Finite Group Databasebeta

← #134244

Subgroups — #134244

Order 512.

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

normal subgroups only

This group has 516 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#20383C211yesmaximal, characteristic
2#12790C212yesmaximal, complemented
3C2×#1321C212yesmaximal
4#7191C212yesmaximal, complemented
5C22×#131C411yescharacteristic
6C2×#408C2211yescharacteristic
7C2×#405C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9#2297C411yescharacteristic
10C2×#364—22no—
11C2×#349C2214yes—
12#1753—22no—
13C8×C22⋊C4—22no—
14#1729C411yescharacteristic
15#1728C2211yescharacteristic
16#1375—22no—
17#1321—24no—
18C22×#41C2×C411yescharacteristic
19C2×#131—22no—
20C2×#131—42no—
21C2×#131—42no—
22C2×#131C814yes—
23C22×M16—21no—
24C22×M16C2×C411yescharacteristic
25C22×M16C2311yescharacteristic
26C23×C8—21no—
27C23×C8C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C2×#117C2×C411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C2×M32—24no—
32#408—22no—
33#408D412yes—
34#408Q812yes—
35C2×C22⋊C8—22no—
36#405—42no—
37C2×#97D412yes—
38C2×C4×C8—21no—
39C2×C4×C8—21no—
40C2×C4×C8—42no—
41C2×C4×C8D412yes—
42#391—21no—
43#390—21no—
44#386—22no—
45C4×C22⋊C4—22no—
46#364—44no—
47#349—24no—
48#349—24no—
49#336—22no—
50#336—22no—
51C2×#41—21no—
52C2×#41—21no—
53C2×#41—41no—
54C2×#41—41no—
55C2×#41—41no—
56C2×#41—41no—
57C2×#41C4⋊C412yes—
58C2×#41C2×C812yes—
59C22×Q8M1611yescharacteristic
60C22×D4—21no—
61C22×D4M1611yescharacteristic
62C23×C4—21no—
63C23×C4M1612yes—
64C23×C4C22×C411yescharacteristic
65#131—48no—
66#131—48no—
67C2×M16—21no—
68C2×M16—24no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16C2×C812yes—
76C2×M16C2×D411yescharacteristic
77C2×M16C2×Q811yescharacteristic
78C2×M16#4112yes—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—24no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C4⋊C412yes—
92C22×C8C2×C812yes—
93C22×C8M1612yes—
94C22×C8C2×D412yes—
95#117—41no—
96#117—41no—
97C2×C4⋊C4—21no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42C22⋊C412yes—
105M32—44no—
106M32—44no—
107C2×C16—44no—
108C22⋊C8—24no—
109C22⋊C8—24no—
110#97—42no—
111C4×C8—24no—
112C4×C8—42no—
113C4×C8—42no—
114#95—24no—
115C24—21no—
116C24C2×M1611yescharacteristic
117#41—42no—
118#41—42no—
119#41—42no—
120#41—42no—
121#41—42no—
122#41—42no—
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8C4⋊C812yes—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4C4⋊C812yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—44no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×C4⋊C411yescharacteristic
154C22×C4#11711yescharacteristic
155C22×C4C22×C811yescharacteristic
156C22×C4C2×M1611yescharacteristic
157C22×C4C2×M1611yescharacteristic
158M16—41no—
159M16—41no—
160M16—42no—
161M16—48no—
162M16—42no—
163M16—42no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—21no—
168C2×C8—24no—
169C2×C8—24no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—48no—
178C2×C8—42no—
179C2×C8C22⋊C812yes—
180C2×C8C22⋊C812yes—
181C2×C8#9912yes—
182C2×C8#10012yes—
183C2×C8#12112yes—
184C2×C8C22⋊Q812yes—
185C4⋊C4—42no—
186C22⋊C4—48no—
187C22⋊C4—42no—
188C42—22no—
189C42—22no—
190C42—42no—
191C42—41no—
192C42—41no—
193C16—84no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—42no—
198C23—44no—
199C23—41no—
200C23—41no—
201C23#40711yescharacteristic
202C23#42311yescharacteristic
203C23#43111yescharacteristic
204Q8—42no—
205Q8—42no—
206D4—42no—
207D4—42no—
208D4—44no—
209D4—44no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4#38011yescharacteristic
235C2×C4C2×C22⋊C811yescharacteristic
236C2×C4C2×#9911yescharacteristic
237C2×C4#41011yescharacteristic
238C2×C4C2×#10011yescharacteristic
239C2×C4#41311yescharacteristic
240C2×C4C2×C4⋊C811yescharacteristic
241C2×C4#43211yescharacteristic
242C8—22no—
243C8—22no—
244C8—22no—
245C8—42no—
246C8—42no—
247C8#32314yes—
248C22—22no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22—82no—
258C22—82no—
259C22#146511yescharacteristic
260C22#146611yescharacteristic
261C22#179711yescharacteristic
262C4—21no—
263C4—21no—
264C4—44no—
265C4—42no—
266C4—82no—
267C4C2×#32311yescharacteristic
268C4#145811yescharacteristic
269C4#179911yescharacteristic
270C4#180011yescharacteristic
271C2—21no—
272C2—21no—
273C2—41no—
274C2—41no—
275C2—82no—
276C2#855711yescharacteristic
277C2#1153712yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC22×M16C23
socleC22#1466
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12790C2semidirect
4. #7191C2semidirect

Maximal subgroups

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

subgroupindex
#203832
#127902
C2×#13212
#71912