Finite Group Databasebeta

← #124575

Subgroups — #124575

Order 512.

457 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 90 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#21374C211yesmaximal, characteristic
2#12785C212yesmaximal, complemented
3#7133C212yesmaximal, complemented
4#7105C212yesmaximal, complemented
5C2×C4×#41C411yescharacteristic
6C2×#431C411yescharacteristic
7C2×C4×M16C2211yescharacteristic
8#2150—22no—
9C2×#349—22no—
10#1985C411yescharacteristic
11#1985C2211yescharacteristic
12#1955C411yescharacteristic
13#1955C2211yescharacteristic
14#1870—22no—
15C4×C22⋊C8—22no—
16#1377C2214yes—
17C22×#41C811yescharacteristic
18C22×M16D412yes—
19C4×#41—41no—
20C4×#41—41no—
21C4×#41—41no—
22C4×#41—41no—
23C2×C4×Q8C811yescharacteristic
24C2×C4×Q8C2×C411yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4—21no—
27C2×C4×D4C811yescharacteristic
28C2×C4×D4C2×C411yescharacteristic
29C2×#117—21no—
30C2×#117C811yescharacteristic
31C22×C42—21no—
32C22×C42C2×C411yescharacteristic
33C2×M32—24no—
34#431—41no—
35#431—41no—
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C22⋊C8—21no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C4×M16—41no—
42C4×M16—41no—
43C2×#97C2×C411yescharacteristic
44C2×#97C2311yescharacteristic
45C2×C4×C8—21no—
46C2×C4×C8D412yes—
47C4⋊C16—42no—
48C4⋊C16—42no—
49#349—44no—
50#336—22no—
51#336—22no—
52C2×#41—44no—
53C2×#41—41no—
54C2×#41—41no—
55C22×Q8C2×C811yescharacteristic
56C22×D4—21no—
57C22×D4C2×C811yescharacteristic
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C2×C811yescharacteristic
63C23×C4M1611yescharacteristic
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—42no—
67C2×M16—42no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8C22⋊C412yes—
71C22×C8C2×D412yes—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×Q8—41no—
75C4×Q8—41no—
76C4×D4—41no—
77C4×D4—41no—
78C4×D4—41no—
79C4×D4—41no—
80C4×D4—42no—
81C4×D4—42no—
82C4×D4—42no—
83C4×D4—42no—
84#117—42no—
85#117—42no—
86#117—41no—
87#117—41no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4C2×C811yescharacteristic
91C2×C4⋊C4C2×C811yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42C22⋊C411yescharacteristic
107C2×C42C2×C811yescharacteristic
108C2×C42M1611yescharacteristic
109C2×C42C22×C411yescharacteristic
110M32—44no—
111M32—44no—
112C2×C16—44no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C22⋊C8—44no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#97—41no—
120#97—41no—
121C4×C8—22no—
122C4×C8—22no—
123C4×C8—41no—
124C4×C8—41no—
125C24—21no—
126C24C22⋊C811yescharacteristic
127#41—44no—
128#41—44no—
129#41—44no—
130#41—44no—
131C2×Q8—42no—
132C2×Q8—41no—
133C2×D4—41no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—42no—
138C22×C4—24no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4C22⋊C811yescharacteristic
163C22×C4C22⋊C811yescharacteristic
164C22×C4C22⋊C811yescharacteristic
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4C22×C811yescharacteristic
167C22×C4C2×M1611yescharacteristic
168M16—44no—
169M16—44no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8AGL(1,ℤ/8)12yes—
180C2×C8#13712yes—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—41no—
195C42—41no—
196C42—41no—
197C42—41no—
198C42—41no—
199C42—41no—
200C42#9911yescharacteristic
201C42#9911yescharacteristic
202C42#10011yescharacteristic
203C42#10011yescharacteristic
204C16—84no—
205C23—22no—
206C23—21no—
207C23—22no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—42no—
214C23C2×C22⋊C811yescharacteristic
215Q8—42no—
216Q8—42no—
217D4—42no—
218D4—48no—
219D4—42no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4#32311yescharacteristic
250C2×C4#32311yescharacteristic
251C2×C4#32311yescharacteristic
252C2×C4#32311yescharacteristic
253C2×C4C2×#9911yescharacteristic
254C2×C4C2×#10011yescharacteristic
255C2×C4#41712yes—
256C2×C4#42112yes—
257C8—22no—
258C8—22no—
259C8—42no—
260C8—81no—
261C8—81no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—82no—
270C22—82no—
271C22—81no—
272C22—81no—
273C22C2×#32311yescharacteristic
274C22#147712yes—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—81no—
282C4—81no—
283C4#132012yes—
284C4#132112yes—
285C4#151312yes—
286C4#153611yescharacteristic
287C4#156011yescharacteristic
288C2—22no—
289C2—81no—
290C2—82no—
291C2#759711yescharacteristic
292C2C2×#132011yescharacteristic
293C2C2×#132111yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#323
prime radicalC22×#41C8
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
2. #12785C2semidirect
3. #7133C2semidirect
4. #7105C2semidirect

Maximal subgroups

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

subgroupindex
#213742
#127852
#71332
#71052