Finite Group Databasebeta

← #114637

Subgroups — #114637

Order 512.

1142 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3152C211yesmaximal, characteristic
2#21840C212yesmaximal, complemented
3#21794C211yesmaximal, characteristic
4#17675C211yesmaximal, characteristic
5#7972C212yesmaximal, complemented
6C22×#137—22no—
7C2×#532C2211yescharacteristic
8C22×C4⋊Q8C2211yescharacteristic
9#3152—24no—
10#3152—24no—
11C2×#497—22no—
12C2×#494C412yescomplemented
13C2×#492C412yescomplemented
14C2×#432C2212yes—
15C2×C4×M16C2211yescharacteristic
16#2283C2212yes—
17#1993—24no—
18#1987—24no—
19#1964—24no—
20#1823—24no—
21#1782—22no—
22#1533—22no—
23#1530—24no—
24#1288—24no—
25C22×#41D412yes—
26C23×Q8—21no—
27C2×#137—216no—
28C2×#137—416no—
29C22×Q16—24no—
30C22×SD16—24no—
31C22×M16—21no—
32#532—22no—
33#532—22no—
34#532—22no—
35#532—22no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8D414yes—
44C2×#127C2×C411yescharacteristic
45C2×#124D412yes—
46C2×#121—22no—
47C2×C4×D4—22no—
48C22×C4⋊C4—22no—
49C22×C42C2311yescharacteristic
50#497—216no—
51#494—24no—
52#494—24no—
53#492—24no—
54#492—24no—
55#432—22no—
56#432—22no—
57#432—22no—
58#432—22no—
59C2×C4⋊C8—22no—
60C2×#99—28no—
61C2×C22⋊C8—22no—
62C4×M16—22no—
63C4×M16D414yes—
64C2×#97—21no—
65C2×#97D414yes—
66C2×C4×C8C2×C412yes—
67#383—22no—
68#382—22no—
69#381—22no—
70#380D414yes—
71C4×C4⋊C4—22no—
72C4×C22⋊C4—22no—
73#376—22no—
74#340—216no—
75#329—28no—
76C2×#41—24no—
77C2×#41—24no—
78C2×#41—48no—
79C22×Q8—22no—
80C22×Q8—24no—
81C22×Q8—48no—
82C22×Q8—42no—
83C22×Q8—41no—
84C22×Q8C22⋊C412yes—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4C22⋊C412yes—
88C23×C4—22no—
89C23×C4—41no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92#137—464no—
93C2×Q16—216no—
94C2×Q16—416no—
95C2×SD16—216no—
96C2×SD16—416no—
97C2×M16—28no—
98C2×M16—24no—
99C22×C8—22no—
100C22×C8—24no—
101C4⋊Q8—22no—
102C4⋊Q8—21no—
103C4⋊Q8—21no—
104C4⋊Q8—24no—
105C4⋊Q8—22no—
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—41no—
110C4⋊Q8—42no—
111C4⋊Q8—41no—
112C4⋊Q8—42no—
113C4⋊Q8C22⋊C414yes—
114#127—21no—
115#127—21no—
116#127—22no—
117#124—22no—
118#124—24no—
119#124—22no—
120#121—42no—
121#121—42no—
122#121—42no—
123#121—42no—
124C4×D4—42no—
125C4×D4—42no—
126C4×D4—42no—
127C4×D4—42no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C42—42no—
139C2×C42C22×C411yescharacteristic
140C2×C42C2×D412yes—
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C2×C42C2×D411yescharacteristic
144C2×C42#4112yes—
145C2×C42#4112yes—
146C4⋊C8—42no—
147C4⋊C8—42no—
148#99—88no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151#97—24no—
152#97—24no—
153#97—24no—
154C4×C8—22no—
155C4×C8C22⋊C414yes—
156#95—24no—
157#95—42no—
158#95—42no—
159#95—42no—
160C24—22no—
161C24#12011yescharacteristic
162#41—416no—
163#41—88no—
164C2×Q8—22no—
165C2×Q8—24no—
166C2×Q8—22no—
167C2×Q8—44no—
168C2×Q8—48no—
169C2×Q8—416no—
170C2×Q8—42no—
171C2×Q8—44no—
172C2×Q8—84no—
173C2×Q8—88no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—44no—
177C2×D4—44no—
178C2×D4—48no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—84no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C4×D412yes—
193C22×C4#12012yes—
194C22×C4#12012yes—
195C22×C4#12112yes—
196C22×C4#12112yes—
197C22×C4#12112yes—
198C22×C4#12112yes—
199C22×C4#12311yescharacteristic
200C22×C4#12712yes—
201Q16—432no—
202SD16—432no—
203M16—216no—
204C2×C8—28no—
205C2×C8—24no—
206C2×C8—48no—
207C2×C8—44no—
208C4⋊C4—28no—
209C4⋊C4—48no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C22⋊C4—48no—
218C42—84no—
219C42C2×C22⋊C411yescharacteristic
220C42C2×C22⋊C411yescharacteristic
221C42C4×D412yes—
222C42#12012yes—
223C42#12011yescharacteristic
224C42#12112yes—
225C42#12112yes—
226C42#12312yes—
227C42#12311yescharacteristic
228C42#12412yes—
229C23—42no—
230C23—42no—
231C23—44no—
232C23—84no—
233C23#38611yescharacteristic
234C23#39212yes—
235C23#45314yes—
236Q8—48no—
237Q8—44no—
238Q8—82no—
239Q8—88no—
240Q8—84no—
241D4—44no—
242D4—82no—
243D4—84no—
244D4—88no—
245D4—84no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—88no—
260C2×C4—84no—
261C2×C4—88no—
262C2×C4—84no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4#37911yescharacteristic
266C2×C4#38611yescharacteristic
267C2×C4#38612yes—
268C2×C4#39012yes—
269C2×C4#39212yes—
270C2×C4#39412yes—
271C2×C4#39412yes—
272C8—28no—
273C8—84no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—44no—
278C22—84no—
279C22—82no—
280C22—164no—
281C22#143811yescharacteristic
282C22#188712yes—
283C22#200214yes—
284C4—22no—
285C4—22no—
286C4—22no—
287C4—21no—
288C4—22no—
289C4—21no—
290C4—82no—
291C4—84no—
292C4—164no—
293C4#190514yes—
294C2—24no—
295C2—22no—
296C2—162no—
297C2#1097511yescharacteristic
298C2#1101312yes—
subgroupG/H
center, socleC22#1438
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21840C2semidirect
5. #7972C2semidirect
12. C2×#494C4semidirect
13. C2×#492C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31522
#218402
#217942
#176752
#79722