Finite Group Databasebeta

← #190540

Subgroups — #190540

Order 512.

1052 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2335C211yesmaximal, characteristic
2#21906C212yesmaximal, complemented
3#8704C212yesmaximal, complemented
4#8562C212yesmaximal, complemented
5#8561C212yesmaximal, complemented
6#8304C212yesmaximal
7#8153C212yesmaximal, complemented
8#8152C212yesmaximal, complemented
9C22×#123C2211yescharacteristic
10#2335—28no—
11C2×#390C411yescharacteristic
12C2×#389C411yescharacteristic
13C2×#388C411yescharacteristic
14C2×#388C411yescharacteristic
15C2×#388C411yescharacteristic
16C2×#388C2211yescharacteristic
17C2×#387C411yescharacteristic
18C2×#387C2211yescharacteristic
19C2×#386C411yescharacteristic
20C2×#385C411yescharacteristic
21C2×#380C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1924—28no—
26#1797C2214yes—
27#1779C2214yes—
28#1749C2214yes—
29C2×#324C2214yes—
30C2×#323C2214yes—
31#1280C2214yes—
32#1280C2214yes—
33C23×D4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2311yescharacteristic
36C2×#123—24no—
37C2×#123—24no—
38C2×#123—24no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2311yescharacteristic
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2311yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8C2318yes—
65C2×#97—24no—
66#390—24no—
67#389—24no—
68#388—24no—
69#388—24no—
70#388—24no—
71#388—24no—
72#387—24no—
73#387—24no—
74#386—24no—
75#385—24no—
76#380—24no—
77C4×C4⋊C4—24no—
78C4×C22⋊C4—24no—
79C4×C22⋊C4D418yes—
80C2×#95C2×C411yescharacteristic
81C2×#95C2×C411yescharacteristic
82C2×#95C2×C411yescharacteristic
83C2×#95C2×C411yescharacteristic
84C2×#95C2×C411yescharacteristic
85C2×#95C2×C411yescharacteristic
86C2×#95C2×C411yescharacteristic
87C2×#95C2×C411yescharacteristic
88C2×#95C2311yescharacteristic
89C2×#95C2311yescharacteristic
90#336—24no—
91#336—24no—
92#324—28no—
93#323—28no—
94C25C22×C411yescharacteristic
95C25C22×C411yescharacteristic
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4—81no—
100C22×D4—81no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—42no—
105C23×C4—81no—
106C23×C4—81no—
107C23×C4—81no—
108C23×C4—81no—
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D414yes—
122C23×C4C2411yescharacteristic
123C22×C8—28no—
124#123—232no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—81no—
137C2×C4⋊C4—81no—
138C2×C4⋊C4—81no—
139C2×C4⋊C4—81no—
140C2×C4⋊C4#4114yes—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—81no—
158C2×C22⋊C4—81no—
159C2×C22⋊C4—81no—
160C2×C22⋊C4—81no—
161C2×C22⋊C4C22⋊C414yes—
162C2×C22⋊C4C22⋊C414yes—
163C2×C22⋊C4#4114yes—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—42no—
168C2×C42—81no—
169C2×C42—81no—
170C2×C42—81no—
171C2×C42—81no—
172C2×C42C2×D414yes—
173C2×C42C2×D414yes—
174C4⋊C8—84no—
175C22⋊C8—28no—
176C22⋊C8—28no—
177C22⋊C8—84no—
178C22⋊C8—84no—
179#97—84no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95C22⋊C414yes—
189#95C22⋊C414yes—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—22no—
195C24—22no—
196C24—22no—
197C24—22no—
198C24—41no—
199C24—41no—
200C24—41no—
201C24—41no—
202C24—41no—
203C24—41no—
204C24—41no—
205C24—41no—
206C24C2×C22⋊C412yes—
207C24#11712yes—
208C24C23×C411yescharacteristic
209C2×D4—28no—
210C2×D4—82no—
211C2×D4—82no—
212C2×D4—84no—
213C2×D4—84no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
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—44no—
246C22×C4—44no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—44no—
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—82no—
265C22×C4—82no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—88no—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4#11712yes—
275C22×C4#11712yes—
276C22×C4#11712yes—
277C22×C4C22×D412yes—
278C22×C4C2×#4112yes—
279C22×C4C2×#4112yes—
280C2×C8—48no—
281C2×C8—48no—
282C2×C8—48no—
283C4⋊C4—28no—
284C4⋊C4—28no—
285C4⋊C4—44no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C22⋊C4—28no—
291C22⋊C4—28no—
292C22⋊C4—28no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—82no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C42—44no—
301C42—44no—
302C42—44no—
303C42—82no—
304C42—82no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—24no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—82no—
336C23—82no—
337C23—82no—
338C23—82no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—82no—
343C23C22×C22⋊C411yescharacteristic
344C23C2×#11711yescharacteristic
345C23C2×#11711yescharacteristic
346C23#56812yes—
347C23#56812yes—
348D4—88no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—88no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—88no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4#52514yes—
408C8—88no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—84no—
428C22—84no—
429C22—84no—
430C22—84no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22#228011yescharacteristic
436C22#287911yescharacteristic
437C22#287911yescharacteristic
438C22#288311yescharacteristic
439C22#288311yescharacteristic
440C22#288811yescharacteristic
441C22#288911yescharacteristic
442C22#292312yes—
443C22#292312yes—
444C4—24no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C2—22no—
458C2—22no—
459C2—82no—
460C2—82no—
461C2#1740511yescharacteristic
462C2#1740611yescharacteristic
463C2#1740711yescharacteristic
464C2#1740811yescharacteristic
465C2#2071911yescharacteristic
466C2#2072411yescharacteristic
467C2#2073111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21906C2semidirect
3. #8704C2semidirect
4. #8562C2semidirect
5. #8561C2semidirect
7. #8153C2semidirect
8. #8152C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23352
#219062
#87042
#85622
#85612
#83042
#81532
#81522