Finite Group Databasebeta

← #391864

Subgroups — #391864

Order 512.

1524 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1524 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3151C211yesmaximal, characteristic
2C2×#1841C212yesmaximal, complemented
3#11014C214yesmaximal, complemented
4#11012C214yesmaximal, complemented
5#10979C212yesmaximal, complemented
6#10975C212yesmaximal, complemented
7C22×AGL(1,ℤ/8)C2214yes—
8C2×#532C2211yescharacteristic
9C22×#127C2211yescharacteristic
10#3151—28no—
11C2×#496C414yescomplemented
12C2×#493C412yescomplemented
13C2×#492C412yescomplemented
14C2×#421C2214yes—
15C2×#418C2214yes—
16C22×#100C2212yes—
17C2×#410C2212yes—
18#2879C2212yes—
19C2×C4×M16C2211yescharacteristic
20#2309C2212yes—
21#1841—28no—
22C2×#344C2214yes—
23C2×#339C2214yes—
24C2×#338C2212yes—
25C2×H(3,ℤ/4)C2212yes—
26C22×#41C2312yes—
27C23×D4—41no—
28C23×D4C2312yes—
29C2×AGL(1,ℤ/8)—28no—
30C2×AGL(1,ℤ/8)—28no—
31C2×AGL(1,ℤ/8)—44no—
32C2×AGL(1,ℤ/8)—44no—
33C2×AGL(1,ℤ/8)—44no—
34C2×AGL(1,ℤ/8)—44no—
35C22×SD16C2×C418yes—
36C22×D8C2×C418yes—
37C22×M16C2314yes—
38C22×M16C2312yes—
39#532—24no—
40#532—42no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#127—24no—
43C2×#127—41no—
44C2×#127—42no—
45C2×#127C2×C412yes—
46C2×#127C2×C411yescharacteristic
47C2×#127C2×C412yes—
48C2×#124C2×C412yes—
49C2×#121—41no—
50C2×C4×D4—41no—
51C2×#117C2314yes—
52C22×C42C2311yescharacteristic
53#496—216no—
54#493—28no—
55#492—28no—
56#421—44no—
57#421—44no—
58C2×#104—44no—
59#418—44no—
60#418—44no—
61C2×#102—44no—
62C2×#100—42no—
63C2×#100—44no—
64#410—44no—
65C2×#99—42no—
66#408—44no—
67#407—42no—
68C4×M16D418yes—
69C2×#97C2×C412yes—
70C2×C4×C8C2×C412yes—
71#390—44no—
72#379—42no—
73#344—28no—
74#339—28no—
75#338—24no—
76H(3,ℤ/4)—24no—
77C25—22no—
78C2×#41—24no—
79C2×#41—24no—
80C2×#41—24no—
81C2×#41—24no—
82C2×#41—24no—
83C2×#41—24no—
84C22×Q8C22×C412yes—
85C22×D4—22no—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—24no—
89C22×D4—24no—
90C22×D4—24no—
91C22×D4—24no—
92C22×D4—22no—
93C22×D4—41no—
94C22×D4—42no—
95C22×D4—42no—
96C22×D4—48no—
97C22×D4—88no—
98C22×D4C22×C412yes—
99C22×D4C22×C414yes—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4C2411yescharacteristic
103AGL(1,ℤ/8)—416no—
104AGL(1,ℤ/8)—416no—
105AGL(1,ℤ/8)—416no—
106AGL(1,ℤ/8)—416no—
107C2×SD16—216no—
108C2×SD16—48no—
109C2×SD16—48no—
110C2×D8—216no—
111C2×D8—48no—
112C2×D8—48no—
113C2×M16—24no—
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16C2×D414yes—
122C2×M16C2×D414yes—
123C22×C8—24no—
124C22×C8C22×C414yes—
125C4⋊Q8—22no—
126C4⋊Q8—41no—
127#127—24no—
128#127—24no—
129#127—22no—
130#127—42no—
131#127—42no—
132#127—41no—
133#127—48no—
134#124—24no—
135#124—42no—
136#121—48no—
137C4×D4—48no—
138#117—24no—
139#117—24no—
140#117—24no—
141#117—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—41no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—48no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—24no—
150C2×C42C22×C411yescharacteristic
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D414yes—
153#104—416no—
154#102—416no—
155#100—48no—
