Finite Group Databasebeta

← #501348

Subgroups — #501348

Order 512.

811 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2187C211yesmaximal, characteristic
2C2×#2140C211yesmaximal, characteristic
3#27927C211yesmaximal, characteristic, complemented
4C22⋊C4×D8C211yesmaximal, characteristic
5#13482C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic, complemented
7#13449C211yesmaximal, characteristic, complemented
8#13449C211yesmaximal, characteristic, complemented
9#13062C211yesmaximal, characteristic, complemented
10#13014C211yesmaximal, characteristic, complemented
11#13014C211yesmaximal, characteristic, complemented
12#12913C211yesmaximal, characteristic, complemented
13#12868C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C22×SD32C2211yescharacteristic
19C22×M32C2211yescharacteristic
20C2×#485—21no—
21C2×#467C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C2×#427C2211yescharacteristic
25#2499—21no—
26D4×C22⋊C4—21no—
27#2187—24no—
28C2×C4⋊C16C2211yescharacteristic, complemented
29C2×C4⋊C16C2211yescharacteristic, complemented
30#2140—24no—
31C2×#358C2211yescharacteristic, complemented
32C2×#358C2211yescharacteristic, complemented
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40#2097—21no—
41#2096—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic, complemented
45#1969C2211yescharacteristic, complemented
46#1879—21no—
47#1879—21no—
48#1846C2211yescharacteristic, complemented
49#1845C2211yescharacteristic, complemented
50#1808C2211yescharacteristic
51#1786—21no—
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic, complemented
54#1380C2211yescharacteristic, complemented
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—21no—
66C22×D8—22no—
67C22×D8—41no—
68C22×D8—41no—
69C22×D8D414yes—
70C22×D8C2311yescharacteristic
71C22×D8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C2×C22⋊Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×#117—21no—
77C22×C22⋊C4—41no—
78C2×SD32—22no—
79C2×SD32—22no—
80C2×SD32—22no—
81C2×SD32—22no—
82C2×SD32—41no—
83C2×SD32—41no—
84C2×M32—22no—
85C2×M32—41no—
86C2×M32—41no—
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91#485—44no—
92#467—22no—
93#467—22no—
94C4×D8—22no—
95C4×D8—22no—
96C4×D8—41no—
97C4×D8—41no—
98#427—22no—
99#427—22no—
100C2×#107—21no—
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#106—21no—
107C2×#106—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×C22⋊C8D411yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#397—41no—
119#397—41no—
120#391—21no—
121#391—21no—
122#390—21no—
123#390—21no—
124#387—41no—
125#387—41no—
126#386—41no—
127#386—41no—
128#384—21no—
129#382—21no—
130#380—21no—
131#379—41no—
132C4×C22⋊C4D411yescharacteristic
133C4⋊C16—41no—
134C4⋊C16—41no—
135#358—41no—
136#358—41no—
137#357—22no—
138#357—22no—
139#357—22no—
140#357—22no—
141#357—22no—
142#357—22no—
143C22⋊C16—22no—
144#340—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147C25—41no—
148C22×Q8—21no—
149C22×D4—24no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—41no—
155C22×D4—42no—
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—28no—
168C2×D8—24no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—44no—
178C2×D8—44no—
179C2×D8—44no—
180C2×D8—42no—
181C2×D8—42no—
182C2×D8C2×D412yes—
183C2×D8C2×D412yes—
184C2×D8#4112yes—
185C2×D8#4112yes—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D412yes—
194C22×C8#4112yes—
195C22×C8C2411yescharacteristic
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—81no—
201C4×D4—81no—
202#117—42no—
203#117—42no—
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×C4⋊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—41no—
221C2×C22⋊C4—84no—
222C2×C22⋊C4D811yescharacteristic
223C2×C22⋊C4D811yescharacteristic
224C2×C42—41no—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227SD32—44no—
228SD32—44no—
229M32—44no—
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×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—42no—
