Finite Group Databasebeta

← #496821

Subgroups — #496821

Order 512.

726 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33516C211yesmaximal, characteristic
2#21544C211yesmaximal, characteristic, complemented
3C2×#1974C211yesmaximal, characteristic, complemented
4#13241C212yesmaximal, complemented
5#13151C212yesmaximal, complemented
6#13134C212yesmaximal, complemented
7#13011C212yesmaximal, complemented
8#12990C211yesmaximal, characteristic, complemented
9#12922C211yesmaximal, characteristic, complemented
10#12734C211yesmaximal, characteristic, complemented
11C4×C22⋊C16C211yesmaximal, characteristic, complemented
12C22×#135C2211yescharacteristic
13C2×#508C2212yes—
14#3050C2211yescharacteristic
15C2×#443C2211yescharacteristic, complemented
16#2929—21no—
17C22×#108C2211yescharacteristic
18C22×#104—21no—
19C2×#408—21no—
20C22×C4×C8C2211yescharacteristic
21#2189C412yescomplemented
22#2188C411yescharacteristic, complemented
23#2187C411yescharacteristic, complemented
24#2186C412yescomplemented
25#2185C411yescharacteristic, complemented
26#2184C411yescharacteristic, complemented
27C2×#368C2212yescomplemented
28C2×#359C2212yes—
29C2×#359C2212yes—
30C2×#359C2212yescomplemented
31#2138C2212yes—
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C4×C16C2212yescomplemented
35#1974—21no—
36#1974—21no—
37#1974C2212yes—
38#1974C2212yes—
39#1974C2211yescharacteristic, complemented
40#1945C2211yescharacteristic, complemented
41#1937—21no—
42#1896—21no—
43#1810C2211yescharacteristic
44#1772C2211yescharacteristic, complemented
45#1764—21no—
46#1382C2212yescomplemented
47#1381C2212yescomplemented
48#1374C2212yescomplemented
49C22×#41—21no—
50C2×#135—21no—
51C2×#135—21no—
52C2×#135—42no—
53C2×#135—42no—
54C2×#135C2×C412yes—
55C2×#135C2311yescharacteristic
56C2×#135C2312yes—
57C22×Q16C2×C411yescharacteristic
58C22×SD16—21no—
59C22×D8C2×C411yescharacteristic
60C2×#131—21no—
61C22×M16—21no—
62C23×C8D411yescharacteristic
63C23×C8C2311yescharacteristic
64#525—21no—
65C22×C42D411yescharacteristic
66#508—42no—
67#508—42no—
68C2×Q32C2×C412yescomplemented
69C2×SD32C2×C412yescomplemented
70C2×SD32C2×C412yescomplemented
71C2×D16C2×C412yescomplemented
72C22×C16C2312yes—
73C22×C16C2312yes—
74#467C2×C411yescharacteristic
75#467C2×C411yescharacteristic
76#466C2×C411yescharacteristic
77#466C2×C411yescharacteristic
78#465—21no—
79#465—21no—
80#443—42no—
81#435—41no—
82#435—41no—
83C8×D4—41no—
84C8×D4—41no—
85C2×#108—21no—
86C2×#108—42no—
87C2×#108C2311yescharacteristic
88C2×#108C2311yescharacteristic
89C2×#108C2312yes—
90#427C2311yescharacteristic
91#424—21no—
92C2×#104—22no—
93C2×#104—22no—
94C2×#104—21no—
95C2×#104—21no—
96C2×#104—44no—
97#416—21no—
98#408—22no—
99#408—22no—
100#408—21no—
101#408—41no—
102#408—41no—
103#407—21no—
104C2×C22⋊C8—41no—
105C2×C22⋊C8—41no—
106#405—21no—
107C2×C4×C8—42no—
108C2×C4×C8D411yescharacteristic
109C2×C4×C8D411yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#368—42no—
113#359—42no—
114#359—42no—
115#359—42no—
116#358C2×C412yescomplemented
117#358C2×C412yescomplemented
118#357C2×C412yescomplemented
119#357C2×C412yescomplemented
120C22⋊C16—21no—
121C22⋊C16—21no—
122C22⋊C16C2×C411yescharacteristic
123C22⋊C16C2×C411yescharacteristic
124C22⋊C16C2×C412yes—
125C4×C16—42no—
126#341—21no—
127#339—21no—
128#339—21no—
129C2×#41—22no—
130C2×#41—21no—
131C2×#41—22no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41—41no—
137C2×#41—41no—
138C22×Q8—21no—
139C22×D4—21no—
140C22×D4—41no—
141C23×C4—21no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144#135—22no—
145#135—22no—
146#135—22no—
147#135—22no—
148#135—22no—
149#135—22no—
150#135—42no—
151#135—42no—
152#135—42no—
153#135—42no—
154#135—44no—
155C2×Q16—21no—
156C2×Q16—42no—
157C2×Q16C22×C412yes—
158C2×Q16C22×C411yescharacteristic
159C2×SD16—22no—
160C2×SD16—22no—
161C2×SD16—21no—
162C2×SD16—44no—
163C2×D8—21no—
164C2×D8—42no—
165C2×D8C22×C412yes—
166C2×D8C22×C411yescharacteristic
167#131—42no—
168#131—42no—
169C2×M16—22no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—41no—
173C2×M16—41no—
174C2×M16—42no—
175C2×M16—42no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—41no—
179C22×C8—41no—
180C22×C8—41no—
181C22×C8—41no—
182C22×C8D811yescharacteristic
183C22×C8D811yescharacteristic
184C22×C8D811yescharacteristic
185C22×C8D811yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8#4111yescharacteristic
192C22×C8#4111yescharacteristic
193C22×C8C2411yescharacteristic
194#123—41no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197#121—21no—
198#121—21no—
199#120—41no—
200C4×D4—41no—
