Finite Group Databasebeta

← #196861

Subgroups — #196861

Order 512.

1981 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C22×#376C211yesmaximal, characteristic, complemented
3#20682C211yesmaximal, characteristic, complemented
4C2×#1504C212yesmaximal
5#9149C214yesmaximal, complemented
6#9145C214yesmaximal, complemented
7#8020C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C23×C42—21no—
10C23×C42C2211yescharacteristic
11C2×#407C2212yes—
12#2855C412yescomplemented
13#2855C2212yes—
14#2854C412yescomplemented
15#2854C2212yes—
16#2853—22no—
17#2849—21no—
18#2849—21no—
19C4×#123—22no—
20C4×#123—22no—
21#2297C412yescomplemented
22#2296C412yescomplemented
23#2280C2211yescharacteristic
24C2×#376—22no—
25C2×#376—28no—
26C2×#376C2214yes—
27C22×#95—21no—
28C22×#95C2212yes—
29#1817—22no—
30#1817C2214yes—
31#1728—22no—
32#1504—28no—
33C2×#328C2214yes—
34C2×#327C2214yes—
35#1294C2218yes—
36C22×#41C2×C411yescharacteristic
37C24×C4—22no—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42#525—24no—
43#525—24no—
44C2×#123—22no—
45C2×#123—22no—
46C2×#123—21no—
47C2×C22⋊Q8C2×C411yescharacteristic
48C2×C22⋊Q8C2311yescharacteristic
49C2×#121C2×C411yescharacteristic
50C2×#121C2311yescharacteristic
51C2×#120—21no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C412yes—
56C22×C42—216no—
57C22×C42—28no—
58C22×C42—24no—
59C22×C42—24no—
60C22×C42—24no—
61C22×C42C2×C412yes—
62C22×C42C2×C412yes—
63C22×C42D412yes—
64C22×C42D412yes—
65C22×C42D412yes—
66C22×C42C2312yes—
67#407—24no—
68#407—24no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8C2314yes—
71#391C2×C414yescomplemented
72#390C2×C414yescomplemented
73#389C2×C414yescomplemented
74#389C2×C414yescomplemented
75#388—24no—
76#388—24no—
77#387—24no—
78#387—24no—
79#386—22no—
80#386—22no—
81#386—22no—
82#386—22no—
83#385—22no—
84#385—22no—
85#385—22no—
86#385—22no—
87#381—22no—
88#381—22no—
89#381—21no—
90#380C2×C412yes—
91#380C2312yes—
92#379—21no—
93C4×C4⋊C4—24no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—24no—
97#376—216no—
98#376—216no—
99C2×#95—28no—
100C2×#95—24no—
101C2×#95—22no—
102C2×#95—24no—
103C2×#95D418yes—
104C2×#95C2314yes—
105#336—28no—
106#328—28no—
107#327—28no—
108C25C2×D411yescharacteristic
109C2×#41—24no—
110C2×#41—44no—
111C22×Q8C22×C411yescharacteristic
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4C22×C411yescharacteristic
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—28no—
119C23×C4—216no—
120C23×C4—28no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—21no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4—48no—
128C23×C4—48no—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D414yes—
141C23×C4#4114yes—
142C23×C4C2411yescharacteristic
143C2×M16—48no—
144C2×M16—44no—
145C22×C8—24no—
146#123—48no—
147#123—48no—
148#123—44no—
149C22⋊Q8—24no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152#121—24no—
153#121—22no—
154#121—22no—
155#120—42no—
156#120—42no—
157C4×D4—42no—
158C4×D4—44no—
159C4×D4—42no—
160#117—24no—
161#117—44no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C414yes—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—21no—
179C2×C42—216no—
180C2×C42—264no—
181C2×C42—232no—
182C2×C42—28no—
183C2×C42—28no—
184C2×C42—24no—
185C2×C42—28no—
186C2×C42—28no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42C22⋊C414yes—
193C2×C42C22⋊C414yes—
194C2×C42C22×C414yes—
195C2×C42C2×D414yes—
196C22⋊C8—24no—
197C22⋊C8—24no—
198C22⋊C8—44no—
199#95—216no—
200#95—22no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—28no—
206#95—24no—
207#95—22no—
208#95C2×D418yes—
209C24—24no—
210C24—24no—
211C24—24no—
212C24—21no—
213C24C2×C22⋊C411yescharacteristic
214C24C2×C22⋊C411yescharacteristic
215C24#12012yes—
216C24#12312yes—
217C24C22×D411yescharacteristic
218#41—88no—
219C2×Q8—22no—
220C2×Q8—42no—
221C2×D4—22no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—44no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—44no—
229C2×D4—44no—
230C22×C4—216no—
231C22×C4—216no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—216no—
242C22×C4—232no—
243C22×C4—22no—
244C22×C4—28no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—464no—
264C22×C4—48no—
265C22×C4—416no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—416no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4#10418yes—
272C22×C4C2×C22⋊C414yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C414yes—
276C22×C4C4×D414yes—
277C22×C4C4×D414yes—
278C22×C4#12012yes—
279C22×C4#12114yes—
280C22×C4#12114yes—
281C22×C4#12312yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D412yes—
285C22×C4C2×#4112yes—
286M16—88no—
287C2×C8—48no—
288C2×C8—44no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4—48no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—48no—
296C22⋊C4—48no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—44no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—48no—
307C42—264no—
308C42—28no—
309C42—28no—
310C42—28no—
311C42—28no—
312C42—44no—
313C42—42no—
314C23—232no—
315C23—24no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—24no—
323C23—24no—
324C23—24no—
325C23—24no—
326C23—24no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—41no—
331C23#38612yes—
332C23#38612yes—
333C23#41014yes—
334C23C2×#10414yes—
335C23C22×C22⋊C411yescharacteristic
336C23C2×#12011yescharacteristic
337C23C2×#12311yescharacteristic
338C23#57514yes—
339Q8—82no—
340D4—82no—
341D4—82no—
342D4—84no—
343D4—84no—
344D4—82no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—216no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—216no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—24no—
357C2×C4—216no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—464no—
368C2×C4—432no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—84no—
385C2×C4#38612yes—
386C2×C4#38612yes—
387C2×C4C2×#10418yes—
388C2×C4C2×C4×D412yes—
389C2×C4C2×#12112yes—
390C8—84no—
391C22—216no—
392C22—216no—
393C22—28no—
394C22—28no—
395C22—28no—
396C22—22no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—84no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#189614yes—
409C22#189914yes—
410C22C2×#38611yescharacteristic
411C22C2×#41012yes—
412C22#289612yes—
413C22C22×#10412yes—
414C22#300512yes—
415C22#309312yes—
416C4—28no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—416no—
422C4—44no—
423C4—84no—
424C4—81no—
425C4—81no—
426C4—81no—
427C2—22no—
428C2—22no—
429C2—24no—
430C2—24no—
431C2—161no—
432C2#1754114yes—
433C2#2091011yescharacteristic
434C2C2×#189611yescharacteristic
435C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#376C2semidirect
3. #20682C2semidirect
5. #9149C2semidirect
6. #9145C2semidirect
7. #8020C2semidirect
12. #2855C4semidirect
14. #2854C4semidirect
21. #2297C4semidirect
22. #2296C4semidirect
71. #391C2×C4semidirect
72. #390C2×C4semidirect
73. #389C2×C4semidirect
74. #389C2×C4semidirect

Maximal subgroups

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

subgroupindex
#366262
C22×#3762
#206822
C2×#15042
#91492
#91452
#80202