Finite Group Databasebeta

← #187758

Subgroups — #187758

Order 512.

2644 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
1C22×#540C211yesmaximal, characteristic
2#20694C211yesmaximal, characteristic, complemented
3#20688C211yesmaximal, characteristic, complemented
4#17577C214yesmaximal, complemented
5#8518C218yesmaximal, complemented
6C2×#540—28no—
7C2×#540—28no—
8C2×#540—22no—
9C2×#540—22no—
10C2×#540C414yescomplemented
11C2×#540C2214yes—
12C22×#127C411yescharacteristic, complemented
13C22×#125C411yescharacteristic, complemented
14C22×#123—22no—
15C22×#121—22no—
16C22×#121C2212yes—
17C22×C4×D4C412yescomplemented
18C22×#117C2211yescharacteristic
19C2×#407C2212yes—
20#2280C2212yes—
21#1822C2214yescomplemented
22#1821—22no—
23#1820C2214yescomplemented
24#1819—22no—
25#1742—22no—
26#1740—22no—
27#1519—28no—
28#1510C22116yescomplemented
29#1507—28no—
30C23×D4—22no—
31C23×D4—22no—
32C23×D4C2×C412yes—
33C24×C4—22no—
34C24×C4C2311yescharacteristic
35C22×M16—22no—
36#540—216no—
37#540—216no—
38#540—216no—
39#540—216no—
40#540—232no—
41#540—232no—
42C2×#127—24no—
43C2×#127—24no—
44C2×#127—22no—
45C2×#127C2×C414yes—
46C2×#125—24no—
47C2×#125—24no—
48C2×#125—22no—
49C2×#125C2×C414yes—
50C2×#123—216no—
51C2×#123—216no—
52C2×#123—28no—
53C2×#123—28no—
54C2×#121—216no—
55C2×#121—216no—
56C2×#121—22no—
57C2×#121—24no—
58C2×#121—24no—
59C2×#121—22no—
60C2×#121—28no—
61C2×#121—28no—
62C2×#121C2×C414yes—
63C2×#121D418yes—
64C2×#121D418yes—
65C2×#121C2314yes—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—24no—
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4C2×C418yes—
72C2×#117—24no—
73C2×#117—24no—
74C2×#117—22no—
75C2×#117C2×C412yes—
76C2×#117D412yes—
77C2×#117D412yes—
78C2×#117C2312yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4D412yes—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D412yes—
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87#407—24no—
88#407—22no—
89#407C2314yes—
90C2×C22⋊C8—24no—
91#381C2314yes—
92#380—24no—
93#379—22no—
94C4×C22⋊C4—24no—
95#343—28no—
96#342—28no—
97#327—232no—
98C25—22no—
99C25C2×D411yescharacteristic
100C22×D4—28no—
101C22×D4—28no—
102C22×D4—24no—
103C22×D4—28no—
104C22×D4—28no—
105C22×D4—24no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—28no—
110C22×D4—48no—
111C22×D4—88no—
112C22×D4—88no—
113C22×D4C22⋊C414yes—
114C22×D4C22⋊C414yes—
115C22×D4C22×C414yes—
116C23×C4—22no—
117C23×C4—28no—
118C23×C4—28no—
119C23×C4—28no—
120C23×C4—44no—
121C23×C4—88no—
122C23×C4C22⋊C412yes—
123C23×C4C22⋊C412yes—
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D414yes—
129C23×C4C2×D414yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C2×M16—24no—
133C2×M16—42no—
134C2×M16—48no—
135C22×C8—42no—
136#127—28no—
137#127—28no—
138#127—216no—
139#125—28no—
140#125—28no—
141#125—216no—
142#123—264no—
143#123—264no—
144#121—264no—
145#121—264no—
146#121—28no—
147#121—28no—
148#121—28no—
149#121—28no—
150#121—28no—
151#121—28no—
152#121C22⋊C4116yes—
153#121C2×D4116yes—
154C4×D4—216no—
155C4×D4—216no—
156C4×D4—216no—
157C4×D4—216no—
158#117—28no—
159#117—24no—
160#117—24no—
161#117—24no—
162#117—24no—
163#117—28no—
164#117—48no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22⋊C414yes—
175C2×C4⋊C4C22⋊C414yes—
176C2×C4⋊C4C22⋊C414yes—
177C2×C4⋊C4C22×C414yes—
178C2×C4⋊C4C2×D414yes—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4C22⋊C414yes—
