Finite Group Databasebeta

← #6629633

Subgroups — #6629633

Order 512.

2378 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic, complemented
3#33097C218yesmaximal, complemented
4#33096C214yesmaximal, complemented
5#33095C212yesmaximal, complemented
6#33094C212yesmaximal, complemented
7#21060C214yesmaximal, complemented
8#21034C214yesmaximal, complemented
9#20912C212yesmaximal
10#20846C212yesmaximal, complemented
11#20682C211yesmaximal, characteristic, complemented
12C22×#123—21no—
13C22×#121C2211yescharacteristic
14C22×#117C2212yes—
15C23×C42C2211yescharacteristic
16C2×#428C2214yes—
17C2×#418C2214yes—
18C2×#407C2212yes—
19#2854—21no—
20#2854—21no—
21#2854—22no—
22#2854C2212yes—
23#2854C2212yes—
24#2854C2214yes—
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#388—22no—
28C2×#386—21no—
29C2×#380C2211yescharacteristic
30C4×#117—22no—
31C4×#117—22no—
32C4×#117C2214yes—
33C4×#117C2214yescomplemented
34C2×C4×C4⋊C4C2211yescharacteristic, complemented
35C2×C4×C4⋊C4C2211yescharacteristic, complemented
36C2×C4×C4⋊C4C2212yescomplemented
37C2×C4×C22⋊C4—21no—
38C2×C4×C22⋊C4—21no—
39C2×#376—21no—
40C2×C43—21no—
41#1926C2218yescomplemented
42#1926C2218yescomplemented
43#1925C2218yescomplemented
44#1925C2218yescomplemented
45#1921C2218yescomplemented
46#1920C2218yescomplemented
47#1919C2218yescomplemented
48#1918C2218yescomplemented
49#1875C2218yes—
50#1874C2214yes—
51#1874C2214yes—
52#1854C2218yescomplemented
53#1853C2214yescomplemented
54#1852C2214yescomplemented
55#1817C2212yescomplemented
56#1817C2212yescomplemented
57#1817C2214yescomplemented
58#1736C2214yes—
59#1736C2214yes—
60#1728C2214yes—
61C23×D4—21no—
62C23×D4C2311yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4D411yescharacteristic
65C24×C4C2311yescharacteristic
66C22×M16C2314yes—
67C2×#123—22no—
68C2×#123—24no—
69C2×#123—22no—
70C2×#123—48no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—28no—
74C2×#121C2312yes—
75C2×#121C2312yes—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—216no—
79C2×#117C2314yes—
80C2×#117C2314yes—
81C22×C4⋊C4—22no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—22no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C42—22no—
92C22×C42—28no—
93C22×C42—22no—
94C22×C42—21no—
95C22×C42—21no—
96C22×C42D411yescharacteristic
97C22×C42D411yescharacteristic
98C22×C42D412yes—
99C22×C42D412yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104#428—216no—
105C2×#107C2318yes—
106C2×#106C2318yes—
107#418—216no—
108C2×#102C2318yes—
109C2×#102C2318yes—
110#407—28no—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113#390—22no—
114#390—22no—
115#390C2314yes—
116#390C2314yes—
117#389—22no—
118#389—22no—
119#389C2314yes—
120#389C2314yes—
121#388—24no—
122#388—24no—
123#388—24no—
124#388—24no—
125#386—22no—
126#386—24no—
127#386—22no—
128#380—21no—
129#380—21no—
130#380C2312yes—
131#380C2312yes—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2314yes—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—24no—
144#376—24no—
145#376—24no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95—21no—
150C2×#95—22no—
151C43—24no—
152C43—24no—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#340C2318yes—
157#336C2318yes—
158#336C2318yes—
159C25—21no—
160C25C2×D411yescharacteristic
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—24no—
165C22×D4—28no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4C2412yes—
170C23×C4—28no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—44no—
179C23×C4—42no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4#4118yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C2×M16—28no—
196C2×M16—28no—
197C2×M16—28no—
198C22×C8C2418yes—
199#123—48no—
200#123—416no—
201#123—48no—
202#121—28no—
203#121—28no—
204#121—44no—
205#121—44no—
206#117—216no—
207#117—216no—
208#117—28no—
209#117—28no—
210#117—28no—
211#117#41116yes—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4C2412yes—
224C2×C4⋊C4C2414yes—
225C2×C4⋊C4C2414yes—
226C2×C4⋊C4C2414yes—
227C2×C22⋊C4—216no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—48no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C42—216no—
243C2×C42—24no—
244C2×C42—28no—
245C2×C42—24no—
