Finite Group Databasebeta

← #241157

Subgroups — #241157

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34642C211yesmaximal, characteristic
2#21986C212yesmaximal
3#13963C211yesmaximal, characteristic
4#10570C212yesmaximal
5#10569C212yesmaximal
6#10568C212yesmaximal
7#10567C212yesmaximal
8#8022C212yesmaximal
9#7910C211yesmaximal, characteristic
10C2×#533—21no—
11C2×#432C2212yes—
12#2660—22no—
13#2660—22no—
14#2659—21no—
15#2659—21no—
16#2659C2212yes—
17#2659C2212yes—
18#2399—24no—
19#2285—21no—
20#2285C2212yes—
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1949—24no—
25#1853C2214yes—
26#1852C2214yes—
27#1847—22no—
28#1836—22no—
29#1444C2211yescharacteristic
30#1444C2211yescharacteristic
31#1442C2211yescharacteristic
32#1442C2211yescharacteristic
33#1436—21no—
34#1436—21no—
35C4⋊#95—21no—
36#1434—21no—
37#1294C2214yes—
38#1294C2214yes—
39#1280C2212yes—
40#1280C2212yes—
41#533—44no—
42#533—44no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×#125—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×#117—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C42—21no—
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59#432—24no—
60#432Q818yescomplemented
61C2×#107—24no—
62C2×#106—24no—
63C2×C4⋊C8—22no—
64C2×C22⋊C8C2314yes—
65C2×#97—22no—
66#398—24no—
67#398—24no—
68#398—24no—
69#398—24no—
70#395—24no—
71#395—24no—
72#395—24no—
73#395—24no—
74#384—21no—
75#384—21no—
76#384D412yes—
77#384D412yes—
78#382—22no—
79#382—22no—
80#382—22no—
81#382—22no—
82#380—22no—
83#380—22no—
84#380—22no—
85#380C2314yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D414yes—
90#376—22no—
91#376—22no—
92#376—22no—
93#376Q814yescomplemented
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107C2×#95C2311yescharacteristic
108C2×#95C2311yescharacteristic
109#340—24no—
110#340—24no—
111C22×Q8—22no—
112C22×Q8—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—24no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127#125—48no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C4×Q8—48no—
133C4×Q8—48no—
134#117—44no—
135#117—44no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—81no—
155C2×C4⋊C4—81no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4—81no—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D414yes—
160C2×C22⋊C4—24no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×Q814yes—
176C2×C42C2×Q812yes—
177C2×C42#4112yes—
178#107—84no—
179#106—84no—
180C4⋊C8—28no—
181C22⋊C8—24no—
182C22⋊C8C2×Q818yes—
183#97—28no—
184#95—24no—
185#95—24no—
186#95—22no—
187#95—22no—
188#95—24no—
189#95—24no—
190#95—22no—
191#95—22no—
192#95—24no—
193#95—21no—
194#95—21no—
195#95—42no—
196#95—42no—
197#95—42no—
198#95—41no—
199#95—41no—
200#95—42no—
201#95—42no—
202#95—81no—
203#95—81no—
204#95C2×Q812yes—
205#95C2×Q812yes—
206C24C22×D411yescharacteristic
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8—82no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4#12012yes—
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4C22⋊Q812yes—
271C22×C4C22⋊Q812yes—
272C22×C4C4⋊Q812yes—
273C22×C4C4⋊Q812yes—
274C22×C4AGL(1,ℤ/8)12yes—
275C22×C4AGL(1,ℤ/8)12yes—
276C22×C4#13712yes—
277C22×C4#13712yes—
278C22×C4C22×D412yes—
279C22×C4C22×Q812yes—
280C22×C4C22×Q811yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C2×C8—28no—
283C2×C8—44no—
284C2×C8—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C42—24no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—48no—
310C42—48no—
311C42C22⋊Q814yes—
312C42C22⋊Q814yes—
313C42C4⋊Q814yes—
314C42C4⋊Q814yes—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23C2×#12011yescharacteristic
320C23C2×C22⋊Q811yescharacteristic
321C23C2×C4⋊Q811yescharacteristic
322C23C2×AGL(1,ℤ/8)11yescharacteristic
323C23C2×AGL(1,ℤ/8)11yescharacteristic
324C23C2×#13711yescharacteristic
325C23C2×#13711yescharacteristic
326Q8—88no—
327Q8—88no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4#39512yes—
400C2×C4#39512yes—
401C2×C4#39514yes—
402C2×C4C2×C22⋊Q812yes—
403C2×C4C2×C4⋊Q812yes—
404C8—48no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22C2×#39511yescharacteristic
418C22#300311yescharacteristic
419C22#300411yescharacteristic
420C22#301811yescharacteristic
421C22#301811yescharacteristic
422C22#302711yescharacteristic
423C22#302811yescharacteristic
424C22#307911yescharacteristic
425C22#307911yescharacteristic
426C22#316511yescharacteristic
427C22#316511yescharacteristic
428C4—22no—
429C4—22no—
430C4—24no—
431C4—22no—
432C4—22no—
433C4—44no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—82no—
439C4—84no—
440C4—84no—
441C4—161no—
442C4—161no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—21no—
447C2#1833211yescharacteristic
448C2#1833311yescharacteristic
449C2#1833911yescharacteristic
450C2#1834011yescharacteristic
451C2#2355211yescharacteristic
452C2#2356011yescharacteristic
453C2#2356611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
60. #432Q8semidirect
93. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#346422
#219862
#139632
#105702
#105692
#105682
#105672
#80222
#79102