Finite Group Databasebeta

← #6642615

Subgroups — #6642615

Order 512.

1772 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic
3#32983C212yesmaximal, complemented
4#32467C214yesmaximal, complemented
5#31794C218yesmaximal, complemented
6#31682C214yesmaximal, complemented
7#29893C212yesmaximal, complemented
8#27601C212yesmaximal, complemented
9#24606C214yesmaximal, complemented
10#24006C212yesmaximal, complemented
11#21790C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#484C2214yes—
16C2×#432C2212yes—
17#2855C2212yes—
18#2848C2212yes—
19#2627—21no—
20#2626—21no—
21#2579—22no—
22#2564—22no—
23#2487—22no—
24#2487—22no—
25#2487C2214yes—
26#2487C2214yes—
27#2487C2212yes—
28#2417C2214yes—
29#2409C2212yes—
30C2×#400—21no—
31C2×#399—21no—
32#2315—21no—
33#2308—21no—
34C2×#389C2212yes—
35C2×#386—21no—
36C2×#385—21no—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40#2086C2218yescomplemented
41#2077C2218yescomplemented
42#2077C2218yescomplemented
43#2066C2218yescomplemented
44#2036C2218yescomplemented
45#2035C2218yescomplemented
46#1955C2214yes—
47#1955C2214yes—
48#1944C2218yescomplemented
49#1935C2218yescomplemented
50#1872C2214yes—
51#1872C2218yescomplemented
52#1853C2214yes—
53#1853C2214yescomplemented
54#1852C2218yescomplemented
55#1852C2214yescomplemented
56#1824C2214yes—
57#1823C2214yescomplemented
58#1787C2214yescomplemented
59#1779C2212yes—
60#1779C2212yes—
61C23×Q8C2311yescharacteristic
62C23×D4D411yescharacteristic
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—44no—
69C2×C22⋊Q8C2312yes—
70C2×C22⋊Q8C2314yes—
71C2×C22⋊Q8C2312yes—
72C2×C4×D4—44no—
73C2×C4×D4D412yes—
74C2×C4×D4D412yes—
75C2×C4×D4C2312yes—
76C2×C4×D4C2312yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—44no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42C2311yescharacteristic
95#484—48no—
96#432—44no—
97C2×#107C2318yes—
98C2×#106C2318yes—
99C2×C4⋊C8C2314yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102C2×C22⋊C8C2314yes—
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2318yes—
105C2×C22⋊C8C2314yes—
106C2×#97C2314yes—
107#400—22no—
108#400—24no—
109#400—24no—
110#399—22no—
111#399—24no—
112#399—24no—
113#398—24no—
114#396—24no—
115#395—22no—
116#393—22no—
117#391C2314yes—
118#389—24no—
119#389C2318yes—
120#389C2314yes—
121#387—24no—
122#386—24no—
123#386—22no—
124#386—22no—
125#386—24no—
126#386—24no—
127#385—24no—
128#385—22no—
129#385—22no—
130#385—22no—
131#385—22no—
132#385—24no—
133#385—24no—
134#381—22no—
135#380—21no—
136#380—21no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380—24no—
141#380C2312yes—
142#380C2314yes—
143#380C2314yes—
144#380C2314yes—
145#380C2314yes—
146#380C2312yes—
147#379—22no—
148C2×#95—21no—
149C2×#95—21no—
150C2×#95—22no—
151C2×#95—22no—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156C25C2×D411yescharacteristic
157C25C2×D411yescharacteristic
158C22×Q8—21no—
159C22×Q8—42no—
160C22×Q8—44no—
161C22×Q8C2412yes—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—42no—
165C22×D4C2×D412yes—
166C22×D4C2×D412yes—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—84no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C22×C8C2418yes—
191C22⋊Q8—28no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C22⋊Q8—48no—
196C4×D4—44no—
197C4×D4—44no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—42no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—84no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2414yes—
226C2×C4⋊C4C2412yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2414yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—84no—
244C2×C22⋊C4C2×D412yes—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C22⋊C4C2×D412yes—
248C2×C42—24no—
249C2×C42C2×D412yes—
