Finite Group Databasebeta

← #138530

Subgroups — #138530

Order 512.

448 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 96 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#29770C211yesmaximal, characteristic
2#21961C211yesmaximal, characteristic, complemented
3#11886C212yesmaximal
4C2×#1322C211yesmaximal, characteristic, complemented
5#7230C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8#2854—21no—
9#2285—21no—
10C2×C22⋊C16C2212yes—
11#2086C412yescomplemented
12#2067—22no—
13#2054C412yescomplemented
14#1947—24no—
15#1893—21no—
16#1808—21no—
17#1808C2211yescharacteristic
18#1808C2212yes—
19#1347—24no—
20#1322—22no—
21C22×D8C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×#123—41no—
25C2×#121—21no—
26C2×#117—41no—
27C22×C42D411yescharacteristic
28C22×C16—22no—
29#466—22no—
30#466—22no—
31C2×#107—22no—
32C2×#107—21no—
33C2×#107—22no—
34C2×#107C2×C411yescharacteristic
35C2×C4⋊C8—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8C2×C412yes—
38C2×C4⋊C8D412yes—
39C2×#102—21no—
40C2×#102—22no—
41#390—22no—
42#389—22no—
43#388—42no—
44#386—41no—
45#384—22no—
46#382—41no—
47#380—22no—
48#380—21no—
49#380—41no—
50C4×C4⋊C4—22no—
51C4×C4⋊C4—41no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54#340—22no—
55#340—22no—
56#340C2×C412yes—
57#340D414yes—
58C22×D4—21no—
59C22×D4—41no—
60C23×C4D811yescharacteristic
61C23×C4D811yescharacteristic
62C23×C4C2×D411yescharacteristic
63C2×D8—22no—
64C2×D8—81no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—81no—
68C22×C8—81no—
69C22×C8C22⋊C412yes—
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D412yes—
72C22×C8C2×D411yescharacteristic
73C22×C8#4112yes—
74#123—82no—
75#123—82no—
76#121—41no—
77#121—41no—
78#121—41no—
79#121—41no—
80#117—82no—
81#117—82no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—41no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—42no—
98C2×C42C22⋊C412yes—
99C2×C42SD1612yes—
100C2×C16—42no—
101C2×C16—82no—
102#107—22no—
103#107—81no—
104#107—82no—
105#107—82no—
106C4⋊C8—24no—
107C4⋊C8—24no—
108#102—42no—
109#102—82no—
110#95—42no—
111#95—42no—
112#95—41no—
113#95—42no—
114C24—41no—
115C24C2×D811yescharacteristic
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—81no—
121C2×D4—81no—
122C2×D4—81no—
123C2×D4—81no—
124C2×D4—81no—
125C2×D4—81no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—44no—
137C22×C4—81no—
138C22×C4#10212yes—
139C22×C4#10212yes—
140C22×C4D1612yes—
141C22×C4SD3212yes—
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4#12011yescharacteristic
144C22×C4#12311yescharacteristic
145C22×C4C2×SD1611yescharacteristic
146C22×C4AGL(1,ℤ/8)11yescharacteristic
147C22×C4#13711yescharacteristic
148D8—82no—
149C2×C8—22no—
150C2×C8—41no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—41no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—81no—
157C2×C8C4×D412yes—
158C2×C8#12112yes—
159C4⋊C4—42no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C4⋊C4—84no—
168C22⋊C4—84no—
169C22⋊C4—82no—
170C22⋊C4—81no—
171C22⋊C4—81no—
172C22⋊C4—82no—
173C22⋊C4—84no—
174C22⋊C4—81no—
175C22⋊C4—81no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—22no—
180C42—84no—
181C16—82no—
182C23—41no—
183C23—41no—
184C23—82no—
185C23—82no—
186C23—81no—
187C23—81no—
188C23—81no—
189C23C2×#10211yescharacteristic
190C23#44711yescharacteristic
191C23#48011yescharacteristic
192C23C2×D1611yescharacteristic
193C23C2×SD3211yescharacteristic
194C23#50911yescharacteristic
195C23#51011yescharacteristic
196D4—82no—
197D4—81no—
198D4—81no—
199D4—161no—
200D4—161no—
201D4—161no—
202D4—161no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—84no—
224C2×C4—82no—
225C2×C4—84no—
226C2×C4—81no—
227C2×C4—82no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—84no—
231C2×C4#35714yes—
232C2×C4#38611yescharacteristic
233C2×C4#41711yescharacteristic
234C2×C4#44811yescharacteristic
235C2×C4#48111yescharacteristic
236C8—22no—
237C8—22no—
238C8—42no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—162no—
250C22—162no—
251C22—161no—
252C22—161no—
253C22—161no—
254C22#187411yescharacteristic
255C22C2×#35711yescharacteristic
256C22#213911yescharacteristic
257C22#218411yescharacteristic
258C22#218511yescharacteristic
259C22#223211yescharacteristic
260C22#223311yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—42no—
266C4—164no—
267C4—162no—
268C4—161no—
269C4#191211yescharacteristic
270C4#191311yescharacteristic
271C2—21no—
272C2—21no—
273C2—21no—
274C2—21no—
275C2—21no—
276C2—21no—
277C2—321no—
278C2#920411yescharacteristic
279C2#1299111yescharacteristic
280C2#1300811yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21961C2semidirect
4. C2×#1322C2semidirect
5. #7230C2semidirect
11. #2086C4semidirect
13. #2054C4semidirect

Maximal subgroups

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

subgroupindex
#297702
#219612
#118862
C2×#13222
#72302