Finite Group Databasebeta

← #6644829

Subgroups — #6644829

Order 512.

1844 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#2547C211yesmaximal, characteristic
3#33098C212yesmaximal, complemented
4#32260C214yesmaximal, complemented
5#31320C218yesmaximal, complemented
6#31262C214yesmaximal, complemented
7#29641C212yesmaximal, complemented
8#27601C212yesmaximal, complemented
9#24551C214yesmaximal, complemented
10#24003C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124—21no—
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#483C2214yes—
17C22×C22⋊C8C2212yes—
18#2855C2212yes—
19#2848C2212yes—
20#2627—21no—
21#2626—21no—
22#2579—22no—
23#2564—22no—
24#2547—22no—
25#2547—22no—
26#2547C2214yes—
27#2547C2214yes—
28#2487C2212yes—
29#2417C2214yes—
30#2409C2212yes—
31C2×#397—21no—
32C2×#397—21no—
33#2315—21no—
34#2308—21no—
35C2×#391—21no—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#2077C2218yescomplemented
41#2077C2218yescomplemented
42#2058C2218yescomplemented
43#2035C2218yescomplemented
44#2035C2218yescomplemented
45#2024C2218yescomplemented
46#1927C2218yescomplemented
47#1920C2218yescomplemented
48#1892C2218yescomplemented
49#1872C2214yes—
50#1869C2214yes—
51#1869C2214yes—
52#1853C2214yes—
53#1853C2214yescomplemented
54#1853C2214yescomplemented
55#1824C2214yescomplemented
56#1823C2214yes—
57#1817C2212yes—
58#1807C2218yescomplemented
59#1787C2214yescomplemented
60#1779C2212yes—
61C23×Q8—21no—
62C23×Q8C2311yescharacteristic
63C23×D4D411yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C23×C8C2314yes—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—44no—
69C2×C4⋊Q8C2314yes—
70C2×#124—24no—
71C2×#124—24no—
72C2×#124—48no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—44no—
76C2×C22⋊Q8C2312yes—
77C2×C22⋊Q8C2314yes—
78C2×C22⋊Q8C2312yes—
79C2×C4×D4—44no—
80C2×C4×D4D412yes—
81C2×C4×D4D412yes—
82C2×C4×D4C2312yes—
83C2×C4×D4C2312yes—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4D411yescharacteristic
96C22×C22⋊C4D411yescharacteristic
97C22×C42—21no—
98C22×C42C2311yescharacteristic
99#483—48no—
100C2×#106C2318yes—
101C2×#106C2318yes—
102C2×#103C2318yes—
103C2×#103C2318yes—
104C2×C22⋊C8—44no—
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2314yes—
107C2×C22⋊C8C2314yes—
108C2×C22⋊C8C2314yes—
109#400—22no—
110#399—22no—
111#398—24no—
112#397—24no—
113#397—24no—
114#397—24no—
115#397—24no—
116#396—24no—
117#395—22no—
118#393—22no—
119#391—24no—
120#391—24no—
121#391C2314yes—
122#389C2314yes—
123#387—24no—
124#386—24no—
125#386—22no—
126#386—22no—
127#385—24no—
128#385—22no—
129#385—22no—
130#385—22no—
131#385—22no—
132#381—22no—
133#380—21no—
134#380—21no—
135#380—22no—
136#380—22no—
137#380—24no—
138#380—24no—
139#380C2314yes—
140#380C2318yes—
141#380C2312yes—
142#380C2314yes—
143#380C2312yes—
144#380C2314yes—
145#379—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4C2314yes—
148C2×#95—21no—
149C2×#95—21no—
150C2×#95—22no—
151C2×#95—22no—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#336C2318yes—
157C25C2×D411yescharacteristic
158C25C2×D411yescharacteristic
159C22×Q8—24no—
160C22×Q8—21no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8C2412yes—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4SD1618yes—
170C22×D4C2×D412yes—
171C22×D4C2×D412yes—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C22×C8—48no—
193C22×C8—44no—
194C22×C8C2418yes—
195C4⋊Q8—44no—
196C4⋊Q8—44no—
197C4⋊Q8—48no—
198#124—416no—
199#124—416no—
200C22⋊Q8—28no—
201C22⋊Q8—44no—
202C22⋊Q8—44no—
203C22⋊Q8—44no—
204C22⋊Q8—48no—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2412yes—
230C2×C4⋊C4C2414yes—
231C2×C4⋊C4C2414yes—
232C2×C4⋊C4C2414yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4C2×D412yes—
