Finite Group Databasebeta

← #194905

Subgroups — #194905

Order 512.

1526 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#21833C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic
5#9028C218yesmaximal, complemented
6#8014C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C22×#124—21no—
9C22×C4×D4C2211yescharacteristic
10C2×#431C2212yes—
11#2547—24no—
12#2547C418yescomplemented
13C2×#397C2214yes—
14#2357—24no—
15#2354—24no—
16#2312C2214yes—
17#2309—21no—
18C2×#391—21no—
19#2285C2212yes—
20#2280—22no—
21C2×#380—22no—
22#2274—21no—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1990—24no—
25#1988—24no—
26#1823C2218yes—
27#1460—24no—
28C2×#323C2214yes—
29#1294C2218yes—
30C23×Q8—21no—
31C23×D4D411yescharacteristic
32C24×C4D412yes—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—24no—
37C2×C4⋊Q8C2×C414yes—
38C2×#124—28no—
39C2×#124—48no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—24no—
43C2×C4×D4C2×C414yes—
44C2×C4×D4D414yes—
45C2×#117—22no—
46C2×#117—24no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—24no—
52C22×C22⋊C4C2312yes—
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#431—24no—
56#431—24no—
57C2×C4⋊C8—22no—
58C2×C22⋊C8C2314yes—
59C2×C4×C8—22no—
60#397—28no—
61#397C2×C4116yes—
62#391—28no—
63#390—28no—
64#388—216no—
65#384D414yes—
66#382—22no—
67#381—24no—
68#380—24no—
69#380—216no—
70#380—22no—
71#380C2314yes—
72#379—24no—
73C4×C4⋊C4—28no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2×C414yes—
77C4×C4⋊C4D414yes—
78C4×C22⋊C4—24no—
79C4×C22⋊C4—24no—
80C2×#95—24no—
81C2×#95C2312yes—
82#340—216no—
83#323—28no—
84C25C2×D412yes—
85C22×Q8—24no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8—44no—
89C22×Q8—44no—
90C22×D4—22no—
91C22×D4—22no—
92C22×D4—24no—
93C22×D4—24no—
94C22×D4C22⋊C414yes—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—48no—
101C23×C4C22⋊C418yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2411yescharacteristic
106C22×C8—24no—
107C4⋊Q8—28no—
108C4⋊Q8—28no—
109C4⋊Q8—44no—
110C4⋊Q8—44no—
111#124—432no—
112C4×D4—28no—
113C4×D4—28no—
114C4×D4—28no—
115C4×D4—28no—
116#117—44no—
117#117—44no—
118#117—88no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C22⋊C414yes—
133C2×C4⋊C4C22×C414yes—
134C2×C4⋊C4C2×D418yes—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—216no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—416no—
142C2×C22⋊C4C22×C414yes—
143C2×C22⋊C4C2×D414yes—
144C2×C42—24no—
145C2×C42—24no—
146C2×C42—28no—
147C2×C42—44no—
148C2×C42C22×C412yes—
149C2×C42C2×D412yes—
150C2×C42C2×D412yes—
151C2×C42C2×D412yes—
152C2×C42#4114yes—
153C4⋊C8—44no—
154C22⋊C8—24no—
155C22⋊C8—24no—
156C4×C8—44no—
157#95—28no—
158#95—216no—
159#95—24no—
160#95—22no—
161#95—24no—
162#95C22×C414yes—
163C24—22no—
164C24—24no—
165C24—22no—
166C24—24no—
167C24—44no—
168C24—44no—
169C24C2×C22⋊C414yes—
170C24C22×D411yescharacteristic
171C2×Q8—48no—
172C2×Q8—48no—
173C2×Q8—48no—
174C2×Q8—84no—
175C2×Q8—84no—
176C2×Q8—88no—
177C2×D4—28no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—48no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—28no—
187C22×C4—28no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—28no—
191C22×C4—42no—
192C22×C4—416no—
193C22×C4—48no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—88no—
203C22×C4—84no—
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C414yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4#12011yescharacteristic
208C22×C4#12012yes—
209C22×C4#12011yescharacteristic
210C22×C4#12312yes—
211C22×C4#12312yes—
212C22×C4#13514yes—
213C22×C4AGL(1,ℤ/8)14yes—
214C22×C4C23×C411yescharacteristic
215C22×C4C22×D412yes—
216C22×C4C22×D411yescharacteristic
217C22×C4C2×#4112yes—
218C2×C8—48no—
219C2×C8—44no—
220C4⋊C4—28no—
221C4⋊C4—28no—
222C4⋊C4—48no—
223C4⋊C4—48no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—48no—
228C4⋊C4—48no—
229C22⋊C4—28no—
230C22⋊C4—28no—
231C22⋊C4—432no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—88no—
235C22⋊C4—88no—
236C42—48no—
237C42—48no—
238C42—88no—
239C42C4×D414yes—
240C42C4×D414yes—
241C42#12114yes—
242C42#12114yes—
243C23—22no—
244C23—22no—
245C23—24no—
246C23—24no—
247C23—24no—
248C23—44no—
249C23—44no—
250C23—416no—
251C23—44no—
252C23—44no—
253C23—44no—
254C23—44no—
255C23—82no—
256C23—82no—
257C23—88no—
258C23C22×C22⋊C411yescharacteristic
259C23C2×#12011yescharacteristic
260C23C2×#12311yescharacteristic
261C23C2×#13512yes—
262C23C2×AGL(1,ℤ/8)12yes—
263Q8—816no—
264D4—416no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—416no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—48no—
284C2×C4—44no—
285C2×C4—88no—
286C2×C4—82no—
287C2×C4—88no—
288C2×C4—84no—
289C2×C4—84no—
290C2×C4#38612yes—
291C2×C4#38614yes—
292C2×C4#38612yes—
293C2×C4#41614yes—
294C2×C4#41814yes—
295C2×C4#44914yes—
296C2×C4#44914yes—
297C2×C4#45514yes—
298C2×C4#48218yes—
299C2×C4C2×C4×D412yes—
300C2×C4C2×#12112yes—
301C8—84no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—24no—
307C22—24no—
308C22—48no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—84no—
313C22—88no—
314C22—88no—
315C22C2×#38611yescharacteristic
316C22C2×#41611yescharacteristic
317C22C2×#41811yescharacteristic
318C22C2×#44912yes—
319C22C2×#45512yes—
320C22#302812yes—
321C22C2×#48212yes—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—44no—
326C4—84no—
327C4—84no—
328C4—88no—
329C4#187518yes—
330C4#188514yes—
331C2—21no—
332C2—22no—
333C2—21no—
334C2—84no—
335C2#1841214yes—
336C2C2×#187511yescharacteristic
337C2C2×#188511yescharacteristic
338C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21833C2semidirect
5. #9028C2semidirect
6. #8014C2semidirect
12. #2547C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
#341862
#218332
C2×#14602
#90282
#80142