Finite Group Databasebeta

← #6519413

Subgroups — #6519413

Order 512.

2026 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#26249C218yesmaximal, complemented
4#26247C218yesmaximal, complemented
5#24010C214yesmaximal, complemented
6#24006C212yesmaximal, complemented
7#24003C212yesmaximal, complemented
8#21811C212yesmaximal, complemented
9#21790C211yesmaximal, characteristic
10#21363C212yesmaximal, complemented
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×C22⋊Q8C2212yes—
14C22×C4×D4C2211yescharacteristic
15C2×#484C2214yescomplemented
16C2×#485C2218yescomplemented
17C2×#483C2214yescomplemented
18C2×#432C2212yes—
19#2542—22no—
20#2542—22no—
21#2542C2214yes—
22#2542C2214yes—
23C2×#399—21no—
24C2×#399—21no—
25#2353—22no—
26#2353—22no—
27#2352C2214yes—
28#2352C2214yes—
29#2313—21no—
30#2312—21no—
31#2308—21no—
32C2×#390—21no—
33C2×#384—21no—
34#2283C2212yes—
35#2280C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38#2273—21no—
39C2×C4×C4⋊C4C2211yescharacteristic
40#2036C2218yescomplemented
41#2036C2218yescomplemented
42#2035C2218yescomplemented
43#2035C2218yescomplemented
44#1981C2218yescomplemented
45#1960C2218yescomplemented
46#1955C2214yes—
47#1955C2214yes—
48#1949C2218yescomplemented
49#1873C2218yescomplemented
50#1872C2214yescomplemented
51#1872C2214yescomplemented
52#1853C2214yes—
53#1852C2214yes—
54#1787C2214yes—
55#1787C2214yes—
56#1779C2212yes—
57#1779C2212yes—
58#1776C2218yescomplemented
59C23×Q8C2312yes—
60C23×D4D411yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—28no—
65C2×C4⋊Q8C2314yes—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—48no—
70C2×C22⋊Q8—48no—
71C2×C22⋊Q8C2314yes—
72C2×C22⋊Q8C2318yes—
73C2×C22⋊Q8C2314yes—
74C2×C4×D4—28no—
75C2×C4×D4D412yes—
76C2×C4×D4D412yes—
77C2×C4×D4C2312yes—
78C2×C4×D4C2312yes—
79C2×#117C2314yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96#485—416no—
97#484—48no—
98#483—48no—
99#432—28no—
100C2×#107C2318yes—
101C2×#106C2318yes—
102C2×C4⋊C8C2314yes—
103C2×#103C2318yes—
104C2×#103C2318yes—
105C2×#103C2318yes—
106C2×#103C2318yes—
107C2×C22⋊C8C2314yes—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2318yes—
110C2×C22⋊C8C2314yes—
111C2×#97C2314yes—
112#399—24no—
113#399—24no—
114#399—24no—
115#399—24no—
116#390—24no—
117#390—24no—
118#388—24no—
119#388—24no—
120#387—24no—
121#387—24no—
122#386—24no—
123#386—24no—
124#384—24no—
125#384—24no—
126#383—22no—
127#382C2314yes—
128#381—22no—
129#381—22no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380C2312yes—
135#380C2312yes—
136#380C2314yes—
137#380C2312yes—
138#380C2312yes—
139#379—22no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4C2314yes—
142C4×C4⋊C4C2314yes—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—24no—
148#376—22no—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95—21no—
152#340C2318yes—
153#340C2318yes—
154C25C2×D411yescharacteristic
155C25C2×D411yescharacteristic
156C22×Q8—22no—
157C22×Q8—28no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8C2414yes—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—42no—
164C22×D4C2×D412yes—
165C22×D4C2×D412yes—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C22×C8C2418yes—
186C4⋊Q8—28no—
187C4⋊Q8—28no—
188C4⋊Q8—48no—
189C22⋊Q8—28no—
190C22⋊Q8—28no—
191C22⋊Q8—216no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C22⋊Q8—48no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197C22⋊Q8—416no—
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—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2414yes—
224C2×C4⋊C4C2414yes—
225C2×C4⋊C4C2414yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4—48no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4C2×D412yes—
242C2×C22⋊C4C2×D412yes—
243C2×C22⋊C4C2×D412yes—
244C2×C22⋊C4C2×D412yes—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—44no—
249C2×C42C2×D412yes—
