Finite Group Databasebeta

← #278051

Subgroups — #278051

Order 512.

887 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33171C211yesmaximal, characteristic, complemented
2#24750C212yesmaximal, complemented
3#21487C211yesmaximal, characteristic, complemented
4C2×#1687C211yesmaximal, characteristic, complemented
5C22×#334C211yesmaximal, characteristic, complemented
6#10587C212yesmaximal, complemented
7#10473C212yesmaximal, complemented
8#9743C211yesmaximal, characteristic, complemented
9#9620C211yesmaximal, characteristic, complemented
10#9133C212yesmaximal, complemented
11#8652C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#466—21no—
14C2×#466—21no—
15C2×#466—22no—
16C2×#459C2212yes—
17C2×#436C2211yescharacteristic
18C2×#435—21no—
19C22×C4⋊C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2856C2212yes—
22C4×#121C2211yescharacteristic
23#2067—22no—
24#2054—22no—
25#1982C2212yes—
26#1970—21no—
27#1970—21no—
28#1957C2212yes—
29#1954C2211yescharacteristic
30#1947C2212yes—
31#1939C2211yescharacteristic
32#1923—21no—
33#1893—22no—
34#1869—21no—
35#1845C2211yescharacteristic
36#1802C2212yes—
37#1797C2211yescharacteristic
38#1770C2211yescharacteristic
39#1753—21no—
40#1687—24no—
41C2×#334—24no—
42C2×#334C2211yescharacteristic
43C2×#334C2212yes—
44C2×#334C2211yescharacteristic
45C2×#331C2212yes—
46C2×#331C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×D8—22no—
50C22×D8—21no—
51C22×D8—21no—
52C22×M16—21no—
53C23×C8—22no—
54C23×C8D411yescharacteristic
55C23×C8D411yescharacteristic
56C2×#127C2312yes—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121D411yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4C2311yescharacteristic
63C22×C42C2311yescharacteristic
64#493—22no—
65#493—22no—
66#466—48no—
67#466—44no—
68#466—44no—
69#459—24no—
70#459—42no—
71#436—22no—
72#436—22no—
73#435—44no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—22no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8D411yescharacteristic
99C2×C4×C8D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#390—22no—
103#390—22no—
104#390—22no—
105#390D411yescharacteristic
106#389D411yescharacteristic
107#388—21no—
108#386—21no—
109#384C2312yes—
110#380D411yescharacteristic
111C4×C4⋊C4C2311yescharacteristic
112C4×C22⋊C4—21no—
113#340—22no—
114#340—22no—
115#336—21no—
116#336—21no—
117#336—21no—
118#334—24no—
119#334—22no—
120#334D414yescomplemented
121#331—24no—
122#331—42no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—42no—
128C22×D4C2×D411yescharacteristic
129C23×C4—21no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C2×D8—44no—
134C2×D8—42no—
135C2×D8—42no—
136C2×D8—44no—
137C2×D8—42no—
138C2×D8—42no—
139C2×D8—42no—
140C2×D8—42no—
141C2×D8—82no—
142C2×M16—42no—
143C2×M16—42no—
144C2×M16—42no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—41no—
154C22×C8—44no—
155C22×C8—44no—
156C22×C8—82no—
157C22×C8D814yescomplemented
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162#127—22no—
163#127—22no—
164#127—22no—
165#127—22no—
166#121—24no—
167#121—44no—
168#121—44no—
169#121—44no—
170#121—44no—
171#121—84no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—42no—
177C4×D4—42no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42C2411yescharacteristic
202#107—44no—
203#107—42no—
204#107—42no—
205C4⋊C8—21no—
206C4⋊C8—22no—
207C4⋊C8—21no—
208C4⋊C8—22no—
209C4⋊C8—21no—
210C4⋊C8—22no—
211C4⋊C8—21no—
212C4⋊C8—44no—
213C4⋊C8—42no—
214C4⋊C8C2×D414yes—
215#102—44no—
216#102—44no—
217#102—44no—
218#102—44no—
219#102—44no—
220#102—82no—
221C22⋊C8—42no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224C22⋊C8—42no—
