Finite Group Databasebeta

← #215586

Subgroups — #215586

Order 512.

1120 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35285C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#9710C211yesmaximal, characteristic, complemented
5#9707C212yesmaximal, complemented
6#9707C211yesmaximal, characteristic, complemented
7#9704C212yesmaximal, complemented
8#9699C211yesmaximal, characteristic, complemented
9#9698C211yesmaximal, characteristic, complemented
10#8343C212yesmaximal, complemented
11#8339C212yesmaximal
12C2×#533C2211yescharacteristic
13C22×#127C2211yescharacteristic
14C2×#433—21no—
15#2858C2211yescharacteristic
16#2858C2211yescharacteristic
17#2858C2211yescharacteristic
18#2858C2211yescharacteristic
19#2641—21no—
20#2641—21no—
21#2548—22no—
22#2548C414yes—
23#2435—22no—
24#2435C414yes—
25#2309—21no—
26#2309—21no—
27C4×#117C2211yescharacteristic
28#1924—22no—
29#1917—22no—
30#1842C2212yescomplemented
31#1842C2211yescharacteristic, complemented
32#1832—21no—
33#1831C2211yescharacteristic, complemented
34#1798—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1768—21no—
38#1767C2211yescharacteristic, complemented
39#1767C2212yescomplemented
40C4×C4⋊C8C2211yescharacteristic, complemented
41C2×#333C2212yescomplemented
42C2×#333C2212yescomplemented
43C2×#332C2212yescomplemented
44C2×#332C2212yescomplemented
45C2×#331C2212yes—
46C2×#331C2212yes—
47C2×#331C2212yes—
48C2×#331C2212yes—
49C23×D4—21no—
50C23×D4—21no—
51C23×D4C2×C412yes—
52#533—42no—
53#533—42no—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×#127—21no—
57C2×#127—21no—
58C2×#127—42no—
59C2×#127—42no—
60C2×#127C2×C412yes—
61C2×#127C2311yescharacteristic
62C2×#127C2311yescharacteristic
63C2×#125—21no—
64C2×#124—22no—
65C2×#124—22no—
66C2×#124C2×C414yes—
67C2×#124C2×C414yes—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8C2×C412yes—
70C2×C4×Q8—21no—
71C2×C4×Q8C2×C412yes—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C42C2311yescharacteristic
76#433—44no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×#97—21no—
87C2×#97—22no—
88C2×C4×C8—21no—
89#394—24no—
90#394—24no—
91#394—24no—
92#394—24no—
93#390—24no—
94#390—24no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390—21no—
103#390—22no—
104#390—22no—
105#390C2×C414yes—
106#390C2×C414yes—
107#379—24no—
108#379—24no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4D412yes—
113#376D411yescharacteristic
114#376D411yescharacteristic
115C43C2311yescharacteristic
116C43C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—21no—
120#336—21no—
121#336—22no—
122#336—22no—
123#333—42no—
124#333—42no—
125#332—42no—
126#332—42no—
127#331—42no—
128#331—42no—
129#331—42no—
130#331—42no—
131C25—22no—
132C22×Q8—21no—
133C22×Q8C22×C412yes—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—44no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—82no—
149C22×D4—82no—
150C22×D4—84no—
151C22×D4C22×C412yes—
152C22×D4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164#127—22no—
165#127—22no—
166#127—22no—
167#127—22no—
168#127—44no—
169#127—42no—
170#127—42no—
171#127—44no—
172#125—44no—
173#124—82no—
174#124—82no—
175#124—82no—
176#124—82no—
177#124—84no—
178#124—84no—
179C22⋊Q8—44no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C4×Q8—44no—
183C4×Q8—42no—
184C4×Q8—42no—
185#117—22no—
186#117—22no—
187#117—42no—
188#117—44no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2411yescharacteristic
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—44no—
232C4⋊C8—44no—
233C4⋊C8—82no—
234C22⋊C8—44no—
235C22⋊C8—82no—
236#97—42no—
237#97—82no—
238C4×C8—42no—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243C24—22no—
244C24—22no—
245C24—44no—
246C24—44no—
247C24—44no—
248C24—84no—
249C24—84no—
250C24C2×C22⋊C411yescharacteristic
251C2×Q8—24no—
252C2×Q8—44no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—44no—
282C2×D4—84no—
283C2×D4—84no—
284C2×D4—88no—
285C2×D4—88no—
286C2×D4—816no—
287C2×D4—84no—
288C2×D4—84no—
289C2×D4—88no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4AGL(1,ℤ/8)12yes—
308C22×C4#13712yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C22⋊C4—24no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C22⋊C4—84no—
339C22⋊C4—84no—
340C22⋊C4—84no—
341C22⋊C4—84no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C42—22no—
345C42—21no—
346C42—21no—
347C42—22no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—42no—
359C42—42no—
360C42—44no—
361C42H(5,𝔽2)12yes—
362C42H(5,𝔽2)12yes—
363C23—21no—
364C23—22no—
365C23—21no—
366C23—22no—
367C23—42no—
368C23—42no—
369C23—44no—
370C23—44no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—44no—
377C23—88no—
378C23—88no—
379C23—88no—
380C23—168no—
381C23—168no—
382C23#41712yes—
383C23C22×C22⋊C411yescharacteristic
384Q8—84no—
385Q8—81no—
386Q8—81no—
387Q8—82no—
388D4—84no—
389D4—84no—
390D4—88no—
391D4—88no—
392D4—88no—
393D4—82no—
394D4—82no—
395D4—82no—
396D4—82no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—82no—
402D4—82no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4#41712yes—
433C2×C4#51812yes—
434C2×C4#53512yes—
435C2×C4#53511yescharacteristic
436C2×C4#53511yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4#57512yes—
440C2×C4#57612yes—
441C2×C4#57712yes—
442C8—82no—
443C8—82no—
444C8—82no—
445C8—82no—
446C22—22no—
447C22—24no—
448C22—41no—
449C22—41no—
450C22—42no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—84no—
458C22—84no—
459C22—84no—
460C22—84no—
461C22—1616no—
462C22#230911yescharacteristic
463C22#288812yes—
464C22C2×#41711yescharacteristic
465C22#289611yescharacteristic
466C22#289712yes—
467C4—22no—
468C4—22no—
469C4—41no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—82no—
477C4#321312yes—
478C4#321512yes—
479C4#322011yescharacteristic
480C4#322211yescharacteristic
481C4#322412yes—
482C2—42no—
483C2—162no—
484C2—162no—
485C2#1776212yes—
486C2#1776612yes—
487C2#2180111yescharacteristic
488C2#2180811yescharacteristic
489C2#2181311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21926C2semidirect
3. #20812C2semidirect
4. #9710C2semidirect
5. #9707C2semidirect
6. #9707C2semidirect
7. #9704C2semidirect
8. #9699C2semidirect
9. #9698C2semidirect
10. #8343C2semidirect
30. #1842C22semidirect
31. #1842C22semidirect
33. #1831C22semidirect
38. #1767C22semidirect
39. #1767C22semidirect
40. C4×C4⋊C8C22semidirect
41. C2×#333C22semidirect
42. C2×#333C22semidirect
43. C2×#332C22semidirect
44. C2×#332C22semidirect

Maximal subgroups

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

subgroupindex
#352852
#219262
#208122
#97102
#97072
#97042
#96992
#96982
#83432
#83392