Finite Group Databasebeta

← #6470345

Subgroups — #6470345

Order 512.

3570 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
1C2×#3189C211yesmaximal, characteristic, complemented
2C23×C22⋊C8C211yesmaximal, characteristic
3C2×#2848C212yesmaximal
4#32931C214yesmaximal, complemented
5#29731C214yesmaximal, complemented
6#24552C218yesmaximal, complemented
7#24380C218yesmaximal, complemented
8#21940C211yesmaximal, characteristic, complemented
9#21759C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C24×C8C2212yes—
12C22×C22⋊Q8C2212yes—
13C22×C22⋊Q8C2214yes—
14#3189—28no—
15C2×#484C2214yescomplemented
16C2×#467C2214yescomplemented
17C2×#467C2218yescomplemented
18C2×#451C2214yescomplemented
19C22×C22⋊C8—28no—
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2218yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2848—22no—
25#2848—22no—
26#2848—24no—
27#2848C2214yes—
28#2848C2218yes—
29#2848C2214yes—
30C2×#385—24no—
31C2×#385—22no—
32C2×#380C2212yes—
33C2×#380C2214yes—
34C2×#379—22no—
35#2076C2218yescomplemented
36#2036C22116yescomplemented
37#2001C2218yescomplemented
38#1892C2218yescomplemented
39#1892C22116yescomplemented
40#1872C2218yescomplemented
41#1852C2218yescomplemented
42#1824C2214yescomplemented
43#1808C2214yescomplemented
44#1808C2218yescomplemented
45#1787C2214yescomplemented
46C26D411yescharacteristic
47C23×Q8C2312yes—
48C24×C4—24no—
49C24×C4—24no—
50C24×C4—48no—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D414yes—
54C24×C4D411yescharacteristic
55C24×C4C2311yescharacteristic
56C24×C4C2314yes—
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C22×Q16C2314yescomplemented
60C23×C8—28no—
61C23×C8—48no—
62C23×C8C2318yes—
63C23×C8C2312yes—
64C23×C8C2312yes—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—416no—
72C2×C22⋊Q8C2314yes—
73C2×C22⋊Q8C2314yes—
74C2×C22⋊Q8C2318yes—
75C2×C22⋊Q8C2318yes—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—22no—
84#484—216no—
85#467—216no—
86#467—416no—
87#451—216no—
88C2×#107C2314yescomplemented
89C2×#107C2318yescomplemented
90C2×#103C2318yescomplemented
91C2×#103C23116yescomplemented
92C2×C22⋊C8—24no—
93C2×C22⋊C8—28no—
94C2×C22⋊C8—232no—
95C2×C22⋊C8D418yes—
96C2×C22⋊C8C2318yes—
97C2×C22⋊C8C2314yes—
98C2×C22⋊C8C2314yes—
99#385—28no—
100#385—24no—
101#385—28no—
102#385—28no—
103#385—28no—
104#385—24no—
105#385—28no—
106#380—24no—
107#380—24no—
108#380—22no—
109#380—22no—
110#380C2314yes—
111#380C2318yes—
112#380C2318yes—
113#380C2314yes—
114#379—24no—
115#379—28no—
116#379—24no—
117C2×#95—24no—
118C2×#95—22no—
119#340C23116yescomplemented
120#340C2318yescomplemented
121C25—24no—
122C25—24no—
123C25—44no—
124C25—44no—
125C25Q1618yescomplemented
126C25C2×D411yescharacteristic
127C25C2×D414yes—
128C25C2×D411yescharacteristic
129C25C2×D411yescharacteristic
130C22×Q8—22no—
131C22×Q8—24no—
132C22×Q8—24no—
133C22×Q8—48no—
134C22×Q8C2414yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—28no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—216no—
141C23×C4—24no—
142C23×C4—28no—
143C23×C4—22no—
144C23×C4—28no—
145C23×C4—22no—
146C23×C4—216no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—44no—
151C23×C4—48no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—48no—
155C23×C4—44no—
156C23×C4—48no—
157C23×C4—44no—
158C23×C4—432no—
159C23×C4—432no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D414yes—
163C23×C4C2×D414yes—
164C23×C4C2×D414yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D414yes—
170C23×C4#4114yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C2×Q16—28no—
176C2×Q16—48no—
177C22×C8—24no—
178C22×C8—216no—
179C22×C8—216no—
180C22×C8—44no—
181C22×C8—48no—
182C22×C8—44no—
183C22×C8—432no—
184C22×C8C2×D418yes—
185C22×C8C2414yes—
186C22×C8C2414yes—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—28no—
190C22⋊Q8—28no—
191C22⋊Q8—232no—
192C22⋊Q8—416no—
193C22⋊Q8—416no—
194C22⋊Q8—416no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—416no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4C2414yes—
209C2×C4⋊C4C2418yes—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—416no—
219C2×C22⋊C4—416no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—48no—
222#107—28no—
223#107—48no—
224#103—216no—
225#103—416no—
226C22⋊C8—28no—
227C22⋊C8—216no—
228C22⋊C8C2×D4116yes—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—28no—
233C24—22no—
234C24—24no—
235C24—22no—
236C24—24no—
237C24—28no—
238C24—22no—
239C24—28no—
240C24—28no—
241C24—22no—
242C24—28no—
243C24—232no—
244C24—42no—
245C24—44no—
246C24—42no—
