Finite Group Databasebeta

← #6477707

Subgroups — #6477707

Order 512.

2080 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2C2×#2920C211yesmaximal, characteristic
3C2×#2855C212yesmaximal
4#33144C214yesmaximal, complemented
5#29783C214yesmaximal, complemented
6#24609C214yesmaximal, complemented
7#24601C214yesmaximal, complemented
8#24508C214yesmaximal, complemented
9#24502C214yesmaximal, complemented
10#21942C211yesmaximal, characteristic
11#21765C212yesmaximal
12C22×C22⋊Q8C2212yes—
13C23×C42C2211yescharacteristic
14#3195—28no—
15C2×#483C2214yes—
16C2×#470C2214yes—
17C2×#451C2214yes—
18#2920—28no—
19C2×#432C2214yes—
20C2×#431C2214yes—
21C2×#423C2212yes—
22C2×#407C2211yescharacteristic
23C2×#407C2211yescharacteristic
24C2×C4×M16C2212yes—
25#2855—22no—
26#2855—22no—
27#2855—22no—
28#2855—22no—
29#2855C2214yes—
30#2855C2214yes—
31#2855C2214yes—
32#2855C2214yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#387—22no—
38C2×#387—22no—
39C2×#385—22no—
40C2×#381—22no—
41C2×#380C2212yes—
42#2088C2218yescomplemented
43#2068C2218yescomplemented
44#2066C2218yescomplemented
45#2056C2218yescomplemented
46#1971C2214yescomplemented
47#1969C2214yescomplemented
48#1943C2214yescomplemented
49#1941C2214yescomplemented
50#1935C2218yescomplemented
51#1933C2218yescomplemented
52#1894C2218yes—
53#1872C2214yes—
54#1872C2214yes—
55#1853C2214yes—
56#1852C2214yes—
57#1823C2214yes—
58#1809C2214yes—
59#1787C2214yes—
60C23×Q8C2312yes—
61C24×C4D412yes—
62C24×C4C2311yescharacteristic
63C22×Q16C2314yes—
64C22×M16C2312yes—
65C22×M16C2312yes—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8C2314yes—
71C2×C22⋊Q8C2314yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C42—24no—
81C22×C42—24no—
82C22×C42D412yes—
83C22×C42D412yes—
84C22×C42D412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2312yes—
88C22×C42C2312yes—
89#483—216no—
90#470—216no—
91#451—216no—
92#432—216no—
93#431—216no—
94C2×#107C2314yes—
95C2×#106C2314yes—
96C2×#106C2314yes—
97#423—28no—
98C2×C4⋊C8C2318yes—
99C2×#103C2318yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102#407—24no—
103#407D418yes—
104C2×C22⋊C8C2314yes—
105C2×C22⋊C8C2314yes—
106C4×M16—28no—
107C2×#97C2314yes—
108C2×C4×C8C2314yes—
109#391—22no—
110#391—22no—
111#391—22no—
112#391—22no—
113#391C2314yes—
114#391C2314yes—
115#391C2314yes—
116#391C2314yes—
117#389—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389C2314yes—
122#389C2314yes—
123#389C2314yes—
124#389C2314yes—
125#387—24no—
126#387—24no—
127#387—24no—
128#387—24no—
129#387—24no—
130#387—24no—
131#387—24no—
132#387—24no—
133#385—24no—
134#385—24no—
135#385—24no—
136#385—24no—
137#381—24no—
138#381—24no—
139#381—24no—
140#381—24no—
141#380—22no—
142#380—22no—
143#380C2314yes—
144#380C2314yes—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155C25C2×D411yescharacteristic
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—44no—
160C22×Q8—44no—
161C22×Q8C2414yes—
162C23×C4—28no—
163C23×C4—28no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D414yes—
180C23×C4#4114yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C2×Q16—28no—
184C2×Q16—44no—
185C2×Q16—44no—
186C2×M16—24no—
187C2×M16—42no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16C2×D418yes—
192C22×C8C2414yes—
193C22×C8C2414yes—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196C22⋊Q8—28no—
197C22⋊Q8—28no—
198C22⋊Q8—48no—
199C22⋊Q8—48no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C4⋊C4C2414yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—48no—
