Finite Group Databasebeta

← #6528931

Subgroups — #6528931

Order 512.

3170 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3502C211yesmaximal, characteristic, complemented
2C2×#3101C212yesmaximal
3C2×#2538C211yesmaximal, characteristic
4#24329C218yesmaximal, complemented
5#24187C218yesmaximal, complemented
6#24063C214yesmaximal, complemented
7#23933C214yesmaximal
8#21860C212yesmaximal, complemented
9#21818C211yesmaximal, characteristic, complemented
10#3502—216no—
11C2×Q82C2214yescomplemented
12C2×#557—24no—
13C2×#557C2214yescomplemented
14C2×#544—22no—
15C2×#544—21no—
16C2×#533C2214yes—
17C2×#533C2212yes—
18C22×C22⋊Q8—21no—
19C2×#519—21no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#3101—216no—
23C2×#483C2214yes—
24C2×#476C2214yes—
25C2×#461C2214yes—
26C2×#450C2214yes—
27C2×C4×SD16C2218yes—
28C2×#431C2212yes—
29#2538—24no—
30#2538C2218yes—
31C2×#396—22no—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#384C2211yescharacteristic
35C2×#380—21no—
36#2058C2218yescomplemented
37#2043C2218yescomplemented
38#2023C2218yes—
39#2013C2218yes—
40#1990C2214yescomplemented
41#1984C2218yescomplemented
42#1956C2214yescomplemented
43#1880C22116yes—
44#1872C2218yes—
45#1865C22116yescomplemented
46#1817C2214yes—
47#1803C2218yescomplemented
48C23×Q8D411yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C22×SD16C2318yes—
52Q82—216no—
53Q82—216no—
54#557—264no—
55#557—216no—
56#557—216no—
57#544—232no—
58#544—216no—
59#533—232no—
60#533—28no—
61#533D4116yes—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8—28no—
64C2×C4⋊Q8C2314yes—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2314yes—
67C2×C4⋊Q8C2312yes—
68C2×#125—24no—
69C2×#125—28no—
70C2×#125—24no—
71C2×#125—22no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—48no—
76C2×C22⋊Q8C2314yes—
77#519—216no—
78C2×C4×Q8—24no—
79C2×C4×Q8—24no—
80C2×C4×Q8—28no—
81C2×C4×Q8—24no—
82C2×C4×Q8—22no—
83C2×C4×Q8D414yes—
84C2×C4×Q8C2318yes—
85C2×C4×Q8C2314yes—
86C2×C4×Q8C2314yes—
87C2×C4×D4—22no—
88C2×C4×D4—28no—
89C2×C4×D4C2314yes—
90C2×#117—24no—
91C2×#117—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4D412yes—
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—22no—
101C22×C22⋊C4—22no—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104#483—216no—
105#476—216no—
106#461—216no—
107#450—216no—
108C4×SD16—232no—
109#431—28no—
110C2×#106C2318yes—
111C2×C4⋊C8C2314yes—
112C2×#103C2318yes—
113C2×#103C23116yes—
114C2×#102C2318yes—
115C2×C22⋊C8C2314yes—
116C2×C4×C8C2314yes—
117#396—216no—
118#390—24no—
119#390C2318yes—
120#389—24no—
121#389C2318yes—
122#384—22no—
123#384C2314yes—
124#380—28no—
125C2×#95—22no—
126#340C23116yes—
127#336C2318yes—
128#336C2318yes—
129C25—21no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—24no—
133C22×Q8—22no—
134C22×Q8SD1618yescomplemented
135C22×Q8C2×D414yes—
136C22×Q8C2414yes—
137C22×D4—24no—
138C22×D4—21no—
139C22×D4—24no—
140C22×D4—42no—
141C22×D4C2412yes—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C2×SD16—216no—
154C2×SD16—216no—
155C2×SD16—48no—
156C22×C8C2418yes—
157C4⋊Q8—264no—
158C4⋊Q8—232no—
159C4⋊Q8—216no—
160C4⋊Q8—216no—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8C2×D418yes—
164C4⋊Q8C2×D418yes—
165#125—232no—
166#125—264no—
167#125—232no—
168#125—216no—
169C22⋊Q8—232no—
170C22⋊Q8—216no—
171C22⋊Q8—28no—
172C22⋊Q8—432no—
173C22⋊Q8C2×D4116yes—
174C4×Q8—232no—
175C4×Q8—232no—
176C4×Q8—232no—
177C4×Q8—216no—
178C4×Q8—216no—
179C4×Q8—216no—
180C4×Q8—216no—
181C4×Q8—28no—
182C4×Q8—28no—
183C4×Q8—28no—
184C4×Q8C2×D4116yes—
185C4×D4—28no—
186C4×D4—28no—
187C4×D4—48no—
188#117—232no—
189#117—216no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D418yes—
213C2×C4⋊C4C2412yes—
214C2×C4⋊C4C2414yes—
215C2×C4⋊C4C2414yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—48no—
