Finite Group Databasebeta

← #510202

Subgroups — #510202

Order 512.

565 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28228C211yesmaximal, characteristic
2#24779C211yesmaximal, characteristic
3#13537C211yesmaximal, characteristic, complemented
4#13494C211yesmaximal, characteristic, complemented
5#13481C211yesmaximal, characteristic, complemented
6#13382C211yesmaximal, characteristic, complemented
7#13306C211yesmaximal, characteristic, complemented
8#13306C211yesmaximal, characteristic, complemented
9#13060C211yesmaximal, characteristic, complemented
10#13060C211yesmaximal, characteristic, complemented
11#13015C211yesmaximal, characteristic, complemented
12#12994C211yesmaximal, characteristic, complemented
13#12976C211yesmaximal, characteristic
14#12938C211yesmaximal, characteristic, complemented
15#12868C211yesmaximal, characteristic, complemented
16C22×Q32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×#467C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C2×#427C2211yescharacteristic
25#2538—21no—
26#2415—21no—
27C2×#367C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37#2066—21no—
38#2036—21no—
39#2036—21no—
40#2035—21no—
41#1969C2211yescharacteristic
42#1942C2211yescharacteristic
43#1873—21no—
44#1845C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1803—21no—
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×Q16C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×D4D411yescharacteristic
69C2×#117—21no—
70C2×Q32—22no—
71C2×Q32—41no—
72C2×Q32—41no—
73C22×C16—41no—
74C22×C16—41no—
75C22×C16—41no—
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80#467—22no—
81#467—22no—
82#467—22no—
83#467—41no—
84#467—41no—
85#467—41no—
86#467—41no—
87#467—41no—
88C4×Q16—41no—
89C4×Q16—41no—
90C8×D4—41no—
91C8×D4—41no—
92#427—22no—
93#427—41no—
94C2×#107—21no—
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#106—21no—
102C2×C4⋊C8D411yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×C22⋊C8D411yescharacteristic
110C2×C22⋊C8D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#397—41no—
113#396—41no—
114#396—41no—
115#394—41no—
116#391—21no—
117#390—41no—
118#389—21no—
119#389—41no—
120#389—41no—
121#388—41no—
122#384—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#367—41no—
127#358—22no—
128#358—41no—
129#358—41no—
130#358—41no—
131#358—41no—
132#358—41no—
133C22⋊C16—41no—
134C22⋊C16—41no—
135C22⋊C16—41no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×D4D811yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16#4112yes—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2411yescharacteristic
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—81no—
179C22⋊Q8—81no—
180C22⋊Q8—81no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×D4—41no—
184C4×D4—41no—
185#117—42no—
186#117—81no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4D811yescharacteristic
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4D811yescharacteristic
211C2×C22⋊C4D811yescharacteristic
212C2×C42—41no—
213C2×C42—41no—
214C2×C42C2×D411yescharacteristic
215Q32—44no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—44no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—21no—
235#107—21no—
236#107—21no—
237#107—21no—
238#107—21no—
239#107—21no—
240#107—81no—
241#107#4112yes—
242#106—42no—
243C4⋊C8—41no—
244#103—42no—
245#103—42no—
246#103—42no—
247#103—42no—
248#103—42no—
249#103—81no—
250C22⋊C8—41no—
251C22⋊C8—41no—
252C4×C8—21no—
253C4×C8—21no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260C24C2×D811yescharacteristic
261C24C2×D811yescharacteristic
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—41no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4C2×D811yescharacteristic
298C22×C4C2×D811yescharacteristic
299C22×C4C2×D811yescharacteristic
300C22×C4C2×D811yescharacteristic
301C22×C4C22×D411yescharacteristic
302Q16—22no—
303Q16—22no—
304Q16—41no—
305Q16—41no—
306Q16—41no—
307Q16—41no—
308Q16—41no—
309Q16—41no—
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—21no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8C2×#4111yescharacteristic
328C2×C8C2×#4111yescharacteristic
329C2×C8H(5,𝔽2)11yescharacteristic
330C2×C8H(5,𝔽2)11yescharacteristic
331C2×C8H(5,𝔽2)11yescharacteristic
332C2×C8H(5,𝔽2)11yescharacteristic
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—161no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—161no—
363C22⋊C4—161no—
364C42—21no—
365C42—21no—
366C42—82no—
367C42—82no—
368C16—41no—
369C16—41no—
370C16—41no—
371C16—41no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23#51012yes—
386C23C22×D811yescharacteristic
387Q8—42no—
388Q8—42no—
389Q8—81no—
390Q8—81no—
391Q8—81no—
392Q8—81no—
393Q8—81no—
394Q8—81no—
395D4—44no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—81no—
418C2×C4—81no—
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—81no—
438C2×C4—81no—
439C2×C4#50812yes—
440C2×C4#53411yescharacteristic
441C2×C4#54611yescharacteristic
442C2×C4#54611yescharacteristic
443C2×C4#57511yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57711yescharacteristic
446C8—41no—
447C8—41no—
448C8—41no—
449C8#53811yescharacteristic
450C8#54011yescharacteristic
451C8#55311yescharacteristic
452C8#56011yescharacteristic
453C22—22no—
454C22—22no—
455C22—22no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22#318611yescharacteristic
468C22#329411yescharacteristic
469C22#329411yescharacteristic
470C22C2×#50811yescharacteristic
471C22C2×#51011yescharacteristic
472C22#341611yescharacteristic
473C22#341711yescharacteristic
474C4—22no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4—82no—
479C4—82no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4#268811yescharacteristic
487C4#320411yescharacteristic
488C4#330711yescharacteristic
489C4#332111yescharacteristic
490C2—42no—
491C2—81no—
492C2#3138211yescharacteristic
493C2#3375611yescharacteristic
494C2#3376711yescharacteristic
495C2#3386911yescharacteristic
496C2#3387111yescharacteristic
497C2#3387911yescharacteristic
498C2#3388311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13537C2semidirect
4. #13494C2semidirect
5. #13481C2semidirect
6. #13382C2semidirect
7. #13306C2semidirect
8. #13306C2semidirect
9. #13060C2semidirect
10. #13060C2semidirect
11. #13015C2semidirect
12. #12994C2semidirect
14. #12938C2semidirect
15. #12868C2semidirect

Maximal subgroups

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

subgroupindex
#282282
#247792
#135372
#134942
#134812
#133822
#133062
#130602
#130152
#129942
#129762
#129382
#128682