125#382—22no—
126#382—22no—
127#382—22no—
128#382C2314yes—
129#380—22no—
130#380—22no—
131#380—22no—
132#380C2314yes—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#336C2314yes—
148#336C2314yes—
149#336C2314yes—
150#336C2314yes—
151C25—21no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8C2×D414yes—
156C22×D4—24no—
157C22×D4—21no—
158C22×D4—24no—
159C22×D4—42no—
160C22×D4C2412yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—44no—
166C23×C4—82no—
167C23×C4—82no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C22×C8C2418yes—
176C4×Q8—216no—
177C4×Q8—28no—
178C4×Q8—28no—
179C4×Q8—416no—
180C4×Q8—416no—
181C4×D4—28no—
182C4×D4—28no—
183C4×D4—48no—
184#117—48no—
185#117—48no—
186#117—48no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2412yes—
216C2×C4⋊C4C2412yes—
217C2×C4⋊C4C2412yes—
218C2×C4⋊C4C2414yes—
219C2×C4⋊C4C2414yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42#4118yes—
240C2×C42C2412yes—
241C2×C42C2412yes—
242C2×C42C2412yes—
243#107—48no—
244#106—48no—
245C4⋊C8—28no—
246#102—216no—
247#102—48no—
248C22⋊C8—28no—
249#97—28no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258C24—24no—
259C24—22no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24C22×D411yescharacteristic
267C2×Q8—232no—
268C2×Q8—28no—
269C2×Q8—28no—
270C2×Q8—28no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—416no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D42−1+414yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—48no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4—816no—
313C22×C4#12314yes—
314C22×C4#12314yes—
315C22×C4AGL(1,ℤ/8)14yes—
316C22×C4#13714yes—
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D412yes—
324C22×C4C2×#4114yes—
325C22×C4C2511yescharacteristic
326C2×C8—28no—
327C2×C8—28no—
328C2×C8—28no—
329C4⋊C4—28no—
330C4⋊C4—28no—
331C4⋊C4—28no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C22⋊C4—48no—
355C22⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C22⋊C4—84no—
359C22⋊C4—84no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—48no—
365C42—48no—
366C42—48no—
367C42#12318yes—
368C42C2×#4114yes—
369C42C2×#4114yes—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—24no—
375C23—48no—
376C23—48no—
377C23—48no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—84no—
385C23—84no—
386C23—84no—
387C23—84no—
388C23C2×#12312yes—
389C23C2×AGL(1,ℤ/8)12yes—
390C23C2×#13712yes—
391C23C23×D411yescharacteristic
392Q8—232no—
393D4—28no—
394D4—48no—
395D4—44no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—28no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—816no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4C2×#12312yes—
439C2×C4C2×#12314yes—
440C2×C4C2×#12314yes—
441C2×C4C2×AGL(1,ℤ/8)14yes—
442C2×C4C2×#13714yes—
443C2×C4C22×#4112yes—
444C2×C4C2×2−1+412yes—
445C2×C4#58512yes—
446C8—48no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—816no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22#309214yes—
462C22C22×#12311yescharacteristic
463C22#348611yescharacteristic
464C22#348711yescharacteristic
465C22C22×AGL(1,ℤ/8)11yescharacteristic
466C22C22×#13711yescharacteristic
467C22#358511yescharacteristic
468C22#358511yescharacteristic
469C4—22no—
470C4—24no—
471C4—24no—
472C4—24no—
473C4—24no—
474C4—42no—
475C4—42no—
476C4—82no—
477C4—84no—
478C4—84no—
479C4—82no—
480C4—82no—
481C4#349514yes—
482C2—24no—
483C2—84no—
484C2#3567211yescharacteristic
485C2C2×#309211yescharacteristic
486C2#6051811yescharacteristic
487C2#6182511yescharacteristic
488C2#6182611yescharacteristic
489C2#6189111yescharacteristic
490C2#6189211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34182C2semidirect
3. #27614C2semidirect
4. #27611C2semidirect
7. #25068C2semidirect
8. #25002C2semidirect
9. #21986C2semidirect
10. #21822C2semidirect
21. Q8×C4⋊C4C22semidirect
22. Q8×C4⋊C4C22semidirect
36. #2100C22semidirect
37. #2099C22semidirect
40. #2084C22semidirect
41. #2083C22semidirect
44. #1985C22semidirect
45. #1949C22semidirect
48. #1921C22semidirect
49. #1920C22semidirect
52. #1847C22semidirect
53. #1836C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
#341822
#276142
#276112
#275852
#275822
#250682
#250022
#219862
#218222
#207382