222C2×C22⋊C4—84no—
223C2×C42—28no—
224C2×C42—24no—
225C2×C42—28no—
226C2×C42C2×D414yes—
227C2×C42C2×D412yes—
228C2×C42C2×D418yes—
229C2×C42C2414yes—
230C2×C42C2412yes—
231#106—216no—
232C4⋊C8—28no—
233#103—216no—
234#103—416no—
235#102—216no—
236C22⋊C8—28no—
237C4×C8—28no—
238#95—28no—
239C24—24no—
240C24—22no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24C22×D411yescharacteristic
248C2×Q8—216no—
249C2×Q8—28no—
250C2×Q8—28no—
251C2×Q8—28no—
252C2×Q8—24no—
253C2×Q8—232no—
254C2×Q8—28no—
255C2×Q8—28no—
256C2×Q8—82no—
257C2×Q8—82no—
258C2×Q8—84no—
259C2×Q8—84no—
260C2×Q8C2×SD16116yes—
261C2×Q8C22×D418yes—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—416no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D42−1+414yes—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—816no—
293C22×C4#12014yes—
294C22×C4#12014yes—
295C22×C4C2×SD1618yes—
296C22×C4C2×SD1614yes—
297C22×C4#13714yes—
298C22×C4C22×D414yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D412yes—
301C22×C4C22×D414yes—
302C22×C4C22×D414yes—
303C22×C4C22×D412yes—
304C22×C4C2511yescharacteristic
305SD16—432no—
306C2×C8—28no—
307C2×C8—28no—
308C2×C8—28no—
309C4⋊C4—216no—
310C4⋊C4—216no—
311C4⋊C4—216no—
312C4⋊C4—232no—
313C4⋊C4—216no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—216no—
318C4⋊C4—28no—
319C4⋊C4—216no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—48no—
326C4⋊C4—416no—
327C4⋊C4—84no—
328C4⋊C4C22×D418yes—
329C22⋊C4—216no—
330C22⋊C4—48no—
331C22⋊C4—416no—
332C22⋊C4—88no—
333C42—232no—
334C42—28no—
335C42—216no—
336C42—28no—
337C42#12018yes—
338C42C22×D414yes—
339C42C22×D414yes—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—24no—
345C23—48no—
346C23—48no—
347C23—48no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—84no—
355C23—84no—
356C23—84no—
357C23—84no—
358C23C2×#12012yes—
359C23C22×SD1612yes—
360C23C2×#13712yes—
361C23C23×D411yescharacteristic
362Q8—232no—
363Q8—216no—
364Q8—48no—
365Q8—88no—
366Q8—816no—
367D4—28no—
368D4—48no—
369D4—44no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—216no—
376C2×C4—24no—
377C2×C4—28no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—28no—
381C2×C4—28no—
382C2×C4—24no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—84no—
395C2×C4—88no—
396C2×C4—84no—
397C2×C4—816no—
398C2×C4#448116yes—
399C2×C4C2×#12012yes—
400C2×C4C2×#12014yes—
401C2×C4C2×#12014yes—
402C2×C4C22×SD1618yes—
403C2×C4C22×SD1614yes—
404C2×C4C2×#13714yes—
405C2×C4C23×D412yes—
406C2×C4C2×2−1+412yes—
407C2×C4#58512yes—
408C8—48no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—44no—
416C22—44no—
417C22—44no—
418C22—816no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22C2×#44814yes—
424C22C22×#12011yescharacteristic
425C22#348611yescharacteristic
426C22#348711yescharacteristic
427C22C23×SD1611yescharacteristic
428C22C22×#13711yescharacteristic
429C22#358511yescharacteristic
430C22#358611yescharacteristic
431C4—22no—
432C4—28no—
433C4—24no—
434C4—48no—
435C4—44no—
436C4—48no—
437C4—84no—
438C4—82no—
439C4C2×#44818yes—
440C4#348914yes—
441C2—24no—
442C2—84no—
443C2#3565911yescharacteristic
444C2C22×#44811yescharacteristic
445C2#6037111yescharacteristic
446C2#6181811yescharacteristic
447C2#6182611yescharacteristic
448C2#6188111yescharacteristic
449C2#6188911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3502C2semidirect
4. #24329C2semidirect
5. #24187C2semidirect
6. #24063C2semidirect
8. #21860C2semidirect
9. #21818C2semidirect
11. C2×Q82C22semidirect
13. C2×#557C22semidirect
36. #2058C22semidirect
37. #2043C22semidirect
40. #1990C22semidirect
41. #1984C22semidirect
42. #1956C22semidirect
45. #1865C22semidirect
47. #1803C22semidirect
134. C22×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#35022
C2×#31012
C2×#25382
#243292
#241872
#240632
#239332
#218602
#218182