Finite Group Databasebeta

← #6487698

Subgroups — #6487698

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2C2×#3101C211yesmaximal, characteristic
3#37641C211yesmaximal, characteristic, complemented
4#34181C211yesmaximal, characteristic
5#32914C214yesmaximal
6#29764C214yesmaximal, complemented
7#24834C214yesmaximal, complemented
8#24816C214yesmaximal, complemented
9#24748C214yesmaximal, complemented
10#24730C214yesmaximal, complemented
11#21954C211yesmaximal, characteristic, complemented
12#21870C211yesmaximal, characteristic, complemented
13#21864C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#533C2211yescharacteristic
16C2×#518C2212yes—
17C22×C4×D4C2211yescharacteristic
18#3110—28no—
19#3101—28no—
20C2×#484C2212yes—
21C2×#483C2212yes—
22C2×#476C2212yes—
23C2×#474C2212yes—
24C2×#461C2212yes—
25C2×#461C2212yes—
26C2×#461C2218yes—
27C2×#450C2212yes—
28C2×#450C2212yes—
29C2×#440C2214yes—
30C2×C4×SD16C2214yes—
31C2×#432C2211yescharacteristic
32C2×#431C2211yescharacteristic
33#2822—22no—
34#2625—22no—
35#2625—22no—
36#2621C2214yes—
37#2582—22no—
38#2568C2212yescomplemented
39#2568C2212yescomplemented
40#2546C2212yescomplemented
41#2546C2212yescomplemented
42#2349—22no—
43#2349C2214yes—
44D4×C4⋊C4—22no—
45D4×C4⋊C4C2214yes—
46#2316C2212yes—
47#2315—22no—
48#2285C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#380—22no—
51#2085C2218yescomplemented
52#2068C2218yescomplemented
53#2066C2218yescomplemented
54#2043C2218yescomplemented
55#1993C2212yescomplemented
56#1990C2212yescomplemented
57#1960C2212yescomplemented
58#1956C2212yescomplemented
59#1940C2214yescomplemented
60#1939C2214yescomplemented
61#1877C2218yes—
62#1877C2218yes—
63#1873C2214yes—
64#1873C2214yes—
65#1805C2214yescomplemented
66#1804C2218yes—
67#1803C2214yescomplemented
68#1798C2214yes—
69C23×D4C2311yescharacteristic
70C24×C4—21no—
71C22×SD16C2314yes—
72C22×SD16C2314yes—
73#533—24no—
74#533—24no—
75#533—24no—
76#533—24no—
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2314yes—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#125—21no—
83C2×#125—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8C2312yes—
87C2×C22⋊Q8C2312yes—
88#518—48no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8C2312yes—
92C2×C4×Q8C2312yes—
93C2×C4×D4—24no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—24no—
97C2×C4×D4D418yes—
98C2×C4×D4C2318yes—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×#117—21no—
102C2×#117—21no—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—24no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—24no—
109C22×C42C2311yescharacteristic
110#484—28no—
111#483—28no—
112#476—28no—
113#474—28no—
114#461—28no—
115#461—28no—
116#461—416no—
117#450—28no—
118#450—28no—
119#440—216no—
120C4×SD16—216no—
121#432—24no—
122#431—24no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2318yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2318yes—
131C2×#103C2318yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138#400—24no—
139#400—24no—
140#398—28no—
141#398—24no—
142#398—24no—
143#397—28no—
144#391—24no—
145#391—24no—
146#391C2314yes—
147#391C2314yes—
148#389—24no—
149#389—24no—
150#389—24no—
151#389C2318yes—
152#385—28no—
153#385—28no—
154#384D412yes—
155#384C2312yes—
156#382D412yes—
157#382C2314yes—
158#382C2312yes—
159#380—21no—
160#380—21no—
161#380—24no—
162#380—24no—
163#380—24no—
164#380D412yes—
165#380D412yes—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4C2312yes—
169C4×C4⋊C4C2312yes—
170C2×#95—24no—
171#340C2318yes—
172#340C2318yes—
173#336C2318yes—
174C25—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8C2412yes—
178C22×Q8C2412yes—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—21no—
182C22×D4—22no—
183C22×D4C2×D414yes—
184C22×D4C2412yes—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—28no—
188C23×C4—22no—
189C23×C4—28no—
190C23×C4—84no—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2411yescharacteristic
194C2×SD16—28no—
195C2×SD16—28no—
196C2×SD16—28no—
197C2×SD16—28no—
198C2×SD16—28no—
199C2×SD16—28no—
200C22×C8C2414yes—
201C22×C8C2414yes—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4⋊Q8—48no—
211#125—44no—
212#125—44no—
213C22⋊Q8—24no—
214C22⋊Q8—24no—
215C22⋊Q8—24no—
216C22⋊Q8—24no—
217C22⋊Q8—44no—
218C22⋊Q8—44no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—44no—
224C4×Q8—44no—
