Finite Group Databasebeta

← #138474

Subgroups — #138474

Order 512.

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

normal subgroups only

This group has 345 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, complemented
2#19264C211yesmaximal, characteristic
3#11856C211yesmaximal, characteristic
4#11744C211yesmaximal, characteristic
5#11639C211yesmaximal, characteristic, complemented
6#7299C211yesmaximal, characteristic, complemented
7#7222C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9#3112—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#1678—22no—
21#1676C412yescomplemented
22#1668C412yescomplemented
23#1568C2211yescharacteristic
24#1551—21no—
25#1347—22no—
26#1347—21no—
27#1347—21no—
28Q8⋊C16—22no—
29#532—21no—
30C2×C4×Q8D411yescharacteristic
31#500C2×C411yescharacteristic
32#493C2×C411yescharacteristic
33#486—21no—
34#480—41no—
35#477—22no—
36#466—21no—
37#466—21no—
38#460—22no—
39#448—41no—
40C8×Q8—21no—
41C8×Q8—21no—
42C8×Q8C2×C412yes—
43C8×Q8D412yes—
44#440—42no—
45C4×D8—21no—
46C4×D8—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#333—22no—
75#331—22no—
76C2×#41—41no—
77C22×Q8D811yescharacteristic
78C2×SD16—42no—
79C2×D8—21no—
80C2×D8—21no—
81C22×C8—21no—
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C4⋊Q8—21no—
85#127—21no—
86#124—41no—
87#121—41no—
88#121—41no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8C22⋊C412yes—
92C4×Q8SD1612yes—
93C4×D4—41no—
94C4×D4—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#107—42no—
112#106—42no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—21no—
117C4⋊C8C22⋊C412yes—
118#103—42no—
119#102—41no—
120#102—41no—
121#102—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—
134#41—161no—
135#41—161no—
136C2×Q8—21no—
137C2×Q8—42no—
138C2×Q8—41no—
139C2×Q8#10212yes—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—41no—
143C2×D4—81no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—81no—
150C22×C4#12011yescharacteristic
151C22×C4C2×D811yescharacteristic
152C22×C4AGL(1,ℤ/8)11yescharacteristic
153SD16—84no—
154D8—81no—
155D8—81no—
156C2×C8—22no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8C4×D411yescharacteristic
162C2×C8C4×D411yescharacteristic
163C2×C8#12111yescharacteristic
164C2×C8#12111yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—82no—
177C4⋊C4—161no—
178C4⋊C4#10212yes—
179C22⋊C4—81no—
180C22⋊C4—81no—
181C42—22no—
182C42—22no—
183C42—82no—
184C42—162no—
185C42C2×C22⋊C411yescharacteristic
186C42#12311yescharacteristic
187C42C2×SD1611yescharacteristic
188C42#13711yescharacteristic
189C16—81no—
190C16—81no—
191C23—81no—
192C23#44711yescharacteristic
193Q8—22no—
194Q8—22no—
195Q8—42no—
196Q8—161no—
197D4—82no—
198D4—81no—
199D4—81no—
200D4—161no—
201D4—161no—
202D4—161no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—81no—
221C2×C4—82no—
222C2×C4—81no—
223C2×C4—162no—
224C2×C4#38611yescharacteristic
225C2×C4C2×#10211yescharacteristic
226C2×C4#41711yescharacteristic
227C2×C4#44811yescharacteristic
228C2×C4#48011yescharacteristic
229C2×C4#48111yescharacteristic
230C8—21no—
231C8—21no—
232C8—21no—
233C8—21no—
234C8—42no—
235C8—82no—
236C22—21no—
237C22—21no—
238C22—161no—
239C22—161no—
240C22—161no—
241C22#187411yescharacteristic
242C22#219211yescharacteristic
243C22#219311yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—82no—
251C4—162no—
252C4—161no—
253C4—161no—
254C4#190511yescharacteristic
255C4#190611yescharacteristic
256C4#214512yes—
257C2—21no—
258C2—21no—
259C2—321no—
260C2#918011yescharacteristic
261C2#1299611yescharacteristic
262C2#1300211yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
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
1. #22739C2semidirect
5. #11639C2semidirect
6. #7299C2semidirect
7. #7222C2semidirect
21. #1676C4semidirect
22. #1668C4semidirect

Maximal subgroups

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

subgroupindex
#227392
#192642
#118562
#117442
#116392
#72992
#72222