Finite Group Databasebeta

← #381216

Subgroups — #381216

Order 512.

1800 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic, complemented
3C42×C22⋊C4C211yesmaximal, characteristic, complemented
4#11359C212yesmaximal, complemented
5#11011C212yesmaximal, complemented
6#10971C214yesmaximal, complemented
7#10923C214yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525C2212yes—
10C23×C42C2211yescharacteristic
11C22×#104C2218yes—
12C2×#407C2212yes—
13#2857C412yescomplemented
14#2856C412yescomplemented
15#2854—22no—
16#2850C414yescomplemented
17#2847—22no—
18C4×#121—22no—
19C4×#121—22no—
20#2297C2212yes—
21#2285C2211yescharacteristic
22C2×C4×C22⋊C4—22no—
23C2×C4×C22⋊C4C2212yes—
24C2×C43C2212yes—
25#1817—22no—
26#1817—22no—
27C2×#339C2218yes—
28#1728C2212yes—
29#1728C2212yes—
30C4×#95—22no—
31C4×#95C2212yes—
32C22×#41C2312yes—
33C24×C4D412yes—
34C24×C4C2311yescharacteristic
35C22×M16C2314yes—
36#532—44no—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×#127C2×C411yescharacteristic
39C2×#124C2×C412yes—
40#525—48no—
41C2×#123—22no—
42C2×C22⋊Q8C2×C414yes—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121—22no—
46C2×#121C2×C414yes—
47C2×#120—22no—
48C2×C4×D4—24no—
49C2×C4×D4—22no—
50C2×#117C2312yes—
51C22×C22⋊C4—22no—
52C22×C42—24no—
53C22×C42—44no—
54C22×C42C2×C412yes—
55C22×C42C2×C412yes—
56C22×C42D412yes—
57C22×C42D412yes—
58C22×C42D412yes—
59C22×C42C2314yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#104—216no—
63C2×#104—48no—
64C2×#104—48no—
65#407—44no—
66C2×C22⋊C8—24no—
67#391C2×C414yes—
68#390—22no—
69#390—22no—
70#390C2×C414yes—
71#389—22no—
72#389—22no—
73#388—24no—
74#388—24no—
75#386—24no—
76#386—24no—
77#386—24no—
78#384C2×C412yes—
79#382C2×C412yes—
80#380—22no—
81#380—22no—
82#380—22no—
83C4×C4⋊C4—22no—
84C4×C22⋊C4—216no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—28no—
88C2×#95—22no—
89C2×#95—24no—
90C43—44no—
91#339—48no—
92#336—24no—
93#336—22no—
94#336—22no—
95C25C2×D411yescharacteristic
96C2×#41—28no—
97C2×#41—28no—
98C2×#41—28no—
99C22×Q8C22×C412yes—
100C22×D4—24no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4C22×C412yes—
104C23×C4—24no—
105C23×C4—28no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—22no—
110C23×C4—44no—
111C23×C4—44no—
112C23×C4C22⋊C412yes—
113C23×C4C22⋊C412yes—
114C23×C4C22⋊C412yes—
115C23×C4C22⋊C412yes—
116C23×C4C22×C412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2411yescharacteristic
122C2×M16—28no—
123C2×M16—24no—
124C2×M16—28no—
125C2×M16—24no—
126C22×C8—24no—
127C4⋊Q8—42no—
128#127—42no—
129#124—44no—
130#123—48no—
131C22⋊Q8—48no—
132#121—48no—
133#121—48no—
134#121—48no—
135#121—48no—
136#120—48no—
137C4×D4—48no—
138C4×D4—416no—
139#117—28no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C22×C414yes—
145C2×C22⋊C4—216no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—28no—
156C2×C42—28no—
157C2×C42—216no—
158C2×C42—22no—
159C2×C42—24no—
160C2×C42—44no—
161C2×C42—44no—
162C2×C42—44no—
163C2×C42—44no—
164C2×C42—44no—
165C2×C42—48no—
166C2×C42—48no—
167C2×C42C22⋊C412yes—
168C2×C42C22⋊C412yes—
169C2×C42C22⋊C412yes—
170C2×C42C22⋊C412yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4118yes—
178#104—432no—
179C22⋊C8—48no—
180#95—216no—
181#95—28no—
182#95—24no—
183#95—24no—
184#95—24no—
185C24—22no—
186C24—24no—
187C24—22no—
188C24—22no—
189C24—22no—
190C24—42no—
191C24C2×C22⋊C412yes—
192C24C22×D411yescharacteristic
193#41—48no—
194#41—48no—
195#41—416no—
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—24no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—48no—
205C2×D4—48no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—48no—
210C2×D4—44no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—216no—
215C22×C4—22no—
216C22×C4—28no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—432no—
242C22×C4—48no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—48no—
246C22×C4—48no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4#10418yescomplemented
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C2×#4114yes—
259C22×C4H(5,𝔽2)14yes—
260C22×C4H(5,𝔽2)14yes—
261M16—48no—
262M16—48no—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—44no—
266C4⋊C4—28no—
267C4⋊C4—44no—
268C4⋊C4—48no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C22⋊C4—232no—
272C22⋊C4—48no—
273C22⋊C4—48no—
274C22⋊C4—48no—
275C22⋊C4—44no—
276C22⋊C4—48no—
277C22⋊C4—48no—
278C42—24no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—24no—
284C42—216no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—432no—
290C42—44no—
291C42—48no—
292C23—22no—
293C23—22no—
294C23—28no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—48no—
302C23—48no—
303C23—44no—
304C23—44no—
305C23—42no—
306C23—42no—
307C23—44no—
308C23—44no—
309C23—42no—
310C23C2×#10414yes—
311C23C22×C22⋊C411yescharacteristic
312C23#52512yes—
313C23#53412yes—
314C23#53512yes—
315Q8—48no—
316D4—48no—
317D4—88no—
318D4—816no—
319D4—88no—
320C2×C4—216no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—216no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—416no—
340C2×C4—416no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—88no—
353C2×C4C2×#10414yes—
354C2×C4C2×#10414yes—
355C2×C4#42114yes—
356C2×C4C4×#4112yes—
357C2×C4#51812yes—
358C2×C4#51812yes—
359C2×C4#53212yes—
360C2×C4#53412yes—
361C2×C4#53512yes—
362C2×C4#53818yes—
363C2×C4#54018yes—
364C8—84no—
365C22—24no—
366C22—24no—
367C22—22no—
368C22—22no—
369C22—24no—
370C22—24no—
371C22—24no—
372C22—416no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—84no—
378C22—84no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22#229711yescharacteristic
383C22#230811yescharacteristic
384C22#230911yescharacteristic
385C22#263514yes—
386C22C22×#10412yes—
387C22C2×#42112yes—
388C4—24no—
389C4—24no—
390C4—24no—
391C4—416no—
392C4—42no—
393C4—42no—
394C4—84no—
395C4—82no—
396C4#233314yes—
397C4#266114yes—
398C2—24no—
399C2—44no—
400C2—162no—
401C2#1429011yescharacteristic
402C2#2141511yescharacteristic
403C2#2141611yescharacteristic
404C2#2177312yes—
405C2#2181512yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radical#36624C2
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
2. #20682C2semidirect
3. C42×C22⋊C4C2semidirect
4. #11359C2semidirect
5. #11011C2semidirect
6. #10971C2semidirect
7. #10923C2semidirect
13. #2857C4semidirect
14. #2856C4semidirect
16. #2850C4semidirect
250. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
#366242
#206822
C42×C22⋊C42
#113592
#110112
#109712
#109232