Finite Group Databasebeta

← #481687

Subgroups — #481687

Order 512.

1293 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35741C211yesmaximal, characteristic
2#20354C211yesmaximal, characteristic, complemented
3#17369C211yesmaximal, characteristic, complemented
4#11757C214yesmaximal, complemented
5#11753C214yesmaximal, complemented
6#11715C212yesmaximal, complemented
7#11693C212yesmaximal, complemented
8C2×#550C411yescharacteristic, complemented
9C2×#518—21no—
10C2×#518C2211yescharacteristic
11#2975—21no—
12#2635C412yescomplemented
13#2619C2211yescharacteristic
14#2580C411yescharacteristic, complemented
15#2541C411yescharacteristic, complemented
16#2539C2212yes—
17#2464—22no—
18#2445—21no—
19#2433—21no—
20#2424—22no—
21#2404C2212yes—
22#2396C2211yescharacteristic
23#2388C412yescomplemented
24#2366—21no—
25#2363—21no—
26#2357C411yescharacteristic, complemented
27#1740—21no—
28#1740C2212yes—
29C2×#342—22no—
30C2×#342C2214yes—
31#1465—22no—
32#1462C2214yes—
33#1460—21no—
34#1458C2212yes—
35#1352C2214yes—
36#1351C2214yes—
37#1343C2218yes—
38C22×#41C2×C412yescomplemented
39C23×D4D411yescharacteristic
40#550—44no—
41C2×#127—21no—
42C2×#127—21no—
43C2×#127C2×C411yescharacteristic, complemented
44C2×#123—22no—
45C2×#121—21no—
46C2×#121—22no—
47C2×#121—22no—
48C2×#121C2×C412yescomplemented
49C2×#120—22no—
50C2×#120—22no—
51#518—28no—
52#518—44no—
53#518—44no—
54C2×C4×Q8C2×C411yescharacteristic, complemented
55C2×C4×D4—22no—
56C2×C4×D4—21no—
57C2×C4×D4—44no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×#117—22no—
65C2×#117D411yescharacteristic
66C2×#117C2311yescharacteristic
67C22×C22⋊C4—24no—
68C22×C22⋊C4D412yes—
69C22×C22⋊C4C2312yes—
70#436—44no—
71#431—22no—
72#407—24no—
73#400C2×C411yescharacteristic, complemented
74#399C2×C412yescomplemented
75#399C2311yescharacteristic
76#398—21no—
77#396—21no—
78#395C2×C411yescharacteristic, complemented
79#394—21no—
80#394—22no—
81#394—22no—
82#394C2×C412yescomplemented
83#393C2×C412yescomplemented
84#393C2311yescharacteristic
85#392—22no—
86#392—22no—
87#392—21no—
88#392C2×C412yescomplemented
89#392C2×C412yescomplemented
90#392C2312yes—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—22no—
95#390—22no—
96#390—21no—
97#389—21no—
98#388—22no—
99#388—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#385—22no—
105#382C2×C411yescharacteristic, complemented
106#381C2×C412yescomplemented
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C411yescharacteristic, complemented
110C4×C22⋊C4—21no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4C2×C412yescomplemented
113#342—22no—
114#342—24no—
115#342C2314yes—
116#324—22no—
117#323—24no—
118#323C2314yes—
119C25C2×D412yes—
120C2×#41—44no—
121C2×#41—84no—
122C22×Q8C22×C411yescharacteristic
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C22⋊C412yes—
135C22×D4C22⋊C411yescharacteristic
136C22×D4C22⋊C411yescharacteristic
137C22×D4C22×C412yes—
138C23×C4—22no—
139C23×C4—44no—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C412yes—
142C23×C4C22×C412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C2×M16—44no—
146#127—42no—
147#127—44no—
148#127—44no—
149#123—84no—
150#121—48no—
151#121—44no—
152#121—42no—
153#121—42no—
154#121—84no—
155#120—84no—
156#120—84no—
157C4×Q8—42no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—42no—
161C4×D4—42no—
162C4×D4—44no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4—432no—
166C4×D4—84no—
167#117—28no—
168#117—42no—
169#117—42no—
170#117—44no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C22⋊C4—216no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22×C412yes—
204C2×C22⋊C4C2×D411yescharacteristic
205C2×C22⋊C4C2×D412yes—
206C2×C22⋊C4C2×D412yes—
207C2×C22⋊C4C2411yescharacteristic
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—42no—
