Finite Group Databasebeta

← #496835

Subgroups — #496835

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33520C211yesmaximal, characteristic
2#21529C211yesmaximal, characteristic, complemented
3#21240C211yesmaximal, characteristic, complemented
4#13249C212yesmaximal, complemented
5#13153C212yesmaximal, complemented
6#13143C212yesmaximal, complemented
7#13037C212yesmaximal, complemented
8#13004C211yesmaximal, characteristic, complemented
9#12928C211yesmaximal, characteristic, complemented
10#12735C211yesmaximal, characteristic, complemented
11#12606C211yesmaximal, characteristic, complemented
12C22×#135C2211yescharacteristic
13C2×#509C2212yes—
14C2×#466C2211yescharacteristic
15C2×#442C2211yescharacteristic, complemented
16C22×#108C2211yescharacteristic
17C2×#421—21no—
18#2879—21no—
19C2×#405C2211yescharacteristic
20#2324—21no—
21#2186C414yescomplemented
22#2185C412yescomplemented
23#2184C412yescomplemented
24C2×#365C2212yescomplemented
25C2×#361C2212yescomplemented
26C2×#360C2212yes—
27C2×#360C2212yes—
28#2139C2212yes—
29#2112C2211yescharacteristic
30#2112C2211yescharacteristic
31C2×#347C2212yescomplemented
32#1975C2212yes—
33#1975C2212yes—
34#1974—21no—
35#1974—21no—
36#1972C2211yescharacteristic, complemented
37#1943C2211yescharacteristic, complemented
38#1934—21no—
39#1895—21no—
40#1807C2211yescharacteristic
41#1776C2211yescharacteristic, complemented
42C4×#102—21no—
43#1387C2212yescomplemented
44#1383C2212yescomplemented
45#1375C2212yescomplemented
46C22×#41—21no—
47C2×#135—21no—
48C2×#135—21no—
49C2×#135—42no—
50C2×#135—42no—
51C2×#135C2×C412yescomplemented
52C2×#135C2312yes—
53C22×Q16C2×C411yescharacteristic, complemented
54C22×SD16—21no—
55C22×D8C2×C411yescharacteristic, complemented
56C22×D8C2311yescharacteristic
57C22×M16—21no—
58C22×M16D411yescharacteristic
59C23×C8C2311yescharacteristic
60C2×#121—21no—
61C2×C4×D4—21no—
62C2×#117D411yescharacteristic
63#509—42no—
64#509—42no—
65C2×SD32C2×C414yes—
66C2×D16C2×C414yes—
67C2×M32C2312yes—
68C2×M32C2312yes—
69#466—21no—
70#466—21no—
71#466C2×C412yes—
72#466C2×C412yes—
73#442—42no—
74C2×#108—21no—
75C2×#108—42no—
76C2×#108C2312yes—
77C2×#107C2311yescharacteristic
78C2×#106C2311yescharacteristic
79C2×#106C2311yescharacteristic
80#421—44no—
81C2×#104—22no—
82C2×#104—22no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86#408—22no—
87#408—22no—
88#408—42no—
89#407—42no—
90#405—42no—
91C2×#97C2311yescharacteristic
92C2×#97C2311yescharacteristic
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95#390—21no—
96#390—41no—
97#389—21no—
98#388—41no—
99#386—41no—
100#385—41no—
101#380—21no—
102C4×C4⋊C4—21no—
103#365—42no—
104#361—42no—
105#360—42no—
106#360—42no—
107#358C2×C414yescomplemented
108#357C2×C414yescomplemented
109C22⋊C16—21no—
110C22⋊C16—21no—
111C22⋊C16C2×C412yescomplemented
112C22⋊C16C2×C412yescomplemented
113#347—42no—
114#340—21no—
115#340—21no—
116#340—21no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—42no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—42no—
123C2×#41—41no—
124C2×#41—41no—
125C22×Q8—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—41no—
129C22×D4—41no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133#135—22no—
134#135—22no—
135#135—22no—
136#135—22no—
137#135—42no—
138#135—42no—
139#135—42no—
140#135—42no—
141#135—44no—
142C2×Q16—21no—
143C2×Q16—42no—
144C2×Q16C22×C412yes—
145C2×SD16—22no—
146C2×SD16—22no—
147C2×SD16—44no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—22no—
151C2×D8—22no—
152C2×D8—42no—
153C2×D8C22×C412yes—
154C2×D8C22×C412yes—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—41no—
159C2×M16—41no—
160C2×M16—41no—
161C2×M16—42no—
162C2×M16—42no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—41no—
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8#4111yescharacteristic
172C22×C8#4111yescharacteristic
173C22×C8C2411yescharacteristic
174#121—22no—
175#121—22no—
176#121—41no—
177#121—41no—
178C4×D4—42no—
179C4×D4—42no—
180#117—22no—
181#117—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C42—41no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196SD32—44no—
197D16—44no—
198M32—22no—
199M32—22no—
200M32—22no—
201M32—22no—
202C2×C16—22no—
203C2×C16C22×C414yes—
