Finite Group Databasebeta

← #196873

Subgroups — #196873

Order 512.

1765 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic, complemented
3#20682C211yesmaximal, characteristic, complemented
4C2×#1504C212yesmaximal
5#9156C214yesmaximal, complemented
6#9152C214yesmaximal, complemented
7#8020C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C23×C4⋊C4—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
23C2×#382—22no—
24C2×#382—28no—
25C2×#382C2214yes—
26#2280C2211yescharacteristic
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×C4⋊C4—28no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—48no—
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#382—216no—
88#382—216no—
89#381—22no—
90#381—22no—
91#381—21no—
92#380C2×C412yes—
93#380C2312yes—
94#379—21no—
95C4×C4⋊C4—24no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—24no—
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—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—28no—
123C23×C4—24no—
124C23×C4—21no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4—48no—
128C23×C4—48no—
129C23×C4—48no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D414yes—
142C23×C4#4114yes—
143C23×C4C2411yescharacteristic
144C2×M16—48no—
145C2×M16—44no—
146C22×C8—24no—
147#123—48no—
148#123—44no—
149#123—44no—
150#123—44no—
151C22⋊Q8—24no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154#121—24no—
155#121—22no—
156#121—22no—
157#120—42no—
158#120—42no—
159C4×D4—42no—
160C4×D4—44no—
161C4×D4—42no—
162#117—24no—
163#117—44no—
164C2×C4⋊C4—216no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—432no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C414yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—21no—
185C2×C42—28no—
186C2×C42—28no—
187C2×C42—24no—
188C2×C42—28no—
189C2×C42—28no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42C22⋊C414yes—
196C2×C42C22⋊C414yes—
197C2×C42C22×C414yes—
198C2×C42C2×D414yes—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—44no—
202#95—216no—
203#95—22no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—28no—
209#95—24no—
210#95—22no—
211#95C2×D418yes—
212C24—22no—
213C24—22no—
214C24—24no—
215C24—24no—
216C24—21no—
217C24C2×C22⋊C411yescharacteristic
218C24C2×C22⋊C411yescharacteristic
219C24#12012yes—
220C24#12312yes—
221C24C22×D411yescharacteristic
222#41—88no—
223C2×Q8—22no—
224C2×Q8—42no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—44no—
234C2×D4—44no—
235C22×C4—28no—
236C22×C4—216no—
237C22×C4—28no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—28no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—28no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—48no—
264C22×C4—44no—
265C22×C4—48no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—416no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—48no—
276C22×C4—42no—
277C22×C4—832no—
278C22×C4C2×C22⋊C414yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C414yes—
282C22×C4C4×D414yes—
283C22×C4C4×D414yes—
284C22×C4#12012yes—
285C22×C4#12114yes—
286C22×C4#12114yes—
287C22×C4#12312yes—
288C22×C4AGL(1,ℤ/8)14yes—
289C22×C4#13714yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D412yes—
293C22×C4C2×#4112yes—
294M16—88no—
295C2×C8—48no—
296C2×C8—44no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—44no—
300C4⋊C4—432no—
301C4⋊C4—48no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—48no—
305C22⋊C4—48no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—48no—
317C42—28no—
318C42—28no—
319C42—28no—
320C42—28no—
321C42—44no—
322C42—42no—
323C23—216no—
324C23—216no—
325C23—24no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—24no—
333C23—24no—
334C23—24no—
335C23—24no—
336C23—22no—
337C23—22no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23#38612yes—
343C23#38612yes—
344C23#41014yes—
345C23#42114yes—
346C23C22×C22⋊C411yescharacteristic
347C23C2×#12011yescharacteristic
348C23C2×#12311yescharacteristic
349C23C2×AGL(1,ℤ/8)12yes—
350C23C2×#13712yes—
351Q8—82no—
352D4—82no—
353D4—82no—
354D4—84no—
355D4—84no—
356D4—81no—
357D4—81no—
358C2×C4—216no—
359C2×C4—216no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—48no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—84no—
398C2×C4—832no—
399C2×C4—816no—
400C2×C4#38612yes—
401C2×C4#38612yes—
402C2×C4#41718yes—
403C2×C4C2×C4×D412yes—
404C2×C4C2×#12112yes—
405C8—84no—
406C22—28no—
407C22—216no—
408C22—28no—
409C22—22no—
410C22—22no—
411C22—24no—
412C22—24no—
413C22—28no—
414C22—28no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—81no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22#189714yes—
426C22#189914yes—
427C22C2×#38611yescharacteristic
428C22C2×#41012yes—
429C22C2×#41712yes—
430C22C2×#42112yes—
431C22#300311yescharacteristic
432C22#300411yescharacteristic
433C22#309212yes—
434C4—28no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—44no—
440C4—88no—
441C4—81no—
442C4—84no—
443C4—81no—
444C4—81no—
445C2—21no—
446C2—22no—
447C2—24no—
448C2—24no—
449C2—21no—
450C2—161no—
451C2#1758814yes—
452C2#2090911yescharacteristic
453C2C2×#189711yescharacteristic
454C2C2×#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×#382C2semidirect
3. #20682C2semidirect
5. #9156C2semidirect
6. #9152C2semidirect
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×#3822
#206822
C2×#15042
#91562
#91522
#80202