Finite Group Databasebeta

← #6487850

Subgroups — #6487850

Order 512.

1798 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#32957C214yesmaximal
4#29969C214yesmaximal
5#26112C218yesmaximal, complemented
6#26046C218yesmaximal, complemented
7#21982C212yesmaximal
8#21951C211yesmaximal, characteristic
9#20734C212yesmaximal
10C2×#544—21no—
11C22×C4×Q8C2211yescharacteristic
12C22×C4×Q8C2212yes—
13C2×#477C2218yes—
14C2×#431C2212yes—
15#2857C2214yes—
16#2659—22no—
17#2552C2214yes—
18#2551C2214yes—
19#2550—24no—
20Q8×C4⋊C4—24no—
21Q8×C4⋊C4C2218yes—
22#2316—21no—
23C2×#389—22no—
24C2×#389C2212yes—
25#2285C2212yes—
26C2×#384—21no—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2100C2218yes—
31#2087C22116yes—
32#2057C2218yes—
33#1947C2218yes—
34#1939C2218yes—
35#1863C22116yes—
36#1852C2218yes—
37#1845C2214yes—
38#1834C2214yes—
39#1824C2214yes—
40#1803C2218yes—
41#1803C2214yes—
42#1803C2214yes—
43#1797C2214yes—
44#1787C2214yes—
45C4×#103C2218yes—
46C23×Q8C2311yescharacteristic
47#544—48no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8C2314yes—
50C2×#125—24no—
51C2×C22⋊Q8—24no—
52C2×C4×Q8—22no—
53C2×C4×Q8—24no—
54C2×C4×Q8—24no—
55C2×C4×Q8—48no—
56C2×C4×Q8Q818yescomplemented
57C2×C4×Q8C2318yes—
58C2×C4×Q8C2314yes—
59C2×#117—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C42—21no—
70C22×C42—22no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73#477—416no—
74#431—28no—
75C2×#107C23116yes—
76C2×C4⋊C8C2314yes—
77C2×C4⋊C8C2318yes—
78C2×#103C2318yes—
79C2×#103C2318yes—
80C2×C22⋊C8C2314yes—
81C2×C4×C8C2314yes—
82#398—28no—
83#395—28no—
84#391—24no—
85#389—24no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389—216no—
90#389C2318yes—
91#389C2318yes—
92#384—28no—
93#384—22no—
94#384C2314yes—
95#384C2312yes—
96#384C2312yes—
97#384C2314yes—
98#382—22no—
99#380—24no—
100#380—22no—
101#380C2314yes—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C4×C4⋊C4C2314yes—
106C4×C4⋊C4C2314yes—
107C2×#95—22no—
108C2×#95D412yes—
109#340C2318yes—
110#340C2318yes—
111#340C23116yes—
112#336C2318yes—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—21no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8C2×Q814yes—
119C22×Q8C2412yes—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C22×C8C2418yes—
129C4⋊Q8—48no—
130C4⋊Q8—416no—
131#125—416no—
132C22⋊Q8—416no—
133C4×Q8—216no—
134C4×Q8—28no—
135C4×Q8—216no—
136C4×Q8—48no—
137C4×Q8—416no—
138C4×Q8C2×Q8116yes—
139#117—48no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—84no—
163C2×C4⋊C4Q1618yes—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×Q814yes—
167C2×C4⋊C4C2414yes—
168C2×C4⋊C4C2414yes—
169C2×C4⋊C4C2414yes—
170C2×C4⋊C4C2412yes—
171C2×C22⋊C4—24no—
172C2×C42—28no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—28no—
176C2×C42—44no—
177C2×C42—48no—
178C2×C42C2×D414yes—
179C2×C42C2×D412yes—
180C2×C42C2×Q814yes—
181C2×C42#4114yes—
182C2×C42C2414yes—
183C2×C42C2412yes—
184#107—416no—
185C4⋊C8—28no—
186C4⋊C8—48no—
187#103—216no—