204#108—22no—
205#108—22no—
206#108—42no—
207#107—21no—
208#107C22×C412yes—
209#106—22no—
210#106—22no—
211#104—44no—
212#104—44no—
213#102—22no—
214#102—22no—
215#102—42no—
216#102—42no—
217C22⋊C8—44no—
218C22⋊C8—44no—
219#97—22no—
220#97—22no—
221C4×C8—41no—
222C4×C8—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226C24—41no—
227C24—41no—
228C24AGL(1,ℤ/8)11yescharacteristic
229#41—44no—
230#41—44no—
231#41—42no—
232#41—42no—
233#41—42no—
234#41—42no—
235#41—82no—
236#41—82no—
237#41—82no—
238#41—82no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—82no—
258C2×D4—82no—
259C2×D4—161no—
260C2×D4—161no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—161no—
277C22×C4—161no—
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4AGL(1,ℤ/8)11yescharacteristic
281C22×C4AGL(1,ℤ/8)11yescharacteristic
282C22×C4AGL(1,ℤ/8)11yescharacteristic
283C22×C4C22×D411yescharacteristic
284Q16—22no—
285Q16—22no—
286Q16—42no—
287SD16—44no—
288SD16—42no—
289SD16—42no—
290D8—22no—
291D8—24no—
292D8—22no—
293D8—42no—
294D8—42no—
295M16—42no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—82no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—42no—
313C2×C8—82no—
314C2×C8C4×D411yescharacteristic
315C2×C8C4×D411yescharacteristic
316C2×C8C23×C411yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8H(5,𝔽2)12yes—
319C2×C8H(5,𝔽2)12yes—
320C4⋊C4—22no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—161no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—82no—
332C22⋊C4—161no—
333C22⋊C4—161no—
334C22⋊C4—161no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—161no—
339C42AGL(1,ℤ/8)12yes—
340C16—24no—
341C16—42no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—81no—
350C23—82no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—161no—
356C23—161no—
357C23#44211yescharacteristic
358C23#44211yescharacteristic
359C23C2×AGL(1,ℤ/8)11yescharacteristic
360Q8—42no—
361Q8—42no—
362Q8—82no—
363D4—42no—
364D4—42no—
365D4—42no—
366D4—42no—
367D4—82no—
368D4—84no—
369D4—82no—
370D4—82no—
371D4—81no—
372D4—81no—
373D4—162no—
374D4—162no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—84no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—164no—
406C2×C4#44211yescharacteristic
407C2×C4#44211yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4#53411yescharacteristic
410C2×C4#53511yescharacteristic
411C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
412C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
413C2×C4#57511yescharacteristic
414C8—21no—
415C8—21no—
416C8—41no—
417C8—41no—
418C8—82no—
419C8#51812yes—
420C8#53812yes—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—81no—
430C22—81no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22—82no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—164no—
442C22C2×#44211yescharacteristic
443C22#319211yescharacteristic
444C22#321311yescharacteristic
445C4—21no—
446C4—21no—
447C4—42no—
448C4—81no—
449C4—82no—
450C4—82no—
451C4—161no—
452C4—161no—
453C4#232111yescharacteristic
454C4C4×AGL(1,ℤ/8)11yescharacteristic
455C4#319411yescharacteristic
456C4#321711yescharacteristic
457C2—21no—
458C2—21no—
459C2—41no—
460C2—41no—
461C2—162no—
462C2—162no—
463C2#2204611yescharacteristic
464C2#3346911yescharacteristic
465C2#3347111yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#33520C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21529C2semidirect
3. #21240C2semidirect
4. #13249C2semidirect
5. #13153C2semidirect
6. #13143C2semidirect
7. #13037C2semidirect
8. #13004C2semidirect
9. #12928C2semidirect
10. #12735C2semidirect
11. #12606C2semidirect
15. C2×#442C22semidirect
21. #2186C4semidirect
22. #2185C4semidirect
23. #2184C4semidirect
24. C2×#365C22semidirect
25. C2×#361C22semidirect
31. C2×#347C22semidirect
36. #1972C22semidirect
37. #1943C22semidirect
41. #1776C22semidirect
43. #1387C22semidirect
44. #1383C22semidirect
45. #1375C22semidirect
51. C2×#135C2×C4semidirect
53. C22×Q16C2×C4semidirect
55. C22×D8C2×C4semidirect
107. #358C2×C4semidirect
108. #357C2×C4semidirect
111. C22⋊C16C2×C4semidirect
112. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
#335202
#215292
#212402
#132492
#131532
#131432
#130372
#130042
#129282
#127352
#126062