Finite Group Databasebeta

← #381217

Subgroups — #381217

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic, complemented
3#13753C211yesmaximal, characteristic, complemented
4#11359C212yesmaximal, complemented
5#11011C212yesmaximal, complemented
6#10971C214yesmaximal, complemented
7#10924C214yesmaximal, 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⋊C4C2212yes—
23C2×#376C2212yes—
24#1817—22no—
25#1817—22no—
26C2×#339C2218yes—
27#1728C2212yes—
28#1728C2212yes—
29#1434—22no—
30C22⋊#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—44no—
53C22×C42C2×C412yes—
54C22×C42C2×C412yes—
55C22×C42D412yes—
56C22×C42D412yes—
57C22×C42D412yes—
58C22×C42C2314yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×#104—216no—
62C2×#104—48no—
63C2×#104—48no—
64#407—44no—
65C2×C22⋊C8—24no—
66#391C2×C414yes—
67#390—22no—
68#390—22no—
69#390C2×C414yes—
70#389—22no—
71#389—22no—
72#388—24no—
73#388—24no—
74#386—24no—
75#386—24no—
76#386—24no—
77#384C2×C412yes—
78#382C2×C412yes—
79#380—22no—
80#380—22no—
81#380—22no—
82C4×C4⋊C4—22no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—28no—
86#376—44no—
87C2×#95—22no—
88C2×#95—24no—
89C2×#95—24no—
90#339—48no—
91#336—24no—
92#336—22no—
93#336—22no—
94C25C2×D411yescharacteristic
95C2×#41—28no—
96C2×#41—28no—
97C2×#41—28no—
98C22×Q8C22×C412yes—
99C22×D4—24no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4C22×C412yes—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—22no—
108C23×C4—44no—
109C23×C4—44no—
110C23×C4—44no—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C412yes—
113C23×C4C22⋊C412yes—
114C23×C4C22⋊C412yes—
115C23×C4C22×C412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2411yescharacteristic
121C2×M16—28no—
122C2×M16—24no—
123C2×M16—28no—
124C2×M16—24no—
125C22×C8—24no—
126C4⋊Q8—42no—
127#127—42no—
128#124—44no—
129#123—48no—
130C22⋊Q8—48no—
131#121—48no—
132#121—48no—
133#121—48no—
134#121—48no—
135#120—48no—
136C4×D4—48no—
137C4×D4—416no—
138#117—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4C22×C414yes—
144C2×C22⋊C4—28no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—48no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—28no—
155C2×C42—28no—
156C2×C42—22no—
157C2×C42—24no—
158C2×C42—44no—
159C2×C42—44no—
160C2×C42—44no—
161C2×C42—44no—
162C2×C42—44no—
163C2×C42C22⋊C412yes—
164C2×C42C22⋊C412yes—
165C2×C42C22⋊C412yes—
166C2×C42C22⋊C412yes—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42#4118yes—
174#104—432no—
175C22⋊C8—48no—
176#95—28no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—48no—
181#95—48no—
182C24—22no—
183C24—24no—
184C24—22no—
185C24—22no—
186C24—22no—
187C24—42no—
188C24C2×C22⋊C412yes—
189C24C22×D411yescharacteristic
190#41—48no—
191#41—48no—
192#41—416no—
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—24no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—48no—
202C2×D4—48no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—48no—
207C2×D4—44no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—28no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—432no—
239C22×C4—48no—
240C22×C4—48no—
241C22×C4—48no—
242C22×C4—48no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C2×#4114yes—
256C22×C4H(5,𝔽2)14yes—
257C22×C4H(5,𝔽2)14yes—
258M16—48no—
259M16—48no—
260C2×C8—44no—
261C2×C8—44no—
262C2×C8—44no—
263C4⋊C4—28no—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C22⋊C4—48no—
269C22⋊C4—48no—
270C22⋊C4—48no—
271C22⋊C4—48no—
272C22⋊C4—416no—
273C22⋊C4—48no—
274C22⋊C4—44no—
275C42—24no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—24no—
281C42—216no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42—42no—
286C42—44no—
287C42—48no—
288C23—22no—
289C23—22no—
290C23—28no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—48no—
298C23—48no—
299C23—44no—
300C23—44no—
301C23—42no—
302C23—42no—
303C23—44no—
304C23—44no—
305C23—42no—
306C23#42114yes—
307C23C22×C22⋊C411yescharacteristic
308C23#52512yes—
309C23#53412yes—
310C23#53512yes—
311Q8—48no—
312D4—48no—
313D4—88no—
314D4—816no—
315D4—88no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—88no—
349C2×C4—88no—
350C2×C4—88no—
351C2×C4#42114yes—
352C2×C4C4×#4112yes—
353C2×C4#51812yes—
354C2×C4#51812yes—
355C2×C4#53212yes—
356C2×C4#53412yes—
357C2×C4#53512yes—
358C2×C4#53818yes—
359C2×C4#54018yes—
360C8—84no—
361C22—24no—
362C22—24no—
363C22—22no—
364C22—22no—
365C22—24no—
366C22—24no—
367C22—24no—
368C22—416no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—84no—
374C22—84no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22#229711yescharacteristic
379C22#230811yescharacteristic
380C22#230911yescharacteristic
381C22#263514yes—
382C22C2×#42112yes—
383C22C2×#42112yes—
384C4—24no—
385C4—42no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—88no—
390C4—84no—
391C4—82no—
392C4#233314yes—
393C4#266114yes—
394C2—24no—
395C2—44no—
396C2—162no—
397C2#1429011yescharacteristic
398C2#2141711yescharacteristic
399C2#2141811yescharacteristic
400C2#2177412yes—
401C2#2181612yes—
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. #13753C2semidirect
4. #11359C2semidirect
5. #11011C2semidirect
6. #10971C2semidirect
7. #10924C2semidirect
13. #2857C4semidirect
14. #2856C4semidirect
16. #2850C4semidirect

Maximal subgroups

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

subgroupindex
#366242
#206822
#137532
#113592
#110112
#109712
#109242