188#103—216no—
189C22⋊C8—28no—
190C4×C8—28no—
191#95—28no—
192#95—24no—
193#95—24no—
194#95—28no—
195#95C2×D418yes—
196C24C22×D411yescharacteristic
197C2×Q8—24no—
198C2×Q8—216no—
199C2×Q8—24no—
200C2×Q8—216no—
201C2×Q8—24no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—48no—
206C2×Q8—84no—
207C2×Q8—84no—
208C2×Q8C22×Q814yes—
209C2×Q8H(5,𝔽2)14yes—
210C2×Q82−1+414yes—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4C22⋊Q814yes—
235C22×C4C22⋊Q814yes—
236C22×C4C2×Q1618yes—
237C22×C4C2×Q1614yes—
238C22×C4AGL(1,ℤ/8)14yes—
239C22×C4C22×D414yes—
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D412yes—
242C22×C4C22×D414yes—
243C22×C4C22×D412yes—
244C22×C4C22×Q812yes—
245C22×C4C2×#4112yes—
246C22×C4C2511yescharacteristic
247C2×C8—28no—
248C2×C8—28no—
249C2×C8—28no—
250C4⋊C4—216no—
251C4⋊C4—24no—
252C4⋊C4—28no—
253C4⋊C4—22no—
254C4⋊C4—28no—
255C4⋊C4—28no—
256C4⋊C4—28no—
257C4⋊C4—416no—
258C4⋊C4—44no—
259C4⋊C4—48no—
260C4⋊C4—416no—
261C4⋊C4—48no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4—88no—
269C4⋊C4—84no—
270C4⋊C4C2×Q16116yes—
271C4⋊C4C22×Q814yes—
272C22⋊C4—48no—
273C42—216no—
274C42—28no—
275C42—44no—
276C42—416no—
277C42—48no—
278C42C22⋊Q818yes—
279C42C22×Q814yes—
280C42C2×#4114yes—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23C2×C22⋊Q812yes—
286C23C22×Q1612yes—
287C23C2×AGL(1,ℤ/8)12yes—
288C23C23×D411yescharacteristic
289Q8—216no—
290Q8—24no—
291Q8—24no—
292Q8—24no—
293Q8—168no—
294Q8#54418yescomplemented
295C2×C4—216no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—21no—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—44no—
312C2×C4—48no—
313C2×C4—48no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—84no—
326C2×C4—88no—
327C2×C4#477116yescomplemented
328C2×C4C2×C22⋊Q812yes—
329C2×C4C2×C22⋊Q814yes—
330C2×C4C2×C22⋊Q814yes—
331C2×C4C22×Q1618yes—
332C2×C4C22×Q1614yes—
333C2×C4C2×AGL(1,ℤ/8)14yes—
334C2×C4C23×Q811yescharacteristic
335C2×C4C22×#4111yescharacteristic
336C2×C4C2×H(5,𝔽2)12yes—
337C2×C4C2×2−1+412yes—
338C8—48no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22C2×#47714yes—
346C22C22×C22⋊Q811yescharacteristic
347C22#348411yescharacteristic
348C22#348611yescharacteristic
349C22C23×Q1611yescharacteristic
350C22C22×AGL(1,ℤ/8)11yescharacteristic
351C22#358511yescharacteristic
352C22#358611yescharacteristic
353C4—24no—
354C4—28no—
355C4—28no—
356C4—22no—
357C4—44no—
358C4—44no—
359C4—84no—
360C4—84no—
361C4—84no—
362C4C2×#47718yes—
363C4C2×#54412yes—
364C4#349212yes—
365C2—24no—
366C2#3522411yescharacteristic
367C2C22×#47711yescharacteristic
368C2#6049211yescharacteristic
369C2#6156311yescharacteristic
370C2#6158711yescharacteristic
371C2#6182111yescharacteristic
372C2#6182811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
5. #26112C2semidirect
6. #26046C2semidirect
56. C2×C4×Q8Q8semidirect
294. Q8#544semidirect
327. C2×C4#477semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×Q8×C4⋊C42
#329572
#299692
#261122
#260462
#219822
#219512
#207342