Finite Group Databasebeta

← #505171

Subgroups — #505171

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2224C211yesmaximal, characteristic, complemented
2C2×#2187C211yesmaximal, characteristic
3#27792C211yesmaximal, characteristic, complemented
4C22⋊C4×D8C211yesmaximal, characteristic
5#13493C211yesmaximal, characteristic, complemented
6#13449C211yesmaximal, characteristic, complemented
7#13383C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic, complemented
9#13238C211yesmaximal, characteristic, complemented
10#13130C211yesmaximal, characteristic, complemented
11#13121C211yesmaximal, characteristic, complemented
12#13014C211yesmaximal, characteristic, complemented
13#13014C211yesmaximal, characteristic, complemented
14#12731C211yesmaximal, characteristic
15#12637C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C22×SD32C2211yescharacteristic
19C2×#500C2211yescharacteristic, complemented
20C2×#467C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23#2492—21no—
24D4×C22⋊C4—21no—
25#2224—24no—
26#2187—24no—
27C2×#366C2211yescharacteristic, complemented
28C2×#366C2211yescharacteristic, complemented
29C2×C4⋊C16C2211yescharacteristic, complemented
30C2×#358C2211yescharacteristic, complemented
31C2×#358C2211yescharacteristic, complemented
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×#346C2211yescharacteristic, complemented
41#2036—21no—
42#2024—21no—
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1969C2211yescharacteristic, complemented
46#1969C2211yescharacteristic, complemented
47#1928—21no—
48#1879—21no—
49#1879—21no—
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52#1786—21no—
53C4×#107C2211yescharacteristic, complemented
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1374C2211yescharacteristic, complemented
60#1374C2211yescharacteristic, complemented
61C23×D4—21no—
62C22×Q16C2311yescharacteristic
63C22×D8—22no—
64C22×D8—22no—
65C22×D8—22no—
66C22×D8—22no—
67C22×D8—21no—
68C22×D8—41no—
69C22×D8—41no—
70C22×D8C2311yescharacteristic
71C22×D8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C2×C4⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C22×C22⋊C4—41no—
78C2×SD32—22no—
79C2×SD32—22no—
80C2×SD32—22no—
81C2×SD32—22no—
82C2×SD32—22no—
83C2×SD32—41no—
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88#500—22no—
89#500—41no—
90#467—22no—
91#467—41no—
92C4×D8—22no—
93C4×D8—22no—
94C4×D8—41no—
95C4×D8—41no—
96C2×#107—21no—
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×C22⋊C8D411yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#400—41no—
113#393—41no—
114#391—21no—
115#391—21no—
116#391—41no—
117#390—21no—
118#390—21no—
119#388—41no—
120#386—41no—
121#386—41no—
122#380—21no—
123#380—21no—
124#379—41no—
125C4×C4⋊C4—21no—
126C4×C22⋊C4D411yescharacteristic
127#366—22no—
128#366—22no—
129C4⋊C16—22no—
130#358—41no—
131#358—41no—
132#357—22no—
133#357—22no—
134#357—22no—
135#357—22no—
136#357—41no—
137#357D414yes—
138C22⋊C16—22no—
139C22⋊C16—41no—
140#346—41no—
141#340—21no—
142#340—21no—
143#340—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146C25—41no—
147C22×Q8—21no—
148C22×Q8—41no—
149C22×D4—24no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—41no—
156C22×D4—81no—
157C22×D4—81no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×D8—24no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—42no—
177C2×D8—42no—
178C2×D8—44no—
179C2×D8—44no—
180C2×D8—44no—
181C2×D8—44no—
182C2×D8C2×D412yes—
183C2×D8#4112yes—
184C2×D8#4112yes—
185C2×D8#4112yes—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2411yescharacteristic
196C4⋊Q8—42no—
197C4⋊Q8—81no—
198C22⋊Q8—42no—
199C22⋊Q8—81no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—81no—
203C4×D4—81no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—84no—
221C2×C22⋊C4D811yescharacteristic
222C2×C22⋊C4D811yescharacteristic
223C2×C42—41no—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226SD32—44no—
227SD32—44no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16C2×D412yes—
240#107—21no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—42no—