156#99—48no—
157C22⋊C8—48no—
158#97C22⋊C418yescomplemented
159C4×C8C22⋊C418yescomplemented
160C24—22no—
161C24—24no—
162C24—44no—
163C24—42no—
164C24—42no—
165C24—44no—
166C24—88no—
167C24H(5,𝔽2)11yescharacteristic
168#41—44no—
169#41—44no—
170#41—44no—
171#41—44no—
172#41—44no—
173#41—44no—
174#41—44no—
175#41—44no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—24no—
179C2×Q8—24no—
180C2×D4—22no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—28no—
187C2×D4—28no—
188C2×D4—44no—
189C2×D4—48no—
190C2×D4—48no—
191C2×D4—44no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—44no—
207C2×D4—84no—
208C2×D4—88no—
209C2×D4—832no—
210C2×D4—88no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4C23×C411yescharacteristic
222C22×C4C22×D412yes—
223C22×C4H(5,𝔽2)14yes—
224C22×C4H(5,𝔽2)14yes—
225C22×C4H(5,𝔽2)12yes—
226C22×C4H(5,𝔽2)11yescharacteristic
227SD16—432no—
228D8—432no—
229M16—28no—
230M16—28no—
231M16—44no—
232M16—44no—
233M16—44no—
234M16—44no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8C2×C22⋊C418yes—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C22⋊C4—48no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—168no—
248C42—21no—
249C42—21no—
250C42—24no—
251C42—44no—
252C42—44no—
253C42C2×C22⋊C412yes—
254C42C2×C22⋊C412yes—
255C23—42no—
256C23—42no—
257C23—44no—
258C23—42no—
259C23—44no—
260C23—42no—
261C23—44no—
262C23—88no—
263C23—84no—
264C23—88no—
265C23—816no—
266C23—84no—
267C23—84no—
268C23—164no—
269C23—168no—
270C23#51811yescharacteristic
271C23#53512yes—
272C23#56012yes—
273C23#56012yes—
274Q8—44no—
275Q8—44no—
276D4—44no—
277D4—48no—
278D4—48no—
279D4—44no—
280D4—88no—
281D4—816no—
282D4—816no—
283D4—84no—
284D4—84no—
285D4—84no—
286D4—84no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—48no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4—88no—
298C2×C4—84no—
299C2×C4C22×C22⋊C411yescharacteristic
300C2×C4#51812yes—
301C2×C4#51812yes—
302C2×C4#51811yescharacteristic
303C2×C4#53512yes—
304C2×C4#53514yes—
305C2×C4#53514yes—
306C2×C4#56012yes—
307C2×C4#56012yes—
308C2×C4#56014yes—
309C2×C4#56014yes—
310C8—28no—
311C8—84no—
312C22—22no—
313C22—24no—
314C22—42no—
315C22—42no—
316C22—88no—
317C22—84no—
318C22—84no—
319C22—88no—
320C22—82no—
321C22—84no—
322C22—162no—
323C22—164no—
324C22—168no—
325C22—168no—
326C22#230911yescharacteristic
327C22#238011yescharacteristic
328C22#238011yescharacteristic
329C22#281814yes—
330C4—24no—
331C4—42no—
332C4—82no—
333C4—164no—
334C4#230912yes—
335C4#238012yes—
336C4#281814yes—
337C2—21no—
338C2—24no—
339C2—21no—
340C2—162no—
341C2—164no—
342C2#1466111yescharacteristic
343C2#2336112yes—
subgroupG/H
center, socleC22#2380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3151C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1841C2semidirect
3. #11014C2semidirect
4. #11012C2semidirect
5. #10979C2semidirect
6. #10975C2semidirect
11. C2×#496C4semidirect
12. C2×#493C4semidirect
13. C2×#492C4semidirect
158. #97C22⋊C4semidirect
159. C4×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31512
C2×#18412
#110142
#110122
#109792
#109752