Finite Group Databasebeta

← #6479537

Subgroups — #6479537

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic
3#33099C212yesmaximal, complemented
4#33098C212yesmaximal, complemented
5#29913C214yesmaximal, complemented
6#27866C218yesmaximal, complemented
7#27616C214yesmaximal, complemented
8#27589C214yesmaximal, complemented
9#21942C212yesmaximal
10#21765C212yesmaximal
11#20682C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C2×#407C2212yes—
15#2855—21no—
16#2855—21no—
17#2855—22no—
18#2855C2212yes—
19#2855C2212yes—
20#2855C2214yes—
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#387—22no—
24C2×#385—21no—
25#2283—22no—
26#2283—22no—
27#2283C2214yes—
28#2283C2214yes—
29C2×#383—21no—
30C2×#382C2212yes—
31C2×#381—21no—
32C2×#381—21no—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4—21no—
38C2×#376—21no—
39#2101C2218yescomplemented
40#2097C2218yescomplemented
41#2085C2218yescomplemented
42#2074C2218yescomplemented
43#1928C2218yescomplemented
44#1927C2218yescomplemented
45#1921C2218yescomplemented
46#1920C2218yescomplemented
47#1894C2218yes—
48#1872C2214yes—
49#1872C2214yes—
50#1853C2214yes—
51#1853C2214yes—
52#1852C2214yes—
53#1852C2214yes—
54#1823C2214yes—
55#1817C2212yes—
56#1817C2214yes—
57#1817C2212yes—
58#1809C2218yes—
59#1787C2214yes—
60#1728C2214yes—
61C23×Q8C2311yescharacteristic
62C24×C4D411yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4C2311yescharacteristic
65C22×M16C2314yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8C2312yes—
70C2×C22⋊Q8C2312yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C42—28no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D412yes—
86C22×C42D412yes—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2312yes—
90C22×C42C2312yes—
91C2×#107C2318yes—
92C2×#106C2318yes—
93C2×#103C2318yes—
94C2×#103C2318yes—
95#407—28no—
96C2×C22⋊C8C2314yes—
97C2×C22⋊C8C2314yes—
98#391—22no—
99#391—22no—
100#391C2314yes—
101#391C2314yes—
102#389—22no—
103#389—22no—
104#389C2314yes—
105#389C2314yes—
106#387—24no—
107#387—24no—
108#387—24no—
109#387—24no—
110#385—22no—
111#385—22no—
112#385—24no—
113#383—24no—
114#383—24no—
115#382—22no—
116#382—22no—
117#382C2314yes—
118#382C2314yes—
119#381—22no—
120#381—24no—
121#381—22no—
122#381—24no—
123#381—24no—
124#380—21no—
125#380—21no—
126#380—22no—
127#380—22no—
128#380C2312yes—
129#380C2314yes—
130#380C2314yes—
131#380C2312yes—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4C2314yes—
135C4×C4⋊C4C2314yes—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—24no—
138#376—24no—
139#376—24no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95—22no—
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2318yes—
154#336C2318yes—
155C25C2×D411yescharacteristic
156C22×Q8—21no—
157C22×Q8—24no—
158C22×Q8—44no—
159C22×Q8C2412yes—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—44no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4118yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C2×M16—28no—
187C2×M16—44no—
188C2×M16—44no—
189C22×C8C2418yes—
190C22⋊Q8—28no—
191C22⋊Q8—28no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4C2412yes—
209C2×C4⋊C4C2414yes—
210C2×C4⋊C4C2414yes—
211C2×C4⋊C4C2414yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—48no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C42—24no—
222C2×C42—28no—
223C2×C42—24no—
224C2×C42—28no—
225C2×C42—24no—
226C2×C42—28no—
227C2×C42—24no—
228C2×C42—28no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—44no—
232C2×C42—44no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2412yes—
240C2×C42C2412yes—
241#107—48no—
242#106—48no—
243#103—48no—
244#103—48no—
245C22⋊C8—28no—
246C22⋊C8—28no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263C24—22no—
264C24—24no—
265C24—22no—
266C24—42no—
267C24#12314yes—
268C24C22×D411yescharacteristic
269C24C22×D411yescharacteristic
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—28no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—88no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—416no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—416no—
310C22×C4—416no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4#12314yes—
325C22×C4#12314yes—
326C22×C4#12314yes—
327C22×C4#13714yes—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C2×#4114yes—
336C22×C4C2×#4114yes—
337C22×C4C2×#4114yes—
338C22×C4C2511yescharacteristic
339M16—88no—
340C2×C8—28no—
341C2×C8—28no—
342C2×C8—28no—
343C4⋊C4—24no—
344C4⋊C4—28no—
345C4⋊C4—28no—
346C4⋊C4—28no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C22⋊C4—48no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—48no—
360C22⋊C4—44no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—24no—
368C42—24no—
369C42—48no—
370C42—48no—
371C23—22no—
372C23—22no—
373C23—21no—
374C23—22no—
375C23—21no—
376C23—28no—
377C23—28no—
378C23—22no—
379C23—21no—
380C23—22no—
381C23—21no—
382C23—44no—
383C23—48no—
384C23—48no—
385C23C2×#12312yes—
386C23C2×#12312yes—
387C23C2×#12312yes—
388C23C2×#13712yes—
389C23#57514yes—
390C23#57512yes—
391C23C23×D411yescharacteristic
392Q8—48no—
393Q8—84no—
394C2×C4—28no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—88no—
436C2×C4—88no—
437C2×C4—88no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4C2×#12312yes—
441C2×C4C2×#12312yes—
442C2×C4C2×#12312yes—
443C2×C4C2×#13714yes—
444C2×C4C22×#4112yes—
445C2×C4#58514yes—
446C8—48no—
447C22—24no—
448C22—24no—
449C22—22no—
450C22—22no—
451C22—24no—
452C22—24no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—416no—
459C22—42no—
460C22—42no—
461C22#309314yes—
462C22#309514yes—
463C22C22×#12311yescharacteristic
464C22#344912yes—
465C22C22×#13711yescharacteristic
466C22C2×#57512yes—
467C22C2×#57511yescharacteristic
468C4—22no—
469C4—22no—
470C4—22no—
471C4—24no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—84no—
476C4—82no—
477C4—84no—
478C4—84no—
479C4—82no—
480C4#349114yes—
481C2—24no—
482C2—24no—
483C2—42no—
484C2#3509711yescharacteristic
485C2C2×#309311yescharacteristic
486C2C2×#309511yescharacteristic
487C2#6095012yes—
488C2#6096412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33099C2semidirect
4. #33098C2semidirect
5. #29913C2semidirect
6. #27866C2semidirect
7. #27616C2semidirect
8. #27589C2semidirect
39. #2101C22semidirect
40. #2097C22semidirect
41. #2085C22semidirect
42. #2074C22semidirect
43. #1928C22semidirect
44. #1927C22semidirect
45. #1921C22semidirect
46. #1920C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#22832
#330992
#330982
#299132
#278662
#276162
#275892
#219422
#217652
#206822