Finite Group Databasebeta

← #505488

Subgroups — #505488

Order 512.

710 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27840C211yesmaximal, characteristic, complemented
2C22⋊C4×D8C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic, complemented
4#13494C211yesmaximal, characteristic, complemented
5#13480C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic, complemented
7#13147C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13120C211yesmaximal, characteristic, complemented
10#12937C211yesmaximal, characteristic, complemented
11#12937C211yesmaximal, characteristic, complemented
12#12811C211yesmaximal, characteristic, complemented
13#12731C211yesmaximal, characteristic
14#12731C211yesmaximal, characteristic
15C4×#357C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2497—21no—
20D4×C22⋊C4—21no—
21C2×#367C2211yescharacteristic, complemented
22C2×#367C2211yescharacteristic, complemented
23C2×C4⋊C16C2211yescharacteristic, complemented
24C2×#357C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic, complemented
35#2100—21no—
36#2076—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1942C2211yescharacteristic, complemented
40#1942C2211yescharacteristic, complemented
41#1921—21no—
42#1879—21no—
43#1879—21no—
44#1845C2211yescharacteristic, complemented
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1786—21no—
49C4×#107C2211yescharacteristic, complemented
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic, complemented
52#1380C2211yescharacteristic, complemented
53#1380C2211yescharacteristic, complemented
54#1380C2211yescharacteristic, complemented
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1374C2211yescharacteristic, complemented
58#1374C2211yescharacteristic, complemented
59C23×D4—21no—
60C22×D8—22no—
61C22×D8—22no—
62C22×D8—22no—
63C22×D8—22no—
64C22×D8—21no—
65C22×D8—41no—
66C22×D8—41no—
67C22×D8C2311yescharacteristic
68C22×D8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C22×C22⋊C4—41no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C4×D8—22no—
78C4×D8—22no—
79C4×D8—41no—
80C4×D8—41no—
81C2×#107—21no—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×C22⋊C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#400—41no—
96#396—41no—
97#390—21no—
98#390—21no—
99#389—21no—
100#389—21no—
101#388—41no—
102#387—41no—
103#386—41no—
104#386—41no—
105#384—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#379—41no—
110C4×C4⋊C4—21no—
111C4×C22⋊C4D411yescharacteristic
112#367—41no—
113#367—41no—
114C4⋊C16—41no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—22no—
119#357—22no—
120#357—22no—
121#357—41no—
122#357—41no—
123C22⋊C16—41no—
124C22⋊C16—41no—
125C4×C16—41no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C25—41no—
135C22×D4—24no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—41no—
142C22×D4—81no—
143C22×D4—81no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C2×D8—24no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—42no—
157C2×D8—42no—
158C2×D8—44no—
159C2×D8—44no—
160C2×D8—44no—
161C2×D8—44no—
162C2×D8#4112yes—
163C2×D8#4112yes—
164C2×D8#4112yes—
165C2×D8#4112yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2411yescharacteristic
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—81no—
179C4×D4—81no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—84no—
200C2×C22⋊C4D811yescharacteristic
201C2×C22⋊C4D811yescharacteristic
202C2×C42—41no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—42no—
230#102—42no—
231#102—42no—
232#102—81no—
233#102—81no—
234C22⋊C8—41no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95D811yescharacteristic
246#95D811yescharacteristic
247C24—44no—
248C24—41no—
249C24—41no—
250C24—82no—
251C24—82no—
252C24—82no—
253C24—82no—
254C24—161no—
255C24—161no—
256C24C2×D811yescharacteristic
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—84no—
274C2×D4—84no—
275C2×D4—84no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—161no—
293C22×C4—161no—
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4C2×D811yescharacteristic
299C22×C4C22×D411yescharacteristic
300D8—22no—
301D8—22no—
302D8—22no—
303D8—22no—
304D8—22no—
305D8—48no—
306D8—41no—
307D8—41no—
308D8#12614yes—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8C2×#4111yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8C2×#4111yescharacteristic
326C2×C8H(5,𝔽2)11yescharacteristic
327C2×C8H(5,𝔽2)11yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—82no—
341C4⋊C4—161no—
342C4⋊C4—161no—
343C4⋊C4—161no—
344C22⋊C4—41no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—161no—
351C22⋊C4—161no—
352C22⋊C4—161no—
353C22⋊C4—161no—
354C22⋊C4SD3214yes—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—161no—
360C16—41no—
361C16—41no—
362C16—41no—
363C16—41no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—88no—
375C23—88no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—164no—
387C23—164no—
388C23—164no—
389C23—164no—
390C23—164no—
391C23C2×SD3212yes—
392C23C22×D811yescharacteristic
393D4—24no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—42no—
398D4—42no—
399D4—88no—
400D4—81no—
401D4—81no—
402C2×C4—21no—
403C2×C4—21no—
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—21no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—164no—
438C2×C4—164no—
439C2×C4C2×SD3212yes—
440C2×C4C2×SD3212yes—
441C2×C4#52511yescharacteristic
442C2×C4#54611yescharacteristic
443C2×C4#54611yescharacteristic
444C2×C4#57511yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57711yescharacteristic
447C8—41no—
448C8—41no—
449C8—41no—
450C8C2×#12611yescharacteristic
451C8#53811yescharacteristic
452C8#55111yescharacteristic
453C8#55111yescharacteristic
454C22—22no—
455C22—22no—
456C22—22no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—44no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—1616no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22#310011yescharacteristic
477C22#329411yescharacteristic
478C22#329411yescharacteristic
479C22C22×SD3211yescharacteristic
480C22#341411yescharacteristic
481C22#341511yescharacteristic
482C22#341511yescharacteristic
483C4—22no—
484C4—22no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—161no—
489C4—161no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4#248011yescharacteristic
495C4#312311yescharacteristic
496C4#331111yescharacteristic
497C4#332311yescharacteristic
498C2—42no—
499C2—84no—
500C2—162no—
501C2#2751811yescharacteristic
502C2#3366111yescharacteristic
503C2#3368211yescharacteristic
504C2#3386211yescharacteristic
505C2#3386211yescharacteristic
506C2#3388611yescharacteristic
507C2#3388911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #27840C2semidirect
3. #13494C2semidirect
4. #13494C2semidirect
5. #13480C2semidirect
6. #13476C2semidirect
7. #13147C2semidirect
8. #13131C2semidirect
9. #13120C2semidirect
10. #12937C2semidirect
11. #12937C2semidirect
12. #12811C2semidirect
15. C4×#357C2semidirect
21. C2×#367C22semidirect
22. C2×#367C22semidirect
23. C2×C4⋊C16C22semidirect
34. C2×C4×C16C22semidirect
39. #1942C22semidirect
40. #1942C22semidirect
44. #1845C22semidirect
49. C4×#107C22semidirect
51. #1380C22semidirect
52. #1380C22semidirect
53. #1380C22semidirect
54. #1380C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1374C22semidirect
58. #1374C22semidirect

Maximal subgroups

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

subgroupindex
#278402
C22⋊C4×D82
#134942
#134802
#134762
#131472
#131312
#131202
#129372
#128112
#127312
C4×#3572