Finite Group Databasebeta

← #6475406

Subgroups — #6475406

Order 512.

2112 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C212yesmaximal
2#33210C214yesmaximal, complemented
3#30298C214yesmaximal, complemented
4#29301C218yesmaximal, complemented
5#27950C214yesmaximal, complemented
6#25811C214yesmaximal, complemented
7#21942C214yesmaximal
8C4×#407C211yesmaximal, characteristic
9C22×C4⋊Q8C2212yes—
10C22×C22⋊Q8—22no—
11C23×C42C2211yescharacteristic
12C2×#498C2214yescomplemented
13C2×#407C2212yes—
14C2×C4×M16C2212yes—
15#2857—22no—
16#2857—22no—
17#2857—22no—
18#2857—22no—
19#2857C2214yes—
20#2857C2214yes—
21#2857C2214yes—
22#2857C2214yes—
23C2×#391—22no—
24C2×#389—22no—
25C2×#389C2214yes—
26C2×#384C2212yes—
27C2×#380—22no—
28C2×#380C2214yes—
29#2083C22116yescomplemented
30#2058C22116yescomplemented
31#1990C2218yescomplemented
32#1943C22116yescomplemented
33#1853C22116yes—
34#1852C22116yes—
35#1845C2214yescomplemented
36#1809C22116yes—
37#1749C2214yes—
38C4×C22⋊C8C2214yes—
39#1728C2212yes—
40C23×Q8—22no—
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C22×M16C2314yes—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—48no—
47C2×C4⋊Q8C2314yes—
48C2×C4⋊Q8C2314yes—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—416no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2314yes—
59C22×C4⋊C4C2314yes—
60C22×C22⋊C4—24no—
61C22×C42—44no—
62C22×C42D412yes—
63C22×C42D412yes—
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C22×C42C2312yes—
69#498—48no—
70C2×#107C23116yes—
71C2×#106C23116yes—
72#407Q8116yescomplemented
73C2×C22⋊C8C2318yes—
74C4×M16—44no—
75C2×#97C2314yes—
76C2×C4×C8C2314yes—
77#391—24no—
78#391—24no—
79#391—24no—
80#391—24no—
81#389—24no—
82#389—24no—
83#389—24no—
84#389—24no—
85#389—24no—
86#389—24no—
87#389C2318yes—
88#389C2318yes—
89#384—22no—
90#384—22no—
91#384C2314yes—
92#384C2314yes—
93#380—24no—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—24no—
99#380C2318yes—
100#380C2318yes—
101C2×#95—24no—
102C2×#95—24no—
103#340C23132yes—
104#336C2318yes—
105C25C2×D411yescharacteristic
106C22×Q8—24no—
107C22×Q8—24no—
108C22×Q8—48no—
109C22×Q8—48no—
110C22×Q8—48no—
111C23×C4—28no—
112C23×C4—28no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—44no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×Q818yes—
127C23×C4C2412yes—
128C23×C4C2411yescharacteristic
129C2×M16—44no—
130C2×M16—44no—
131C2×M16C2×Q8116yescomplemented
132C22×C8C2418yes—
133C4⋊Q8—216no—
134C4⋊Q8—48no—
135C4⋊Q8—48no—
136C4⋊Q8—48no—
137C22⋊Q8—416no—
138C22⋊Q8—416no—
139C22⋊Q8—416no—
140C22⋊Q8—416no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—48no—
152C2×C4⋊C4—48no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4—416no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4C2418yes—
158C2×C4⋊C4C2418yes—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—416no—
162C2×C42—28no—
163C2×C42—24no—
164C2×C42—44no—
165C2×C42—44no—
166C2×C42—44no—
167C2×C42—44no—
168C2×C42—42no—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2412yes—
177C2×C42C2411yescharacteristic
178C2×C42C2411yescharacteristic
179#107—416no—
180#106—416no—
181C22⋊C8C2×Q8132yescomplemented
