Finite Group Databasebeta

← #138476

Subgroups — #138476

Order 512.

334 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 82 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#22739C211yesmaximal, characteristic
2#19265C211yesmaximal, characteristic
3#11857C211yesmaximal, characteristic
4#11745C211yesmaximal, characteristic
5#11639C211yesmaximal, characteristic
6#7299C211yesmaximal, characteristic
7#7222C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#3113—21no—
10C2×C8×Q8C2211yescharacteristic
11#2355—21no—
12#2163C2211yescharacteristic
13#2162C2211yescharacteristic
14#1949—22no—
15#1947—21no—
16#1835—21no—
17#1834C2211yescharacteristic
18C4×#107—21no—
19C4×#106C2211yescharacteristic
20#1680—22no—
21#1674C412yescomplemented
22#1670C412yescomplemented
23#1568C2211yescharacteristic
24#1552—21no—
25#1348—22no—
26#1348—21no—
27#1348—21no—
28Q8⋊C16—22no—
29#533—21no—
30C2×C4×Q8D411yescharacteristic
31#500C2×C411yescharacteristic
32#494C2×C411yescharacteristic
33#487—21no—
34#483—41no—
35#475—22no—
36#467—21no—
37#467—21no—
38#462—22no—
39#451—41no—
40C8×Q8—21no—
41C8×Q8—21no—
42C8×Q8C2×C412yes—
43C8×Q8D412yes—
44#441—42no—
45C4×Q16—21no—
46C4×Q16—21no—
47#432—21no—
48C2×#107—22no—
49C2×#106—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8D411yescharacteristic
52C2×C4×C8—21no—
53C2×C4×C8C2311yescharacteristic
54#391—41no—
55#387—42no—
56#384—21no—
57#382—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—21no—
60C4×C4⋊C4—41no—
61C4⋊C16—21no—
62C4⋊C16—21no—
63C22⋊C16—21no—
64C22⋊C16—21no—
65#346—21no—
66C4×C16—21no—
67#340—22no—
68#340—21no—
69#340D412yes—
70#340D411yescharacteristic
71#340D411yescharacteristic
72#335C2×C412yes—
73#334—21no—
74#332—22no—
75#332—22no—
76C22×Q8D811yescharacteristic
77C2×Q16—21no—
78C2×Q16—21no—
79C2×Q16—42no—
80C22×C8—21no—
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C4⋊Q8—21no—
84C4⋊Q8—21no—
85#125—41no—
86C22⋊Q8—41no—
87C22⋊Q8—41no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8—41no—
91C4×Q8—41no—
92C4×Q8C22⋊C412yes—
93C4×Q8SD1612yes—
94#117—41no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4D811yescharacteristic
102C2×C42—21no—
103C2×C42—41no—
104C2×C42—42no—
105C2×C42C2×D411yescharacteristic
106C2×C16—41no—
107C2×C16—41no—
108#107—21no—
109#107—21no—
110#107—44no—
111#106—42no—
112#106—42no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—21no—
117C4⋊C8C22⋊C412yes—
118#103—42no—
119#103—41no—
120#103—41no—
121#103—42no—
122C22⋊C8—41no—
123#97—41no—
124C4×C8—22no—
125C4×C8—22no—
126C4×C8C22×C411yescharacteristic
127C4×C8C2×D411yescharacteristic
128C4×C8#4111yescharacteristic
129C4×C8#4111yescharacteristic
130#95—42no—
131#95—42no—
132#95—42no—
133#95—41no—
134C2×Q8—21no—
135C2×Q8—42no—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8#10212yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4#12011yescharacteristic
145C22×C4C2×D811yescharacteristic
146C22×C4AGL(1,ℤ/8)11yescharacteristic
147Q16—81no—
148Q16—81no—
149Q16—84no—
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8C4×D411yescharacteristic
156C2×C8C4×D411yescharacteristic
157C2×C8#12111yescharacteristic
158C2×C8#12111yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—82no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—161no—
176C4⋊C4#10212yes—
177C22⋊C4—81no—
178C42—22no—
179C42—22no—
180C42—82no—
181C42—81no—
182C42—162no—
183C42C2×C22⋊C411yescharacteristic
184C42#12311yescharacteristic
185C42C2×SD1611yescharacteristic
186C42#13711yescharacteristic
187C16—81no—
188C16—81no—
189C23#44711yescharacteristic
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—82no—
194Q8—81no—
195Q8—81no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4#38611yescharacteristic
218C2×C4C2×#10211yescharacteristic
219C2×C4#41711yescharacteristic
220C2×C4#44811yescharacteristic
221C2×C4#48011yescharacteristic
222C2×C4#48111yescharacteristic
223C8—21no—
224C8—21no—
225C8—21no—
226C8—21no—
227C8—42no—
228C8—82no—
229C22—21no—
230C22—21no—
231C22#187411yescharacteristic
232C22#219411yescharacteristic
233C22#219511yescharacteristic
234C4—21no—
235C4—21no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—42no—
240C4—82no—
241C4—162no—
242C4—161no—
243C4—161no—
244C4—161no—
245C4#190511yescharacteristic
246C4#190611yescharacteristic
247C4#214512yes—
248C2—21no—
249C2—21no—
250C2#918011yescharacteristic
251C2#1299711yescharacteristic
252C2#1300211yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#447
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1674C4semidirect
22. #1670C4semidirect

Maximal subgroups

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

subgroupindex
#227392
#192652
#118572
#117452
#116392
#72992
#72222