225C4×D4—216no—
226C4×D4—28no—
227C4×D4—28no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—48no—
231C4×D4—416no—
232C4×D4C2×D418yes—
233C4×D4C2×D418yes—
234#117—44no—
235#117—44no—
236#117—44no—
237#117—44no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—88no—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—416no—
271C2×C22⋊C4C2×D414yes—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D414yes—
278C2×C42#4114yes—
279C2×C42C2414yes—
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282#107—28no—
283#106—28no—
284C4⋊C8—24no—
285C4⋊C8—24no—
286C4⋊C8—48no—
287#103—28no—
288#103—28no—
289#103—48no—
290#103—48no—
291#102—28no—
292#102—28no—
293C22⋊C8—24no—
294C22⋊C8—24no—
295#97—24no—
296C4×C8—24no—
297#95—22no—
298#95—28no—
299#95—22no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95C2×D414yes—
304#95C2×D414yes—
305C24—24no—
306C24—24no—
307C24—22no—
308C24—24no—
309C24—82no—
310C24—82no—
311C24C22×D411yescharacteristic
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—44no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—44no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—216no—
327C2×D4—216no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4C22×D414yes—
333C2×D4H(5,𝔽2)14yes—
334C2×D42−1+414yes—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—416no—
349C22×C4—48no—
350C22×C4—816no—
351C22×C4—84no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4—82no—
355C22×C4#12114yes—
356C22×C4#12114yes—
357C22×C4C22×D418yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4C2511yescharacteristic
366SD16—416no—
367SD16—416no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C4⋊C4—24no—
375C4⋊C4—28no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—28no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—82no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4C22×D414yes—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—44no—
402C22⋊C4—416no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C42—28no—
407C42—44no—
408C42—44no—
409C42—44no—
410C42#12114yes—
411C42#12114yes—
412C42C22×D412yes—
413C42C22×D412yes—
414C42C2×#4112yes—
415C42C2×#4112yes—
416C23—216no—
417C23—22no—
418C23—24no—
419C23—21no—
420C23—24no—
421C23—21no—
422C23—24no—
423C23—22no—
424C23—44no—
425C23—44no—
426C23—84no—
427C23—84no—
428C23—88no—
429C23—88no—
430C23C2×#12112yes—
431C23C23×D411yescharacteristic
432Q8—44no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—84no—
437Q8—84no—
438D4—216no—
439D4—24no—
440D4—24no—
441D4—24no—
442D4#53718yes—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—416no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—88no—
472C2×C4—84no—
473C2×C4—1616no—
474C2×C4C2×#12112yes—
475C2×C4C2×#12112yes—
476C2×C4C2×#12112yes—
477C2×C4C2×#12112yes—
478C2×C4C2×#12112yes—
479C2×C4#57714yes—
480C2×C4C23×D411yescharacteristic
481C2×C4C22×#4111yescharacteristic
482C2×C4C2×H(5,𝔽2)12yes—
483C2×C4C2×2−1+412yes—
484C8—44no—
485C8—44no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—28no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—48no—
494C22—48no—
495C22—42no—
496C22—816no—
497C22C22×#12111yescharacteristic
498C22#348411yescharacteristic
499C22#348611yescharacteristic
500C22C2×#57712yes—
501C22#358612yes—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—44no—
507C4—48no—
508C4—82no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—84no—
513C4#304214yes—
514C4C2×#53712yes—
515C4#349212yes—
516C2—42no—
517C2—48no—
518C2#3522311yescharacteristic
519C2C2×#304211yescharacteristic
520C2#6042711yescharacteristic
521C2#6168012yes—
522C2#6184812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #37641C2semidirect
6. #29764C2semidirect
7. #24834C2semidirect
8. #24816C2semidirect
9. #24748C2semidirect
10. #24730C2semidirect
11. #21954C2semidirect
12. #21870C2semidirect
13. #21864C2semidirect
38. #2568C22semidirect
39. #2568C22semidirect
40. #2546C22semidirect
41. #2546C22semidirect
51. #2085C22semidirect
52. #2068C22semidirect
53. #2066C22semidirect
54. #2043C22semidirect
55. #1993C22semidirect
56. #1990C22semidirect
57. #1960C22semidirect
58. #1956C22semidirect
59. #1940C22semidirect
60. #1939C22semidirect
65. #1805C22semidirect
67. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C2×#31012
#376412
#341812
#329142
#297642
#248342
#248162
#247482
#247302
#219542
#218702
#218642