Finite Group Databasebeta

← #6512617

Subgroups — #6512617

Order 512.

3350 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#3104C212yesmaximal, complemented
3C2×#2538C211yesmaximal, characteristic
4#24243C218yesmaximal, complemented
5#24122C218yesmaximal, complemented
6#24006C214yesmaximal, complemented
7#23921C214yesmaximal, complemented
8#21860C212yesmaximal, complemented
9#21786C211yesmaximal, characteristic
10#3501—216no—
11C2×#554—24no—
12C2×#554C2214yes—
13C2×D4×Q8C2214yes—
14C2×#544—21no—
15C2×#543—22no—
16C2×#533C2212yes—
17C22×C22⋊Q8C2214yes—
18C22×C22⋊Q8C2212yes—
19C2×#518—21no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#3104—216no—
23C2×#484C2214yes—
24C2×#477C2214yescomplemented
25C2×#462C2214yescomplemented
26C2×#451C2214yes—
27C2×#451C2218yescomplemented
28C2×C4×Q16C2218yescomplemented
29C2×#431C2212yes—
30#2538—24no—
31#2538C2218yes—
32C2×#396—22no—
33C2×#390—21no—
34C2×#389—21no—
35C2×#384C2211yescharacteristic
36C2×#380C2212yes—
37#2036C22116yescomplemented
38#2001C22116yescomplemented
39#1990C2214yescomplemented
40#1956C2214yescomplemented
41#1954C2218yes—
42#1872C2218yes—
43#1872C2218yescomplemented
44#1852C2218yescomplemented
45#1803C2218yescomplemented
46#1787C2218yescomplemented
47#1779C2214yes—
48C23×Q8C2311yescharacteristic
49C23×Q8C2314yes—
50C23×D4D411yescharacteristic
51C24×C4C2312yes—
52C22×Q16C2318yes—
53#554—264no—
54#554—216no—
55#554—216no—
56D4×Q8—216no—
57D4×Q8—216no—
58#544—216no—
59#543—232no—
60#533—28no—
61#533D4116yes—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×#125—24no—
66C2×#125—22no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—232no—
71C2×C22⋊Q8—24no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—48no—
75C2×C22⋊Q8C2318yes—
76C2×C22⋊Q8C2318yes—
77C2×C22⋊Q8C2314yes—
78#518—216no—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8—28no—
82C2×C4×Q8C2314yes—
83C2×C4×Q8C2314yes—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4—28no—
87C2×C4×D4D414yes—
88C2×C4×D4C2314yes—
89C2×#117—22no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—22no—
101C22×C22⋊C4—22no—
102C22×C22⋊C4D412yes—
103C22×C42—21no—
104C22×C42C2311yescharacteristic
105#484—216no—
106#477—216no—
107#462—216no—
108#451—216no—
109#451—416no—
110C4×Q16—232no—
111#431—28no—
112C2×#107C2318yes—
113C2×C4⋊C8C2314yes—
114C2×#103C2318yes—
115C2×#103C2318yes—
116C2×#103C23116yes—
117C2×C22⋊C8C2314yes—
118C2×C22⋊C8C2318yes—
119C2×C22⋊C8C2318yes—
120C2×C4×C8C2314yes—
121#396—216no—
122#390—28no—
123#389—28no—
124#384—22no—
125#384C2314yes—
126#380—24no—
127#380C2318yes—
128C2×#95—22no—
129#340C23116yes—
130C25C2×D412yes—
131C22×Q8—24no—
132C22×Q8—216no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8—42no—
138C22×Q8—48no—
139C22×Q8—48no—
140C22×Q8C2414yes—
141C22×Q8C2412yes—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4Q1618yescomplemented
145C22×D4C2×D414yes—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—28no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4C2×D412yes—
155C23×C4C2×D414yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2414yes—
159C23×C4C2411yescharacteristic
160C2×Q16—216no—
161C2×Q16—216no—
162C2×Q16—48no—
163C22×C8C2418yes—
164C4⋊Q8—232no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8C2×D418yes—
168C4⋊Q8C2×D418yes—
169#125—232no—
170#125—216no—
171C22⋊Q8—232no—
172C22⋊Q8—232no—
173C22⋊Q8—264no—
174C22⋊Q8—264no—
175C22⋊Q8—216no—
176C22⋊Q8—216no—
177C22⋊Q8—28no—
178C22⋊Q8—216no—
179C22⋊Q8—216no—
180C22⋊Q8—48no—
181C22⋊Q8—416no—
182C22⋊Q8C2×D4116yes—
183C4×Q8—216no—
184C4×Q8—28no—
185C4×Q8—28no—
186C4×Q8—28no—
187C4×Q8—48no—
188C4×Q8C2×D4116yes—
189C4×D4—232no—
190C4×D4—232no—
191C4×D4—216no—
192C4×D4—28no—
193C4×D4—28no—
194#117—216no—
195C2×C4⋊C4—232no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—216no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2412yes—
219C2×C4⋊C4C2414yes—
220C2×C4⋊C4C2414yes—
221C2×C22⋊C4—232no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4—84no—