212C2×C42C22⋊C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C4⋊C8—42no—
216C22⋊C8—24no—
217C22⋊C8—44no—
218C4×C8—42no—
219#95—21no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95C22×C411yescharacteristic
227#95C22×C412yes—
228#95C22×C411yescharacteristic
229C24—24no—
230C24—22no—
231C24—22no—
232C24—24no—
233C24—22no—
234C24—82no—
235C24—82no—
236C24C2×C22⋊C412yes—
237C24C2×C22⋊C412yes—
238C24C22×D411yescharacteristic
239#41—88no—
240#41—88no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×D4—216no—
244C2×D4—216no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4—82no—
275C2×D4#9914yes—
276C2×D4#10414yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—416no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—164no—
298C22×C4—164no—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4AGL(1,ℤ/8)12yes—
306C22×C4#13712yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—164no—
310C2×C8—44no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—48no—
321C4⋊C4—42no—
322C4⋊C4—82no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—28no—
326C22⋊C4—41no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—416no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—22no—
346C42—22no—
347C42—42no—
348C42—48no—
349C42—42no—
350C42—42no—
351C42—82no—
352C23—28no—
353C23—28no—
354C23—21no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—21no—
359C23—22no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—84no—
374C23—84no—
375C23—164no—
376C23—162no—
377C23—162no—
378C23C2×#9914yes—
379C23C2×#10414yes—
380C23C22×C22⋊C411yescharacteristic
381C23C2×AGL(1,ℤ/8)11yescharacteristic
382C23C2×#13711yescharacteristic
383Q8—42no—
384Q8—42no—
385D4—216no—
386D4—28no—
387D4—28no—
388D4—88no—
389D4—88no—
390D4—84no—
391D4—84no—
392D4—162no—
393D4—164no—
394D4—162no—
395D4—162no—
396D4—164no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—88no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—162no—
435C2×C4—162no—
436C2×C4—1616no—
437C2×C4C2×#9912yes—
438C2×C4#41712yes—
439C2×C4#41712yes—
440C2×C4C2×#10412yes—
441C8—84no—
442C22—21no—
443C22—21no—
444C22—24no—
445C22—24no—
446C22—21no—
447C22—21no—
448C22—24no—
449C22—24no—
450C22—44no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—41no—
456C22—42no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—81no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—88no—
466C22—88no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22C22×#9911yescharacteristic
471C22C2×#41711yescharacteristic
472C22C22×#10411yescharacteristic
473C4—22no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—41no—
478C4—41no—
479C4—41no—
480C4—42no—
481C4—41no—
482C4—82no—
483C4—81no—
484C4—88no—
485C4—81no—
486C4#150414yes—
487C2—21no—
488C2—21no—
489C2—41no—
490C2—44no—
491C2—44no—
492C2—41no—
493C2—162no—
494C2C2×#150411yescharacteristic
495C2#3334811yescharacteristic
496C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35741C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20354C2semidirect
3. #17369C2semidirect
4. #11757C2semidirect
5. #11753C2semidirect
6. #11715C2semidirect
7. #11693C2semidirect
8. C2×#550C4semidirect
12. #2635C4semidirect
14. #2580C4semidirect
15. #2541C4semidirect
23. #2388C4semidirect
26. #2357C4semidirect
38. C22×#41C2×C4semidirect
43. C2×#127C2×C4semidirect
48. C2×#121C2×C4semidirect
54. C2×C4×Q8C2×C4semidirect
73. #400C2×C4semidirect
74. #399C2×C4semidirect
78. #395C2×C4semidirect
82. #394C2×C4semidirect
83. #393C2×C4semidirect
88. #392C2×C4semidirect
89. #392C2×C4semidirect
105. #382C2×C4semidirect
106. #381C2×C4semidirect
109. C4×C4⋊C4C2×C4semidirect
112. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#357412
#203542
#173692
#117572
#117532
#117152
#116932