Finite Group Databasebeta

← #393517

Subgroups — #393517

Order 512.

1917 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic, complemented
3C2×#1897C212yesmaximal
4#20682C211yesmaximal, characteristic, complemented
5#10924C212yesmaximal, complemented
6#10923C212yesmaximal, complemented
7#10879C212yesmaximal, complemented
8#10878C212yesmaximal, complemented
9#10720C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C22×#117C2212yes—
12C23×C42C2211yescharacteristic
13C22×#104C2214yes—
14C2×#407C2212yes—
15#2855C414yescomplemented
16#2854C414yescomplemented
17#2853—22no—
18#2849—22no—
19C4×#123—24no—
20#2297C2211yescharacteristic
21#2297C2211yescharacteristic
22#2296C2211yescharacteristic
23#2296C2211yescharacteristic
24#2280C2211yescharacteristic
25C4×#117—28no—
26C2×C4×C4⋊C4—22no—
27C2×C4×C22⋊C4—22no—
28C2×#376C2212yes—
29C2×C43C2212yes—
30#1897—28no—
31#1817—24no—
32C2×#339C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1728C2212yes—
36#1728C2212yes—
37C22×#41C2311yescharacteristic
38C24×C4D412yes—
39C24×C4C2311yescharacteristic
40C22×M16C2314yes—
41#525—24no—
42#525—24no—
43C2×#123—24no—
44C2×#123—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×#121C2×C412yes—
47C2×#120—21no—
48C2×C4×D4—22no—
49C2×#117—24no—
50C2×#117—28no—
51C2×#117—28no—
52C2×#117D418yes—
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42—24no—
59C22×C42—24no—
60C22×C42—24no—
61C22×C42C2×C414yes—
62C22×C42D412yes—
63C22×C42D414yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2314yes—
67C2×#104—28no—
68C2×#104—28no—
69C2×#104—28no—
70#407—24no—
71#407—24no—
72C2×C22⋊C8—24no—
73#391C2×C412yes—
74#391C2×C412yes—
75#390C2×C412yes—
76#390C2×C412yes—
77#389C2×C414yes—
78#389C2×C414yes—
79#388—24no—
80#388—24no—
81#387—24no—
82#387—24no—
83#386—24no—
84#386—22no—
85#386—22no—
86#385—24no—
87#385—22no—
88#385—22no—
89#381—24no—
90#381—21no—
91#380C2×C414yes—
92#379—21no—
93C4×C4⋊C4—216no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—216no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100#376—28no—
101C2×#95—24no—
102C43—28no—
103#339—28no—
104#339—28no—
105#339—28no—
106#336—24no—
107#336—24no—
108C25C2×D411yescharacteristic
109C2×#41—22no—
110C2×#41—22no—
111C2×#41—24no—
112C2×#41—24no—
113C22×Q8C22×C411yescharacteristic
114C22×D4—22no—
115C22×D4—21no—
116C22×D4C22×C411yescharacteristic
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—21no—
123C23×C4—48no—
124C23×C4C22⋊C414yes—
125C23×C4C22⋊C414yes—
126C23×C4C22×C412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D414yes—
131C23×C4C2×D412yes—
132C23×C4#4114yes—
133C23×C4C2411yescharacteristic
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—24no—
140C22×C8—24no—
141#123—416no—
142#123—42no—
143#123—42no—
144C22⋊Q8—28no—
145#121—28no—
146#120—42no—
147#120—42no—
148C4×D4—48no—
149#117—232no—
150#117—232no—
151#117—22no—
152#117—22no—
153#117—24no—
154#117—24no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C42—216no—
176C2×C42—24no—
177C2×C42—216no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—216no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—416no—
191C2×C42C22⋊C414yes—
192C2×C42C22⋊C414yes—
193C2×C42C22×C412yes—
194C2×C42C22×C412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D418yes—
198C2×C42#4118yes—
199C2×C42#4118yes—
200#104—416no—
201C22⋊C8—48no—
202#95—216no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208C24—22no—
209C24—22no—
210C24—21no—
211C24—44no—
212C24C2×C22⋊C412yes—
213C24#12012yes—
214C24#12312yes—
215C24C22×D411yescharacteristic
216#41—44no—
217#41—44no—
218#41—44no—
219#41—44no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—48no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—44no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—28no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—28no—
245C22×C4—22no—
246C22×C4—28no—
247C22×C4—22no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—28no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—416no—
262C22×C4—416no—
263C22×C4—416no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C2×C22⋊C414yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C414yes—
270C22×C4C4×D418yes—
271C22×C4#12012yes—
272C22×C4#12118yes—
273C22×C4#12312yes—
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D412yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4112yes—
278C22×C4C2×#4114yes—
279C22×C4C2×#4114yes—
280M16—44no—
281M16—44no—
282M16—44no—
283M16—44no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—44no—
287C4⋊C4—232no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—48no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C22⋊C4—232no—
295C22⋊C4—48no—
296C22⋊C4—48no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—48no—
304C42—24no—
305C42—24no—
306C42—24no—
307C42—24no—
308C42—28no—
309C42—28no—
310C42—28no—
311C42—28no—
312C42—28no—
313C42—432no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42#123116yescomplemented
318C23—24no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—22no—
324C23—22no—
325C23—44no—
326C23—44no—
327C23—48no—
328C23—48no—
329C23—44no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23#38614yes—
334C23C22×C22⋊C411yescharacteristic
335C23C2×#12011yescharacteristic
336C23C2×#12311yescharacteristic
337C23#52512yes—
338C23#52512yes—
339Q8—44no—
340D4—44no—
341D4—88no—
342D4—82no—
343D4—82no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—216no—
353C2×C4—216no—
354C2×C4—216no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—416no—
359C2×C4—416no—
360C2×C4—416no—
361C2×C4—416no—
362C2×C4—416no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—84no—
375C2×C4#38614yes—
376C2×C4#42118yes—
377C2×C4C2×C4×D412yes—
378C2×C4C4×#4112yes—
379C2×C4C4×#4112yes—
380C2×C4C2×#12112yes—
381C2×C4C2×#12314yes—
382C2×C4#52514yes—
383C2×C4#52914yes—
384C2×C4#52914yes—
385C2×C4#53212yes—
386C2×C4#53212yes—
387C8—84no—
388C22—24no—
389C22—24no—
390C22—22no—
391C22—22no—
392C22—24no—
393C22—24no—
394C22—24no—
395C22—48no—
396C22—48no—
397C22—44no—
398C22—44no—
399C22—81no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22C2×#38611yescharacteristic
404C22#229711yescharacteristic
405C22#229711yescharacteristic
406C22#284912yes—
407C22#285312yes—
408C22C2×#42114yes—
409C4—24no—
410C4—28no—
411C4—416no—
412C4—44no—
413C4—81no—
414C4—82no—
415C4—82no—
416C4—82no—
417C4C4×#12314yes—
418C4#285414yes—
419C2—22no—
420C2—22no—
421C2—44no—
422C2—161no—
423C2#1470611yescharacteristic
424C2#2097912yes—
425C2#2141612yes—
426C2#2141812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
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. C2×C4×#117C2semidirect
4. #20682C2semidirect
5. #10924C2semidirect
6. #10923C2semidirect
7. #10879C2semidirect
8. #10878C2semidirect
9. #10720C2semidirect
15. #2855C4semidirect
16. #2854C4semidirect
317. C42#123semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×C4×#1172
C2×#18972
#206822
#109242
#109232
#108792
#108782
#107202