Finite Group Databasebeta

← #6487665

Subgroups — #6487665

Order 512.

2430 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C212yesmaximal
2#35173C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#32905C214yesmaximal
5#29756C214yesmaximal, complemented
6#24815C218yesmaximal, complemented
7#24730C218yesmaximal, complemented
8#21951C211yesmaximal, characteristic, complemented
9#21860C212yesmaximal, complemented
10C2×#533C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2212yes—
13C23×C4⋊C4—21no—
14#3101—216no—
15C2×#483C2214yes—
16C2×#476C2214yes—
17C2×#461C2214yes—
18C2×#461C2218yes—
19C2×#450C2214yes—
20C2×C4×SD16C2218yes—
21C2×#431C2212yes—
22#2857C2214yes—
23#2659—22no—
24#2584—24no—
25#2567C2214yescomplemented
26#2546C2214yescomplemented
27D4×C4⋊C4—24no—
28D4×C4⋊C4C2218yes—
29#2316—21no—
30#2316—21no—
31C2×#389C2212yes—
32C2×#385—22no—
33C2×#384C2211yescharacteristic
34C2×#380—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#2084C2218yescomplemented
37#2066C22116yescomplemented
38#2043C2218yescomplemented
39#1990C2214yescomplemented
40#1956C2214yescomplemented
41#1939C2218yescomplemented
42#1877C22116yes—
43#1872C2218yes—
44#1803C2218yescomplemented
45#1802C2218yes—
46#1797C2214yes—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—22no—
50C22×SD16C2318yes—
51#533—28no—
52#533—28no—
53C2×C4⋊Q8C2312yes—
54C2×C4⋊Q8C2314yes—
55C2×C4⋊Q8C2312yes—
56C2×#125—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8C2314yes—
59C2×C4×Q8—22no—
60C2×C4×Q8C2314yes—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—48no—
65C2×C4×D4D418yes—
66C2×C4×D4C2318yes—
67C2×C4×D4C2314yes—
68C2×#117—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—48no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#483—216no—
83#476—216no—
84#461—216no—
85#461—416no—
86#450—216no—
87C4×SD16—232no—
88#431—28no—
89C2×#106C2318yes—
90C2×C4⋊C8C2314yes—
91C2×C4⋊C8C2318yes—
92C2×#103C2318yes—
93C2×#103C23116yes—
94C2×#102C2318yes—
95C2×C22⋊C8C2314yes—
96C2×C4×C8C2314yes—
97#398—28no—
98#395—28no—
99#391C2318yes—
100#389—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389—24no—
105#389—24no—
106#389—24no—
107#389C2318yes—
108#385—216no—
109#384—22no—
110#384C2314yes—
111#384C2314yes—
112#380—24no—
113#380—24no—
114#380—28no—
115C4×C4⋊C4D414yes—
116C4×C4⋊C4C2314yes—
117C2×#95—22no—
118C2×#95D412yes—
119#340C23116yes—
120#336C2318yes—
121C25—21no—
122C22×Q8—22no—
123C22×Q8C2414yes—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4C2×D414yes—
129C22×D4C2412yes—
130C23×C4—21no—
131C23×C4—28no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—28no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—48no—
138C23×C4—48no—
139C23×C4—84no—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C2×SD16—216no—
146C2×SD16—216no—
147C2×SD16—216no—
148C22×C8C2418yes—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—48no—
154#125—48no—
155C22⋊Q8—28no—
156C22⋊Q8—28no—
157C22⋊Q8—48no—
158C4×Q8—28no—
159C4×Q8—28no—
160C4×Q8—48no—
161C4×D4—216no—
162C4×D4—216no—
163C4×D4—28no—
164C4×D4—48no—
165C4×D4—416no—
166C4×D4C2×D4116yes—
167#117—48no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—432no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4SD1618yescomplemented
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2412yes—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—48no—
201C2×C42—24no—
202C2×C42—44no—
203C2×C42C2×D414yes—
204C2×C42C2×D412yes—
205C2×C42C2×D414yes—
206C2×C42#4114yes—
207C2×C42C2414yes—
208C2×C42C2412yes—
209#106—216no—
210C4⋊C8—28no—
211C4⋊C8—48no—
212#103—216no—
213#103—416no—
214#102—216no—
215C22⋊C8—28no—
216C4×C8—28no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95C2×D418yes—
