Finite Group Databasebeta

← #138376

Subgroups — #138376

Order 512.

525 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 114 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#33129C211yesmaximal, characteristic
2C2×#1834C211yesmaximal, characteristic, complemented
3#12844C211yesmaximal, characteristic, complemented
4C2×#1347C212yesmaximal
5#7258C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2854—21no—
9C2×#384—21no—
10C2×C22⋊C16C2212yes—
11#1970C412yescomplemented
12#1942C412yescomplemented
13#1934—22no—
14#1893—21no—
15#1834—24no—
16#1808C2211yescharacteristic
17C2×#340—21no—
18C2×#340C2212yes—
19#1374—22no—
20#1347—24no—
21C22×D8C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×#123—41no—
25C2×#121—21no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4—41no—
29C22×C42D411yescharacteristic
30C22×C16—22no—
31#466—22no—
32#466—22no—
33C2×#107C2×C411yescharacteristic
34C2×#107C2×C412yes—
35C2×#102—22no—
36C2×#102—21no—
37C2×C4×C8—22no—
38C2×C4×C8—22no—
39C2×C4×C8C2×C412yes—
40C2×C4×C8D412yes—
41#390—22no—
42#389—22no—
43#388—42no—
44#386—41no—
45#384—24no—
46#384—42no—
47#380—21no—
48C22⋊C16—22no—
49C22⋊C16—22no—
50#340—24no—
51#340—22no—
52#340—21no—
53#340—22no—
54#340D414yes—
55C22×D4—21no—
56C22×D4—41no—
57C23×C4—42no—
58C23×C4D811yescharacteristic
59C23×C4D811yescharacteristic
60C23×C4C2×D411yescharacteristic
61C2×D8—22no—
62C2×D8—42no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—42no—
66C22×C8—42no—
67C22×C8C22⋊C412yes—
68C22×C8C22×C411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C22×C8C2×D412yes—
71C22×C8#4112yes—
72#123—82no—
73#123—82no—
74#121—41no—
75#121—41no—
76#121—41no—
77#121—41no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—82no—
88C2×C4⋊C4—84no—
89C2×C4⋊C4—82no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—41no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42C22⋊C412yes—
99C2×C42SD1612yes—
100C2×C16—44no—
101C2×C16—42no—
102#107—22no—
103#107—42no—
104#102—44no—
105#102—42no—
106C4×C8—24no—
107C4×C8—24no—
108#95—42no—
109#95—41no—
110C24—41no—
111C24C2×D811yescharacteristic
112C2×D4—41no—
113C2×D4—42no—
114C2×D4—41no—
115C2×D4—42no—
116C2×D4—81no—
117C2×D4—81no—
118C2×D4—81no—
119C2×D4—81no—
120C2×D4—81no—
121C2×D4—81no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—42no—
132C22×C4—41no—
133C22×C4—44no—
134C22×C4—84no—
135C22×C4—84no—
136C22×C4—82no—
137C22×C4—82no—
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—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8C4×D412yes—
159C2×C8#12112yes—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—41no—
163C4⋊C4—42no—
164C4⋊C4—41no—
165C4⋊C4—42no—
166C4⋊C4—84no—
167C4⋊C4—88no—
168C4⋊C4—88no—
169C4⋊C4—84no—
170C22⋊C4—81no—
171C22⋊C4—81no—
172C22⋊C4—84no—
173C22⋊C4—84no—
174C22⋊C4—81no—
175C22⋊C4—81no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—22no—
180C16—82no—
181C23—22no—
182C23—22no—
183C23—81no—
184C23—81no—
185C23—81no—
186C23—82no—
187C23—82no—
188C23C2×#10211yescharacteristic
189C23#44711yescharacteristic
190C23#48011yescharacteristic
191C23C2×D1611yescharacteristic
192C23C2×SD3211yescharacteristic
193C23#50911yescharacteristic
194C23#51011yescharacteristic
195D4—81no—
196D4—81no—
197D4—82no—
198D4—161no—
199D4—161no—
200D4—161no—
201D4—161no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—84no—
220C2×C4—84no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—84no—
227C2×C4—82no—
228C2×C4—164no—
229C2×C4—164no—
230C2×C4#35714yes—
231C2×C4#36014yes—
232C2×C4#38611yescharacteristic
233C2×C4#41012yes—
234C2×C4#41711yescharacteristic
235C2×C4#42112yes—
236C2×C4#44811yescharacteristic
237C2×C4#48111yescharacteristic
238C8—22no—
239C8—22no—
240C8—42no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—162no—
250C22—162no—
251C22—161no—
252C22—161no—
253C22—161no—
254C22#151712yes—
255C22#187411yescharacteristic
256C22C2×#35711yescharacteristic
257C22#213911yescharacteristic
258C22C2×#36012yes—
259C22#218411yescharacteristic
260C22#218511yescharacteristic
261C22#223211yescharacteristic
262C22#223311yescharacteristic
263C4—24no—
264C4—21no—
265C4—21no—
266C4—24no—
267C4—164no—
268C4—162no—
269C4—161no—
270C4#151212yes—
271C4#189711yescharacteristic
272C4#189911yescharacteristic
273C2—21no—
274C2—21no—
275C2—21no—
276C2—21no—
277C2—321no—
278C2#915611yescharacteristic
279C2#1185612yes—
280C2#1186012yes—
281C2#1299111yescharacteristic
282C2#1300111yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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. C2×#1834C2semidirect
3. #12844C2semidirect
5. #7258C2semidirect
11. #1970C4semidirect
12. #1942C4semidirect

Maximal subgroups

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

subgroupindex
#331292
C2×#18342
#128442
C2×#13472
#72582