249C2×C22⋊C4C2×D412yes—
250C2×C22⋊C4C2×D412yes—
251C2×C22⋊C4C2×D412yes—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—44no—
255C2×C42C2×D412yes—
256C2×C42C2412yes—
257#106—48no—
258#106—48no—
259#103—48no—
260#103—48no—
261C22⋊C8—28no—
262C22⋊C8—44no—
263C22⋊C8—44no—
264C22⋊C8—44no—
265#95—24no—
266#95—22no—
267#95—28no—
268#95—24no—
269#95—24no—
270#95—22no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24C2×SD1614yes—
284C24C2×SD1614yes—
285C24C22×D412yes—
286C24C22×D411yescharacteristic
287C24C22×D411yescharacteristic
288C24C22×D411yescharacteristic
289C2×Q8—24no—
290C2×Q8—28no—
291C2×Q8—42no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—48no—
295C2×Q8—82no—
296C2×Q8—816no—
297C2×Q8—88no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—416no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4C2×SD1618yes—
304C2×D4C2×SD1618yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4C2×SD1614yes—
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4H(5,𝔽2)14yes—
355C22×C4C2511yescharacteristic
356C2×C8—28no—
357C2×C8—28no—
358C2×C8—28no—
359C2×C8—88no—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—48no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—84no—
373C4⋊C4—82no—
374C4⋊C4—84no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—416no—
383C22⋊C4—48no—
384C22⋊C4—48no—
385C22⋊C4—44no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C42—24no—
389C42—22no—
390C42—22no—
391C42—48no—
392C23—22no—
393C23—22no—
394C23—24no—
395C23—22no—
396C23—22no—
397C23—21no—
398C23—21no—
399C23—22no—
400C23—24no—
401C23—22no—
402C23—22no—
403C23—21no—
404C23—21no—
405C23—22no—
406C23—28no—
407C23—28no—
408C23—48no—
409C23—48no—
410C23—82no—
411C23—82no—
412C23—84no—
413C23—84no—
414C23—82no—
415C23—82no—
416C23#53412yes—
417C23C22×SD1614yes—
418C23C22×SD1612yes—
419C23C22×SD1612yes—
420C23C22×SD1612yes—
421C23#57512yes—
422C23C23×D411yescharacteristic
423Q8—48no—
424Q8—816no—
425Q8—84no—
426D4—48no—
427D4—48no—
428C2×C4—28no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—82no—
460C2×C4—84no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—88no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—88no—
468C2×C4—164no—
469C2×C4#53412yes—
470C2×C4C22×SD1614yes—
471C2×C4C2×H(5,𝔽2)12yes—
472C2×C4#58514yes—
473C8—48no—
474C22—24no—
475C22—22no—
476C22—22no—
477C22—24no—
478C22—22no—
479C22—22no—
480C22—24no—
481C22—28no—
482C22—24no—
483C22—24no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—41no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—42no—
492C22—41no—
493C22—88no—
494C22—88no—
495C22#318714yes—
496C22C2×#53411yescharacteristic
497C22#348512yes—
498C22C23×SD1611yescharacteristic
499C22C2×#57511yescharacteristic
500C22#358612yes—
501C4—24no—
502C4—22no—
503C4—42no—
504C4—41no—
505C4—42no—
506C4—41no—
507C4—84no—
508C4—84no—
509C4—82no—
510C4—82no—
511C4—84no—
512C4#354514yes—
513C2—24no—
514C2—42no—
515C2—42no—
516C2—44no—
517C2#3711911yescharacteristic
518C2C2×#318711yescharacteristic
519C2#6075411yescharacteristic
520C2#6169512yes—
521C2#6171212yes—
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. #33098C2semidirect
4. #32260C2semidirect
5. #31320C2semidirect
6. #31262C2semidirect
7. #29641C2semidirect
8. #27601C2semidirect
9. #24551C2semidirect
10. #24003C2semidirect
40. #2077C22semidirect
41. #2077C22semidirect
42. #2058C22semidirect
43. #2035C22semidirect
44. #2035C22semidirect
45. #2024C22semidirect
46. #1927C22semidirect
47. #1920C22semidirect
48. #1892C22semidirect
53. #1853C22semidirect
54. #1853C22semidirect
55. #1824C22semidirect
58. #1807C22semidirect
59. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#25472
#330982
#322602
#313202
#312622
#296412
#276012
#245512
#240032
#217382