250C2×C42C2412yes—
251#107—48no—
252#106—48no—
253C4⋊C8—44no—
254#103—48no—
255#103—48no—
256C22⋊C8—28no—
257C22⋊C8—44no—
258C22⋊C8—48no—
259C22⋊C8—44no—
260#97—44no—
261#95—22no—
262#95—24no—
263#95—24no—
264#95—22no—
265#95—24no—
266#95—28no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24#13714yes—
280C24#13714yes—
281C24C22×D411yescharacteristic
282C24C22×D411yescharacteristic
283C24C22×D412yes—
284C24C22×D411yescharacteristic
285C2×Q8—24no—
286C2×Q8—28no—
287C2×Q8—42no—
288C2×Q8—82no—
289C2×Q8—88no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—416no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—28no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—416no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—48no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—88no—
337C22×C4—82no—
338C22×C4—88no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—816no—
343C22×C4C22×D412yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D412yes—
347C22×C4C22×D412yes—
348C22×C4C22×D411yescharacteristic
349C22×C4H(5,𝔽2)14yes—
350C22×C4C2511yescharacteristic
351C2×C8—28no—
352C2×C8—28no—
353C2×C8—28no—
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—84no—
367C4⋊C4—82no—
368C4⋊C4—88no—
369C4⋊C4—84no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—48no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—84no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—24no—
384C42—22no—
385C42—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—24no—
391C23—21no—
392C23—22no—
393C23—21no—
394C23—28no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—28no—
402C23—48no—
403C23—48no—
404C23—44no—
405C23—44no—
406C23—48no—
407C23—82no—
408C23—84no—
409C23—82no—
410C23#53412yes—
411C23C2×#13714yes—
412C23C2×#13712yes—
413C23C2×#13712yes—
414C23C2×#13712yes—
415C23C2×#13712yes—
416C23C23×D411yescharacteristic
417Q8—48no—
418Q8—84no—
419D4—48no—
420D4—48no—
421C2×C4—22no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—48no—
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—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—88no—
459C2×C4—88no—
460C2×C4—84no—
461C2×C4—816no—
462C2×C4—84no—
463C2×C4—164no—
464C2×C4#53412yes—
465C2×C4C2×H(5,𝔽2)12yes—
466C2×C4#58514yes—
467C8—48no—
468C22—24no—
469C22—24no—
470C22—22no—
471C22—22no—
472C22—28no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—22no—
477C22—22no—
478C22—416no—
479C22—42no—
480C22—42no—
481C22—44no—
482C22—42no—
483C22—41no—
484C22—41no—
485C22—42no—
486C22—41no—
487C22—41no—
488C22—88no—
489C22#319514yes—
490C22C2×#53411yescharacteristic
491C22#348512yes—
492C22C22×#13711yescharacteristic
493C22C22×#13711yescharacteristic
494C22#358512yes—
495C4—22no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—41no—
500C4—41no—
501C4—84no—
502C4—84no—
503C4—82no—
504C4—84no—
505C4—82no—
506C4#353314yes—
507C2—24no—
508C2—44no—
509C2—42no—
510C2—42no—
511C2#3696511yescharacteristic
512C2C2×#319511yescharacteristic
513C2#6075311yescharacteristic
514C2#6170812yes—
515C2#6170912yes—
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
3. #32983C2semidirect
4. #32467C2semidirect
5. #31794C2semidirect
6. #31682C2semidirect
7. #29893C2semidirect
8. #27601C2semidirect
9. #24606C2semidirect
10. #24006C2semidirect
40. #2086C22semidirect
41. #2077C22semidirect
42. #2077C22semidirect
43. #2066C22semidirect
44. #2036C22semidirect
45. #2035C22semidirect
48. #1944C22semidirect
49. #1935C22semidirect
51. #1872C22semidirect
53. #1853C22semidirect
54. #1852C22semidirect
55. #1852C22semidirect
57. #1823C22semidirect
58. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#24872
#329832
#324672
#317942
#316822
#298932
#276012
#246062
#240062
#217902