Finite Group Databasebeta

← #124284

Subgroups — #124284

Order 512.

384 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 102 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#21146C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic
3C4×#349C211yesmaximal, characteristic
4#7124C214yesmaximal, complemented
5C2×#423—21no—
6C2×#423C411yescharacteristic
7C2×C4×M16C2211yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9#2150—21no—
10C2×#349—21no—
11C2×#349—21no—
12C4×M32—21no—
13#1940C412yes—
14#1940C2212yes—
15#1939—21no—
16#1939—21no—
17#1728—21no—
18#1377C2212yes—
19#1377C2212yes—
20C22×M16—21no—
21C22×M16D411yescharacteristic
22C22×M16D411yescharacteristic
23C22×C4⋊C4—21no—
24C22×C42—21no—
25C22×C42C2×C411yescharacteristic
26C2×M32—22no—
27C2×M32—22no—
28#423—42no—
29#423—42no—
30#423—41no—
31#423—41no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8C2×C412yes—
36C4×M16—41no—
37C4×M16—41no—
38C2×#97C2×C411yescharacteristic
39C2×#97C2311yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C4×C4⋊C4—21no—
43C4×C4⋊C4—21no—
44C4×C4⋊C4C811yescharacteristic
45C4×C4⋊C4C811yescharacteristic
46C4×C4⋊C4C2×C412yes—
47C2×#95C812yes—
48C4⋊C16—41no—
49C4⋊C16—41no—
50#349—24no—
51#349—42no—
52#346—41no—
53C4×C16—41no—
54#336—22no—
55#336—22no—
56C23×C4—22no—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C2×C811yescharacteristic
59C23×C4M1611yescharacteristic
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16D812yes—
69C2×M16SD1612yes—
70C22×C8—22no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—44no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—22no—
86C2×C42—42no—
87C2×C42—42no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C2×C811yescharacteristic
90C2×C42M1611yescharacteristic
91C2×C42C22×C411yescharacteristic
92M32—24no—
93M32—42no—
94M32—42no—
95M32—42no—
96C2×C16—42no—
97C2×C16—42no—
98C4⋊C8—42no—
99C4⋊C8—44no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104#97—41no—
105#97—41no—
106C4×C8—21no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—21no—
110#95—22no—
111#95C2×C812yes—
112#95C2×C812yes—
113C24C22⋊C811yescharacteristic
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—42no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—82no—
128C22×C4—81no—
129C22×C4C22⋊C811yescharacteristic
130C22×C4C22⋊C811yescharacteristic
131C22×C4C22⋊C811yescharacteristic
132C22×C4#10212yes—
133C22×C4#10412yes—
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4C22×C811yescharacteristic
136C22×C4C2×M1611yescharacteristic
137M16—42no—
138M16—42no—
139M16—44no—
140M16—84no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8#10212yes—
154C2×C8C2×D811yescharacteristic
155C2×C8C2×SD1611yescharacteristic
156C2×C8#13712yes—
157C4⋊C4—24no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—42no—
167C42—42no—
168C42—42no—
169C42—42no—
170C42#9911yescharacteristic
171C42#9911yescharacteristic
172C42#10111yescharacteristic
173C42#10111yescharacteristic
174C16—44no—
175C16—82no—
176C23—21no—
177C23—21no—
178C23—41no—
179C23—41no—
180C23D4⋊C812yes—
181C23C2×C22⋊C811yescharacteristic
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—88no—
205C2×C4#32411yescharacteristic
206C2×C4#32411yescharacteristic
207C2×C4#32411yescharacteristic
208C2×C4#32411yescharacteristic
209C2×C4D4⋊C812yes—
210C2×C4C2×#9911yescharacteristic
211C2×C4C2×#10111yescharacteristic
212C2×C4C2×#10211yescharacteristic
213C2×C4#41911yescharacteristic
214C2×C4C2×#10411yescharacteristic
215C2×C4#42111yescharacteristic
216C8—22no—
217C8—21no—
218C8—22no—
219C8—21no—
220C8—41no—
221C8—82no—
222C22—21no—
223C22—21no—
224C22—22no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—41no—
229C22C2×#32411yescharacteristic
230C22C2×D4⋊C811yescharacteristic
231C22#147911yescharacteristic
232C4—22no—
233C4—21no—
234C4—21no—
235C4—21no—
236C4—44no—
237C4—41no—
238C4—41no—
239C4—82no—
240C4#132712yes—
241C4#150811yescharacteristic
242C4#151911yescharacteristic
243C4#152811yescharacteristic
244C4#154611yescharacteristic
245C2—21no—
246C2—21no—
247C2—42no—
248C2#757311yescharacteristic
249C2C2×#132711yescharacteristic
250C2#1145811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#324
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7124C2semidirect

Maximal subgroups

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

subgroupindex
#211462
#127852
C4×#3492
#71242