Finite Group Databasebeta

← #498128

Subgroups — #498128

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2184C211yesmaximal, characteristic
2C2×C4×D16C211yesmaximal, characteristic, complemented
3C22×#357C211yesmaximal, characteristic, complemented
4#23025C211yesmaximal, characteristic, complemented
5C22⋊C4×D8C211yesmaximal, characteristic, complemented
6#13235C211yesmaximal, characteristic, complemented
7#13147C211yesmaximal, characteristic, complemented
8#13130C211yesmaximal, characteristic, complemented
9#13059C211yesmaximal, characteristic, complemented
10#13012C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic, complemented
12#12935C211yesmaximal, characteristic, complemented
13#12919C211yesmaximal, characteristic, complemented
14C4×#357C211yesmaximal, characteristic, complemented
15C16×C22⋊C4C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C22×D16C2211yescharacteristic
19C23×C16C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic, complemented
24C22×#107C2211yescharacteristic
25#2361—21no—
26D4×C22⋊C4—21no—
27#2184C418yescomplemented
28C4×D16—24no—
29C2×#366C2211yescharacteristic, complemented
30C2×#366C2211yescharacteristic, complemented
31C2×#357—24no—
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic, complemented
39C2×#357C2211yescharacteristic, complemented
40C2×C22⋊C16C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic, complemented
42C2×C4×C16C2211yescharacteristic, complemented
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic, complemented
46#1942C2211yescharacteristic, complemented
47#1925—21no—
48#1879—21no—
49#1879—21no—
50#1874—21no—
51#1862—21no—
52#1808C2211yescharacteristic
53#1786—21no—
54C4×#107C2211yescharacteristic, complemented
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1380C2211yescharacteristic, complemented
60#1374C2211yescharacteristic, complemented
61#1374C2211yescharacteristic, complemented
62C23×D4—21no—
63C22×D8—22no—
64C22×D8—22no—
65C22×D8—21no—
66C22×D8—22no—
67C22×D8—41no—
68C22×D8—41no—
69C22×D8C2×C414yescomplemented
70C22×D8C2311yescharacteristic
71C22×D8C2311yescharacteristic
72C22×D8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C2×#121—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—41no—
79C22×C4⋊C4—21no—
80C22×C22⋊C4—41no—
81C2×D16—22no—
82C2×D16—22no—
83C2×D16—41no—
84C2×D16—41no—
85C2×D16C2×C414yescomplemented
86C2×D16C2×C414yescomplemented
87C22×C16—22no—
88C22×C16—22no—
89C22×C16—41no—
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92C22×C16C2311yescharacteristic
93C22×C16C2311yescharacteristic
94#466—22no—
95#466C2×C414yescomplemented
96C4×D8—22no—
97C4×D8—22no—
98C4×D8—22no—
99C4×D8—41no—
100C4×D8—41no—
101C4×D8—41no—
102C2×#107—22no—
103C2×#107—21no—
104C2×#107—41no—
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×C22⋊C8D411yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#390—21no—
121#390—21no—
122#390—21no—
123#389—21no—
124#388—41no—
125#386—41no—
126#386—41no—
127#386—41no—
128#385—41no—
129#380—21no—
130#379—41no—
131C4×C4⋊C4—21no—
132C4×C22⋊C4D411yescharacteristic
133#366—22no—
134#366—41no—
135#357—22no—
136#357—22no—
137#357—22no—
138#357—22no—
139#357—41no—
140#357C2×C414yescomplemented
141#357C2×C414yescomplemented
142#357D414yescomplemented
143C22⋊C16C2×C414yescomplemented
144C4×C16—22no—
145C4×C16—41no—
146#340—21no—
147#340—21no—
148#340—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151C25—41no—
152C22×D4—24no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—42no—
161C22×D4—81no—
162C22×D4—81no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4—41no—
166C23×C4—41no—
167C23×C4C2×D411yescharacteristic
168C2×D8—28no—
169C2×D8—24no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—44no—
184C2×D8—44no—
185C2×D8—44no—
186C2×D8—42no—
187C2×D8—42no—
188C2×D8C22×C412yes—
189C2×D8C22×C412yes—
190C2×D8C22×C412yes—
191C2×D8C2×D412yes—
192C2×D8#4112yes—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8C22×C412yes—
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8#4112yes—
202C22×C8C2411yescharacteristic
203#121—24no—
204#121—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—81no—
209C4×D4—81no—
210C4×D4—81no—
211C4×D4—84no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—81no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—84no—
233C2×C22⋊C4D811yescharacteristic
234C2×C22⋊C4D811yescharacteristic
235C2×C42—41no—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238D16—28no—
239D16—44no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—44no—
250C2×C16C22×C412yes—
251C2×C16C22×C412yes—
252C2×C16C2×D412yes—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—21no—
259#107—21no—
260#107—21no—
261#107—42no—
262#107C22×C412yes—
263#107C2×D412yes—
264#102—24no—
265#102—42no—
266#102—42no—
267#102—42no—
268#102—81no—
269#102—81no—
270#102—81no—