250#107C2×D412yes—
251#103—42no—
252#103—42no—
253#103—81no—
254#102—42no—
255#102—42no—
256#102—81no—
257#102—81no—
258C22⋊C8—41no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95D811yescharacteristic
271#95D811yescharacteristic
272C24—44no—
273C24—41no—
274C24—41no—
275C24—82no—
276C24—82no—
277C24—82no—
278C24—82no—
279C24—161no—
280C24—161no—
281C24C2×D811yescharacteristic
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—82no—
289C2×Q8—82no—
290C2×Q8—161no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—44no—
296C2×D4—48no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—82no—
306C2×D4—82no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—84no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—161no—
327C22×C4—161no—
328C22×C4C2×D811yescharacteristic
329C22×C4C2×D811yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4C2×D811yescharacteristic
332C22×C4C2×D811yescharacteristic
333C22×C4C22×D411yescharacteristic
334Q16—41no—
335Q16—41no—
336Q16—41no—
337Q16—41no—
338D8—22no—
339D8—22no—
340D8—22no—
341D8—22no—
342D8—22no—
343D8—48no—
344D8—41no—
345D8—41no—
346D8#12414yes—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8C22×D411yescharacteristic
361C2×C8C2×#4111yescharacteristic
362C2×C8C2×#4111yescharacteristic
363C2×C8C2×#4111yescharacteristic
364C2×C8H(5,𝔽2)11yescharacteristic
365C2×C8H(5,𝔽2)11yescharacteristic
366C4⋊C4—22no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—161no—
387C22⋊C4—161no—
388C22⋊C4—161no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—161no—
394C16—22no—
395C16—41no—
396C16—41no—
397C16—41no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—88no—
409C23—88no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—164no—
421C23—164no—
422C23—164no—
423C23—164no—
424C23—164no—
425C23#51012yes—
426C23C22×D811yescharacteristic
427Q8—81no—
428Q8—81no—
429Q8—81no—
430Q8—81no—
431Q8—164no—
432D4—24no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—88no—
439D4—81no—
440D4—81no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—164no—
478C2×C4—164no—
479C2×C4#50912yes—
480C2×C4#52511yescharacteristic
481C2×C4D4211yescharacteristic
482C2×C4#54611yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57611yescharacteristic
485C2×C4#57711yescharacteristic
486C8—41no—
487C8—41no—
488C8—41no—
489C8C2×#12411yescharacteristic
490C8#53811yescharacteristic
491C8#54611yescharacteristic
492C8#55111yescharacteristic
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—1616no—
511C22—161no—
512C22—161no—
513C22—161no—
514C22—161no—
515C22#310011yescharacteristic
516C22D4×D811yescharacteristic
517C22#329411yescharacteristic
518C22C2×#50911yescharacteristic
519C22C2×#51011yescharacteristic
520C22#341511yescharacteristic
521C22#341611yescharacteristic
522C4—22no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—82no—
528C4—161no—
529C4—161no—
530C4—161no—
531C4—161no—
532C4—161no—
533C4#247211yescharacteristic
534C4#311811yescharacteristic
535C4#327511yescharacteristic
536C4#331911yescharacteristic
537C2—42no—
538C2—84no—
539C2—162no—
540C2#2722211yescharacteristic
541C2#3367011yescharacteristic
542C2#3367811yescharacteristic
543C2#3383011yescharacteristic
544C2#3383211yescharacteristic
545C2#3388111yescharacteristic
546C2#3389111yescharacteristic
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. C2×#2224C2semidirect
3. #27792C2semidirect
5. #13493C2semidirect
6. #13449C2semidirect
7. #13383C2semidirect
8. #13363C2semidirect
9. #13238C2semidirect
10. #13130C2semidirect
11. #13121C2semidirect
12. #13014C2semidirect
13. #13014C2semidirect
15. #12637C2semidirect
19. C2×#500C22semidirect
27. C2×#366C22semidirect
28. C2×#366C22semidirect
29. C2×C4⋊C16C22semidirect
30. C2×#358C22semidirect
31. C2×#358C22semidirect
40. C2×#346C22semidirect
45. #1969C22semidirect
46. #1969C22semidirect
53. C4×#107C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1374C22semidirect
60. #1374C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22242
C2×#21872
#277922
C22⋊C4×D82
#134932
#134492
#133832
#133632
#132382
#131302
#131212
#130142
#127312
#126372