Finite Group Databasebeta

← #6487860

Subgroups — #6487860

Order 512.

2020 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
1#35167C211yesmaximal, characteristic, complemented
2C2×#2349C211yesmaximal, characteristic
3#32956C212yesmaximal
4#32953C212yesmaximal
5#29969C212yesmaximal, complemented
6#29967C212yesmaximal, complemented
7#26124C218yesmaximal, complemented
8#26058C218yesmaximal, complemented
9#21986C212yesmaximal, complemented
10#21955C211yesmaximal, characteristic, complemented
11#20738C212yesmaximal
12C2×#544—21no—
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#476C2214yes—
17C2×#474C2214yes—
18C2×#432C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2659—21no—
22#2659—21no—
23#2552C2214yescomplemented
24#2551C2214yescomplemented
25#2550—24no—
26#2349—24no—
27#2349C2218yes—
28#2316—21no—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#385—21no—
32C2×#385—21no—
33#2285C2212yes—
34C2×#382C2211yescharacteristic
35C2×#380—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37C22×#95—21no—
38#2100C2214yescomplemented
39#2099C2214yescomplemented
40#2088C2218yescomplemented
41#2087C2218yescomplemented
42#2058C2214yescomplemented
43#2057C2214yescomplemented
44#1949C2218yescomplemented
45#1940C2218yescomplemented
46#1864C2218yes—
47#1862C2218yes—
48#1853C2214yes—
49#1852C2214yes—
50#1847C2214yescomplemented
51#1836C2214yescomplemented
52#1824C2214yes—
53#1802C2214yes—
54#1802C2214yes—
55#1802C2214yes—
56#1802C2214yes—
57#1797C2212yes—
58#1797C2212yes—
59#1786C2214yes—
60#1762C2218yes—
61C23×D4C2311yescharacteristic
62C24×C4—21no—
63C24×C4—21no—
64C24×C4—21no—
65#544—48no—
66C2×C4⋊Q8—24no—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×#125—24no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—24no—
76C2×C4×D4—44no—
77C2×C4×D4—44no—
78C2×C4×D4Q818yescomplemented
79C2×C4×D4C2314yes—
80C2×C4×D4C2314yes—
81C2×C4×D4C2314yes—
82C2×#117—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#476—48no—
97#474—48no—
98#432—28no—
99C2×#107C2318yes—
100C2×#106C2318yes—
101C2×C4⋊C8C2314yes—
102C2×C4⋊C8C2314yes—
103C2×C4⋊C8C2314yes—
104C2×#102C2318yes—
105C2×#102C2318yes—
106C2×C22⋊C8C2314yes—
107C2×#97C2314yes—
108#398—24no—
109#398—24no—
110#395—24no—
111#395—24no—
112#391—22no—
113#391—22no—
114#389—24no—
115#389—24no—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389C2314yes—
121#389C2314yes—
122#389C2314yes—
123#389C2314yes—
124#385—28no—
125#385—28no—
126#384C2312yes—
127#384C2312yes—
128#384C2312yes—
129#384C2312yes—
130#382—22no—
131#382—22no—
132#382C2314yes—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—28no—
137#380C2314yes—
138C4×C4⋊C4—22no—
139C4×C4⋊C4D414yes—
140C4×C4⋊C4C2314yes—
141C4×C4⋊C4C2314yes—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—28no—
147C2×#95D411yescharacteristic
148C2×#95D411yescharacteristic
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2314yes—
154#336C2314yes—
155C25—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4C2×Q814yes—
163C22×D4C2412yes—
164C23×C4—21no—
165C23×C4—28no—
166C23×C4—24no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—82no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4—82no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2411yescharacteristic
186C22×C8C2418yes—
187C4⋊Q8—44no—
188C4⋊Q8—44no—
189C4⋊Q8—416no—
190#125—416no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—216no—
196C4×D4—28no—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—48no—
200C4×D4—48no—
201C4×D4C2×Q8116yes—
202#117—48no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×Q814yes—
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2412yes—
232C2×C4⋊C4C2414yes—
233C2×C4⋊C4C2414yes—
234C2×C4⋊C4C2412yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C42—24no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×Q814yes—
249C2×C42#4114yes—
250C2×C42C2412yes—
251C2×C42C2412yes—
252C2×C42C2412yes—