250C2×C42#4118yes—
251C2×C42C2412yes—
252#107—216no—
253#106—216no—
254C4⋊C8—28no—
255#103—48no—
256#103—48no—
257#103—48no—
258#103—48no—
259C22⋊C8—28no—
260C22⋊C8—44no—
261C22⋊C8—44no—
262C22⋊C8—48no—
263#97—28no—
264#95—22no—
265#95—22no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24#13714yes—
283C24C22×D412yes—
284C24C22×D411yescharacteristic
285C24C22×D411yescharacteristic
286C24C22×D411yescharacteristic
287C2×Q8—28no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—28no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—816no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—416no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—28no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—88no—
336C22×C4—88no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4#12314yes—
342C22×C4#12314yes—
343C22×C4C22×D412yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D412yes—
347C22×C4C22×D412yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4114yes—
350C22×C4C2511yescharacteristic
351C2×C8—28no—
352C2×C8—28no—
353C2×C8—28no—
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—28no—
360C4⋊C4—416no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4H(5,𝔽2)18yes—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—416no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—84no—
380C22⋊C4—84no—
381C42—48no—
382C42—84no—
383C42—84no—
384C42#12318yes—
385C42C2×#4114yes—
386C42C2×#4114yes—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—28no—
398C23—24no—
399C23—22no—
400C23—24no—
401C23—22no—
402C23—22no—
403C23—48no—
404C23—48no—
405C23—44no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—84no—
411C23—84no—
412C23C2×#12312yes—
413C23C2×#13712yes—
414C23C2×#13712yes—
415C23C2×#13712yes—
416C23#57512yes—
417C23C23×D411yescharacteristic
418Q8—48no—
419Q8—48no—
420Q8—88no—
421D4—48no—
422D4—48no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—28no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—24no—
433C2×C4—28no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—44no—
437C2×C4—416no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
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—82no—
454C2×C4—82no—
455C2×C4—84no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—88no—
459C2×C4—164no—
460C2×C4—164no—
461C2×C4C2×#12312yes—
462C2×C4C2×#12314yes—
463C2×C4C2×#12314yes—
464C2×C4C22×#4112yes—
465C2×C4C2×H(5,𝔽2)14yes—
466C8—48no—
467C22—22no—
468C22—24no—
469C22—24no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—24no—
474C22—24no—
475C22—41no—
476C22—41no—
477C22—42no—
478C22—42no—
479C22—41no—
480C22—41no—
481C22—42no—
482C22—42no—
483C22—44no—
484C22—44no—
485C22—42no—
486C22—88no—
487C22—88no—
488C22#309514yes—
489C22C22×#12311yescharacteristic
490C22#348412yes—
491C22C22×#13711yescharacteristic
492C22C2×#57511yescharacteristic
493C22#358612yes—
494C4—24no—
495C4—24no—
496C4—22no—
497C4—48no—
498C4—42no—
499C4—41no—
500C4—41no—
501C4—84no—
502C4—84no—
503C4—84no—
504C4#350814yes—
505C2—24no—
506C2—42no—
507C2—42no—
508C2—82no—
509C2#3555411yescharacteristic
510C2C2×#309511yescharacteristic
511C2#6051911yescharacteristic
512C2#6157412yes—
513C2#6157512yes—
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. #26249C2semidirect
4. #26247C2semidirect
5. #24010C2semidirect
6. #24006C2semidirect
7. #24003C2semidirect
8. #21811C2semidirect
10. #21363C2semidirect
15. C2×#484C22semidirect
16. C2×#485C22semidirect
17. C2×#483C22semidirect
40. #2036C22semidirect
41. #2036C22semidirect
42. #2035C22semidirect
43. #2035C22semidirect
44. #1981C22semidirect
45. #1960C22semidirect
48. #1949C22semidirect
49. #1873C22semidirect
50. #1872C22semidirect
51. #1872C22semidirect
58. #1776C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#341852
#262492
#262472
#240102
#240062
#240032
#218112
#217902
#213632