247C24—432no—
248C24—42no—
249C24—44no—
250C24—42no—
251C24—416no—
252C24—416no—
253C24—432no—
254C24#12114yes—
255C24C2×Q1614yes—
256C24C2×Q16116yes—
257C24C2×Q1614yes—
258C24C2×Q1614yes—
259C24#13514yes—
260C24C22×D412yes—
261C24C22×D411yescharacteristic
262C24C22×D412yes—
263C24C22×D412yes—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—216no—
267C2×Q8—48no—
268C2×Q8—44no—
269C2×Q8—88no—
270C2×Q8—88no—
271C22×C4—216no—
272C22×C4—216no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—28no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—28no—
284C22×C4—22no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—28no—
294C22×C4—24no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—432no—
307C22×C4—432no—
308C22×C4—416no—
309C22×C4—464no—
310C22×C4—416no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—416no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—48no—
321C22×C4#12118yes—
322C22×C4#12114yes—
323C22×C4C22×D412yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D412yes—
326C22×C4C22×D414yes—
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C2×#4114yes—
330C22×C4C2×#4112yes—
331C22×C4H(5,𝔽2)18yes—
332C22×C4C2511yescharacteristic
333Q16—88no—
334C2×C8—28no—
335C2×C8—24no—
336C2×C8—28no—
337C2×C8—48no—
338C2×C8—432no—
339C2×C8—48no—
340C2×C8C22×D418yes—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—216no—
344C4⋊C4—48no—
345C4⋊C4—416no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C4⋊C4—48no—
349C4⋊C4—416no—
350C22⋊C4—416no—
351C22⋊C4—416no—
352C22⋊C4—48no—
353C22⋊C4—48no—
354C22⋊C4—416no—
355C22⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—416no—
358C23—28no—
359C23—28no—
360C23—216no—
361C23—22no—
362C23—216no—
363C23—24no—
364C23—22no—
365C23—22no—
366C23—28no—
367C23—22no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—232no—
373C23—24no—
374C23—232no—
375C23—28no—
376C23—22no—
377C23—24no—
378C23—44no—
379C23—44no—
380C23—48no—
381C23—416no—
382C23—48no—
383C23—48no—
384C23—416no—
385C23—416no—
386C23—416no—
387C23—416no—
388C23—44no—
389C23—4128no—
390C23—44no—
391C23—48no—
392C23#467116yescomplemented
393C23C2×#12112yes—
394C23C2×#12114yes—
395C23#53414yes—
396C23C22×Q1614yes—
397C23C22×Q1614yes—
398C23C22×Q1612yes—
399C23C22×Q1614yes—
400C23C2×#13512yes—
401C23C2×#13514yes—
402C23#57514yes—
403C23C23×D411yescharacteristic
404Q8—48no—
405Q8—48no—
406Q8—88no—
407C2×C4—216no—
408C2×C4—28no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—416no—
424C2×C4—416no—
425C2×C4—416no—
426C2×C4—48no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—48no—
437C2×C4—88no—
438C2×C4—816no—
439C2×C4—88no—
440C2×C4C2×#12112yes—
441C2×C4C2×#12114yes—
442C2×C4#53414yes—
443C2×C4C23×D411yescharacteristic
444C2×C4C22×#4111yescharacteristic
445C2×C4C2×H(5,𝔽2)14yes—
446C8—28no—
447C8—44no—
448C22—232no—
449C22—24no—
450C22—24no—
451C22—22no—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—28no—
456C22—28no—
457C22—24no—
458C22—24no—
459C22—24no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—28no—
465C22—416no—
466C22—464no—
467C22—416no—
468C22—42no—
469C22—44no—
470C22—42no—
471C22C2×#46714yes—
472C22C2×#46718yes—
473C22#318918yescomplemented
474C22C22×#12111yescharacteristic
475C22C2×#53412yes—
476C22C23×Q1611yescharacteristic
477C22C22×#13511yescharacteristic
478C22C2×#57512yes—
479C4—22no—
480C4—24no—
481C4—24no—
482C4—42no—
483C4—84no—
484C4—88no—
485C4—84no—
486C4#348412yes—
487C4#349012yes—
488C2—28no—
489C2—24no—
490C2—28no—
491C2—44no—
492C2#3498111yescharacteristic
493C2C22×#46711yescharacteristic
494C2#6044411yescharacteristic
495C2C2×#318912yes—
496C2#6073412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3189C2semidirect
4. #32931C2semidirect
5. #29731C2semidirect
6. #24552C2semidirect
7. #24380C2semidirect
8. #21940C2semidirect
9. #21759C2semidirect
15. C2×#484C22semidirect
16. C2×#467C22semidirect
17. C2×#467C22semidirect
18. C2×#451C22semidirect
35. #2076C22semidirect
36. #2036C22semidirect
37. #2001C22semidirect
38. #1892C22semidirect
39. #1892C22semidirect
40. #1872C22semidirect
41. #1852C22semidirect
42. #1824C22semidirect
43. #1808C22semidirect
44. #1808C22semidirect
45. #1787C22semidirect
59. C22×Q16C23semidirect
88. C2×#107C23semidirect
89. C2×#107C23semidirect
90. C2×#103C23semidirect
91. C2×#103C23semidirect
119. #340C23semidirect
120. #340C23semidirect
125. C25Q16semidirect
392. C23#467semidirect
473. C22#3189semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C23×C22⋊C82
C2×#28482
#329312
#297312
#245522
#243802
#219402
#217592