224C2×C22⋊C4—48no—
225C2×C42—28no—
226C2×C42—28no—
227C2×C42—28no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2411yescharacteristic
244C2×C42C2411yescharacteristic
245C2×C42C2412yes—
246#107—44no—
247#106—28no—
248#106—44no—
249C4⋊C8—216no—
250#103—216no—
251#103—48no—
252#103—48no—
253C22⋊C8—28no—
254C22⋊C8C2×D4116yes—
255#97—28no—
256C4×C8—28no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271C24—24no—
272C24—24no—
273C24—41no—
274C24—41no—
275C24#12114yes—
276C24C22×D411yescharacteristic
277C24C22×D412yes—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—28no—
281C2×Q8—28no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×Q8—88no—
286C2×Q8—88no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—432no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4#12118yes—
332C22×C4#12114yes—
333C22×C4#13718yes—
334C22×C4C22×D412yes—
335C22×C4C22×D412yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D412yes—
338C22×C4C22×D414yes—
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C2×#4114yes—
343C22×C4C2×#4112yes—
344C22×C4C2511yescharacteristic
345Q16—88no—
346M16—48no—
347M16—48no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8C22×D418yes—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C42—22no—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—22no—
377C42—24no—
378C42—21no—
379C42—21no—
380C42—22no—
381C42—21no—
382C42—22no—
383C42—21no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—21no—
389C23—21no—
390C23—216no—
391C23—22no—
392C23—22no—
393C23—44no—
394C23—48no—
395C23—44no—
396C23—44no—
397C23C2×#12112yes—
398C23C2×#12114yes—
399C23C2×#13714yes—
400C23#57514yes—
401C23C23×D411yescharacteristic
402Q8—48no—
403Q8—48no—
404Q8—88no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—216no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—48no—
425C2×C4—44no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—48no—
429C2×C4—48no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4—88no—
446C2×C4—88no—
447C2×C4C2×#12112yes—
448C2×C4C2×#12114yes—
449C2×C4C2×#13714yes—
450C2×C4C2×#13714yes—
451C2×C4C23×D411yescharacteristic
452C2×C4C22×#4111yescharacteristic
453C2×C4#58512yes—
454C2×C4#58512yes—
455C8—28no—
456C8—44no—
457C22—24no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—24no—
463C22—22no—
464C22—24no—
465C22—416no—
466C22—42no—
467C22—41no—
468C22—41no—
469C22#305618yes—
470C22C22×#12111yescharacteristic
471C22#344911yescharacteristic
472C22#344911yescharacteristic
473C22C22×#13712yes—
474C22C2×#57512yes—
475C4—22no—
476C4—24no—
477C4—24no—
478C4—24no—
479C4—42no—
480C4—42no—
481C4—84no—
482C4—84no—
483C4—84no—
484C4—84no—
485C4#348512yes—
486C4#349112yes—
487C2—28no—
488C2—42no—
489C2#3508911yescharacteristic
490C2C2×#305612yes—
491C2#6095012yes—
492C2#6096411yescharacteristic
493C2#6097111yescharacteristic
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
4. #33144C2semidirect
5. #29783C2semidirect
6. #24609C2semidirect
7. #24601C2semidirect
8. #24508C2semidirect
9. #24502C2semidirect
42. #2088C22semidirect
43. #2068C22semidirect
44. #2066C22semidirect
45. #2056C22semidirect
46. #1971C22semidirect
47. #1969C22semidirect
48. #1943C22semidirect
49. #1941C22semidirect
50. #1935C22semidirect
51. #1933C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C2×#29202
C2×#28552
#331442
#297832
#246092
#246012
#245082
#245022
#219422
#217652