Finite Group Databasebeta

← #192146

Subgroups — #192146

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34178C211yesmaximal, characteristic
2#21944C212yesmaximal
3C2×#1463C211yesmaximal, characteristic
4#14339C211yesmaximal, characteristic
5#8857C212yesmaximal, complemented
6#8856C212yesmaximal, complemented
7#8855C214yesmaximal, complemented
8#8016C212yesmaximal
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2345—24no—
12#2344—24no—
13#2316C2211yescharacteristic
14#2316C2211yescharacteristic
15#2313—21no—
16#2313C2212yes—
17#2280—21no—
18#2280—21no—
19#2280C2212yes—
20C2×#381—21no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#2274—21no—
24#1854—24no—
25#1853C2214yes—
26#1852C2214yes—
27#1809—24no—
28#1463C418yescomplemented
29C2×#324C2212yes—
30C2×#324C2212yes—
31#1441—21no—
32#1441—21no—
33#1441C2212yes—
34#1441C2212yes—
35C22⋊#95—21no—
36#1294C2214yes—
37#1294C2214yes—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42C2×#117—21no—
43C2×#117—21no—
44C2×#117—22no—
45C2×#117D414yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2×C414yes—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4Q814yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C2×#107—24no—
60C2×#106—24no—
61#407—24no—
62#407C2×C418yes—
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2314yes—
65#389—24no—
66#389—24no—
67#387—24no—
68#387—24no—
69#387—24no—
70#387—24no—
71#381—22no—
72#381—22no—
73#381—22no—
74#381—44no—
75#380—22no—
76#380—22no—
77#380—22no—
78#380—21no—
79#380—21no—
80#380—24no—
81#380—22no—
82#380—22no—
83#380D414yes—
84#380C2312yes—
85#380C2312yes—
86#379—22no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—24no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95C2312yes—
101#340—24no—
102#340—24no—
103#324C2×C418yes—
104#324C2×C418yes—
105C25C2×D411yescharacteristic
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—41no—
117C23×C4—41no—
118C23×C4—82no—
119C23×C4—82no—
120C23×C4C4⋊C414yes—
121C23×C4C4⋊C414yes—
122C23×C4C22×C412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×Q812yes—
131C23×C4#4114yes—
132C23×C4C2411yescharacteristic
133C2×M16—28no—
134C2×M16—84no—
135C22×C8—24no—
136#117—42no—
137#117—42no—
138#117—42no—
139#117—42no—
140#117—84no—
141#117—84no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C4⋊C414yes—
163C2×C4⋊C4C4⋊C414yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C2×D414yes—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×Q812yes—
169C2×C4⋊C4C2×Q812yes—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—42no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—24no—
184C2×C42—24no—
185#107—84no—
186#106—84no—
187C22⋊C8—28no—
188C22⋊C8—24no—
189C22⋊C8C22×C418yes—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—22no—
197#95—22no—
198#95—24no—
199#95—44no—
200#95—42no—
201#95—44no—
202#95—82no—
203#95—82no—
204C24—21no—
205C24—21no—
206C24—21no—
207C24—21no—
208C24—42no—
209C24—42no—
210C24C2×C4⋊C412yes—
211C24#12011yescharacteristic
212C24#12011yescharacteristic
213C24#12312yes—
214C24C22×D411yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—28no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—82no—
269C22×C4—84no—
270C22×C4C2×C4⋊C412yes—
271C22×C4C2×C4⋊C412yes—
272C22×C4C2×C4⋊C412yes—
273C22×C4C2×C4⋊C412yes—
274C22×C4C2×C4⋊C412yes—
275C22×C4C4×D414yes—
276C22×C4C4×D414yes—
277C22×C4#12012yes—
278C22×C4C22⋊Q814yes—
279C22×C4C22⋊Q814yes—
280C22×C4#12312yes—
281C22×C4AGL(1,ℤ/8)12yes—
282C22×C4#13712yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D412yes—
285C22×C4C22×Q811yescharacteristic
286C22×C4C2×#4112yes—
287M16—88no—
288C2×C8—28no—
289C2×C8—44no—
290C2×C8—44no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—84no—
306C22⋊C4—48no—
307C22⋊C4—48no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—84no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C42—44no—
316C42—44no—
317C42—84no—
318C42—84no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—44no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23#38512yes—
342C23#38512yes—
343C23C22×C4⋊C411yescharacteristic
344C23C2×#12011yescharacteristic
345C23C2×#12311yescharacteristic
346C23C2×AGL(1,ℤ/8)11yescharacteristic
347C23C2×#13711yescharacteristic
348C23#57511yescharacteristic
349C23#57511yescharacteristic
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#38514yes—
407C2×C4#42814yes—
408C2×C4C2×C4×D412yes—
409C2×C4C2×C22⋊Q812yes—
410C8—48no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22C2×#38511yescharacteristic
434C22C2×#42811yescharacteristic
435C22#291111yescharacteristic
436C22#300611yescharacteristic
437C22#300711yescharacteristic
438C22#302012yes—
439C22#302612yes—
440C22#309411yescharacteristic
441C22#309511yescharacteristic
442C4—24no—
443C4—22no—
444C4—22no—
445C4—24no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—84no—
450C4—84no—
451C4—162no—
452C4—162no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—42no—
458C2—42no—
459C2#1834812yes—
460C2#1834912yes—
461C2#2084611yescharacteristic
462C2#2085711yescharacteristic
463C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8857C2semidirect
6. #8856C2semidirect
7. #8855C2semidirect
28. #1463C4semidirect

Maximal subgroups

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

subgroupindex
#341782
#219442
C2×#14632
#143392
#88572
#88562
#88552
#80162