Finite Group Databasebeta

← #6500757

Subgroups — #6500757

Order 512.

2430 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#27586C212yesmaximal, complemented
4#27583C212yesmaximal, complemented
5#24892C218yesmaximal, complemented
6#24888C218yesmaximal, complemented
7#24010C214yesmaximal, complemented
8#21944C212yesmaximal, complemented
9#21742C211yesmaximal, characteristic
10#20599C212yesmaximal, complemented
11C24×Q8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C2×#485C2218yescomplemented
15C2×#419C2218yescomplemented
16C2×#407C2212yes—
17#2409—22no—
18#2409—22no—
19#2409C2214yes—
20#2409C2214yes—
21C2×#395—21no—
22C2×#393—21no—
23#2357—22no—
24#2357—22no—
25#2354C2214yes—
26#2354C2214yes—
27#2312—21no—
28#2312—21no—
29#2309—21no—
30C2×#391C2212yes—
31C2×#386—21no—
32C2×#385—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35#2280C2212yes—
36#2280C2212yes—
37C2×#380C2211yescharacteristic
38#2274—21no—
39#2096C2214yescomplemented
40#2095C2214yescomplemented
41#2072C2214yescomplemented
42#2072C2214yescomplemented
43#2036C2218yescomplemented
44#2035C2218yescomplemented
45#1928C2218yescomplemented
46#1927C2218yescomplemented
47#1873C2218yescomplemented
48#1870C2214yes—
49#1870C2214yes—
50#1854C2218yescomplemented
51#1853C2214yes—
52#1852C2214yes—
53#1817C2212yes—
54#1809C2218yescomplemented
55#1787C2214yes—
56#1787C2214yes—
57#1787C2218yescomplemented
58#1779C2212yes—
59C23×Q8—21no—
60C23×Q8—22no—
61C23×Q8—28no—
62C23×Q8—48no—
63C23×Q8C2314yes—
64C23×Q8C2312yes—
65C23×D4D411yescharacteristic
66C24×C4—21no—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C22×M16C2314yes—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—48no—
72C2×C22⋊Q8C2318yes—
73C2×C4×D4—28no—
74C2×C4×D4D412yes—
75C2×C4×D4D412yes—
76C2×C4×D4C2312yes—
77C2×C4×D4C2312yes—
78C2×#117—21no—
79C2×#117—21no—
80C2×#117C2314yes—
81C2×#117C2314yes—
82C22×C4⋊C4—22no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4D411yescharacteristic
88C22×C22⋊C4D411yescharacteristic
89C22×C42C2311yescharacteristic
90#485—416no—
91C2×#107C2318yes—
92C2×#106C2318yes—
93#419—416no—
94C2×#103C2318yes—
95C2×#103C2318yes—
96C2×#103C2318yes—
97C2×#103C2318yes—
98#407—28no—
99C2×C22⋊C8C2314yes—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2318yes—
102#395—24no—
103#395—24no—
104#393—24no—
105#393—24no—
106#391—22no—
107#391—22no—
108#391C2314yes—
109#391C2314yes—
110#390—24no—
111#390—24no—
112#388—24no—
113#388—24no—
114#388—24no—
115#388—24no—
116#386—24no—
117#386—24no—
118#385—24no—
119#385—24no—
120#384C2312yes—
121#384C2312yes—
122#382—21no—
123#382—21no—
124#381—22no—
125#381—22no—
126#380—21no—
127#380—21no—
128#380—22no—
129#380C2314yes—
130#380C2314yes—
131#380C2314yes—
132#380C2312yes—
133#380C2312yes—
134#379—22no—
135#379—22no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2312yes—
141C4×C4⋊C4C2312yes—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149#340C2318yes—
150#340C2318yes—
151#336C2314yes—
152#336C2314yes—
153C25C2×D411yescharacteristic
154C25C2×D411yescharacteristic
155C22×Q8—28no—
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—216no—
159C22×Q8—28no—
160C22×Q8—28no—
161C22×Q8—28no—
162C22×Q8—44no—
163C22×Q8—48no—
164C22×Q8—48no—
165C22×Q8—464no—
166C22×Q8—416no—
167C22×Q8—44no—
168C22×Q8—48no—
169C22×Q8C2414yes—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—42no—
173C22×D4C2×D412yes—
174C22×D4C2×D412yes—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—48no—
182C23×C4—44no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4#4118yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C2×M16—28no—
194C2×M16—48no—
195C22×C8C2418yes—
196C22⋊Q8—216no—
197C22⋊Q8—48no—
198C22⋊Q8—416no—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205#117—44no—
206#117—44no—
207#117—48no—
208#117—48no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2414yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2414yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—48no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4C2×D412yes—
