Finite Group Databasebeta

← #479760

Subgroups — #479760

Order 512.

629 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11387C212yesmaximal
2#11386C211yesmaximal, characteristic
3#11370C211yesmaximal, characteristic
4#11359C211yesmaximal, characteristic
5#11321C212yesmaximal
6#11275C212yesmaximal
7#11275C212yesmaximal
8#11262C212yesmaximal
9#11212C212yesmaximal
10C2×#533C2211yescharacteristic
11#2905C2212yes—
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#344C2212yes—
15C2×#344C2212yes—
16#1740C2212yes—
17#1738C2212yes—
18#1733C2212yes—
19#1733C2212yes—
20C2×#339C2212yes—
21C2×#339C2212yes—
22C2×#339C2212yes—
23C2×#339C2212yes—
24C2×#339C2212yes—
25C2×#339C2212yes—
26C2×H(3,ℤ/4)C2212yes—
27#1728C2211yescharacteristic
28#1728C2211yescharacteristic
29#1728C2211yescharacteristic
30C4×#95C2211yescharacteristic
31C22×M16C2312yes—
32C22×M16C2312yes—
33C22×M16C2312yes—
34#533—24no—
35#533—42no—
36C2×C4⋊Q8—21no—
37C2×#125—21no—
38C2×C22⋊Q8—41no—
39C2×C4×Q8—41no—
40C2×#117C2312yes—
41C2×#117C2312yes—
42C2×#117C2312yes—
43C22×C42C2311yescharacteristic
44C22×C42C2311yescharacteristic
45C22×C42C2311yescharacteristic
46#424—42no—
47#424—42no—
48#423—42no—
49#384—21no—
50#384Q811yescharacteristic, complemented
51#384Q811yescharacteristic, complemented
52#382—21no—
53#382Q812yescomplemented
54#380—41no—
55#380—41no—
56C4×C4⋊C4—41no—
57C4×C4⋊C4—41no—
58C2×#95—41no—
59#344—24no—
60#344—42no—
61#342—42no—
62#341—42no—
63#339—24no—
64#339—22no—
65#339—22no—
66#339—22no—
67#339—22no—
68#339—42no—
69#339—42no—
70#339—42no—
71#339—42no—
72#339—42no—
73H(3,ℤ/4)—42no—
74#336—41no—
75#336—41no—
76#336—41no—
77C22×Q8—41no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4C2411yescharacteristic
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C4⋊Q8—44no—
104#125—44no—
105C22⋊Q8—44no—
106C22⋊Q8—44no—
107C4×Q8—48no—
108#117—22no—
109#117—22no—
110#117—22no—
111#117—22no—
112#117—22no—
113#117—22no—
114#117—22no—
115#117—22no—
116#117—22no—
117#117—22no—
118#117—22no—
119#117#4114yes—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4C2×Q812yes—
129C2×C4⋊C4C2×Q812yes—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—22no—
145C2×C42C2×Q811yescharacteristic
146C2×C42C2×Q811yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C4⋊C8—48no—
152#95—42no—
153#95—42no—
154#95—44no—
155C24H(5,𝔽2)11yescharacteristic
156C2×Q8—44no—
157C2×Q8—82no—
158C2×Q8—82no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4C22×Q811yescharacteristic
182C22×C4C2×#4112yes—
183C22×C4C2×#4111yescharacteristic
184C22×C4C2×#4111yescharacteristic
185C22×C4H(5,𝔽2)11yescharacteristic
186C22×C4H(5,𝔽2)12yes—
187C22×C4H(5,𝔽2)11yescharacteristic
188C22×C4H(5,𝔽2)11yescharacteristic
189C22×C42−1+412yes—
190C22×C42−1+412yes—
191M16—42no—
192M16—42no—
193M16—42no—
194M16—42no—
195M16—42no—
196M16—42no—
197M16—42no—
198M16—42no—
199M16—42no—
200M16—42no—
201M16—42no—
202M16—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C4⋊C4—24no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—48no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C22⋊C4—24no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C42—24no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—42no—
240C42—42no—
241C42—42no—
242C42—42no—
243C42—42no—
244C42—42no—
245C42—42no—
246C42—42no—
247C23—41no—
248C23—41no—
249C23#53811yescharacteristic
250C23#53811yescharacteristic
251C23#54311yescharacteristic
252C23#55111yescharacteristic
253C23#56011yescharacteristic
254C23#56311yescharacteristic
255C23#56311yescharacteristic
256Q8—88no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—44no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4#53812yes—
292C2×C4#54112yes—
293C2×C4#54212yes—
294C2×C4#54212yes—
295C2×C4#54212yes—
296C2×C4#54412yes—
297C2×C4#55112yes—
298C2×C4#55212yes—
299C2×C4#55411yescharacteristic
300C2×C4#55411yescharacteristic
301C2×C4#55712yes—
302C2×C4#55911yescharacteristic
303C2×C4H(3,𝔽4)11yescharacteristic
304C2×C4#56212yes—
305C2×C4#56212yes—
306C2×C4#56312yes—
307C8—82no—
308C8—82no—
309C8—82no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22#273511yescharacteristic
321C22#276411yescharacteristic
322C22#279911yescharacteristic
323C22#280411yescharacteristic
324C22#283011yescharacteristic
325C22#283311yescharacteristic
326C22#284111yescharacteristic
327C4—42no—
328C4—42no—
329C4—42no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—84no—
334C4—82no—
335C4—82no—
336C4—162no—
337C4#275412yes—
338C4#275512yes—
339C4#277412yes—
340C4#283212yes—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2#1723511yescharacteristic
348C2#3270411yescharacteristic
349C2#3270511yescharacteristic
subgroupG/H
centerC2×C4H(3,𝔽4)
commutator, Frattini, cosocleC23×C4C24
socleC22#2804
prime radicalC24H(5,𝔽2)
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
50. #384Q8semidirect
51. #384Q8semidirect
53. #382Q8semidirect

Maximal subgroups

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

subgroupindex
#113872
#113862
#113702
#113592
#113212
#112752
#112622
#112122