225#97—42no—
226C4×C8—22no—
227C4×C8—22no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8C2×D412yes—
231C4×C8#4112yes—
232#95—21no—
233#95—21no—
234#95—21no—
235C24—21no—
236C24—42no—
237C24#12111yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—82no—
260C2×D4—82no—
261C2×D4—84no—
262C2×D4—84no—
263C2×D4—84no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4C2×D812yes—
289C22×C4AGL(1,ℤ/8)12yes—
290C22×C4AGL(1,ℤ/8)14yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4111yescharacteristic
294D8—84no—
295D8—88no—
296D8—84no—
297M16—84no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—28no—
303C2×C8—21no—
304C2×C8—24no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—44no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—88no—
320C2×C8#12112yes—
321C2×C8#12112yes—
322C2×C8C2×D812yes—
323C2×C8C2×D812yes—
324C2×C8#13512yes—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—82no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—84no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—41no—
351C42—41no—
352C42C22×D411yescharacteristic
353C42C2×#4111yescharacteristic
354C42H(5,𝔽2)11yescharacteristic
355C422−1+411yescharacteristic
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23#45912yes—
375C23#46912yes—
376C23#47212yes—
377C23C2×#12111yescharacteristic
378D4—42no—
379D4—42no—
380D4—84no—
381D4—82no—
382D4—82no—
383D4—82no—
384D4—82no—
385D4—82no—
386D4—82no—
387D4—82no—
388D4—82no—
389D4—162no—
390D4—162no—
391D4—168no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4#45912yes—
426C2×C4#46912yes—
427C2×C4C2×#12111yescharacteristic
428C2×C4#53711yescharacteristic
429C2×C4D4211yescharacteristic
430C2×C4#54711yescharacteristic
431C2×C4#54811yescharacteristic
432C2×C4#55311yescharacteristic
433C2×C4C22×D811yescharacteristic
434C2×C4C2×#13511yescharacteristic
435C2×C4C2×AGL(1,ℤ/8)12yes—
436C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
437C2×C4#57511yescharacteristic
438C2×C4#57712yes—
439C8—22no—
440C8—22no—
441C8—44no—
442C8—41no—
443C8—81no—
444C8—82no—
445C8—81no—
446C8#46614yescomplemented
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—162no—
461C22—162no—
462C22—162no—
463C22—162no—
464C22—162no—
465C22—162no—
466C22—162no—
467C22#168714yescomplemented
468C22#241611yescharacteristic
469C22C2×#45911yescharacteristic
470C22#303511yescharacteristic
471C22C2×#46911yescharacteristic
472C22#306011yescharacteristic
473C22C2×#47211yescharacteristic
474C22#306711yescharacteristic
475C4—22no—
476C4—22no—
477C4—21no—
478C4—21no—
479C4—41no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—162no—
484C4C2×#46611yescharacteristic
485C4#305511yescharacteristic
486C4#324511yescharacteristic
487C4#324711yescharacteristic
488C4#326512yes—
489C4#334512yes—
490C2—24no—
491C2—161no—
492C2—322no—
493C2C2×#168711yescharacteristic
494C2#1915711yescharacteristic
495C2#1918911yescharacteristic
496C2#1922711yescharacteristic
497C2#2475011yescharacteristic
498C2#2478211yescharacteristic
499C2#2480311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33171C2semidirect
2. #24750C2semidirect
3. #21487C2semidirect
4. C2×#1687C2semidirect
5. C22×#334C2semidirect
6. #10587C2semidirect
7. #10473C2semidirect
8. #9743C2semidirect
9. #9620C2semidirect
10. #9133C2semidirect
11. #8652C2semidirect
120. #334D4semidirect
157. C22×C8D8semidirect
446. C8#466semidirect
467. C22#1687semidirect

Maximal subgroups

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

subgroupindex
#331712
#247502
#214872
C2×#16872
C22×#3342
#105872
#104732
#97432
#96202
#91332
#86522