Finite Group Databasebeta

← #122284

Subgroups — #122284

Order 512.

433 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 78 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#11460C212yesmaximal, complemented
3#7061C212yesmaximal, complemented
4#7060C212yesmaximal, complemented
5C2×C4×#41C411yescharacteristic
6C2×#431C2211yescharacteristic
7C2×C4×M16C411yescharacteristic
8#2163—22no—
9#1985C411yescharacteristic
10#1985C2211yescharacteristic
11#1955C411yescharacteristic
12#1955C2211yescharacteristic
13#1870—22no—
14C4×C22⋊C8—22no—
15#1322C2214yes—
16#1314—22no—
17C22×#41C811yescharacteristic
18C22×M16—21no—
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
33#431—41no—
34#431—41no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8C2311yescharacteristic
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8D412yes—
40C4×M16—41no—
41C4×M16—41no—
42C2×#97C2×C411yescharacteristic
43C2×#97C2×C411yescharacteristic
44C2×C4×C8—21no—
45C2×C4×C8D412yes—
46C4⋊C16—42no—
47C22⋊C16—24no—
48#346—42no—
49#336—22no—
50#336—22no—
51C2×#41—44no—
52C2×#41—41no—
53C2×#41—41no—
54C22×Q8C2×C811yescharacteristic
55C22×D4—21no—
56C22×D4C2×C811yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C2×C811yescharacteristic
62C23×C4M1611yescharacteristic
63C2×M16—42no—
64C2×M16—42no—
65C2×M16—42no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8C22⋊C412yes—
69C22×C8C2×D412yes—
70C4×Q8—41no—
71C4×Q8—41no—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×D4—41no—
75C4×D4—41no—
76C4×D4—41no—
77C4×D4—41no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—42no—
81C4×D4—42no—
82#117—42no—
83#117—42no—
84#117—41no—
85#117—41no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4C2×C811yescharacteristic
89C2×C4⋊C4C2×C811yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—41no—
101C2×C42—41no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C2×C811yescharacteristic
106C2×C42M1611yescharacteristic
107C2×C42C22×C411yescharacteristic
108C2×C16—44no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C22⋊C8—42no—
112C22⋊C8—44no—
113C22⋊C8—44no—
114#97—41no—
115#97—41no—
116C4×C8—22no—
117C4×C8—22no—
118C4×C8—41no—
119C4×C8—41no—
120C24—21no—
121C24C22⋊C811yescharacteristic
122#41—44no—
123#41—44no—
124#41—44no—
125#41—44no—
126C2×Q8—42no—
127C2×Q8—41no—
128C2×D4—41no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—42no—
133C22×C4—24no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4C22⋊C811yescharacteristic
158C22×C4C22⋊C811yescharacteristic
159C22×C4C22⋊C811yescharacteristic
160C22×C4C2×C22⋊C411yescharacteristic
161C22×C4C22×C811yescharacteristic
162C22×C4C2×M1611yescharacteristic
163M16—48no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—42no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—41no—
188C42—41no—
189C42—41no—
190C42—41no—
191C42—41no—
192C42—41no—
193C42#9911yescharacteristic
194C42#9911yescharacteristic
195C42#10011yescharacteristic
196C42#10011yescharacteristic
197C16—84no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23—41no—
202C23—42no—
203C23—41no—
204C23—41no—
205C23—42no—
206C23—41no—
207C23C2×C22⋊C811yescharacteristic
208Q8—42no—
209Q8—42no—
210D4—42no—
211D4—48no—
212D4—42no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4#32311yescharacteristic
245C2×C4#32311yescharacteristic
246C2×C4#32311yescharacteristic
247C2×C4#32311yescharacteristic
248C2×C4C2×#9911yescharacteristic
249C2×C4C2×#10011yescharacteristic
250C8—42no—
251C8—42no—
252C8—42no—
253C8—42no—
254C22—21no—
255C22—21no—
256C22—44no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—42no—
261C22—82no—
262C22—82no—
263C22—81no—
264C22—81no—
265C22C2×#32311yescharacteristic
266C22#211712yes—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—21no—
271C4—41no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—81no—
276C4—81no—
277C4#132012yes—
278C4#212611yescharacteristic
279C4#213211yescharacteristic
280C2—22no—
281C2—81no—
282C2—82no—
283C2#1142111yescharacteristic
284C2#1143111yescharacteristic
285C2C2×#132011yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#323
prime radicalC22×#41C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11460C2semidirect
3. #7061C2semidirect
4. #7060C2semidirect

Maximal subgroups

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

subgroupindex
#213742
#114602
#70612
#70602