Finite Group Databasebeta

← #503307

Subgroups — #503307

Order 512.

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

normal subgroups only

This group has 774 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×#2254C211yesmaximal, characteristic, complemented
3C2×#2224C211yesmaximal, characteristic
4#28675C211yesmaximal, characteristic, complemented
5#13453C214yesmaximal, complemented
6#13276C211yesmaximal, characteristic, complemented
7#13263C211yesmaximal, characteristic, complemented
8#12937C214yesmaximal
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#2546—21no—
22#2254—24no—
23#2224—24no—
24C2×#367C2214yes—
25C2×#366C2212yes—
26C2×C4⋊C16C2211yescharacteristic
27C2×#357C2212yes—
28C2×#357C2212yes—
29C2×C4×C16C2211yescharacteristic
30#2086—22no—
31#1988—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×C4⋊Q8—41no—
43C2×#125—22no—
44C2×C22⋊Q8—41no—
45C2×C22⋊Q8—42no—
46C2×C4×Q8D411yescharacteristic
47C2×C4×D4—22no—
48C2×C4×D4—21no—
49C22×C4⋊C4—41no—
50C22×C16C2312yes—
51C22×C16C2312yes—
52#500—22no—
53#500—22no—
54#500—22no—
55#500D414yes—
56#499—24no—
57#499—24no—
58#478—216no—
59#476—28no—
60C8×Q8—22no—
61C8×Q8—22no—
62C4×D8—24no—
63C4×D8—24no—
64C4×D8—22no—
65C4×D8D414yes—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107C2312yes—
69C2×#107C2311yescharacteristic
70C2×#107C2312yes—
71C2×#107C2312yes—
72C2×#106—22no—
73C2×C4⋊C8D412yes—
74C2×C4⋊C8D411yescharacteristic
75C2×#102—21no—
76C2×#102—22no—
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79#398—42no—
80#391—41no—
81#389—24no—
82C4×C4⋊C4—21no—
83#367—28no—
84#366—24no—
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—
108C4⋊Q8—84no—
109#125—28no—
110#125—44no—
111C22⋊Q8—416no—
112C22⋊Q8—48no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×D4—28no—
116C4×D4—24no—
117C4×D4—44no—
118C4×D4—42no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—48no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4D811yescharacteristic
130C2×C4⋊C4D812yes—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—41no—
133C2×C42—41no—
134C2×C42C2×D412yes—
135C2×C42C2×D411yescharacteristic
136C2×C16—22no—
137C2×C16—22no—
138C2×C16—22no—
139C2×C16—24no—
140C2×C16C2×Q814yes—
141#107—24no—
142#107—21no—
143#107—22no—
144#107—22no—
145#107—22no—
146#107—22no—
147#107—42no—
148#107C2×D412yes—
149#107C2×D414yes—
150#107C2×Q814yes—
151#106—28no—
152C4⋊C8—24no—
153C4⋊C8—22no—
154#102—28no—
155#102—24no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8C2×D412yes—
159C4×C8C2×D412yes—
160#95—41no—
161#95—44no—
162C24—41no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—44no—
167C2×Q8—82no—
168C2×Q8—82no—
169C2×Q8—82no—
170C2×Q8—162no—
171C2×D4—22no—
172C2×D4—24no—
173C2×D4—41no—
174C2×D4—42no—
175C22×C4—48no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—164no—
184C22×C4C2×D812yes—
185C22×C4C2×D812yes—
186C22×C4C2×D811yescharacteristic
187C22×C4C2×D811yescharacteristic
188C22×C4C22×D411yescharacteristic
189D8—22no—
190D8—24no—
191D8C4⋊Q814yescomplemented
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—24no—
195C2×C8#12012yes—
196C2×C8C22×D411yescharacteristic
197C2×C8C22×D411yescharacteristic
198C2×C8C22×Q812yes—
199C2×C82−1+412yes—
200C4⋊C4—22no—
201C4⋊C4—24no—
202C4⋊C4—22no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—48no—
206C4⋊C4—48no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—88no—
214C4⋊C4—82no—
215C22⋊C4—48no—
216C22⋊C4—44no—
217C42—22no—
218C42—22no—
219C42—82no—
220C42#12012yes—
221C42AGL(1,ℤ/8)12yes—
222C16—24no—
223C16—42no—
224C23—44no—
225C23—42no—
226C23—81no—
227C23—81no—
228C23—82no—
229C23C22×D811yescharacteristic
230Q8—44no—
231Q8—164no—
232Q8—164no—
233D4—24no—
234D4—44no—
235D4—42no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—81no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—84no—
250C2×C4—88no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—168no—
255C2×C4#44712yes—
256C2×C4#44712yes—
257C2×C4#50812yes—
258C2×C4#50912yes—
259C2×C4C2×#12011yescharacteristic
260C2×C4D4×Q812yes—
261C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
262C2×C4#57812yes—
263C8—22no—
264C8—42no—
265C8C2×C4⋊Q811yescharacteristic
266C8#53611yescharacteristic
267C8D4×Q812yes—
268C22—44no—
269C22—84no—
270C22—84no—
271C22—82no—
272C22—82no—
273C22C2×#44711yescharacteristic
274C22Q8×D812yes—
275C22C2×#50811yescharacteristic
276C22C2×#50911yescharacteristic
277C22#341612yes—
278C4—22no—
279C4—42no—
280C4—42no—
281C4—84no—
282C4—82no—
283C4—84no—
284C4—162no—
285C4#218614yes—
286C4#243411yescharacteristic
287C4#301111yescharacteristic
288C4#338612yes—
289C2—84no—
290C2#2583811yescharacteristic
291C2C2×#218611yescharacteristic
292C2#3352311yescharacteristic
293C2#3394812yes—
294C2#3395112yes—
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×#2254C2semidirect
4. #28675C2semidirect
5. #13453C2semidirect
6. #13276C2semidirect
7. #13263C2semidirect
87. #357Q8semidirect
191. D8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#33802
C2×#22542
C2×#22242
#286752
#134532
#132762
#132632
#129372
C4×#3572