Finite Group Databasebeta

← #129504

Subgroups — #129504

Order 512.

858 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×C4×Q16C211yesmaximal, characteristic
2#12599C211yesmaximal, characteristic
3C2×#1334C214yesmaximal, complemented
4#8595C211yesmaximal, characteristic
5C23×Q16C411yescharacteristic
6C22×C4×Q8—21no—
7C2×C4×Q16—28no—
8C2×C4×Q16—21no—
9C2×C4×Q16—21no—
10C2×C4×Q16C412yes—
11C2×C4×Q16C2212yes—
12C22×#107C411yescharacteristic
13C22×#103—21no—
14C22×C4×C8C2211yescharacteristic
15C2×C22⋊C16—21no—
16C2×#346C2212yes—
17#1797—21no—
18C2×#334C2212yes—
19#1374—21no—
20#1334—28no—
21#1277—21no—
22C23×Q8—21no—
23C22×Q16—24no—
24C22×Q16—24no—
25C22×Q16—21no—
26C22×Q16—22no—
27C22×Q16C814yescomplemented
28C22×Q16C2×C412yes—
29C23×C8C2×C411yescharacteristic
30C23×C8D411yescharacteristic
31C2×C4×Q8—28no—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—44no—
35C2×C4×Q8—41no—
36C2×C4×Q8—41no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—41no—
39C22×C42—41no—
40C22×C42D411yescharacteristic
41C22×C16—22no—
42C4×Q16—28no—
43C4×Q16—28no—
44C4×Q16—24no—
45C4×Q16—24no—
46C4×Q16—24no—
47C4×Q16D418yes—
48C2×#107—21no—
49C2×#107—24no—
50C2×#107C2×C412yes—
51C2×C4⋊C8—22no—
52C2×#103—28no—
53C2×#103—22no—
54C2×#103—22no—
55C2×C22⋊C8—41no—
56C2×C4×C8—24no—
57C2×C4×C8C2×C411yescharacteristic
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8C2311yescharacteristic
61C22⋊C16—24no—
62#346—24no—
63#336—41no—
64#334—24no—
65C22×Q8—24no—
66C22×Q8—24no—
67C22×Q8—24no—
68C22×Q8—22no—
69C22×Q8—42no—
70C22×Q8—42no—
71C22×Q8—41no—
72C22×Q8—42no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4C22⋊C411yescharacteristic
76C23×C4D811yescharacteristic
77C23×C4SD1611yescharacteristic
78C2×Q16—24no—
79C2×Q16—216no—
80C2×Q16—24no—
81C2×Q16—216no—
82C2×Q16—24no—
83C2×Q16—22no—
84C2×Q16—22no—
85C2×Q16—22no—
86C2×Q16C22⋊C414yes—
87C2×Q16C2×C814yes—
88C2×Q16M1614yes—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—42no—
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C2×C812yes—
98C22×C8M1612yes—
99C22×C8C22×C411yescharacteristic
100C22×C8C2×D411yescharacteristic
101C4×Q8—28no—
102C4×Q8—28no—
103C4×Q8—44no—
104C4×Q8—48no—
105C4×Q8—48no—
106C4×Q8—44no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—48no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—48no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42C22⋊C411yescharacteristic
120C2×C42D811yescharacteristic
121C2×C42SD1611yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C16—24no—
124C2×C16—42no—
125C2×C16—42no—
126#107—24no—
127#107—22no—
128#107C22⋊C414yes—
129C4⋊C8—42no—
130C4⋊C8—42no—
131#103—28no—
132#103—28no—
133C22⋊C8—82no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8C2×D414yes—
138C24#10211yescharacteristic
139C2×Q8—216no—
140C2×Q8—24no—
141C2×Q8—24no—
142C2×Q8—24no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—48no—
146C2×Q8—48no—
147C2×Q8—48no—
148C2×Q8—44no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—41no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4C22⋊C812yes—
171C22×C4#10211yescharacteristic
172C22×C4#10211yescharacteristic
173C22×C4#10211yescharacteristic
174C22×C4#10412yes—
175C22×C4C2×C22⋊C411yescharacteristic
176C22×C4C2×D811yescharacteristic
177C22×C4C2×SD1611yescharacteristic
178Q16—216no—
179Q16—24no—
180Q16—24no—
181Q16—24no—
182Q16C22⋊C818yescomplemented
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—24no—
188C2×C8—24no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—44no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8C22⋊C812yes—
197C2×C8C2×C22⋊C412yes—
198C2×C8#12012yes—
199C2×C8C22×C811yescharacteristic
200C2×C8C2×M1612yes—
201C2×C8C2×M1611yescharacteristic
202C4⋊C4—24no—
203C4⋊C4—44no—
204C4⋊C4—48no—
205C4⋊C4—42no—
206C4⋊C4—48no—
207C42—21no—
208C42—21no—
209C42—22no—
210C42—22no—
211C42—48no—
212C42—48no—
213C42#12012yes—
214C42AGL(1,ℤ/8)12yes—
215C16—44no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23D4⋊C812yes—
221C23#36012yes—
222C23#36212yes—
223C23C2×#10211yescharacteristic
224Q8—28no—
225Q8—416no—
226Q8—44no—
227Q8—44no—
228Q8—44no—
229C2×C4—28no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—44no—
246C2×C4—416no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—88no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4D4⋊C812yes—
255C2×C4#36212yes—
256C2×C4#37911yescharacteristic
257C2×C4C2×C22⋊C811yescharacteristic
258C2×C4#40711yescharacteristic
259C2×C4#41811yescharacteristic
260C2×C4C2×#10411yescharacteristic
261C2×C4#42111yescharacteristic
262C2×C4#44712yes—
263C2×C4#45012yes—
264C8—22no—
265C8—22no—
266C8—41no—
267C8—162no—
268C8C2×C22⋊C812yes—
269C8#40712yes—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22#133414yescomplemented
277C22C2×D4⋊C811yescharacteristic
278C22#148611yescharacteristic
279C22#178611yescharacteristic
280C22C2×#36011yescharacteristic
281C22#214511yescharacteristic
282C22C2×#36211yescharacteristic
283C22C2×#36211yescharacteristic
284C4—22no—
285C4—22no—
286C4—22no—
287C4—22no—
288C4—48no—
289C4—41no—
290C4—84no—
291C4#147811yescharacteristic
292C4#149011yescharacteristic
293C4#177911yescharacteristic
294C4#178911yescharacteristic
295C2—24no—
296C2#831011yescharacteristic
297C2C2×#133411yescharacteristic
298C2#1160011yescharacteristic
299C2#1166411yescharacteristic
300C2#1166811yescharacteristic
301C2#1273511yescharacteristic
302C2#1273711yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC2×C8C22×C8
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1334C2semidirect
27. C22×Q16C8semidirect
182. Q16C22⋊C8semidirect
276. C22#1334semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q162
#125992
C2×#13342
#85952