Finite Group Databasebeta

← #278069

Subgroups — #278069

Order 512.

836 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33171C211yesmaximal, characteristic
2#24753C212yesmaximal, complemented
3#21597C211yesmaximal, characteristic, complemented
4C2×#1688C211yesmaximal, characteristic, complemented
5C22×#334C211yesmaximal, characteristic, complemented
6#10595C212yesmaximal, complemented
7#10475C212yesmaximal, complemented
8#9743C211yesmaximal, characteristic, complemented
9#9621C211yesmaximal, characteristic, complemented
10#9133C212yesmaximal, complemented
11#8653C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#466—21no—
14C2×#466—21no—
15C2×#465—22no—
16C2×#460C2212yes—
17C2×#446C2211yescharacteristic
18C22×C4⋊C8C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2856C2212yes—
21C4×C22⋊Q8C2211yescharacteristic
22#2065—22no—
23#2055—22no—
24#1984C2211yescharacteristic
25#1982C2212yes—
26#1970—21no—
27#1970—21no—
28#1957C2212yes—
29#1948C2212yes—
30#1939C2211yescharacteristic
31#1916—21no—
32#1893—22no—
33#1850—21no—
34#1845C2211yescharacteristic
35#1803C2212yes—
36#1797C2211yescharacteristic
37#1771—21no—
38#1770C2211yescharacteristic
39#1752—21no—
40#1688—24no—
41C2×#334—24no—
42C2×#334C2211yescharacteristic
43C2×#334C2212yes—
44C2×#334C2211yescharacteristic
45C2×#331C2212yes—
46C2×#331C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×SD16—22no—
50C22×D8—21no—
51C22×D8—21no—
52C23×C8—22no—
53C23×C8D411yescharacteristic
54C23×C8D411yescharacteristic
55C2×#127C2312yes—
56C2×C22⋊Q8D411yescharacteristic
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×C4×Q8C2311yescharacteristic
61C22×C42C2311yescharacteristic
62#493—22no—
63#493—22no—
64#466—44no—
65#466—44no—
66#465—48no—
67#460—24no—
68#460—42no—
69#446—22no—
70#446—22no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#106—22no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—42no—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×#103—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×C22⋊C8—21no—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8D411yescharacteristic
95C2×C4×C8D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#391D411yescharacteristic
99#390—22no—
100#390—22no—
101#390—22no—
102#389D411yescharacteristic
103#387—21no—
104#385—21no—
105#384C2312yes—
106#380D411yescharacteristic
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2311yescharacteristic
109C4×C22⋊C4—21no—
110#340—22no—
111#340—22no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#334—24no—
118#334—22no—
119#334D414yescomplemented
120#331—24no—
121#331—42no—
122C22×Q8C2×D411yescharacteristic
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—42no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130C2×SD16—44no—
131C2×SD16—44no—
132C2×SD16—82no—
133C2×D8—42no—
134C2×D8—42no—
135C2×D8—42no—
136C2×D8—42no—
137C2×D8—42no—
138C2×D8—42no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—41no—
148C22×C8—44no—
149C22×C8—44no—
150C22×C8—82no—
151C22×C8SD1614yescomplemented
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156#127—22no—
157#127—22no—
158#127—22no—
159#127—22no—
160C22⋊Q8—24no—
161#121—44no—
162#121—44no—
163#121—44no—
164#121—44no—
165#121—84no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42C2411yescharacteristic
194#107—42no—
195#107—42no—
196#106—44no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—21no—
200C4⋊C8—22no—
201C4⋊C8—21no—
202C4⋊C8—22no—
203C4⋊C8—21no—
204C4⋊C8—44no—
205C4⋊C8—42no—
206C4⋊C8—42no—
207C4⋊C8C2×D414yes—
208#103—44no—
209#102—44no—
210#102—44no—
211#102—44no—
212#102—44no—
213#102—82no—
214C22⋊C8—42no—
215#97—42no—
216#97—42no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8C2×D412yes—
222C4×C8#4112yes—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228C24—42no—
229C24#12111yescharacteristic
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—84no—
251C2×D4—84no—
252C2×D4—84no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4C2×SD1612yes—
273C22×C4AGL(1,ℤ/8)12yes—
274C22×C4AGL(1,ℤ/8)14yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4111yescharacteristic
278SD16—88no—
279D8—84no—
280D8—84no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—28no—
286C2×C8—21no—
287C2×C8—24no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—44no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—88no—
303C2×C8#12112yes—
304C2×C8#12112yes—
305C2×C8C2×SD1612yes—
306C2×C8C2×SD1612yes—
307C2×C8#13512yes—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—82no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C22⋊C4—84no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42C22×D411yescharacteristic
337C42C2×#4111yescharacteristic
338C42H(5,𝔽2)11yescharacteristic
339C422−1+411yescharacteristic
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23—82no—
350C23—82no—
351C23#46012yes—
352C23#46912yes—
353C23#47212yes—
354C23C2×#12111yescharacteristic
355Q8—42no—
356Q8—42no—
357D4—82no—
358D4—82no—
359D4—82no—
360D4—82no—
361D4—82no—
362D4—82no—
363D4—82no—
364D4—82no—
365D4—162no—
366D4—162no—
367D4—168no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#46012yes—
401C2×C4#46912yes—
402C2×C4C2×#12111yescharacteristic
403C2×C4#53711yescharacteristic
404C2×C4D4211yescharacteristic
405C2×C4#54711yescharacteristic
406C2×C4#54811yescharacteristic
407C2×C4#55311yescharacteristic
408C2×C4C22×SD1611yescharacteristic
409C2×C4C2×#13511yescharacteristic
410C2×C4C2×AGL(1,ℤ/8)12yes—
411C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
412C2×C4#57511yescharacteristic
413C2×C4#57712yes—
414C8—22no—
415C8—22no—
416C8—44no—
417C8—41no—
418C8—81no—
419C8—82no—
420C8—81no—
421C8#46514yescomplemented
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—162no—
429C22—162no—
430C22—162no—
431C22—162no—
432C22—162no—
433C22—162no—
434C22—162no—
435C22#168814yescomplemented
436C22#241611yescharacteristic
437C22C2×#46011yescharacteristic
438C22#303511yescharacteristic
439C22C2×#46911yescharacteristic
440C22#306011yescharacteristic
441C22C2×#47211yescharacteristic
442C22#306711yescharacteristic
443C4—22no—
444C4—22no—
445C4—21no—
446C4—21no—
447C4—41no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—162no—
453C4C2×#46511yescharacteristic
454C4#305511yescharacteristic
455C4#324611yescharacteristic
456C4#324711yescharacteristic
457C4#326512yes—
458C4#334512yes—
459C2—24no—
460C2—322no—
461C2C2×#168811yescharacteristic
462C2#1915711yescharacteristic
463C2#1920511yescharacteristic
464C2#1922711yescharacteristic
465C2#2475311yescharacteristic
466C2#2478211yescharacteristic
467C2#2480311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#33171C2
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
2. #24753C2semidirect
3. #21597C2semidirect
4. C2×#1688C2semidirect
5. C22×#334C2semidirect
6. #10595C2semidirect
7. #10475C2semidirect
8. #9743C2semidirect
9. #9621C2semidirect
10. #9133C2semidirect
11. #8653C2semidirect
119. #334D4semidirect
151. C22×C8SD16semidirect
421. C8#465semidirect
435. C22#1688semidirect

Maximal subgroups

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

subgroupindex
#331712
#247532
#215972
C2×#16882
C22×#3342
#105952
#104752
#97432
#96212
#91332
#86532