Finite Group Databasebeta

← #149905

Subgroups — #149905

Order 512.

315 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 56 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#27919C211yesmaximal, characteristic
2#11511C211yesmaximal, characteristic
3C2×#1324C211yesmaximal, characteristic
4#7372C212yesmaximal, complemented
5#7353C212yesmaximal, complemented
6C2×#484—21no—
7C2×#483—21no—
8#2857C411yescharacteristic
9C2×#382C2211yescharacteristic
10#2097—21no—
11#2097—21no—
12#1960C411yescharacteristic
13#1960C2211yescharacteristic
14#1847C411yescharacteristic
15#1847C2211yescharacteristic
16#1749C411yescharacteristic
17#1463—21no—
18C2×#324—21no—
19#1324—21no—
20#1324—21no—
21#1324C2212yes—
22#1324C2212yes—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×C22⋊Q8—21no—
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C22×C42C2×C411yescharacteristic
28#484—42no—
29#484—42no—
30#483—42no—
31#483—42no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#106—21no—
35C2×#106—21no—
36C2×#103—21no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40#407—41no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—41no—
43C2×#97C2×C411yescharacteristic
44C2×#97C2×C411yescharacteristic
45#391—21no—
46#389—41no—
47#384C2×C411yescharacteristic
48#382—41no—
49#382C2×C411yescharacteristic
50#382D411yescharacteristic
51#382D411yescharacteristic
52#382C2311yescharacteristic
53#380—41no—
54C2×#95—41no—
55#336—21no—
56#324—22no—
57#324—22no—
58#324—41no—
59#324—41no—
60C22×Q8—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C22⋊C411yescharacteristic
65C2×M16—81no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—81no—
69C4⋊Q8—22no—
70C4⋊Q8—41no—
71C22⋊Q8—41no—
72C22⋊Q8—41no—
73C22⋊Q8—41no—
74C22⋊Q8—41no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4C22⋊C411yescharacteristic
86C2×C4⋊C4C22⋊C411yescharacteristic
87C2×C4⋊C4C2×D411yescharacteristic
88C2×C4⋊C4C2×D411yescharacteristic
89C2×C22⋊C4—41no—
90C2×C42—22no—
91C2×C42—21no—
92C2×C42—41no—
93C2×C42C22⋊C411yescharacteristic
94C2×C42C22×C411yescharacteristic
95#107—42no—
96#107—42no—
97#106—42no—
98#106—42no—
99#103—42no—
100#103—42no—
101#103—42no—
102#103—42no—
103C22⋊C8—22no—
104C22⋊C8—22no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—81no—
108C22⋊C8—81no—
109#97—22no—
110#97—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114#95—41no—
115#95—42no—
116C24#9911yescharacteristic
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—81no—
143C22×C4—82no—
144C22×C4—82no—
145C22×C4#9911yescharacteristic
146C22×C4C2×C22⋊C411yescharacteristic
147M16—162no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—82no—
155C2×C8—82no—
156C2×C8—162no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—82no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—82no—
169C22⋊C4—81no—
170C22⋊C4—81no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—42no—
175C42#13112yes—
176C23—21no—
177C23—22no—
178C23—21no—
179C23—41no—
180C23#35311yescharacteristic
181C23#35311yescharacteristic
182C23C2×#9911yescharacteristic
183Q8—81no—
184Q8—81no—
185Q8—81no—
186Q8—81no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—84no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—82no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—161no—
219C2×C4—161no—
220C2×C4#40811yescharacteristic
221C2×C4#41311yescharacteristic
222C8—42no—
223C8—42no—
224C8—162no—
225C8—162no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—42no—
234C22—42no—
235C22#140712yes—
236C22#146811yescharacteristic
237C22C2×#35311yescharacteristic
238C22#213311yescharacteristic
239C4—21no—
240C4—21no—
241C4—42no—
242C4—41no—
243C4—81no—
244C4—81no—
245C4—81no—
246C4—81no—
247C4—81no—
248C4—161no—
249C4—161no—
250C2—21no—
251C2—21no—
252C2—22no—
253C2—41no—
254C2#1154911yescharacteristic
255C2C2×#140711yescharacteristic
256C2#1357811yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7372C2semidirect
5. #7353C2semidirect

Maximal subgroups

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

subgroupindex
#279192
#115112
C2×#13242
#73722
#73532