221C24—24no—
222C24—24no—
223C24—24no—
224C24—22no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24C22×D411yescharacteristic
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—28no—
233C2×Q8—28no—
234C2×Q8—44no—
235C2×Q8—82no—
236C2×Q8—82no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—216no—
240C2×D4—216no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4C22×D414yes—
246C2×D4H(5,𝔽2)14yes—
247C2×D42−1+414yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—216no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—416no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—432no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—416no—
273C22×C4—88no—
274C22×C4—832no—
275C22×C4#12114yes—
276C22×C4#12114yes—
277C22×C4C2×SD1618yes—
278C22×C4C2×SD1614yes—
279C22×C4#13714yes—
280C22×C4C22×D414yes—
281C22×C4C22×D412yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D412yes—
284C22×C4C22×D414yes—
285C22×C4C22×D412yes—
286C22×C4C2×#4112yes—
287C22×C4C2511yescharacteristic
288SD16—432no—
289C2×C8—28no—
290C2×C8—28no—
291C2×C8—28no—
292C4⋊C4—24no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—28no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—48no—
302C4⋊C4—48no—
303C4⋊C4—44no—
304C4⋊C4—416no—
305C4⋊C4—48no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4C2×SD16116yes—
312C4⋊C4C22×D414yes—
313C22⋊C4—216no—
314C22⋊C4—416no—
315C22⋊C4—48no—
316C42—28no—
317C42—44no—
318C42—48no—
319C42#12118yes—
320C42C22×D414yes—
321C42C2×#4114yes—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—216no—
327C23—24no—
328C23—24no—
329C23—24no—
330C23—48no—
331C23—48no—
332C23—48no—
333C23—48no—
334C23—48no—
335C23—42no—
336C23—42no—
337C23—48no—
338C23—42no—
339C23—42no—
340C23C2×#12112yes—
341C23C22×SD1612yes—
342C23C2×#13712yes—
343C23C23×D411yescharacteristic
344Q8—48no—
345Q8—48no—
346Q8—88no—
347D4—216no—
348D4—24no—
349D4—24no—
350D4—24no—
351D4#53718yes—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—432no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—832no—
380C2×C4—84no—
381C2×C4—88no—
382C2×C4#461116yescomplemented
383C2×C4C2×#12112yes—
384C2×C4C2×#12114yes—
385C2×C4C2×#12114yes—
386C2×C4C22×SD1618yes—
387C2×C4C22×SD1614yes—
388C2×C4C2×#13714yes—
389C2×C4C23×D411yescharacteristic
390C2×C4C22×#4111yescharacteristic
391C2×C4C2×H(5,𝔽2)12yes—
392C2×C4C2×2−1+412yes—
393C8—48no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—28no—
401C22—432no—
402C22—44no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22C2×#46114yes—
409C22C22×#12111yescharacteristic
410C22#348411yescharacteristic
411C22#348611yescharacteristic
412C22C23×SD1611yescharacteristic
413C22C22×#13711yescharacteristic
414C22#358511yescharacteristic
415C22#358611yescharacteristic
416C4—24no—
417C4—28no—
418C4—22no—
419C4—44no—
420C4—44no—
421C4—84no—
422C4—84no—
423C4—84no—
424C4C2×#46118yes—
425C4C2×#53712yes—
426C4#349212yes—
427C2—24no—
428C2—48no—
429C2#3522311yescharacteristic
430C2C22×#46111yescharacteristic
431C2#6042511yescharacteristic
432C2#6156011yescharacteristic
433C2#6158611yescharacteristic
434C2#6181811yescharacteristic
435C2#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
2. #35173C2semidirect
5. #29756C2semidirect
6. #24815C2semidirect
7. #24730C2semidirect
8. #21951C2semidirect
9. #21860C2semidirect
25. #2567C22semidirect
26. #2546C22semidirect
36. #2084C22semidirect
37. #2066C22semidirect
38. #2043C22semidirect
39. #1990C22semidirect
40. #1956C22semidirect
41. #1939C22semidirect
44. #1803C22semidirect
189. C2×C4⋊C4SD16semidirect
382. C2×C4#461semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
#351732
C2×D4×C4⋊C42
#329052
#297562
#248152
#247302
#219512
#218602