246C2×C42—28no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—28no—
252C2×C42—28no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—44no—
258C2×C42—48no—
259C2×C42—44no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2412yes—
267C2×C42C2412yes—
268#107—216no—
269#106—216no—
270#102—216no—
271#102—216no—
272C22⋊C8—28no—
273C22⋊C8—28no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—22no—
279#95—22no—
280#95—24no—
281#95—24no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—42no—
287C24—42no—
288C24—44no—
289C24—42no—
290C24—42no—
291C24—44no—
292C24#12314yes—
293C24C22×D411yescharacteristic
294C24C22×D411yescharacteristic
295C24C22×D411yescharacteristic
296C2×D4—28no—
297C2×D4—24no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—42no—
305C2×D4—44no—
306C2×D4—416no—
307C2×D4—816no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—232no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—416no—
342C22×C4—416no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4#12314yes—
354C22×C4#12314yes—
355C22×C4#12314yes—
356C22×C4AGL(1,ℤ/8)14yes—
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4114yes—
365C22×C4C2×#4114yes—
366C22×C4C2×#4114yes—
367C22×C4C2×#4118yes—
368C22×C4C2511yescharacteristic
369M16—416no—
370C2×C8—28no—
371C2×C8—28no—
372C2×C8—28no—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4C2×#4118yes—
384C4⋊C4C2×#4118yes—
385C22⋊C4—216no—
386C22⋊C4—216no—
387C22⋊C4—48no—
388C22⋊C4—48no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C42—216no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—24no—
402C42—216no—
403C42—24no—
404C42—24no—
405C42—48no—
406C42—48no—
407C42—48no—
408C42—48no—
409C23—22no—
410C23—21no—
411C23—22no—
412C23—22no—
413C23—21no—
414C23—28no—
415C23—28no—
416C23—22no—
417C23—21no—
418C23—22no—
419C23—21no—
420C23—44no—
421C23—48no—
422C23—48no—
423C23—42no—
424C23—44no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23—84no—
429C23—84no—
430C23—88no—
431C23—88no—
432C23—84no—
433C23C2×#12312yes—
434C23C2×#12312yes—
435C23C2×#12312yes—
436C23#52514yes—
437C23C2×AGL(1,ℤ/8)12yes—
438C23#57512yes—
439C23C23×D411yescharacteristic
440D4—48no—
441D4—88no—
442D4—88no—
443D4—84no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—48no—
469C2×C4—416no—
470C2×C4—416no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—88no—
484C2×C4—84no—
485C2×C4C2×#12312yes—
486C2×C4C2×#12312yes—
487C2×C4C2×#12312yes—
488C2×C4#52514yes—
489C2×C4#529116yes—
490C2×C4C2×AGL(1,ℤ/8)14yes—
491C2×C4#57514yes—
492C2×C4C22×#4112yes—
493C2×C4C22×#4114yes—
494C8—48no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—24no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—416no—
507C22—42no—
508C22—42no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—84no—
513C22—84no—
514C22—168no—
515C22#285314yes—
516C22#309414yes—
517C22C22×#12311yescharacteristic
518C22C2×#52512yes—
519C22C22×AGL(1,ℤ/8)11yescharacteristic
520C22C2×#57511yescharacteristic
521C22C2×#57512yes—
522C4—22no—
523C4—22no—
524C4—24no—
525C4—24no—
526C4—42no—
527C4—42no—
528C4—48no—
529C4—48no—
530C4—82no—
531C4—84no—
532C4#285314yes—
533C4C2×#52914yes—
534C2—24no—
535C2—24no—
536C2—42no—
537C2—162no—
538C2C2×#285311yescharacteristic
539C2C2×#309411yescharacteristic
540C2#6051111yescharacteristic
541C2#6056112yes—
542C2#6056312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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×C4×#117C2semidirect
3. #33097C2semidirect
4. #33096C2semidirect
5. #33095C2semidirect
6. #33094C2semidirect
7. #21060C2semidirect
8. #21034C2semidirect
10. #20846C2semidirect
11. #20682C2semidirect
33. C4×#117C22semidirect
34. C2×C4×C4⋊C4C22semidirect
35. C2×C4×C4⋊C4C22semidirect
36. C2×C4×C4⋊C4C22semidirect
41. #1926C22semidirect
42. #1926C22semidirect
43. #1925C22semidirect
44. #1925C22semidirect
45. #1921C22semidirect
46. #1920C22semidirect
47. #1919C22semidirect
48. #1918C22semidirect
52. #1854C22semidirect
53. #1853C22semidirect
54. #1852C22semidirect
55. #1817C22semidirect
56. #1817C22semidirect
57. #1817C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×C4×#1172
#330972
#330962
#330952
#330942
#210602
#210342
#209122
#208462
#206822