Finite Group Databasebeta

← #6519030

Subgroups — #6519030

Order 512.

2054 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#27603C214yesmaximal, complemented
4#25016C218yesmaximal, complemented
5#24971C218yesmaximal, complemented
6#24006C214yesmaximal, complemented
7#21982C212yesmaximal, complemented
8#21786C211yesmaximal, characteristic
9#20734C212yesmaximal, complemented
10C22×C22⋊Q8C2212yes—
11C2×#518—21no—
12C22×C4×Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#484C2218yescomplemented
15C2×#431C2212yes—
16#2538—24no—
17#2538C2218yes—
18C2×#396—22no—
19#2349—24no—
20D4×C4⋊C4C2214yes—
21D4×C4⋊C4C2214yes—
22#2316—21no—
23#2315—22no—
24C2×#390—21no—
25C2×#389—21no—
26#2285C2212yes—
27C2×#384C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2214yes—
30#2076C22116yescomplemented
31#2036C22116yescomplemented
32#1954C2218yes—
33#1947C2218yescomplemented
34#1872C2218yescomplemented
35#1852C2218yes—
36#1852C2218yescomplemented
37#1845C2214yescomplemented
38#1834C2214yescomplemented
39#1824C2214yes—
40#1824C2218yescomplemented
41#1803C2214yescomplemented
42#1803C2214yescomplemented
43#1787C2214yes—
44#1779C2214yes—
45C4×#103C2218yescomplemented
46C23×Q8C2311yescharacteristic
47C23×D4D411yescharacteristic
48C24×C4C2312yes—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—48no—
51C2×C22⋊Q8C2318yes—
52#518—48no—
53C2×C4×Q8—22no—
54C2×C4×Q8—28no—
55C2×C4×Q8C2314yes—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—28no—
59C2×C4×D4D414yes—
60C2×C4×D4C2314yes—
61C2×#117—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—24no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#484—416no—
78#431—28no—
79C2×#107C23116yes—
80C2×C4⋊C8C2314yes—
81C2×#103C2318yes—
82C2×#103C2318yes—
83C2×C22⋊C8C2314yes—
84C2×C22⋊C8C2318yes—
85C2×C22⋊C8C2318yes—
86C2×C4×C8C2314yes—
87#396—216no—
88#390—28no—
89#389—24no—
90#389—24no—
91#389—28no—
92#385—28no—
93#385—28no—
94#384—22no—
95#384C2312yes—
96#384C2312yes—
97#384C2314yes—
98#382—22no—
99#380—22no—
100#380—24no—
101#380—24no—
102#380—24no—
103#380C2318yes—
104#380C2318yes—
105#380C2314yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2314yes—
108C2×#95—24no—
109C2×#95—22no—
110#340C2318yes—
111#340C2318yes—
112#340C23116yes—
113C25C2×D412yes—
114C22×Q8—24no—
115C22×Q8—21no—
116C22×Q8—24no—
117C22×Q8—42no—
118C22×Q8C2412yes—
119C22×D4—24no—
120C22×D4—24no—
121C22×D4Q1618yes—
122C22×D4C2×D414yes—
123C23×C4—21no—
124C23×C4—28no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4C2×D412yes—
132C23×C4C2×D414yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2414yes—
136C23×C4C2411yescharacteristic
137C22×C8C2418yes—
138C22⋊Q8—216no—
139C22⋊Q8—48no—
140C22⋊Q8—416no—
141C4×Q8—28no—
142C4×Q8—28no—
143C4×Q8—48no—
144C4×D4—216no—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—416no—
148C4×D4—416no—
149#117—48no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2414yes—
173C2×C4⋊C4C2414yes—
174C2×C4⋊C4C2412yes—
175C2×C4⋊C4C2414yes—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—48no—
182C2×C22⋊C4—416no—
183C2×C22⋊C4—84no—
184C2×C22⋊C4C2×D418yes—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—44no—
188C2×C42C2×D412yes—
189C2×C42#4118yes—
190C2×C42C2412yes—
191#107—416no—
192C4⋊C8—28no—
193#103—216no—
194#103—48no—
195C22⋊C8—28no—
196C22⋊C8—48no—
197C22⋊C8—48no—
198C4×C8—28no—
199#95—24no—
200#95—28no—
201#95—28no—
202#95—24no—
203C24—24no—
