Finite Group Databasebeta

← #503308

Subgroups — #503308

Order 512.

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

normal subgroups only

This group has 726 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×#3383C211yesmaximal, characteristic
2C2×#2254C211yesmaximal, characteristic
3C2×#2227C211yesmaximal, characteristic
4#28676C211yesmaximal, characteristic
5#13453C214yesmaximal
6#13275C211yesmaximal, characteristic
7#13263C211yesmaximal, characteristic
8#12938C214yesmaximal
9C4×#358C211yesmaximal, characteristic
10C2×#557—21no—
11#3383—24no—
12#3383—24no—
13C2×#500C2211yescharacteristic
14C2×#500C2211yescharacteristic
15C2×#499C2212yes—
16C2×#479—22no—
17C2×#475—21no—
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2212yes—
20C2×C4×Q16C2211yescharacteristic
21#2567—21no—
22#2254—24no—
23#2227—24no—
24C2×#367C2214yes—
25C2×#366C2212yes—
26C2×C4⋊C16C2211yescharacteristic
27C2×#358C2212yes—
28C2×#358C2212yes—
29C2×C4×C16C2211yescharacteristic
30#2087—22no—
31#1987—21no—
32#1942C2214yes—
33C4×#107C2211yescharacteristic
34#1380C2212yes—
35#1380C2214yes—
36#1374C2214yes—
37C22×Q16C2311yescharacteristic
38#557—28no—
39#557—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—41no—
43C2×#125—22no—
44C2×#125—42no—
45C2×C4×Q8—21no—
46C2×C4×Q8—22no—
47C2×C4×Q8—41no—
48C2×C4×Q8D411yescharacteristic
49C22×C16C2312yes—
50C22×C16C2312yes—
51#500—22no—
52#500—22no—
53#500—22no—
54#500D414yes—
55#499—24no—
56#499—24no—
57#479—216no—
58#475—28no—
59C8×Q8—22no—
60C8×Q8—22no—
61C4×Q16—24no—
62C4×Q16—24no—
63C4×Q16—22no—
64C4×Q16D414yes—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107C2312yes—
68C2×#107C2311yescharacteristic
69C2×#107C2312yes—
70C2×#107C2312yes—
71C2×#106—22no—
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8D412yes—
74C2×#103—22no—
75C2×#103—21no—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78#395—42no—
79#391—41no—
80#389—24no—
81#384—41no—
82C4×C4⋊C4—21no—
83#367—28no—
84#366—24no—
85C4⋊C16—22no—
86#358—24no—
87#358Q818yescomplemented
88C4×C16—22no—
89#340—21no—
90#340—24no—
91C22×Q8—21no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8D811yescharacteristic
95C2×Q16—21no—
96C2×Q16—22no—
97C2×Q16C2×D412yes—
98C2×Q16C2×Q814yes—
99C22×C8C2×D412yes—
100C22×C8C2×D411yescharacteristic
101C22×C8C2411yescharacteristic
102C4⋊Q8—24no—
103C4⋊Q8—48no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107#125—28no—
108#125—44no—
109#125—416no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×Q8—28no—
113C4×Q8—24no—
114C4×Q8—44no—
115C4×Q8—48no—
116C4×Q8—42no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4D811yescharacteristic
129C2×C4⋊C4D812yes—
130C2×C42—42no—
131C2×C42—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×D412yes—
148#107C2×D414yes—
149#107C2×Q814yes—
150#106—28no—
151C4⋊C8—24no—
152C4⋊C8—22no—
153#103—28no—
154#103—24no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8C2×D412yes—
158C4×C8C2×D412yes—
159#95—41no—
160#95—44no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—24no—
166C2×Q8—41no—
167C2×Q8—42no—
168C2×Q8—44no—
169C2×Q8—82no—
170C2×Q8—82no—
171C2×Q8—82no—
172C2×Q8—162no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4C2×D812yes—
179C22×C4C2×D812yes—
180C22×C4C2×D811yescharacteristic
181C22×C4C2×D811yescharacteristic
182C22×C4C22×D411yescharacteristic
183Q16—22no—
184Q16—24no—
185Q16C4⋊Q814yes—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—24no—
189C2×C8#12012yes—
190C2×C8C22×D411yescharacteristic
191C2×C8C22×D411yescharacteristic
192C2×C8C22×Q812yes—
193C2×C82−1+412yes—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—48no—
200C4⋊C4—48no—
201C4⋊C4—48no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—82no—
210C4⋊C4—164no—
211C42—22no—
212C42—22no—
213C42—48no—
214C42—44no—
215C42—82no—
216C42#12012yes—
217C42AGL(1,ℤ/8)12yes—
218C16—24no—
219C16—42no—
220C23C22×D811yescharacteristic
221Q8—24no—
222Q8—44no—
223Q8—44no—
224Q8—42no—
225Q8—88no—
226Q8—164no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—84no—
244C2×C4—82no—
245C2×C4#44712yes—
246C2×C4#44712yes—
247C2×C4#50812yes—
248C2×C4#51012yes—
249C2×C4C2×#12011yescharacteristic
250C2×C4D4×Q812yes—
251C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
252C2×C4#57812yes—
253C8—22no—
254C8—42no—
255C8C2×C4⋊Q811yescharacteristic
256C8#53611yescharacteristic
257C8D4×Q812yes—
258C22C2×#44711yescharacteristic
259C22Q8×D812yes—
260C22C2×#50811yescharacteristic
261C22C2×#51011yescharacteristic
262C22#341612yes—
263C4—22no—
264C4—42no—
265C4—44no—
266C4—42no—
267C4—84no—
268C4—84no—
269C4—82no—
270C4—162no—
271C4#218914yes—
272C4#243411yescharacteristic
273C4#301111yescharacteristic
274C4#338612yes—
275C2#2583811yescharacteristic
276C2C2×#218911yescharacteristic
277C2#3352411yescharacteristic
278C2#3394812yes—
279C2#3395412yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
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
87. #358Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#33832
C2×#22542
C2×#22272
#286762
#134532
#132752
#132632
#129382
C4×#3582