Finite Group Databasebeta

← #278219

Subgroups — #278219

Order 512.

709 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24659C212yesmaximal
3#21487C211yesmaximal, characteristic
4C2×#1693C211yesmaximal, characteristic, complemented
5C2×#1568C211yesmaximal, characteristic, complemented
6#10594C212yesmaximal, complemented
7#10476C212yesmaximal, complemented
8#9740C211yesmaximal, characteristic, complemented
9#9620C211yesmaximal, characteristic, complemented
10#9140C212yesmaximal, complemented
11#8652C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#461C2212yes—
14C2×#436C2211yescharacteristic
15C2×#435—21no—
16C2×#432C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2857C2212yes—
19C4×#121C2211yescharacteristic
20#2036—22no—
21#2000—22no—
22#1990C2212yes—
23#1981C2212yes—
24#1973—22no—
25#1954C2211yescharacteristic
26#1947C2212yes—
27#1942—21no—
28#1942—21no—
29#1939C2211yescharacteristic
30#1935—22no—
31#1923—21no—
32#1869—21no—
33#1845C2211yescharacteristic
34#1808—21no—
35#1808—21no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38#1770C2211yescharacteristic
39#1753—21no—
40#1693—24no—
41#1568—24no—
42C2×#335C2212yes—
43C2×#334C2211yescharacteristic
44C2×#334C2211yescharacteristic
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×SD16—22no—
50C22×M16—21no—
51C23×C8D411yescharacteristic
52C23×C8D411yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×C22⋊Q8—22no—
55C2×#121D411yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#500—22no—
60#500—22no—
61#461—24no—
62#461—42no—
63#436—22no—
64#436—22no—
65#435—44no—
66#432—24no—
67#432—42no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#102—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×#97—21no—
88C2×#97—22no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391—22no—
96#390D411yescharacteristic
97#389—22no—
98#389—22no—
99#389D411yescharacteristic
100#388—21no—
101#386—21no—
102#384C2312yes—
103#380—22no—
104#380—22no—
105#380D411yescharacteristic
106C4×C4⋊C4C2311yescharacteristic
107C4×C22⋊C4—21no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#336—21no—
113#336—21no—
114#336—21no—
115#335—24no—
116#334—22no—
117#334D414yescomplemented
118#332—24no—
119#332—42no—
120C22×Q8—22no—
121C22×D4—21no—
122C22×D4C2×D411yescharacteristic
123C23×C4—21no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4#4111yescharacteristic
127C2×SD16—44no—
128C2×SD16—44no—
129C2×SD16—82no—
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—42no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—41no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C22⋊Q8—82no—
152C22⋊Q8—82no—
153#121—42no—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—42no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42C2411yescharacteristic
185#107—42no—
186#107—42no—
187#107—42no—
188#107—42no—
189#107—82no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—42no—
198C4⋊C8C2×D414yes—
199#103—44no—
200#103—44no—
201#103—82no—
202#102—82no—
203C22⋊C8—44no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206C22⋊C8—42no—
207C22⋊C8—42no—
208#97—44no—
209#97—42no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8C2×D412yes—
215C4×C8#4112yes—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—42no—
220#95—42no—
221C24—21no—
222C24#12111yescharacteristic
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C2×#4111yescharacteristic
267SD16—88no—
268M16—84no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—24no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—44no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—44no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8#12112yes—
291C2×C8#12112yes—
292C2×C8AGL(1,ℤ/8)12yes—
293C2×C8#13712yes—
294C2×C8#13712yes—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—82no—
309C4⋊C4—84no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—84no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—41no—
324C42—41no—
325C42C22×D411yescharacteristic
326C42C2×#4111yescharacteristic
327C42H(5,𝔽2)11yescharacteristic
328C422−1+411yescharacteristic
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23#47312yes—
343C23C2×#12111yescharacteristic
344Q8—82no—
345Q8—82no—
346Q8—82no—
347Q8—82no—
348D4—42no—
349D4—42no—
350D4—84no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4C2×#12111yescharacteristic
390C2×C4#53711yescharacteristic
391C2×C4D4211yescharacteristic
392C2×C4#54711yescharacteristic
393C2×C4#54811yescharacteristic
394C2×C4#55311yescharacteristic
395C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
396C2×C4C2×#13711yescharacteristic
397C2×C4C2×#13711yescharacteristic
398C2×C4#57512yes—
399C2×C4#57511yescharacteristic
400C2×C4#57612yes—
401C8—22no—
402C8—22no—
403C8—44no—
404C8—41no—
405C8—81no—
406C8—82no—
407C8—81no—
408C8#46814yes—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22#241611yescharacteristic
424C22#303611yescharacteristic
425C22#303911yescharacteristic
426C22#305411yescharacteristic
427C22#305811yescharacteristic
428C22C2×#47311yescharacteristic
429C22#306711yescharacteristic
430C4—21no—
431C4—21no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—81no—
437C4—81no—
438C4—162no—
439C4—162no—
440C4C2×#46811yescharacteristic
441C4#305611yescharacteristic
442C4#325211yescharacteristic
443C4#325411yescharacteristic
444C4#326812yes—
445C4#334712yes—
446C2—42no—
447C2—161no—
448C2#1928411yescharacteristic
449C2#1929811yescharacteristic
450C2#1933511yescharacteristic
451C2#1935811yescharacteristic
452C2#2475911yescharacteristic
453C2#2478411yescharacteristic
454C2#2480411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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. #33209C2semidirect
4. C2×#1693C2semidirect
5. C2×#1568C2semidirect
6. #10594C2semidirect
7. #10476C2semidirect
8. #9740C2semidirect
9. #9620C2semidirect
10. #9140C2semidirect
11. #8652C2semidirect
117. #334D4semidirect

Maximal subgroups

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

subgroupindex
#332092
#246592
#214872
C2×#16932
C2×#15682
#105942
#104762
#97402
#96202
#91402
#86522