Finite Group Databasebeta

← #139020

Subgroups — #139020

Order 512.

297 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 78 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#33805C211yesmaximal, characteristic
2#18092C211yesmaximal, characteristic
3#11977C211yesmaximal, characteristic
4C2×#1364C211yesmaximal, characteristic, complemented
5#9810C211yesmaximal, characteristic, complemented
6#7292C211yesmaximal, characteristic, complemented
7#7210C211yesmaximal, characteristic, complemented
8#3168C2211yescharacteristic
9C2×#500C2211yescharacteristic
10#2994—21no—
11#2255—21no—
12#2243—21no—
13#2236C412yes—
14#2233C412yes—
15#2225—21no—
16#2224—21no—
17#2163C2211yescharacteristic
18C2×C4⋊C16C2211yescharacteristic
19#1997—21no—
20#1592—21no—
21#1584—21no—
22#1581—21no—
23SD16⋊C8—21no—
24#1566C2211yescharacteristic
25C2×#335C2211yescharacteristic
26#1549—21no—
27#1483—21no—
28#1451C2211yescharacteristic
29#1364—21no—
30#1364—21no—
31#1363—21no—
32#1360—21no—
33C2×C4⋊Q8—21no—
34#529—21no—
35C22×C16—21no—
36#500—21no—
37#500—21no—
38#500—21no—
39#500—21no—
40#499D411yescharacteristic
41#495D411yescharacteristic
42#488—21no—
43#467C2×C411yescharacteristic
44#466C2×C411yescharacteristic
45#465—21no—
46#446—21no—
47C8×Q8—21no—
48C4×Q16D411yescharacteristic
49C4×SD16—21no—
50C4×D8D411yescharacteristic
51#436—21no—
52#435—41no—
53C8×D4—21no—
54C8×D4—41no—
55#433—41no—
56#431D411yescharacteristic
57C2×#107—21no—
58C2×#107C2×C412yes—
59#424—41no—
60#423—21no—
61C2×C4⋊C8—21no—
62C4×M16D411yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#367—22no—
65#366—22no—
66C4⋊C16—21no—
67C4⋊C16—21no—
68C4⋊C16—21no—
69#358—22no—
70#357—22no—
71C22⋊C16C2×C412yes—
72#346—21no—
73#343—42no—
74#335—21no—
75#335D411yescharacteristic
76#335D411yescharacteristic
77#334—21no—
78#334—21no—
79#332—41no—
80#332—41no—
81Q8⋊C8—21no—
82Q8⋊C8—21no—
83D4⋊C8—21no—
84D4⋊C8—21no—
85#322—21no—
86#322—21no—
87C22×Q8—41no—
88C2×Q16C22⋊C411yescharacteristic
89C2×SD16—21no—
90C2×D8C22⋊C411yescharacteristic
91C2×M16—21no—
92C2×M16—41no—
93C2×M16—41no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8C22×C411yescharacteristic
97C22×C8#4111yescharacteristic
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102#126—41no—
103#125—21no—
104#124—21no—
105#123—41no—
106C22⋊Q8—21no—
107#121—21no—
108C4×Q8—21no—
109C4×D4—21no—
110C4×D4—41no—
111#117—41no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C42C2×D411yescharacteristic
115C2×C16—22no—
116C2×C16—41no—
117C2×C16—81no—
118#107—22no—
119#107—22no—
120#107—41no—
121#107—41no—
122#107C22⋊C411yescharacteristic
123#107C22⋊C411yescharacteristic
124#106—21no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8D811yescharacteristic
133C4⋊C8D811yescharacteristic
134#103—21no—
135#103—21no—
136#102—21no—
137#102—21no—
138C22⋊C8—21no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141#97—21no—
142#97—41no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8—41no—
146C4×C8C2×D411yescharacteristic
147C4×C8C2×D411yescharacteristic
148C4×C8C2×D411yescharacteristic
149C4×C8#4111yescharacteristic
150C2×Q8—21no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8—161no—
154C2×Q8—161no—
155C2×D4—21no—
156C2×D4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4C4×D411yescharacteristic
160C22×C4AGL(1,ℤ/8)11yescharacteristic
161C22×C4AGL(1,ℤ/8)11yescharacteristic
162Q16—41no—
163SD16—41no—
164SD16—41no—
165D8—41no—
166M16—42no—
167M16—82no—
168M16—161no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—81no—
178C2×C8—161no—
179C2×C8—161no—
180C2×C8C2×C22⋊C411yescharacteristic
181C2×C8C4×D411yescharacteristic
182C2×C8#12111yescharacteristic
183C2×C8#12311yescharacteristic
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C42—41no—
200C42#12011yescharacteristic
201C42#12111yescharacteristic
202C42C2×D811yescharacteristic
203C42#13511yescharacteristic
204C16—42no—
205C16—81no—
206C23—41no—
207C23#44211yescharacteristic
208Q8—41no—
209Q8—41no—
210Q8—162no—
211Q8—161no—
212Q8—161no—
213D4—41no—
214D4—41no—
215D4—82no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—82no—
227C2×C4—162no—
228C2×C4#38611yescharacteristic
229C2×C4C4×D811yescharacteristic
230C2×C4#44711yescharacteristic
231C2×C4#44911yescharacteristic
232C2×C4#45911yescharacteristic
233C2×C4#46311yescharacteristic
234C8—21no—
235C8—21no—
236C8—21no—
237C8—41no—
238C8—41no—
239C8—81no—
240C8—81no—
241C8—81no—
242C8—161no—
243C8#47111yescharacteristic
244C8#47311yescharacteristic
245C22—21no—
246C22—21no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22#187911yescharacteristic
251C22#218012yes—
252C4—21no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—81no—
257C4—81no—
258C4—81no—
259C4—162no—
260C4#190811yescharacteristic
261C4#191511yescharacteristic
262C4#219211yescharacteristic
263C4#219311yescharacteristic
264C2—22no—
265C2—161no—
266C2#927111yescharacteristic
267C2#1302211yescharacteristic
268C2#1303911yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#466C2×C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1364C2semidirect
5. #9810C2semidirect
6. #7292C2semidirect
7. #7210C2semidirect

Maximal subgroups

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

subgroupindex
#338052
#180922
#119772
C2×#13642
#98102
#72922
#72102