Finite Group Databasebeta

← #6489531

Subgroups — #6489531

Order 512.

4162 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic
2C2×#3229C212yesmaximal, complemented
3#36625C211yesmaximal, characteristic
4#32982C214yesmaximal
5#29641C214yesmaximal, complemented
6#25902C214yesmaximal, complemented
7#25811C214yesmaximal, complemented
8#24301C218yesmaximal, complemented
9#21833C212yesmaximal, complemented
10#20714C211yesmaximal, characteristic
11C24×Q8—21no—
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8—22no—
14C22×C4⋊Q8—21no—
15C22×C4⋊Q8—21no—
16C22×C4⋊Q8—22no—
17C22×C4⋊Q8—216no—
18C22×C4⋊Q8C2214yes—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2214yes—
21C22×C4⋊Q8C2212yes—
22C23×C4⋊C4—21no—
23C23×C4⋊C4C2212yes—
24C23×C42C2211yescharacteristic
25#3229—216no—
26C2×#498C2214yescomplemented
27C2×#487C2214yescomplemented
28C2×#483C2218yes—
29C2×#475C2218yescomplemented
30C2×#431C2212yes—
31#2857C2212yes—
32#2857C2212yes—
33#2855—21no—
34#2855C2212yes—
35#2851—22no—
36#2848—21no—
37#2848C2212yes—
38C4×C22⋊Q8—22no—
39C4×C22⋊Q8C2214yes—
40#2296—21no—
41#2285C2211yescharacteristic
42#2280—21no—
43#2058C22116yescomplemented
44#2024C22116yescomplemented
45#1990C2218yescomplemented
46#1988C2218yescomplemented
47#1872C22116yes—
48#1853C22116yes—
49#1823C2218yes—
50#1817C2212yes—
51#1797C2214yes—
52#1779C2212yes—
53C4×C22⋊C8C2214yes—
54C23×Q8—216no—
55C23×Q8—24no—
56C23×Q8—24no—
57C23×Q8—22no—
58C23×Q8—22no—
59C23×Q8—48no—
60C23×Q8—48no—
61C24×C4—22no—
62C24×C4D411yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4C2311yescharacteristic
65#533—28no—
66C2×C4⋊Q8—216no—
67C2×C4⋊Q8—216no—
68C2×C4⋊Q8—232no—
69C2×C4⋊Q8—232no—
70C2×C4⋊Q8—232no—
71C2×C4⋊Q8—232no—
72C2×C4⋊Q8—216no—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8D4116yes—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2318yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2314yes—
86C2×C4⋊Q8C2314yes—
87C2×#125—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2314yes—
92C2×C4×Q8—21no—
93C2×C4×Q8C2312yes—
94C2×#117—21no—
95C2×#117—21no—
96C22×C4⋊C4—216no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—216no—
104C22×C4⋊C4C2318yes—
105C22×C4⋊C4C2314yes—
106C22×C42—28no—
107C22×C42D412yes—
108C22×C42D411yescharacteristic
109C22×C42D411yescharacteristic
110C22×C42D412yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#498—216no—
116#487—216no—
117#483—232no—
118#475—232no—
119#431—28no—
120C2×#106C23116yes—
121C2×C4⋊C8C2314yes—
122C2×#103C23116yes—
123C2×C22⋊C8C2318yes—
124C2×C22⋊C8C2314yes—
125C2×C4×C8C2314yes—
126#391—22no—
127#391C2314yes—
128#389—22no—
129#389—22no—
130#389—22no—
131#389C2314yes—
132#387—24no—
133#387—24no—
134#385—24no—
135#385—22no—
136#385—24no—
137#385—22no—
138#384C2312yes—
139#382—21no—
140#381—22no—
141#380—21no—
142#380—22no—
143#380—22no—
144#380—22no—
145#380C2312yes—
146#380C2314yes—
147#379—22no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4C2312yes—
151C4×C22⋊C4—24no—
152#340C23132yes—
153#336C2318yes—
154C25C2×D411yescharacteristic
155C22×Q8—216no—
156C22×Q8—232no—
157C22×Q8—232no—
158C22×Q8—216no—
159C22×Q8—28no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—21no—
163C22×Q8—48no—
164C22×Q8—416no—
165C22×Q8—416no—
166C22×Q8—4128no—
167C22×Q8—416no—
168C22×Q8—416no—
169C22×Q8—48no—
170C22×Q8—48no—
171C22×Q8—48no—
172C22×Q8C2412yes—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—28no—
177C23×C4—24no—
178C23×C4—28no—
179C23×C4—216no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—48no—
183C23×C4—48no—
184C23×C4SD1618yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C22×C8C2418yes—
198C4⋊Q8—232no—
199C4⋊Q8—232no—
200C4⋊Q8—232no—
201C4⋊Q8—232no—
202C4⋊Q8—232no—
203C4⋊Q8—232no—
204C4⋊Q8—216no—
205C4⋊Q8—216no—
206C4⋊Q8—28no—
207C4⋊Q8—28no—
208C4⋊Q8—24no—
209C4⋊Q8—24no—
210C4⋊Q8C2×D4116yes—
211C4⋊Q8C2×D4116yes—
212#125—44no—
213C22⋊Q8—28no—
214C22⋊Q8—216no—
215C22⋊Q8—44no—
216C22⋊Q8—48no—
217C4×Q8—28no—
218C4×Q8—44no—
219#117—44no—
220#117—44no—
221C2×C4⋊C4—216no—
222C2×C4⋊C4—216no—
223C2×C4⋊C4—232no—
224C2×C4⋊C4—232no—
225C2×C4⋊C4—232no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—232no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4C2×D4116yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2418yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—28no—