191C2×C22⋊C4C2×D414yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—42no—
197C2×C42C22⋊C412yes—
198C2×C42C22⋊C412yes—
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C22⋊C8—28no—
202C22⋊C8—44no—
203#95—28no—
204C24—24no—
205C24—24no—
206C24—24no—
207C24—42no—
208C24—44no—
209C24—48no—
210C24—48no—
211C24—48no—
212C24C2×C22⋊C411yescharacteristic
213C24C2×C22⋊C411yescharacteristic
214C24#12014yes—
215C24C22×D411yescharacteristic
216C2×D4—232no—
217C2×D4—216no—
218C2×D4—216no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—48no—
223C2×D4—48no—
224C2×D4—48no—
225C2×D4—48no—
226C2×D4—48no—
227C2×D4—48no—
228C2×D4—816no—
229C2×D4—88no—
230C2×D4—816no—
231C2×D4—88no—
232C2×D4—816no—
233C2×D4—816no—
234C2×D4—816no—
235C2×D4C2×C22⋊C418yes—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—232no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—816no—
257C22×C4—816no—
258C22×C4—816no—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C414yes—
263C22×C4C2×C22⋊C414yes—
264C22×C4C2×C22⋊C414yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4#12012yes—
267C22×C4#12012yes—
268C22×C4#12012yes—
269C22×C4#12012yes—
270C22×C4#12014yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D414yes—
274M16—48no—
275M16—84no—
276C2×C8—44no—
277C2×C8—84no—
278C4⋊C4—232no—
279C4⋊C4—24no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—84no—
285C4⋊C4C2×C22⋊C418yes—
286C22⋊C4—232no—
287C22⋊C4—28no—
288C22⋊C4—28no—
289C22⋊C4—216no—
290C22⋊C4—216no—
291C22⋊C4—48no—
292C22⋊C4—88no—
293C42—24no—
294C42—24no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—84no—
300C23—24no—
301C23—22no—
302C23—28no—
303C23—22no—
304C23—22no—
305C23—28no—
306C23—44no—
307C23—42no—
308C23—44no—
309C23—42no—
310C23—48no—
311C23—48no—
312C23—48no—
313C23—48no—
314C23—48no—
315C23—48no—
316C23—48no—
317C23—48no—
318C23—48no—
319C23—88no—
320C23—832no—
321C23#37912yes—
322C23#37912yes—
323C23C22×C22⋊C411yescharacteristic
324C23C2×#12012yes—
325D4—48no—
326D4—48no—
327D4—48no—
328D4—48no—
329D4—816no—
330D4—816no—
331D4—832no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—832no—
357C2×C4—84no—
358C2×C4—88no—
359C2×C4#37912yes—
360C2×C4#37911yescharacteristic
361C2×C4#37911yescharacteristic
362C2×C4#41014yes—
363C2×C4C22×C22⋊C412yes—
364C2×C4C2×#12012yes—
365C8—84no—
366C22—28no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—24no—
373C22—24no—
374C22—48no—
375C22—48no—
376C22—48no—
377C22—88no—
378C22—82no—
379C22—82no—
380C22—816no—
381C22—816no—
382C22—816no—
383C22#179514yes—
384C22C2×#37911yescharacteristic
385C22C2×#41012yes—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—48no—
390C4—44no—
391C4—44no—
392C4—164no—
393C4#178212yes—
394C4#178212yes—
395C2—24no—
396C2—22no—
397C2—42no—
398C2—42no—
399C2—88no—
400C2C2×#178211yescharacteristic
401C2C2×#179512yes—
subgroupG/H
center, socleC22C2×#379
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#540C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20694C2semidirect
3. #20688C2semidirect
4. #17577C2semidirect
5. #8518C2semidirect
10. C2×#540C4semidirect
12. C22×#127C4semidirect
13. C22×#125C4semidirect
17. C22×C4×D4C4semidirect
21. #1822C22semidirect
23. #1820C22semidirect
28. #1510C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5402
#206942
#206882
#175772
#85182