Finite Group Databasebeta

← #364024

Subgroups — #364024

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34012C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic
3#13756C211yesmaximal, characteristic
4#11370C212yesmaximal, complemented
5#11359C212yesmaximal, complemented
6#10808C214yesmaximal, complemented
7#10720C214yesmaximal
8C22×#117C2212yes—
9C23×C42C2211yescharacteristic
10C2×#407C2212yes—
11#2285C412yescomplemented
12#2285C412yescomplemented
13#2285C2211yescharacteristic
14#2285C2211yescharacteristic
15#2283—22no—
16#2283—22no—
17#2283C414yescomplemented
18#2280—22no—
19#2280C2212yes—
20C4×#117—22no—
21C4×#117C2212yes—
22#1817—22no—
23#1817—22no—
24C2×#339C2218yes—
25C2×#339C2218yes—
26#1728C2212yes—
27#1728C2212yes—
28C4⋊#95—22no—
29C22⋊#95—22no—
30C4×#95C2212yes—
31C4×#95C2212yes—
32C24×C4D412yes—
33C24×C4C2311yescharacteristic
34C22×M16C2314yes—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—28no—
38C2×#117—28no—
39C2×#117—44no—
40C2×#117C2312yes—
41C2×#117C2312yes—
42C22×C4⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C42—28no—
45C22×C42C2×C412yes—
46C22×C42C2×C412yes—
47C22×C42D412yes—
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42C2314yes—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53#407—28no—
54C2×C22⋊C8—24no—
55#384C2×C412yes—
56#384C2×C411yescharacteristic
57#384C2×C411yescharacteristic
58#383—24no—
59#382—22no—
60#382—22no—
61#382C2×C412yes—
62#382C2×C412yes—
63#381—24no—
64#381—22no—
65#380—22no—
66#380—22no—
67#380—22no—
68#380—22no—
69#380C2×C414yes—
70#380C2×C414yes—
71#379—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2×C414yes—
77C4×C4⋊C4C2×C414yes—
78C4×C22⋊C4—24no—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—24no—
81#376—24no—
82#376—22no—
83C2×#95—24no—
84C2×#95—24no—
85C43—22no—
86#339—216no—
87#339—48no—
88#336—22no—
89#336—22no—
90#336—24no—
91C25C2×D411yescharacteristic
92C23×C4—24no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—28no—
96C23×C4—24no—
97C23×C4—44no—
98C23×C4—44no—
99C23×C4C22⋊C412yes—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C412yes—
102C23×C4C22⋊C412yes—
103C23×C4C22×C412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4#4118yes—
109C23×C4C2411yescharacteristic
110C2×M16—28no—
111C2×M16—28no—
112C2×M16—24no—
113C2×M16—24no—
114C22×C8—24no—
115#117—28no—
116#117—24no—
117#117—24no—
118#117—432no—
119#117—48no—
120#117—48no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4C22×C412yes—
129C2×C4⋊C4C22×C412yes—
130C2×C4⋊C4C22×C414yes—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4—44no—
138C2×C42—28no—
139C2×C42—28no—
140C2×C42—28no—
141C2×C42—28no—
142C2×C42—28no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—28no—
146C2×C42—28no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—24no—
152C2×C42C22⋊C412yes—
153C2×C42C22⋊C412yes—
154C2×C42C22⋊C412yes—
155C2×C42C22⋊C412yes—
156C2×C42C22×C412yes—
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D412yes—
162C2×C42#4118yes—
163C22⋊C8—48no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—48no—
171#95—48no—
172C24—22no—
173C24—21no—
174C24—21no—
175C24—42no—
176C24—42no—
177C24C2×C22⋊C412yes—
178C24#12314yes—
179C24C22×D411yescharacteristic
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—216no—
189C22×C4—216no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—28no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—432no—
212C22×C4—48no—
213C22×C4—48no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4#11718yes—
222C22×C4#12318yes—
223C22×C4#12314yes—
224C22×C4C23×C411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4C2×#4114yes—
227C22×C4C2×#4114yes—
228C22×C4H(5,𝔽2)12yes—
229C22×C42−1+412yes—
230M16—48no—
231M16—48no—
232C2×C8—44no—
233C2×C8—44no—
234C2×C8—44no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—28no—
238C4⋊C4—416no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C22⋊C4—416no—
244C22⋊C4—48no—
245C22⋊C4—48no—
246C22⋊C4—44no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C42—22no—
250C42—24no—
251C42—22no—
252C42—22no—
253C42—24no—
254C42—22no—
255C42—24no—
256C42—24no—
257C42—24no—
258C42—24no—
259C42—216no—
260C42—48no—
261C42—44no—
262C42—44no—
263C42—48no—
264C42—48no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—24no—
274C23—22no—
275C23—48no—
276C23—44no—
277C23—44no—
278C23—44no—
279C23—44no—
280C23#38114yes—
281C23C22×C22⋊C411yescharacteristic
282C23C2×#12312yes—
283C23#52512yes—
284C23#53511yescharacteristic
285C23#53611yescharacteristic
286C2×C4—216no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—216no—
296C2×C4—216no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—88no—
319C2×C4—88no—
320C2×C4#38114yes—
321C2×C4#42114yes—
322C2×C4#42114yes—
323C2×C4C2×#11712yes—
324C2×C4C4×#4112yes—
325C2×C4#52012yes—
326C2×C4C2×#12312yes—
327C2×C4#53212yes—
328C2×C4#53712yes—
329C2×C4#55118yes—
330C2×C4#55218yes—
331C8—84no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—24no—
337C22—24no—
338C22—24no—
339C22—24no—
340C22—48no—
341C22—48no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22C2×#38111yescharacteristic
347C22#229711yescharacteristic
348C22#231111yescharacteristic
349C22#248212yes—
350C22#248412yes—
351C22C2×#42112yes—
352C22C2×#42112yes—
353C4—24no—
354C4—24no—
355C4—24no—
356C4—42no—
357C4—42no—
358C4—88no—
359C4—84no—
360C4—82no—
361C4—82no—
362C4#234314yes—
363C4#249814yes—
364C2—22no—
365C2—22no—
366C2—44no—
367C2#1395811yescharacteristic
368C2#2070512yes—
369C2#2141611yescharacteristic
370C2#2141811yescharacteristic
371C2#2190012yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11370C2semidirect
5. #11359C2semidirect
6. #10808C2semidirect
11. #2285C4semidirect
12. #2285C4semidirect
17. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#340122
#206822
#137562
#113702
#113592
#108082
#107202