Finite Group Databasebeta

← #503296

Subgroups — #503296

Order 512.

717 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 190 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
1#60660C211yesmaximal, characteristic
2#33732C212yesmaximal, complemented
3#33214C211yesmaximal, characteristic
4#13454C214yesmaximal, complemented
5#13287C212yesmaximal, complemented
6#12923C214yesmaximal
7#12602C211yesmaximal, characteristic
8#3458—21no—
9#3164—24no—
10#3163—24no—
11#3162—21no—
12#3162C2212yes—
13#3160—21no—
14#3160C2212yes—
15C2×#499C2211yescharacteristic
16#2911C2212yes—
17C2×#427—21no—
18C2×#405C2211yescharacteristic
19#2629—21no—
20#2255—24no—
21#2253—24no—
22#2159C2218yes—
23C2×#349C2212yes—
24C4×M32C2212yes—
25#1992—21no—
26#1944C2212yes—
27#1941—21no—
28#1844C2211yescharacteristic
29#1809C2212yes—
30#1807—21no—
31#1386C2218yes—
32#1377C2212yes—
33C22×M16C2311yescharacteristic
34C23×C8D411yescharacteristic
35#544—28no—
36#544—42no—
37#543—42no—
38#533—22no—
39#533—22no—
40#533—44no—
41C2×#125—21no—
42C2×#117—22no—
43C2×#117D411yescharacteristic
44C2×M32C2314yes—
45#501—28no—
46#501—28no—
47#500—22no—
48#500D414yes—
49#499—22no—
50#499—22no—
51#499—22no—
52#499—22no—
53#499—22no—
54#499D414yes—
55#498—22no—
56#498—22no—
57#428—28no—
58#427—22no—
59#427—22no—
60#427—22no—
61#427—22no—
62#427—22no—
63#427C2314yes—
64C2×#107—22no—
65C2×#107C2312yes—
66C2×#106—22no—
67C2×#106—21no—
68C2×#106—21no—
69C2×#106C2312yes—
70#405—22no—
71#405—22no—
72C2×#97D411yescharacteristic
73C2×#97C2311yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#398—42no—
77#396—42no—
78#389—22no—
79#384—21no—
80#380—22no—
81#367—28no—
82#366—28no—
83#349Q818yescomplemented
84#346—22no—
85C4×C16—22no—
86#340—22no—
87#340—22no—
88C23×C4C2×D411yescharacteristic
89C2×M16—22no—
90C2×M16—22no—
91C2×M16C2×Q814yes—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—22no—
95C22×C8Q1614yes—
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C22×C8C2411yescharacteristic
100C4⋊Q8—24no—
101C4⋊Q8—24no—
102C4⋊Q8—48no—
103#125—24no—
104#125—24no—
105#125—48no—
106#125—42no—
107#125—42no—
108C22⋊Q8—48no—
109C22⋊Q8—48no—
110C22⋊Q8—44no—
111C22⋊Q8—44no—
112C4×Q8—44no—
113C4×Q8—44no—
114#117—22no—
115#117—22no—
116#117—24no—
117#117—24no—
118#117—42no—
119#117—42no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4D812yes—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4D812yes—
130C2×C42—42no—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133M32—24no—
134M32C2×Q818yes—
135C2×C16—24no—
136#107—24no—
137#107—24no—
138#107—24no—
139#107—24no—
140#107C2×D418yes—
141#106—24no—
142#106—24no—
143#106—24no—
144#106—24no—
145#106—24no—
146#106—41no—
147#106—41no—
148#106—42no—
149#97—22no—
150#97—22no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8C2×D412yes—
154C4×C8C2×D412yes—
155#95—42no—
156#95—42no—
157C24C2×D811yescharacteristic
158C2×Q8—44no—
159C2×Q8—44no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4C4⋊Q812yes—
168C22×C4C2×D811yescharacteristic
169C22×C4C2×D812yes—
170C22×C4C2×D812yes—
171C22×C4C2×Q1612yes—
172C22×C4C22×D411yescharacteristic
173M16—28no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8#12011yescharacteristic
181C2×C8#12011yescharacteristic
182C2×C8C4⋊Q812yes—
183C2×C8C2×Q1612yes—
184C2×C8C2×Q1612yes—
185C2×C8#13712yes—
186C2×C8C22×D412yes—
187C2×C8C22×Q812yes—
188C2×C82−1+412yes—
189C4⋊C4—24no—
190C4⋊C4—28no—
191C4⋊C4—44no—
192C4⋊C4—48no—
193C4⋊C4—44no—
194C4⋊C4—48no—
195C4⋊C4—44no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—84no—
201C4⋊C4—162no—
202C4⋊C4—162no—
203C22⋊C4—24no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—162no—
207C42—21no—
208C42—21no—
209C42—44no—
210C42—44no—
211C42#12012yes—
212C42AGL(1,ℤ/8)12yes—
213C16—28no—
214C23—21no—
215C23—21no—
216C23—42no—
217C23#50012yes—
218C23C22×D811yescharacteristic
219Q8—164no—
220Q8—164no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—48no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—164no—
238C2×C4#44714yes—
239C2×C4#50012yes—
240C2×C4C2×#12011yescharacteristic
241C2×C4C2×C4⋊Q811yescharacteristic
242C2×C4#53611yescharacteristic
243C2×C4C22×Q1611yescharacteristic
244C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
245C2×C4C2×#13711yescharacteristic
246C8—22no—
247C8—22no—
248C8#45112yes—
249C8#45112yes—
250C8D4×Q814yes—
251C22—22no—
252C22—22no—
253C22—42no—
254C22—42no—
255C22—81no—
256C22C2×#44711yescharacteristic
257C22C2×#50011yescharacteristic
258C22#322611yescharacteristic
259C4—22no—
260C4—24no—
261C4—42no—
262C4—88no—
263C4—162no—
264C4—162no—
265C4#243411yescharacteristic
266C4C2×#45111yescharacteristic
267C4#316511yescharacteristic
268C4#323011yescharacteristic
269C2—21no—
270C2—21no—
271C2—42no—
272C2#2581311yescharacteristic
273C2#3353312yes—
subgroupG/H
center, socleC22C2×#447
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33732C2semidirect
4. #13454C2semidirect
5. #13287C2semidirect
83. #349Q8semidirect

Maximal subgroups

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

subgroupindex
#606602
#337322
#332142
#134542
#132872
#129232
#126022