204C24—24no—
205C24—42no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24C2×Q1618yes—
210C24C22×D414yes—
211C24C22×D411yescharacteristic
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—416no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8H(5,𝔽2)14yes—
220C2×D4—28no—
221C2×D4—28no—
222C2×D4—232no—
223C2×D4C2×Q16116yes—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—816no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4—84no—
258C22×C4#12314yes—
259C22×C4#12314yes—
260C22×C4C2×Q1614yes—
261C22×C4AGL(1,ℤ/8)14yes—
262C22×C4C22×D414yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D412yes—
265C22×C4C22×D412yes—
266C22×C4C2×#4114yes—
267C22×C4C2511yescharacteristic
268C2×C8—28no—
269C2×C8—28no—
270C2×C8—28no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—28no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—416no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—84no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C22⋊C4—216no—
289C22⋊C4—416no—
290C22⋊C4—416no—
291C22⋊C4—48no—
292C22⋊C4—88no—
293C42—48no—
294C42—48no—
295C42#12318yes—
296C42C2×#4114yes—
297C42C2×#4114yes—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—24no—
303C23—24no—
304C23—24no—
305C23—24no—
306C23—24no—
307C23—24no—
308C23—216no—
309C23—48no—
310C23—48no—
311C23—84no—
312C23—84no—
313C23—84no—
314C23—84no—
315C23C2×#12312yes—
316C23C22×Q1618yes—
317C23C22×Q1612yes—
318C23C2×AGL(1,ℤ/8)12yes—
319C23C23×D411yescharacteristic
320Q8—28no—
321Q8—48no—
322Q8—44no—
323D4—232no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—28no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—416no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—88no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—88no—
354C2×C4—168no—
355C2×C4—168no—
356C2×C4#484116yescomplemented
357C2×C4C2×#12312yes—
358C2×C4C2×#12314yes—
359C2×C4C2×#12314yes—
360C2×C4C22×Q1614yes—
361C2×C4C2×AGL(1,ℤ/8)14yes—
362C2×C4C22×#4112yes—
363C2×C4C2×H(5,𝔽2)12yes—
364C2×C4#58512yes—
365C8—48no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—28no—
373C22—28no—
374C22—28no—
375C22—44no—
376C22—44no—
377C22—44no—
378C22—44no—
379C22—816no—
380C22C2×#48414yes—
381C22C22×#12311yescharacteristic
382C22#348411yescharacteristic
383C22#348511yescharacteristic
384C22C23×Q1611yescharacteristic
385C22C22×AGL(1,ℤ/8)11yescharacteristic
386C22#358511yescharacteristic
387C22#358611yescharacteristic
388C4—24no—
389C4—22no—
390C4—44no—
391C4—44no—
392C4—84no—
393C4—82no—
394C4—84no—
395C4—84no—
396C4C2×#48418yes—
397C4#349414yes—
398C2—24no—
399C2—48no—
400C2#3554011yescharacteristic
401C2C22×#48411yescharacteristic
402C2#6052011yescharacteristic
403C2#6156311yescharacteristic
404C2#6158711yescharacteristic
405C2#6169811yescharacteristic
406C2#6172211yescharacteristic
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. #27603C2semidirect
4. #25016C2semidirect
5. #24971C2semidirect
6. #24006C2semidirect
7. #21982C2semidirect
9. #20734C2semidirect
14. C2×#484C22semidirect
30. #2076C22semidirect
31. #2036C22semidirect
33. #1947C22semidirect
34. #1872C22semidirect
36. #1852C22semidirect
37. #1845C22semidirect
38. #1834C22semidirect
40. #1824C22semidirect
41. #1803C22semidirect
42. #1803C22semidirect
45. C4×#103C22semidirect
356. C2×C4#484semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#341812
#276032
#250162
#249712
#240062
#219822
#217862
#207342