236C2×C22⋊C4C2×D412yes—
237C2×C22⋊C4C2×D412yes—
238C2×C22⋊C4C2×D412yes—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2411yescharacteristic
247C2×C42C2411yescharacteristic
248#107—48no—
249#106—48no—
250#103—216no—
251#103—216no—
252#103—48no—
253#103—48no—
254C22⋊C8—28no—
255C22⋊C8—28no—
256C22⋊C8—48no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—22no—
263#95—22no—
264#95—24no—
265#95—24no—
266#95—22no—
267#95—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24#12314yes—
278C24C22×D412yes—
279C24C22×D412yes—
280C24C22×D411yescharacteristic
281C2×Q8—28no—
282C2×Q8—24no—
283C2×Q8—232no—
284C2×Q8—28no—
285C2×Q8—28no—
286C2×Q8—416no—
287C2×Q8—432no—
288C2×Q8—432no—
289C2×Q8—44no—
290C2×Q8—416no—
291C2×Q8—432no—
292C2×Q8—464no—
293C2×Q8—44no—
294C2×Q8—48no—
295C2×Q8H(5,𝔽2)18yes—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—416no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—28no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—416no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—416no—
325C22×C4—44no—
326C22×C4—432no—
327C22×C4—44no—
328C22×C4—48no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—88no—
333C22×C4#12314yes—
334C22×C4#12318yes—
335C22×C4#13714yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4114yes—
344C22×C4C2×#4114yes—
345C22×C4C2×#4114yes—
346C22×C4C2511yescharacteristic
347M16—88no—
348C2×C8—28no—
349C2×C8—28no—
350C2×C8—28no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—416no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—416no—
367C22⋊C4—48no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—84no—
371C22⋊C4—84no—
372C22⋊C4—84no—
373C22⋊C4—84no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—44no—
379C42—44no—
380C42—84no—
381C42—84no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—28no—
386C23—24no—
387C23—22no—
388C23—22no—
389C23—24no—
390C23—22no—
391C23—24no—
392C23—22no—
393C23—22no—
394C23—44no—
395C23—44no—
396C23—48no—
397C23—48no—
398C23—48no—
399C23—44no—
400C23—82no—
401C23—82no—
402C23—84no—
403C23C2×#12314yes—
404C23C2×#12312yes—
405C23C2×#13712yes—
406C23#57512yes—
407C23C23×D411yescharacteristic
408Q8—216no—
409Q8—416no—
410Q8—432no—
411Q8—48no—
412D4—48no—
413D4—48no—
414C2×C4—28no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—48no—
432C2×C4—44no—
433C2×C4—416no—
434C2×C4—432no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
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—84no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—88no—
452C2×C4C2×#12312yes—
453C2×C4C2×#12314yes—
454C2×C4C2×#13712yes—
455C2×C4C2×#13712yes—
456C2×C4C22×#4112yes—
457C2×C4C2×H(5,𝔽2)14yes—
458C8—48no—
459C22—22no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—416no—
467C22—44no—
468C22—44no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—44no—
473C22—44no—
474C22—88no—
475C22#309514yes—
476C22C22×#12311yescharacteristic
477C22#348412yes—
478C22C22×#13711yescharacteristic
479C22C2×#57511yescharacteristic
480C22#358612yes—
481C4—22no—
482C4—24no—
483C4—22no—
484C4—22no—
485C4—42no—
486C4—48no—
487C4—41no—
488C4—41no—
489C4—84no—
490C4—84no—
491C4—84no—
492C4#349014yes—
493C2—24no—
494C2—44no—
495C2—82no—
496C2#3535011yescharacteristic
497C2C2×#309511yescharacteristic
498C2#6051911yescharacteristic
499C2#6157812yes—
500C2#6158412yes—
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. #27586C2semidirect
4. #27583C2semidirect
5. #24892C2semidirect
6. #24888C2semidirect
7. #24010C2semidirect
8. #21944C2semidirect
10. #20599C2semidirect
14. C2×#485C22semidirect
15. C2×#419C22semidirect
39. #2096C22semidirect
40. #2095C22semidirect
41. #2072C22semidirect
42. #2072C22semidirect
43. #2036C22semidirect
44. #2035C22semidirect
45. #1928C22semidirect
46. #1927C22semidirect
47. #1873C22semidirect
50. #1854C22semidirect
54. #1809C22semidirect
57. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
#341862
#275862
#275832
#248922
#248882
#240102
#219442
#217422
#205992