Finite Group Databasebeta

← #503303

Subgroups — #503303

Order 512.

764 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 186 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×#3380C211yesmaximal, characteristic
2C2×#2252C211yesmaximal, characteristic, complemented
3C2×#2226C211yesmaximal, characteristic
4#28676C211yesmaximal, characteristic, complemented
5#13453C214yesmaximal, complemented
6#13276C211yesmaximal, characteristic, complemented
7#13263C211yesmaximal, characteristic, complemented
8#12935C214yesmaximal
9C4×#357C211yesmaximal, characteristic
10C2×#554—21no—
11#3380—24no—
12#3380—24no—
13C2×#500C2211yescharacteristic
14C2×#500C2211yescharacteristic
15C2×#499C2212yes—
16C2×#478—22no—
17C2×#476—21no—
18C2×C8×Q8C2211yescharacteristic
19C2×C4×D8C2212yes—
20C2×C4×D8C2211yescharacteristic
21#2567—21no—
22#2252—24no—
23#2226—24no—
24C2×#367C2212yes—
25C2×#366C2214yes—
26C2×C4⋊C16C2211yescharacteristic
27C2×#357C2212yes—
28C2×#357C2212yes—
29C2×C4×C16C2211yescharacteristic
30#2087—22no—
31#1987—21no—
32#1942C2214yes—
33C4×#107C2211yescharacteristic
34#1380C2212yes—
35#1380C2214yes—
36#1374C2214yes—
37C22×D8C2311yescharacteristic
38#554—28no—
39#554—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×#125—22no—
43C2×C22⋊Q8—41no—
44C2×C22⋊Q8—42no—
45C2×C4×Q8D411yescharacteristic
46C2×C4×D4—22no—
47C2×C4×D4—21no—
48C22×C4⋊C4—41no—
49C22×C16C2312yes—
50C22×C16C2312yes—
51#500—22no—
52#500—22no—
53#500—22no—
54#500D414yes—
55#499—24no—
56#499—24no—
57#478—216no—
58#476—28no—
59C8×Q8—22no—
60C8×Q8—22no—
61C4×D8—24no—
62C4×D8—24no—
63C4×D8—22no—
64C4×D8D414yes—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107C2312yes—
68C2×#107C2311yescharacteristic
69C2×#107C2312yes—
70C2×#107C2312yes—
71C2×#106—22no—
72C2×C4⋊C8D412yes—
73C2×C4⋊C8D411yescharacteristic
74C2×#102—21no—
75C2×#102—22no—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78#395—42no—
79#391—41no—
80#389—24no—
81#384—41no—
82C4×C4⋊C4—21no—
83#367—24no—
84#366—28no—
85C4⋊C16—22no—
86#357—24no—
87#357Q818yescomplemented
88C4×C16—22no—
89#340—21no—
90#340—24no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8D811yescharacteristic
94C22×D4—21no—
95C23×C4—41no—
96C23×C4—42no—
97C2×D8—21no—
98C2×D8—22no—
99C2×D8C2×D412yes—
100C2×D8C2×Q814yes—
101C22×C8C2×D412yes—
102C22×C8C2×D411yescharacteristic
103C22×C8C2411yescharacteristic
104C4⋊Q8—24no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108#125—28no—
109#125—44no—
110C22⋊Q8—416no—
111C22⋊Q8—48no—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×D4—28no—
115C4×D4—24no—
116C4×D4—44no—
117C4×D4—42no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—48no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4D811yescharacteristic
129C2×C4⋊C4D812yes—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—41no—
132C2×C42—41no—
133C2×C42C2×D412yes—
134C2×C42C2×D411yescharacteristic
135C2×C16—22no—
136C2×C16—22no—
137C2×C16—22no—
138C2×C16—24no—
139C2×C16C2×Q814yes—
140#107—24no—
141#107—21no—
142#107—22no—
143#107—22no—
144#107—22no—
145#107—22no—
146#107—42no—
147#107C2×D414yes—
148#107C2×D412yes—
149#107C2×Q814yes—
150#106—28no—
151C4⋊C8—24no—
152C4⋊C8—22no—
153#102—28no—
154#102—24no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8C2×D412yes—
158C4×C8C2×D412yes—
159#95—41no—
160#95—44no—
161C24—41no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—44no—
166C2×Q8—82no—
167C2×Q8—162no—
168C2×Q8—162no—
169C2×D4—22no—
170C2×D4—24no—
171C2×D4—41no—
172C2×D4—42no—
173C22×C4—48no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—164no—
182C22×C4C2×D812yes—
183C22×C4C2×D812yes—
184C22×C4C2×D811yescharacteristic
185C22×C4C2×D811yescharacteristic
186C22×C4C22×D411yescharacteristic
187D8—22no—
188D8—24no—
189D8C4⋊Q814yes—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8#12012yes—
194C2×C8C22×D411yescharacteristic
195C2×C8C22×D411yescharacteristic
196C2×C8C22×Q812yes—
197C2×C82−1+412yes—
198C4⋊C4—22no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—48no—
204C4⋊C4—48no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C4⋊C4—164no—
213C22⋊C4—48no—
214C22⋊C4—44no—
215C42—22no—
216C42—22no—
217C42—82no—
218C42#12012yes—
219C42AGL(1,ℤ/8)12yes—
220C16—24no—
221C16—42no—
222C23—44no—
223C23—42no—
224C23—81no—
225C23—81no—
226C23—82no—
227C23C22×D811yescharacteristic
228Q8—44no—
229Q8—164no—
230Q8—164no—
231D4—24no—
232D4—44no—
233D4—42no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—82no—
247C2×C4—84no—
248C2×C4—88no—
249C2×C4—84no—
250C2×C4—82no—
251C2×C4—168no—
252C2×C4#44712yes—
253C2×C4#44712yes—
254C2×C4#50812yes—
255C2×C4#51012yes—
256C2×C4C2×#12011yescharacteristic
257C2×C4D4×Q812yes—
258C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
259C2×C4#57812yes—
260C8—22no—
261C8—42no—
262C8C2×C4⋊Q811yescharacteristic
263C8#53611yescharacteristic
264C8D4×Q812yes—
265C22—44no—
266C22—84no—
267C22—84no—
268C22—82no—
269C22—82no—
270C22C2×#44711yescharacteristic
271C22Q8×D812yes—
272C22C2×#50811yescharacteristic
273C22C2×#51011yescharacteristic
274C22#341512yes—
275C4—22no—
276C4—42no—
277C4—42no—
278C4—84no—
279C4—82no—
280C4—84no—
281C4—162no—
282C4#218914yes—
283C4#243411yescharacteristic
284C4#301111yescharacteristic
285C4#338612yes—
286C2—84no—
287C2#2583811yescharacteristic
288C2C2×#218911yescharacteristic
289C2#3352211yescharacteristic
290C2#3394712yes—
291C2#3395212yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2252C2semidirect
4. #28676C2semidirect
5. #13453C2semidirect
6. #13276C2semidirect
7. #13263C2semidirect
87. #357Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#33802
C2×#22522
C2×#22262
#286762
#134532
#132762
#132632
#129352
C4×#3572