253#107—48no—
254#106—48no—
255C4⋊C8—28no—
256C4⋊C8—44no—
257C4⋊C8—44no—
258#102—216no—
259#102—216no—
260C22⋊C8—28no—
261#97—28no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95C2×D414yes—
271#95C2×D414yes—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—22no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24C22×D411yescharacteristic
281C2×Q8—44no—
282C2×Q8—44no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—82no—
286C2×Q8—82no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—216no—
290C2×D4—216no—
291C2×D4—24no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4C22×Q814yes—
296C2×D4H(5,𝔽2)14yes—
297C2×D42−1+414yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—216no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4—816no—
338C22×C4—816no—
339C22×C4—816no—
340C22×C4C22⋊Q814yes—
341C22×C4C22⋊Q814yes—
342C22×C4AGL(1,ℤ/8)14yes—
343C22×C4#13714yes—
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C22×D412yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×D412yes—
351C22×C4C22×Q812yes—
352C22×C4C2×#4112yes—
353C22×C4C2511yescharacteristic
354C2×C8—28no—
355C2×C8—28no—
356C2×C8—28no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—28no—
360C4⋊C4—22no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—28no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—48no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4C22×Q814yes—
382C22⋊C4—216no—
383C22⋊C4—48no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C42—24no—
387C42—24no—
388C42—42no—
389C42—42no—
390C42—48no—
391C42C22⋊Q818yes—
392C42C22×Q814yes—
393C42C2×#4114yes—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—216no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—48no—
403C23—48no—
404C23—48no—
405C23—48no—
406C23—48no—
407C23—42no—
408C23—42no—
409C23—48no—
410C23—42no—
411C23—42no—
412C23C2×C22⋊Q812yes—
413C23C2×AGL(1,ℤ/8)12yes—
414C23C2×#13712yes—
415C23C23×D411yescharacteristic
416Q8—164no—
417Q8—164no—
418D4—216no—
419D4—24no—
420D4—24no—
421D4—24no—
422D4#54418yescomplemented
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—816no—
462C2×C4—816no—
463C2×C4—816no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4C2×C22⋊Q812yes—
469C2×C4C2×C22⋊Q814yes—
470C2×C4C2×C22⋊Q814yes—
471C2×C4C2×AGL(1,ℤ/8)14yes—
472C2×C4C2×#13714yes—
473C2×C4C23×Q811yescharacteristic
474C2×C4C22×#4111yescharacteristic
475C2×C4C2×H(5,𝔽2)12yes—
476C2×C4C2×2−1+412yes—
477C8—48no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—28no—
485C22—432no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22#307914yes—
493C22C22×C22⋊Q811yescharacteristic
494C22#348411yescharacteristic
495C22#348611yescharacteristic
496C22C22×AGL(1,ℤ/8)11yescharacteristic
497C22C22×#13711yescharacteristic
498C22#358511yescharacteristic
499C22#358511yescharacteristic
500C4—24no—
501C4—24no—
502C4—24no—
503C4—24no—
504C4—22no—
505C4—44no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—84no—
510C4—84no—
511C4—82no—
512C4C2×#54412yes—
513C4#349212yes—
514C2—24no—
515C2—48no—
516C2#3522411yescharacteristic
517C2C2×#307911yescharacteristic
518C2#6049011yescharacteristic
519C2#6156911yescharacteristic
520C2#6157011yescharacteristic
521C2#6182511yescharacteristic
522C2#6182611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35167C2semidirect
5. #29969C2semidirect
6. #29967C2semidirect
7. #26124C2semidirect
8. #26058C2semidirect
9. #21986C2semidirect
10. #21955C2semidirect
23. #2552C22semidirect
24. #2551C22semidirect
38. #2100C22semidirect
39. #2099C22semidirect
40. #2088C22semidirect
41. #2087C22semidirect
42. #2058C22semidirect
43. #2057C22semidirect
44. #1949C22semidirect
45. #1940C22semidirect
50. #1847C22semidirect
51. #1836C22semidirect
78. C2×C4×D4Q8semidirect
422. D4#544semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#23492
#329562
#329532
#299692
#299672
#261242
#260582
#219862
#219552
#207382