182#97—44no—
183C4×C8—28no—
184#95—28no—
185#95—28no—
186#95—28no—
187#95—28no—
188C24—24no—
189C24—24no—
190C24—41no—
191C24—41no—
192C24C4⋊Q814yes—
193C24C22×D411yescharacteristic
194C24C22×D412yes—
195C2×Q8—48no—
196C2×Q8—48no—
197C2×Q8—48no—
198C2×Q8—48no—
199C2×Q8—48no—
200C2×Q8—48no—
201C2×Q8—832no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—28no—
220C22×C4—28no—
221C22×C4—416no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—432no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—48no—
236C22×C4C4⋊Q814yes—
237C22×C4C4⋊Q818yes—
238C22×C4AGL(1,ℤ/8)18yes—
239C22×C4#13718yes—
240C22×C4C22×D412yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D412yes—
243C22×C4C22×D412yes—
244C22×C4C22×D412yes—
245C22×C4C22×D412yes—
246C22×C4C22×Q818yes—
247C22×C4C22×Q814yes—
248C22×C4C2511yescharacteristic
249M16—416no—
250C2×C8—28no—
251C2×C8—28no—
252C2×C8C22×Q8116yes—
253C4⋊C4—216no—
254C4⋊C4—216no—
255C4⋊C4—48no—
256C4⋊C4—48no—
257C4⋊C4—416no—
258C4⋊C4—48no—
259C4⋊C4—48no—
260C4⋊C4—48no—
261C4⋊C4—48no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C4⋊C4—416no—
265C22⋊C4—416no—
266C22⋊C4—416no—
267C42—24no—
268C42—24no—
269C42—24no—
270C42—22no—
271C42—24no—
272C42—41no—
273C42—42no—
274C42—42no—
275C42—41no—
276C42H(5,𝔽2)14yes—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—216no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—44no—
287C23—48no—
288C23—44no—
289C23—44no—
290C23C2×C4⋊Q812yes—
291C23C2×C4⋊Q814yes—
292C23C2×AGL(1,ℤ/8)14yes—
293C23C2×#13714yes—
294C23C23×D411yescharacteristic
295Q8—816no—
296Q8—816no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—216no—
306C2×C4—216no—
307C2×C4—28no—
308C2×C4—216no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—48no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—48no—
324C2×C4—816no—
325C2×C4—816no—
326C2×C4C2×C4⋊Q812yes—
327C2×C4C2×C4⋊Q814yes—
328C2×C4#53414yes—
329C2×C4C2×AGL(1,ℤ/8)14yes—
330C2×C4C2×AGL(1,ℤ/8)14yes—
331C2×C4C2×#13714yes—
332C2×C4C2×#13714yes—
333C2×C4C23×Q812yes—
334C2×C4C2×H(5,𝔽2)12yes—
335C2×C4#58512yes—
336C8—216no—
337C22—24no—
338C22—24no—
339C22—24no—
340C22—24no—
341C22—24no—
342C22—24no—
343C22—22no—
344C22—24no—
345C22—416no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22#316518yes—
350C22C22×C4⋊Q811yescharacteristic
351C22C2×#53411yescharacteristic
352C22#344911yescharacteristic
353C22C22×AGL(1,ℤ/8)12yes—
354C22C22×#13712yes—
355C4—24no—
356C4—24no—
357C4—24no—
358C4—22no—
359C4—42no—
360C4—88no—
361C4—88no—
362C4#319318yes—
363C4#350114yes—
364C2—28no—
365C2—42no—
366C2#3501111yescharacteristic
367C2C2×#316512yes—
368C2C2×#319312yes—
369C2#6096811yescharacteristic
370C2#6097111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33210C2semidirect
3. #30298C2semidirect
4. #29301C2semidirect
5. #27950C2semidirect
6. #25811C2semidirect
12. C2×#498C22semidirect
29. #2083C22semidirect
30. #2058C22semidirect
31. #1990C22semidirect
32. #1943C22semidirect
35. #1845C22semidirect
72. #407Q8semidirect
131. C2×M16C2×Q8semidirect
181. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#332102
#302982
#293012
#279502
#258112
#219422
C4×#4072