Finite Group Databasebeta

← #127334

Subgroups — #127334

Order 512.

730 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21373C211yesmaximal, characteristic
2#20632C212yesmaximal
3#7237C214yesmaximal, complemented
4C2×C4×#41C2211yescharacteristic
5C2×#431C411yescharacteristic, complemented
6C2×#410—22no—
7C22×C4×C8C411yescharacteristic, complemented
8#2285—22no—
9#1984C412yescomplemented
10#1954C2212yes—
11#1869—22no—
12#1799—22no—
13#1799—22no—
14#1799C414yescomplemented
15#1799C2214yes—
16C4×C22⋊C8—22no—
17#1740C414yescomplemented
18C22×#41D411yescharacteristic
19C23×C8—21no—
20C4×#41—41no—
21C4×#41—41no—
22C4×#41—41no—
23C4×#41—41no—
24C2×C4×Q8C2×C411yescharacteristic
25C2×C4×Q8Q811yescharacteristic
26C2×C4×D4C2×C412yes—
27C2×C4×D4D412yes—
28C2×C4×D4D411yescharacteristic
29C2×C4×D4C2311yescharacteristic
30C2×#117—24no—
31C2×#117C2×C412yes—
32C2×#117Q811yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D412yes—
35#431—41no—
36#431—41no—
37C2×C4⋊C8C2×C412yes—
38#410—44no—
39#410—44no—
40C2×#99—24no—
41C2×#99—24no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—24no—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8—41no—
47C2×C4×C8—41no—
48C2×C4×C8C2×C412yes—
49#384—42no—
50#382—42no—
51#380—24no—
52#380—24no—
53C4×C4⋊C4—42no—
54C4×C4⋊C4—42no—
55#342—24no—
56#342C2×C418yes—
57#336—24no—
58C2×#41—41no—
59C2×#41—41no—
60C2×#41—42no—
61C2×#41D812yes—
62C2×#41SD1614yes—
63C2×#41Q1612yes—
64C22×Q8C4⋊C411yescharacteristic
65C22×D4C22⋊C412yes—
66C22×D4C2×D411yescharacteristic
67C23×C4C4211yescharacteristic
68C23×C4C22⋊C412yes—
69C23×C4C22⋊C412yes—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C4⋊C411yescharacteristic
72C23×C4C2×D412yes—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—44no—
76C22×C8—42no—
77C4×Q8—41no—
78C4×Q8—41no—
79C4×Q8—41no—
80C4×Q8—41no—
81C4×D4—42no—
82C4×D4—42no—
83C4×D4—41no—
84C4×D4—41no—
85C4×D4—42no—
86C4×D4—42no—
87C4×D4—41no—
88C4×D4—41no—
89#117—42no—
90#117—42no—
91#117—41no—
92#117—41no—
93#117—48no—
94#117—48no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4C4212yes—
99C2×C4⋊C4C4⋊C412yes—
100C2×C4⋊C4C4⋊C411yescharacteristic
101C2×C4⋊C4C2×Q811yescharacteristic
102C2×C22⋊C4—28no—
103C2×C22⋊C4C22⋊C412yes—
104C2×C22⋊C4C4⋊C412yes—
105C2×C22⋊C4C22×C412yes—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—44no—
115C2×C42C4211yescharacteristic
116C2×C42C22⋊C412yes—
117C2×C42C22⋊C411yescharacteristic
118C2×C42C4⋊C412yes—
119C2×C42C4⋊C412yes—
120C2×C42C4⋊C411yescharacteristic
121C2×C42C22×C411yescharacteristic
122C4⋊C8—42no—
123#99—48no—
124#99—44no—
125#99—44no—
126C22⋊C8—48no—
127C22⋊C8—42no—
128C4×C8—42no—
129C4×C8—41no—
130C4×C8—41no—
131C4×C8—41no—
132C4×C8—41no—
133#95—44no—
134#95—44no—
135C24#9511yescharacteristic
136C24C2×C22⋊C412yes—
137#41—42no—
138#41—42no—
139#41—44no—
140#41—44no—
141#41—42no—
142#41—42no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8#10612yes—
146C2×Q8#10712yes—
147C2×D4—42no—
148C2×D4—41no—
149C2×D4—42no—
150C2×D4—41no—
151C2×D4#10214yes—
152C2×D4#10314yes—
153C2×D4C2×D811yescharacteristic
154C2×D4C2×SD1612yes—
155C2×D4C2×Q1611yescharacteristic
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—48no—
175C22×C4#9512yes—
176C22×C4#9511yescharacteristic
177C22×C4#9511yescharacteristic
178C22×C4#9512yes—
179C22×C4#9511yescharacteristic
180C22×C4#10214yes—
181C22×C4#10314yes—
182C22×C4#10612yes—
183C22×C4#10712yes—
184C22×C4C2×C4211yescharacteristic
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C2×C4⋊C411yescharacteristic
187C22×C4C2×C4⋊C412yes—
188C2×C8—24no—
189C2×C8—42no—
190C2×C8—48no—
191C2×C8—42no—
192C2×C8—44no—
193C2×C8—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—88no—
199C4⋊C4—88no—
200C4⋊C4—88no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—48no—
205C22⋊C4—48no—
206C42—21no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—22no—
214C42—21no—
215C42—21no—
216C42—22no—
217C42—22no—
218C42—88no—
219C42#9911yescharacteristic
220C42#9911yescharacteristic
221C42#9911yescharacteristic
222C42#9911yescharacteristic
223C23—21no—
224C23—22no—
225C23—21no—
226C23—22no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23—42no—
231C23#34012yes—
232C23C2×#9511yescharacteristic
233C23C2×#10212yes—
234C23C2×#10312yes—
235Q8—42no—
236Q8—42no—
237D4—44no—
238D4—44no—
239D4—41no—
240D4—42no—
241D4—41no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—88no—
273C2×C4—88no—
274C2×C4—88no—
275C2×C4H(3,ℤ/4)12yes—
276C2×C4#34014yes—
277C2×C4#34012yes—
278C2×C4#34211yescharacteristic
279C2×C4#34211yescharacteristic
280C2×C4#34211yescharacteristic
281C2×C4#34211yescharacteristic
282C2×C4C2×#9911yescharacteristic
283C2×C4C2×#9911yescharacteristic
284C2×C4C2×#10611yescharacteristic
285C2×C4C2×#10711yescharacteristic
286C8—44no—
287C8—82no—
288C22—21no—
289C22—21no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—42no—
299C22—41no—
300C22—42no—
301C22—82no—
302C22C2×H(3,ℤ/4)11yescharacteristic
303C22C2×#34011yescharacteristic
304C22C2×#34211yescharacteristic
305C4—21no—
306C4—21no—
307C4—24no—
308C4—21no—
309C4—21no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—42no—
314C4—168no—
315C4#128912yes—
316C4#128912yes—
317C4#139112yes—
318C4#212612yes—
319C2—21no—
320C2—21no—
321C2—42no—
322C2—42no—
323C2—42no—
324C2C2×#128911yescharacteristic
325C2C2×#139111yescharacteristic
326C2#1258111yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7237C2semidirect
5. C2×#431C4semidirect
7. C22×C4×C8C4semidirect
9. #1984C4semidirect
14. #1799C4semidirect
17. #1740C4semidirect

Maximal subgroups

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

subgroupindex
#213732
#206322
#72372