229C2×C22⋊C4C2×D418yes—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—44no—
233C2×C42C2×D412yes—
234C2×C42C2×D418yes—
235C2×C42C2412yes—
236#107—216no—
237C4⋊C8—28no—
238#103—216no—
239#103—216no—
240#103—416no—
241C22⋊C8—28no—
242C22⋊C8—48no—
243C22⋊C8—48no—
244C4×C8—28no—
245#95—28no—
246C24—24no—
247C24—24no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24C2×Q1618yes—
253C24C22×D414yes—
254C24C22×D411yescharacteristic
255C2×Q8—24no—
256C2×Q8—216no—
257C2×Q8—28no—
258C2×Q8—24no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—28no—
262C2×Q8—24no—
263C2×Q8—28no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—84no—
270C2×Q8—84no—
271C2×Q8—832no—
272C2×Q8C22×D418yes—
273C2×Q8H(5,𝔽2)14yes—
274C2×D4—28no—
275C2×D4—28no—
276C2×D4—232no—
277C2×D4C2×Q16116yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—216no—
285C22×C4—216no—
286C22×C4—24no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—816no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4#12014yes—
313C22×C4#12014yes—
314C22×C4C2×Q1614yes—
315C22×C4#13714yes—
316C22×C4C22×D414yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D412yes—
319C22×C4C22×D414yes—
320C22×C4C22×D412yes—
321C22×C4C2511yescharacteristic
322Q16—432no—
323C2×C8—28no—
324C2×C8—28no—
325C2×C8—28no—
326C4⋊C4—216no—
327C4⋊C4—232no—
328C4⋊C4—216no—
329C4⋊C4—28no—
330C4⋊C4—28no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—216no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—416no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—48no—
341C4⋊C4—48no—
342C4⋊C4—84no—
343C4⋊C4C22×D418yes—
344C22⋊C4—232no—
345C22⋊C4—216no—
346C22⋊C4—216no—
347C22⋊C4—416no—
348C22⋊C4—88no—
349C42—216no—
350C42—48no—
351C42#12018yes—
352C42C22×D414yes—
353C42C22×D414yes—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—216no—
365C23—48no—
366C23—48no—
367C23—84no—
368C23—84no—
369C23—84no—
370C23—84no—
371C23C2×#12012yes—
372C23C22×Q1618yes—
373C23C22×Q1612yes—
374C23C2×#13712yes—
375C23C23×D411yescharacteristic
376Q8—28no—
377Q8—216no—
378Q8—48no—
379Q8—44no—
380Q8—48no—
381Q8—88no—
382Q8—816no—
383D4—232no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—44no—
396C2×C4—48no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—416no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—88no—
411C2×C4—84no—
412C2×C4—816no—
413C2×C4—816no—
414C2×C4—88no—
415C2×C4—84no—
416C2×C4#451116yescomplemented
417C2×C4C2×#12012yes—
418C2×C4C2×#12014yes—
419C2×C4C2×#12014yes—
420C2×C4C22×Q1614yes—
421C2×C4C2×#13714yes—
422C2×C4C23×D412yes—
423C2×C4C2×H(5,𝔽2)12yes—
424C2×C4#58512yes—
425C8—48no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—28no—
433C22—28no—
434C22—28no—
435C22—44no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—816no—
440C22C2×#45114yes—
441C22C22×#12011yescharacteristic
442C22#348411yescharacteristic
443C22#348511yescharacteristic
444C22C23×Q1611yescharacteristic
445C22C22×#13711yescharacteristic
446C22#358511yescharacteristic
447C22#358611yescharacteristic
448C4—22no—
449C4—24no—
450C4—44no—
451C4—44no—
452C4—48no—
453C4—48no—
454C4—84no—
455C4—82no—
456C4C2×#45118yes—
457C4#348814yes—
458C2—24no—
459C2—48no—
460C2#3551211yescharacteristic
461C2C22×#45111yescharacteristic
462C2#6037111yescharacteristic
463C2#6156311yescharacteristic
464C2#6158811yescharacteristic
465C2#6169811yescharacteristic
466C2#6172311yescharacteristic
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
2. C2×#3104C2semidirect
4. #24243C2semidirect
5. #24122C2semidirect
6. #24006C2semidirect
7. #23921C2semidirect
8. #21860C2semidirect
24. C2×#477C22semidirect
25. C2×#462C22semidirect
27. C2×#451C22semidirect
28. C2×C4×Q16C22semidirect
37. #2036C22semidirect
38. #2001C22semidirect
39. #1990C22semidirect
40. #1956C22semidirect
43. #1872C22semidirect
44. #1852C22semidirect
45. #1803C22semidirect
46. #1787C22semidirect
144. C22×D4Q16semidirect
416. C2×C4#451semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#31042
C2×#25382
#242432
#241222
#240062
#239212
#218602
#217862