Finite Group Databasebeta

← #6508967

Subgroups — #6508967

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27617C214yesmaximal, complemented
4#27604C214yesmaximal, complemented
5#24947C218yesmaximal, complemented
6#24943C218yesmaximal, complemented
7#21940C211yesmaximal, characteristic, complemented
8#21938C211yesmaximal, characteristic, complemented
9#21738C211yesmaximal, characteristic
10C2×#1824C211yesmaximal, characteristic, complemented
11C2×#1824C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2487—22no—
17#2487—22no—
18#2487C2214yes—
19#2487C2214yes—
20C2×#400—21no—
21C2×#399—21no—
22#2353—22no—
23#2353—22no—
24#2352C2214yes—
25#2352C2214yes—
26#2313—21no—
27#2312—21no—
28#2308—21no—
29C2×#389C2212yes—
30C2×#386—21no—
31C2×#385—21no—
32#2283C2212yes—
33#2280C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38#2273—21no—
39#2101C2218yescomplemented
40#2085C2218yescomplemented
41#2077C2218yescomplemented
42#2075C2218yescomplemented
43#1919C2218yescomplemented
44#1918C2218yescomplemented
45#1869C2214yes—
46#1869C2214yes—
47#1854C2218yescomplemented
48#1853C2214yes—
49#1853C2214yescomplemented
50#1852C2214yes—
51#1852C2214yescomplemented
52#1825C2218yescomplemented
53#1824C2214yes—
54#1824C2214yes—
55#1824C2214yescomplemented
56#1824C2214yescomplemented
57#1817C2212yes—
58#1808C2214yescomplemented
59#1807C2214yescomplemented
60#1779C2212yes—
61C2×#340C2214yescomplemented
62C2×#340C2214yescomplemented
63C23×D4D411yescharacteristic
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C23×C8C2312yes—
68C23×C8C2312yes—
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4D412yes—
72C2×C4×D4D412yes—
73C2×C4×D4C2312yes—
74C2×C4×D4C2312yes—
75C2×#117C2314yes—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4D411yescharacteristic
94C22×C22⋊C4D411yescharacteristic
95C22×C42C2311yescharacteristic
96C2×#107C2318yes—
97C2×#106C2318yes—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8C2312yes—
101C2×C22⋊C8C2312yes—
102C2×C22⋊C8C2312yes—
103C2×C22⋊C8C2312yes—
104C2×C22⋊C8C2318yes—
105#400—24no—
106#400—24no—
107#399—24no—
108#399—24no—
109#389—24no—
110#389C2318yes—
111#388—24no—
112#388—24no—
113#387—24no—
114#387—24no—
115#386—24no—
116#386—24no—
117#386—24no—
118#386—24no—
119#385—24no—
120#385—24no—
121#383—22no—
122#382C2314yes—
123#381—22no—
124#381—22no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—24no—
130#380—21no—
131#380—21no—
132#380C2312yes—
133#380C2312yes—
134#380C2312yes—
135#380C2312yes—
136#380C2314yes—
137#380C2318yes—
138#380C2312yes—
139#380C2312yes—
140#379—22no—
141C4×C4⋊C4C2314yes—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147#376—22no—
148C2×#95—21no—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95—22no—
152C2×#95—22no—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#340C2318yes—
157#340C2318yes—
158#340C2318yes—
159#336C2318yes—
160C25C2×D411yescharacteristic
161C25C2×D411yescharacteristic
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4C2×D412yes—
166C22×D4C2×D412yes—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—84no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4114yes—
187C23×C4#4114yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—44no—
195C22×C8—44no—
196C22×C8C2414yes—
197C22×C8C2414yes—
198C4×D4—28no—
199C4×D4—28no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—24no—
204#117—48no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—84no—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2414yes—
229C2×C4⋊C4C2414yes—
230C2×C4⋊C4C2414yes—
231C2×C4⋊C4C2414yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—84no—
247C2×C22⋊C4C2×D412yes—