271C22⋊C8—22no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95D811yescharacteristic
280#95D811yescharacteristic
281C24—41no—
282C24—41no—
283C24—41no—
284C24—44no—
285C24—82no—
286C24—82no—
287C24—82no—
288C24—82no—
289C24—161no—
290C24—161no—
291C24C2×D811yescharacteristic
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—48no—
298C2×D4—44no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—44no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4—82no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—161no—
338C22×C4—161no—
339C22×C4—161no—
340C22×C4—161no—
341C22×C4C4×D412yes—
342C22×C4C2×D811yescharacteristic
343C22×C4C2×D811yescharacteristic
344C22×C4C2×D811yescharacteristic
345C22×C4C2×D811yescharacteristic
346C22×C4C2×D811yescharacteristic
347C22×C4#13512yes—
348C22×C4C22×D411yescharacteristic
349D8—22no—
350D8—22no—
351D8—22no—
352D8—22no—
353D8—22no—
354D8—22no—
355D8—22no—
356D8—48no—
357D8—41no—
358D8—41no—
359D8—41no—
360D8—41no—
361D8C4×D414yescomplemented
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8C4×D412yes—
375C2×C8C23×C411yescharacteristic
376C2×C8C22×D411yescharacteristic
377C2×C8C2×#4111yescharacteristic
378C2×C8C2×#4111yescharacteristic
379C2×C8H(5,𝔽2)11yescharacteristic
380C2×C8H(5,𝔽2)11yescharacteristic
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C22⋊C4—22no—
394C22⋊C4—44no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—161no—
403C22⋊C4—161no—
404C22⋊C4—161no—
405C22⋊C4D1614yescomplemented
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—82no—
411C16—22no—
412C16—22no—
413C16—22no—
414C16—41no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—88no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—88no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—164no—
443C23—164no—
444C23—164no—
445C23—164no—
446C23—164no—
447C23C4×D812yes—
448C23C2×D1612yes—
449C23#50812yes—
450C23C22×D811yescharacteristic
451D4—24no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—88no—
460D4—88no—
461D4—81no—
462D4—81no—
463D4—81no—
464D4—81no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—164no—
497C2×C4—164no—
498C2×C4—164no—
499C2×C4—164no—
500C2×C4C4×D812yes—
501C2×C4C2×D1612yes—
502C2×C4C2×D1612yes—
503C2×C4C2×C4×D411yescharacteristic
504C2×C4D4211yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4C2×#13511yescharacteristic
507C2×C4#57611yescharacteristic
508C2×C4#57711yescharacteristic
509C8—22no—
510C8—41no—
511C8—41no—
512C8C2×C4×D411yescharacteristic
513C8#51811yescharacteristic
514C8#54611yescharacteristic
515C8#55111yescharacteristic
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—44no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—1616no—
536C22—161no—
537C22—161no—
538C22—161no—
539C22—161no—
540C22—161no—
541C22—161no—
542C22—161no—
543C22—161no—
544C22C4×D1614yescomplemented
545C22C2×C4×D811yescharacteristic
546C22D4×D811yescharacteristic
547C22#329411yescharacteristic
548C22C22×D1611yescharacteristic
549C22C2×#50811yescharacteristic
550C22#341511yescharacteristic
551C22#341611yescharacteristic
552C4—22no—
553C4—22no—
554C4—22no—
555C4—41no—
556C4—41no—
557C4—82no—
558C4—82no—
559C4—161no—
560C4—161no—
561C4—161no—
562C4#235811yescharacteristic
563C4#294611yescharacteristic
564C4#328211yescharacteristic
565C4#329711yescharacteristic
566C2—24no—
567C2—84no—
568C2—162no—
569C2—162no—
570C2#2287011yescharacteristic
571C2C2×C4×D1611yescharacteristic
572C2#3345411yescharacteristic
573C2D4×D1611yescharacteristic
574C2#3384811yescharacteristic
575C2#3386111yescharacteristic
576C2#3386411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2184C2
LCSG ▷ C2×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
2. C2×C4×D16C2semidirect
3. C22×#357C2semidirect
4. #23025C2semidirect
5. C22⋊C4×D8C2semidirect
6. #13235C2semidirect
7. #13147C2semidirect
8. #13130C2semidirect
9. #13059C2semidirect
10. #13012C2semidirect
11. #13012C2semidirect
12. #12935C2semidirect
13. #12919C2semidirect
14. C4×#357C2semidirect
15. C16×C22⋊C4C2semidirect
23. C2×C4×D8C22semidirect
27. #2184C4semidirect
29. C2×#366C22semidirect
30. C2×#366C22semidirect
38. C2×#357C22semidirect
39. C2×#357C22semidirect
41. C2×C4×C16C22semidirect
42. C2×C4×C16C22semidirect
45. #1970C22semidirect
46. #1942C22semidirect
54. C4×#107C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1380C22semidirect
60. #1374C22semidirect
61. #1374C22semidirect
69. C22×D8C2×C4semidirect
85. C2×D16C2×C4semidirect
86. C2×D16C2×C4semidirect
95. #466C2×C4semidirect
140. #357C2×C4semidirect
141. #357C2×C4semidirect
142. #357D4semidirect
143. C22⋊C16C2×C4semidirect
361. D8C4×D4semidirect
405. C22⋊C4D16semidirect
544. C22C4×D16semidirect

Maximal subgroups

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

subgroupindex
C2×#21842
C2×C4×D162
C22×#3572
#230252
C22⋊C4×D82
#132352
#131472
#131302
#130592
#130122
#129352
#129192
C4×#3572
C16×C22⋊C42