Finite Group Databasebeta

← #6508612

Subgroups — #6508612

Order 512.

3236 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C22×#484C212yesmaximal, complemented
3C2×#2487C211yesmaximal, characteristic
4#24098C218yesmaximal, complemented
5#24064C214yesmaximal, complemented
6#24042C218yesmaximal, complemented
7#24006C214yesmaximal, complemented
8#21759C212yesmaximal, complemented
9#21738C211yesmaximal, characteristic
10#3501—216no—
11C2×#554—24no—
12C2×#554C2214yes—
13C2×D4×Q8C2214yes—
14C2×#544—21no—
15C2×#543—21no—
16C2×#543—21no—
17C2×#533C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C2×#518—21no—
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C2×#484—216no—
24C2×#484C2218yes—
25C2×#484C2218yescomplemented
26C22×#107C2214yescomplemented
27C22×#103C2214yescomplemented
28C22×C22⋊C8C2212yes—
29#2487—24no—
30#2487C2218yes—
31C2×#400—21no—
32C2×#399—21no—
33C2×#389C2212yes—
34C2×#386—21no—
35C2×#385—21no—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38#2057C2218yescomplemented
39#2043C2218yescomplemented
40#2036C22116yescomplemented
41#1892C2218yescomplemented
42#1872C2218yes—
43#1869C2218yes—
44#1863C22116yescomplemented
45#1852C2218yescomplemented
46#1824C2214yescomplemented
47#1817C2212yes—
48#1787C2218yescomplemented
49#1787C2214yescomplemented
50#1779C2212yes—
51C23×Q8C2312yes—
52C23×Q8C2312yes—
53C23×D4D411yescharacteristic
54C24×C4—21no—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C23×C8C2314yes—
58#554—264no—
59#554—216no—
60#554—216no—
61D4×Q8—216no—
62D4×Q8—216no—
63#544—216no—
64#543—216no—
65#543—216no—
66#533—216no—
67C2×C4⋊Q8—24no—
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2312yes—
70C2×#125—24no—
71C2×#125—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—216no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8D4116yes—
84C2×C22⋊Q8C2314yes—
85C2×C22⋊Q8C2314yes—
86C2×C22⋊Q8C2314yes—
87#518—216no—
88C2×C4×Q8—22no—
89C2×C4×Q8C2314yes—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—28no—
93C2×C4×D4D414yes—
94C2×C4×D4C2314yes—
95C2×#117—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4D411yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4—22no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—22no—
112C22×C22⋊C4D411yescharacteristic
113C22×C22⋊C4D411yescharacteristic
114C22×C42C2311yescharacteristic
115#484—232no—
116#484—232no—
117C2×#107—216no—
118C2×#107C23116yes—
119C2×#103—216no—
120C2×#103C23116yes—
121C2×#103C23116yes—
122C2×C22⋊C8—28no—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2318yes—
126#400—28no—
127#399—28no—
128#389—24no—
129#389C2318yes—
130#386—28no—
131#385—28no—
132#380—24no—
133#380—22no—
134#380C2314yes—
135#380C2318yes—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139#340C23116yes—
140#336C2318yes—
141C25C2×D411yescharacteristic
142C25C2×D411yescharacteristic
143C22×Q8—28no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—48no—
149C22×Q8—48no—
150C22×Q8C2×D418yes—
151C22×Q8C2414yes—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4Q1618yes—
155C22×D4C2×D414yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—84no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D418yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2412yes—
178C22×C8—28no—
179C22×C8—48no—
180C22×C8C2418yes—
181C4⋊Q8—232no—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184#125—216no—
185#125—232no—
186C22⋊Q8—216no—
187C22⋊Q8—216no—
188C22⋊Q8—216no—
189C22⋊Q8—232no—
190C22⋊Q8—232no—
191C22⋊Q8—232no—
192C22⋊Q8—232no—
193C22⋊Q8—28no—
194C22⋊Q8—216no—
195C22⋊Q8—216no—
196C22⋊Q8—28no—
197C22⋊Q8—28no—
198C22⋊Q8—28no—
199C22⋊Q8C2×D4116yes—
200C22⋊Q8C2×D4116yes—
201C4×Q8—216no—
202C4×Q8—28no—
203C4×Q8—28no—
204C4×D4—232no—
205C4×D4—232no—
206C4×D4—216no—
207C4×D4—28no—
208C4×D4—28no—
209#117—216no—
210C2×C4⋊C4—216no—
211C2×C4⋊C4—216no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—84no—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D418yes—