201C4×D4—41no—
202#117—21no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208Q32—42no—
209SD32—42no—
210SD32—42no—
211D16—42no—
212C2×C16—22no—
213C2×C16—22no—
214C2×C16—22no—
215C2×C16—22no—
216C2×C16—22no—
217C2×C16C22×C412yes—
218C2×C16C22×C412yes—
219#108—22no—
220#108—22no—
221#108—22no—
222#108—22no—
223#108—42no—
224#107C22×C411yescharacteristic
225#107C22×C411yescharacteristic
226#106—21no—
227C4⋊C8—41no—
228C4⋊C8—41no—
229#104—44no—
230#104—42no—
231#104—42no—
232#104—44no—
233#103—21no—
234#103—21no—
235#102—21no—
236#102—21no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243C22⋊C8—41no—
244C22⋊C8—41no—
245#97—41no—
246C4×C8—22no—
247C4×C8—22no—
248C4×C8—41no—
249C4×C8—41no—
250C4×C8—41no—
251C24—41no—
252C24#13511yescharacteristic
253#41—44no—
254#41—44no—
255#41—41no—
256#41—42no—
257#41—42no—
258#41—41no—
259#41—42no—
260#41—41no—
261#41—41no—
262#41—42no—
263#41—82no—
264#41—82no—
265#41—82no—
266#41—82no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—82no—
283C2×D4—82no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4#13511yescharacteristic
304C22×C4C22×D411yescharacteristic
305Q16—22no—
306Q16—22no—
307Q16—22no—
308Q16—42no—
309SD16—44no—
310SD16—42no—
311SD16—42no—
312SD16—42no—
313D8—22no—
314D8—22no—
315D8—22no—
316D8—42no—
317M16—41no—
318M16—42no—
319M16—41no—
320M16—42no—
321M16—41no—
322M16—41no—
323M16—82no—
324M16—82no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—82no—
343C2×C8—82no—
344C2×C8C4×D411yescharacteristic
345C2×C8C4×D411yescharacteristic
346C2×C8C2×D811yescharacteristic
347C2×C8C2×D811yescharacteristic
348C2×C8C2×D811yescharacteristic
349C2×C8C2×D811yescharacteristic
350C2×C8C23×C411yescharacteristic
351C2×C8C2×#4111yescharacteristic
352C2×C8H(5,𝔽2)12yes—
353C2×C8H(5,𝔽2)12yes—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—42no—
367C42—41no—
368C42AGL(1,ℤ/8)12yes—
369C16—22no—
370C16—22no—
371C16—42no—
372C23—21no—
373C23—22no—
374C23—21no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23C4×D811yescharacteristic
382C23C4×D811yescharacteristic
383C23C2×#13511yescharacteristic
384Q8—42no—
385Q8—41no—
386Q8—42no—
387Q8—41no—
388Q8—82no—
389D4—42no—
390D4—41no—
391D4—42no—
392D4—41no—
393D4—82no—
394D4—82no—
395D4—84no—
396D4—82no—
397D4—81no—
398D4—82no—
399D4—82no—
400D4—81no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—48no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4C4×D811yescharacteristic
429C2×C4C4×D811yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4#53411yescharacteristic
432C2×C4#53511yescharacteristic
433C2×C4C22×D811yescharacteristic
434C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
435C2×C4#57511yescharacteristic
436C8—21no—
437C8—21no—
438C8—21no—
439C8—21no—
440C8—21no—
441C8—21no—
442C8—81no—
443C8—82no—
444C8—81no—
445C8#51812yes—
446C8#53812yes—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—42no—
455C22—42no—
456C22—82no—
457C22—81no—
458C22—81no—
459C22—82no—
460C22—81no—
461C22—82no—
462C22—164no—
463C22#217812yes—
464C22C2×C4×D811yescharacteristic
465C22#318811yescharacteristic
466C22#321111yescharacteristic
467C4—24no—
468C4—21no—
469C4—21no—
470C4—42no—
471C4—82no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4#232111yescharacteristic
476C4C4×AGL(1,ℤ/8)11yescharacteristic
477C4#318611yescharacteristic
478C4#320511yescharacteristic
479C2—21no—
480C2—22no—
481C2—21no—
482C2—41no—
483C2—161no—
484C2—162no—
485C2#2204011yescharacteristic
486C2C2×#217811yescharacteristic
487C2#3346311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radical#33516C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21544C2semidirect
3. C2×#1974C2semidirect
4. #13241C2semidirect
5. #13151C2semidirect
6. #13134C2semidirect
7. #13011C2semidirect
8. #12990C2semidirect
9. #12922C2semidirect
10. #12734C2semidirect
11. C4×C22⋊C16C2semidirect
15. C2×#443C22semidirect
21. #2189C4semidirect
22. #2188C4semidirect
23. #2187C4semidirect
24. #2186C4semidirect
25. #2185C4semidirect
26. #2184C4semidirect
27. C2×#368C22semidirect
30. C2×#359C22semidirect
34. C2×C4×C16C22semidirect
39. #1974C22semidirect
40. #1945C22semidirect
44. #1772C22semidirect
46. #1382C22semidirect
47. #1381C22semidirect
48. #1374C22semidirect
68. C2×Q32C2×C4semidirect
69. C2×SD32C2×C4semidirect
70. C2×SD32C2×C4semidirect
71. C2×D16C2×C4semidirect
116. #358C2×C4semidirect
117. #358C2×C4semidirect
118. #357C2×C4semidirect
119. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
#335162
#215442
C2×#19742
#132412
#131512
#131342
#130112
#129902
#129222
#127342
C4×C22⋊C162