248C2×C22⋊C4C2×D412yes—
249C2×C22⋊C4C2×D412yes—
250C2×C22⋊C4C2×D412yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42C2×D412yes—
256C2×C42C2412yes—
257#107—48no—
258#106—48no—
259C22⋊C8—24no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8—24no—
263C22⋊C8—48no—
264#95—22no—
265#95—22no—
266#95—28no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24#12314yes—
284C24#13514yes—
285C24#13714yes—
286C24C22×D412yes—
287C24C22×D411yescharacteristic
288C24C22×D411yescharacteristic
289C24C22×D411yescharacteristic
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—416no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—28no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—816no—
340C22×C4#12314yes—
341C22×C4#12314yes—
342C22×C4#12314yes—
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4112yes—
350C22×C4C2×#4112yes—
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4112yes—
354C22×C4C2×#4112yes—
355C22×C4C2511yescharacteristic
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—28no—
359C2×C8—28no—
360C2×C8—84no—
361C2×C8—84no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—28no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—88no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—84no—
387C22⋊C4—88no—
388C22⋊C4—84no—
389C42—24no—
390C42—22no—
391C42—22no—
392C42—84no—
393C42—84no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—24no—
401C23—22no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—44no—
406C23—44no—
407C23—48no—
408C23—48no—
409C23—48no—
410C23—44no—
411C23—82no—
412C23—84no—
413C23—82no—
414C23#485116yes—
415C23C2×#12312yes—
416C23C2×#12312yes—
417C23C2×#12312yes—
418C23C2×#13512yes—
419C23C2×#13512yes—
420C23C2×#13512yes—
421C23C2×#13712yes—
422C23C2×#13712yes—
423C23C2×#13712yes—
424C23C23×D411yescharacteristic
425D4—48no—
426D4—48no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—84no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—88no—
465C2×C4—88no—
466C2×C4—88no—
467C2×C4—88no—
468C2×C4—164no—
469C2×C4—164no—
470C2×C4C2×#12312yes—
471C2×C4C2×#12312yes—
472C2×C4C2×#12312yes—
473C2×C4C22×#4111yescharacteristic
474C2×C4C22×#4111yescharacteristic
475C2×C4#58514yes—
476C8—44no—
477C8—44no—
478C22—24no—
479C22—24no—
480C22—24no—
481C22—24no—
482C22—24no—
483C22—24no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—416no—
491C22—44no—
492C22—44no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—88no—
497C22C2×#48514yes—
498C22C2×#48514yes—
499C22C2×#48514yes—
500C22C22×#12311yescharacteristic
501C22#348512yes—
502C22C22×#13511yescharacteristic
503C22C22×#13711yescharacteristic
504C22#358512yes—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—84no—
512C4—84no—
513C4—84no—
514C4—84no—
515C4#349112yes—
516C4#349112yes—
517C2—24no—
518C2—24no—
519C2—24no—
520C2—82no—
521C2#3544811yescharacteristic
522C2C22×#48511yescharacteristic
523C2#6051811yescharacteristic
524C2#6169912yes—
525C2#6170912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27617C2semidirect
4. #27604C2semidirect
5. #24947C2semidirect
6. #24943C2semidirect
7. #21940C2semidirect
8. #21938C2semidirect
10. C2×#1824C2semidirect
11. C2×#1824C2semidirect
39. #2101C22semidirect
40. #2085C22semidirect
41. #2077C22semidirect
42. #2075C22semidirect
43. #1919C22semidirect
44. #1918C22semidirect
47. #1854C22semidirect
49. #1853C22semidirect
51. #1852C22semidirect
52. #1825C22semidirect
55. #1824C22semidirect
56. #1824C22semidirect
58. #1808C22semidirect
59. #1807C22semidirect
61. C2×#340C22semidirect
62. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#341852
#276172
#276042
#249472
#249432
#219402
#219382
#217382
C2×#18242