Finite Group Databasebeta

← #110704

Subgroups — #110704

Order 512.

624 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 150 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#21417C211yesmaximal, characteristic
2#17571C212yesmaximal
3#17325C211yesmaximal, characteristic, complemented
4#8050C211yesmaximal, characteristic, complemented
5#8013C212yesmaximal, complemented
6#3444C2211yescharacteristic
7#2920C2211yescharacteristic
8C2×#431C2212yes—
9#2283C2212yes—
10C2×#376C2211yescharacteristic
11#1995C412yescomplemented
12#1963C412yescomplemented
13#1896—22no—
14#1759—22no—
15#1518C414yescomplemented
16#1517C414yescomplemented
17#1516—24no—
18#1500—22no—
19#1500—22no—
20#1453—22no—
21#1452—22no—
22#1450—22no—
23#1297—22no—
24#1294—24no—
25#533C2×C411yescharacteristic
26#533C2×C412yes—
27#532C2×C411yescharacteristic
28#532C2×C412yes—
29#529—22no—
30#529—22no—
31#529—22no—
32#525—21no—
33#525—22no—
34C4×#41—21no—
35C4×#41—22no—
36C22×C42—21no—
37C22×C42C2311yescharacteristic
38#432—22no—
39#431—22no—
40#431—22no—
41#431—22no—
42#431C2×C414yes—
43#431D412yes—
44#431D412yes—
45#423C2×C412yes—
46C2×C4⋊C8—22no—
47C2×#104—24no—
48#407D412yes—
49C2×C22⋊C8D414yes—
50C4×M16D412yes—
51C2×C4×C8—22no—
52#383—22no—
53#382C2×C414yes—
54#381—22no—
55#380—22no—
56C4×C4⋊C4—22no—
57C4×C22⋊C4—22no—
58#376—22no—
59#376—22no—
60#376—21no—
61#376C2×C412yes—
62C2×#95—22no—
63#339—24no—
64#328—24no—
65#328—24no—
66#327—24no—
67#327—24no—
68#322—24no—
69#322—22no—
70C82—22no—
71C2×#41—21no—
72C2×#41—22no—
73C23×C4—41no—
74C23×C4C2×D412yes—
75C23×C4C2×D411yescharacteristic
76C2×M16C22⋊C414yes—
77C22×C8—24no—
78C4⋊Q8C4211yescharacteristic
79C4⋊Q8C4212yes—
80#127C4211yescharacteristic
81#126—42no—
82#125—21no—
83#125—22no—
84#124—21no—
85#124—22no—
86#123—42no—
87#123—41no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91#121—22no—
92#121—22no—
93#121—22no—
94#120—41no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—42no—
102C4×D4—42no—
103C4×D4—42no—
104#117—21no—
105#117—22no—
106#117—41no—
107#117—42no—
108C2×C4⋊C4—24no—
109C2×C22⋊C4—42no—
110C2×C42—22no—
111C2×C42—44no—
112C2×C42—41no—
113C2×C42—41no—
114C2×C42—42no—
115C2×C42C22×C411yescharacteristic
116C2×C42C22×C412yes—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D412yes—
119C2×C42#4112yes—
120C2×C42#4112yes—
121C2×C42#4112yes—
122C4⋊C8—24no—
123C4⋊C8—24no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126#104—48no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8C22⋊C414yes—
130C22⋊C8C22⋊C414yes—
131#97—22no—
132C4×C8—24no—
133C4×C8—22no—
134C4×C8—22no—
135C4×C8—22no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—42no—
140#95—42no—
141#95—42no—
142C24#12711yescharacteristic
143#41—44no—
144#41—84no—
145C2×Q8—21no—
146C2×Q8—22no—
147C2×D4—21no—
148C2×D4—22no—
149C2×D4—41no—
150C2×D4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—84no—
159C22×C4C2×C22⋊C411yescharacteristic
160C22×C4C2×C22⋊C412yes—
161C22×C4C4×D412yes—
162C22×C4C4×D412yes—
163C22×C4C4×D412yes—
164C22×C4#12112yes—
165C22×C4#12112yes—
166C22×C4#12112yes—
167C22×C4#12412yes—
168C22×C4#12411yescharacteristic
169M16—44no—
170C2×C8—24no—
171C2×C8—24no—
172C2×C8—24no—
173C2×C8—48no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—84no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—84no—
187C42—24no—
188C42—42no—
189C42—44no—
190C42—41no—
191C42—42no—
192C42—84no—
193C42C2×C4211yescharacteristic
194C42#11712yes—
195C42#11711yescharacteristic
196C42C4×D412yes—
197C42C4×D412yes—
198C42C4×D412yes—
199C42#12011yescharacteristic
200C42#12312yes—
201C42#12311yescharacteristic
202C42#12612yes—
203C23—24no—
204C23—41no—
205C23—41no—
206C23#39011yescharacteristic
207C23#39012yes—
208Q8—42no—
209Q8—82no—
210D4—42no—
211D4—82no—
212D4—82no—
213D4—84no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4—88no—
230C2×C4—82no—
231C2×C4—84no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4C4×C22⋊C412yes—
235C2×C4C4×C22⋊C411yescharacteristic
236C2×C4#38612yes—
237C2×C4#38611yescharacteristic
238C2×C4#38812yes—
239C2×C4#38812yes—
240C2×C4#38812yes—
241C8—44no—
242C8—44no—
243C8—44no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—21no—
248C22—44no—
249C22—82no—
250C22—81no—
251C22#143711yescharacteristic
252C22#197614yes—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—22no—
257C4—21no—
258C4—22no—
259C4—84no—
260C4—82no—
261C4—81no—
262C4—164no—
263C4#188412yes—
264C2—24no—
265C2—41no—
266C2—161no—
267C2#1094412yes—
268C2#1095911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17325C2semidirect
4. #8050C2semidirect
5. #8013C2semidirect
11. #1995C4semidirect
12. #1963C4semidirect
15. #1518C4semidirect
16. #1517C4semidirect

Maximal subgroups

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

subgroupindex
#214172
#175712
#173252
#80502
#80132