256C2×C42—28no—
257C2×C42—28no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D418yes—
268C2×C42C2412yes—
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271#106—232no—
272C4⋊C8—28no—
273#103—232no—
274C22⋊C8—28no—
275C22⋊C8—216no—
276C4×C8—28no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—24no—
282#95—22no—
283C24—22no—
284C24—24no—
285C24—22no—
286C24—24no—
287C24C2×SD1614yes—
288C24C22×D411yescharacteristic
289C24C22×D412yes—
290C2×Q8—232no—
291C2×Q8—216no—
292C2×Q8—216no—
293C2×Q8—24no—
294C2×Q8—28no—
295C2×Q8—464no—
296C2×Q8—416no—
297C2×Q8—4128no—
298C2×Q8—4128no—
299C2×Q8—42no—
300C2×Q8—416no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—44no—
306C2×Q8—464no—
307C22×C4—232no—
308C22×C4—232no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—232no—
332C22×C4—232no—
333C22×C4—216no—
334C22×C4—28no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—464no—
339C22×C4—416no—
340C22×C4—416no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4#12014yes—
344C22×C4#12014yes—
345C22×C4C2×SD1614yes—
346C22×C4C2×SD1618yes—
347C22×C4C2×SD1618yes—
348C22×C4C2×SD1614yes—
349C22×C4AGL(1,ℤ/8)14yes—
350C22×C4#13718yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D414yes—
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C42−1+418yes—
361C22×C4C2511yescharacteristic
362C2×C8—28no—
363C2×C8—216no—
364C4⋊C4—232no—
365C4⋊C4—232no—
366C4⋊C4—216no—
367C4⋊C4—216no—
368C4⋊C4—216no—
369C4⋊C4—28no—
370C4⋊C4—24no—
371C4⋊C4—28no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—48no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—48no—
381C4⋊C4C22×D4116yes—
382C22⋊C4—44no—
383C22⋊C4—48no—
384C22⋊C4—44no—
385C22⋊C4—48no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—24no—
392C42—44no—
393C42—44no—
394C42#12014yes—
395C42#12014yes—
396C42C22×D414yes—
397C42C22×D414yes—
398C23—28no—
399C23—28no—
400C23—216no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—216no—
405C23—28no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23C2×#12012yes—
410C23#53614yes—
411C23C22×SD1614yes—
412C23C22×SD1612yes—
413C23C2×AGL(1,ℤ/8)12yes—
414C23C2×#13714yes—
415C23C23×D411yescharacteristic
416Q8—464no—
417Q8—432no—
418Q8—432no—
419Q8—48no—
420Q8—84no—
421C2×C4—216no—
422C2×C4—28no—
423C2×C4—216no—
424C2×C4—28no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—28no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—28no—
435C2×C4—216no—
436C2×C4—216no—
437C2×C4—216no—
438C2×C4—28no—
439C2×C4—216no—
440C2×C4—216no—
441C2×C4—464no—
442C2×C4—432no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4#450116yes—
453C2×C4C2×#12012yes—
454C2×C4C2×#12012yes—
455C2×C4C2×#12012yes—
456C2×C4C2×#12012yes—
457C2×C4C2×#12012yes—
458C2×C4#53614yes—
459C2×C4C22×SD1614yes—
460C2×C4C22×SD1612yes—
461C2×C4C22×SD1612yes—
462C2×C4C2×AGL(1,ℤ/8)12yes—
463C2×C4C2×AGL(1,ℤ/8)12yes—
464C2×C4C2×#13714yes—
465C2×C4C2×#13714yes—
466C2×C4C23×D412yes—
467C2×C4C2×2−1+414yes—
468C8—48no—
469C22—232no—
470C22—24no—
471C22—24no—
472C22—22no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—24no—
477C22—24no—
478C22—22no—
479C22—24no—
480C22C2×#45014yes—
481C22#322918yescomplemented
482C22C22×#12011yescharacteristic
483C22C2×#53612yes—
484C22C23×SD1611yescharacteristic
485C22C22×AGL(1,ℤ/8)11yescharacteristic
486C22C22×#13712yes—
487C4—22no—
488C4—24no—
489C4—24no—
490C4—22no—
491C4—24no—
492C4—416no—
493C4—82no—
494C4—84no—
495C4—82no—
496C4C2×#45014yes—
497C4C2×#45014yes—
498C4#300414yes—
499C4#348614yes—
500C2—28no—
501C2—24no—
502C2#3524211yescharacteristic
503C2C22×#45011yescharacteristic
504C2C2×#300411yescharacteristic
505C2C2×#322912yes—
506C2#6086012yes—
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
2. C2×#3229C2semidirect
5. #29641C2semidirect
6. #25902C2semidirect
7. #25811C2semidirect
8. #24301C2semidirect
9. #21833C2semidirect
26. C2×#498C22semidirect
27. C2×#487C22semidirect
29. C2×#475C22semidirect
43. #2058C22semidirect
44. #2024C22semidirect
45. #1990C22semidirect
46. #1988C22semidirect
481. C22#3229semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C2×#32292
#366252
#329822
#296412
#259022
#258112
#243012
#218332
#207142