237C2×C4⋊C4C2414yes—
238C2×C4⋊C4C2414yes—
239C2×C4⋊C4C2414yes—
240C2×C22⋊C4—216no—
241C2×C22⋊C4—216no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—84no—
251C2×C22⋊C4C2×D414yes—
252C2×C22⋊C4C2×D414yes—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42C2×D412yes—
256C2×C42C2412yes—
257#107—232no—
258#103—232no—
259#103—416no—
260C22⋊C8—28no—
261C22⋊C8—28no—
262C22⋊C8—48no—
263#95—28no—
264#95—24no—
265#95—28no—
266C24—22no—
267C24—22no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24#12014yes—
273C24C2×Q1614yes—
274C24C2×Q1614yes—
275C24#13714yes—
276C24C22×D412yes—
277C24C22×D412yes—
278C24C22×D411yescharacteristic
279C2×Q8—28no—
280C2×Q8—28no—
281C2×Q8—28no—
282C2×Q8—24no—
283C2×Q8—216no—
284C2×Q8—416no—
285C2×Q8—84no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8—88no—
289C2×Q8—816no—
290C2×Q8C22×D418yes—
291C2×D4—28no—
292C2×D4—28no—
293C2×D4—232no—
294C2×D4C2×Q16116yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—416no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—48no—
324C22×C4—44no—
325C22×C4—88no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4—88no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—816no—
332C22×C4#12014yes—
333C22×C4#12018yes—
334C22×C4C2×Q1614yes—
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D414yes—
340C22×C4C22×D414yes—
341C22×C4C22×D414yes—
342C22×C4C22×D412yes—
343C22×C4C2511yescharacteristic
344C2×C8—28no—
345C2×C8—216no—
346C2×C8—416no—
347C4⋊C4—28no—
348C4⋊C4—216no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—216no—
352C4⋊C4—28no—
353C4⋊C4—216no—
354C4⋊C4—216no—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—28no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—88no—
364C4⋊C4C22×D418yes—
365C22⋊C4—216no—
366C22⋊C4—216no—
367C22⋊C4—216no—
368C22⋊C4—28no—
369C22⋊C4—28no—
370C22⋊C4—48no—
371C22⋊C4—88no—
372C42—216no—
373C42—24no—
374C42—24no—
375C23—22no—
376C23—24no—
377C23—22no—
378C23—24no—
379C23—24no—
380C23—22no—
381C23—24no—
382C23—22no—
383C23—28no—
384C23—44no—
385C23—44no—
386C23—48no—
387C23—48no—
388C23—48no—
389C23—82no—
390C23—84no—
391C23—82no—
392C23#451116yes—
393C23C2×#12012yes—
394C23C2×#12014yes—
395C23C22×Q1614yes—
396C23C22×Q1612yes—
397C23C22×Q1614yes—
398C23C2×#13714yes—
399C23C2×#13712yes—
400C23C23×D411yescharacteristic
401Q8—216no—
402Q8—48no—
403Q8—88no—
404Q8—88no—
405Q8—88no—
406D4—232no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—28no—
411C2×C4—22no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—88no—
436C2×C4—88no—
437C2×C4—88no—
438C2×C4—88no—
439C2×C4—88no—
440C2×C4—88no—
441C2×C4—816no—
442C2×C4C2×#12012yes—
443C2×C4C2×#12014yes—
444C2×C4C22×Q1614yes—
445C2×C4C23×D412yes—
446C2×C4#58514yes—
447C8—48no—
448C22—24no—
449C22—28no—
450C22—24no—
451C22—28no—
452C22—24no—
453C22—22no—
454C22—24no—
455C22—22no—
456C22—28no—
457C22—24no—
458C22—416no—
459C22—44no—
460C22—42no—
461C22—42no—
462C22—88no—
463C22C2×#45114yes—
464C22C2×#45118yes—
465C22C22×#12011yescharacteristic
466C22#348512yes—
467C22C23×Q1611yescharacteristic
468C22C22×#13711yescharacteristic
469C22#358512yes—
470C4—22no—
471C4—24no—
472C4—24no—
473C4—42no—
474C4—48no—
475C4—42no—
476C4—48no—
477C4—84no—
478C4—84no—
479C4#348514yes—
480C2—24no—
481C2—28no—
482C2—44no—
483C2#3544611yescharacteristic
484C2C22×#45111yescharacteristic
485C2#6036911yescharacteristic
486C2#6169812yes—
487C2#6170812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#484C2semidirect
4. #24098C2semidirect
5. #24064C2semidirect
6. #24042C2semidirect
7. #24006C2semidirect
8. #21759C2semidirect
25. C2×#484C22semidirect
26. C22×#107C22semidirect
27. C22×#103C22semidirect
38. #2057C22semidirect
39. #2043C22semidirect
40. #2036C22semidirect
41. #1892C22semidirect
44. #1863C22semidirect
45. #1852C22semidirect
46. #1824C22semidirect
48. #1787C22semidirect
49. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C22×#4842
C2×#24872
#240982
#240642
#240422
#240062
#217592
#217382