253#106—42no—
254#106—42no—
255#103—42no—
256#103—42no—
257#103—42no—
258#102—42no—
259#102—42no—
260#102—81no—
261#102—81no—
262C22⋊C8—22no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95D811yescharacteristic
275#95D811yescharacteristic
276C24—41no—
277C24—41no—
278C24—44no—
279C24—82no—
280C24—82no—
281C24—82no—
282C24—82no—
283C24—161no—
284C24—161no—
285C24C2×D811yescharacteristic
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
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—44no—
306C2×D4—84no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—82no—
310C2×D4—82no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—161no—
326C22×C4—161no—
327C22×C4#12112yes—
328C22×C4C2×D811yescharacteristic
329C22×C4C2×D811yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4C2×D811yescharacteristic
332C22×C4C2×D811yescharacteristic
333C22×C4AGL(1,ℤ/8)12yes—
334C22×C4C22×D411yescharacteristic
335Q16—41no—
336Q16—41no—
337Q16—41no—
338Q16—41no—
339D8—22no—
340D8—22no—
341D8—22no—
342D8—22no—
343D8—22no—
344D8—48no—
345D8—41no—
346D8—41no—
347D8#12314yes—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8#12112yes—
361C2×C8C22×D411yescharacteristic
362C2×C8C2×#4111yescharacteristic
363C2×C8C2×#4111yescharacteristic
364C2×C8C2×#4111yescharacteristic
365C2×C8H(5,𝔽2)11yescharacteristic
366C2×C8H(5,𝔽2)11yescharacteristic
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—41no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C22⋊C4—22no—
381C22⋊C4—41no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—161no—
390C22⋊C4—161no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—82no—
396C16—41no—
397C16—41no—
398C16—41no—
399C16—41no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—88no—
411C23—88no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—164no—
423C23—164no—
424C23—164no—
425C23—164no—
426C23—164no—
427C23#45912yes—
428C23C22×D811yescharacteristic
429Q8—81no—
430Q8—81no—
431Q8—81no—
432Q8—81no—
433D4—24no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—88no—
440D4—81no—
441D4—81no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—81no—
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#45912yes—
480C2×C4#50912yes—
481C2×C4C2×#12111yescharacteristic
482C2×C4#54611yescharacteristic
483C2×C4#54611yescharacteristic
484C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
485C2×C4#57711yescharacteristic
486C2×C4#57711yescharacteristic
487C8—22no—
488C8—41no—
489C8—41no—
490C8C2×#12311yescharacteristic
491C8#53411yescharacteristic
492C8#55111yescharacteristic
493C8#55111yescharacteristic
494C22—22no—
495C22—22no—
496C22—22no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—44no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—1616no—
512C22—161no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22C2×#45911yescharacteristic
517C22#329411yescharacteristic
518C22#329411yescharacteristic
519C22C2×#50911yescharacteristic
520C22#341411yescharacteristic
521C22#341611yescharacteristic
522C22#341611yescharacteristic
523C4—22no—
524C4—22no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—161no—
529C4—161no—
530C4—161no—
531C4—161no—
532C4—161no—
533C4—161no—
534C4#241211yescharacteristic
535C4#304211yescharacteristic
536C4#330611yescharacteristic
537C4#330611yescharacteristic
538C2—42no—
539C2—84no—
540C2—162no—
541C2#2452611yescharacteristic
542C2#3358011yescharacteristic
543C2#3358611yescharacteristic
544C2#3387711yescharacteristic
545C2#3387811yescharacteristic
546C2#3389011yescharacteristic
547C2#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
3. #27927C2semidirect
5. #13482C2semidirect
6. #13480C2semidirect
7. #13449C2semidirect
8. #13449C2semidirect
9. #13062C2semidirect
10. #13014C2semidirect
11. #13014C2semidirect
12. #12913C2semidirect
13. #12868C2semidirect
14. #12813C2semidirect
15. #12811C2semidirect
28. C2×C4⋊C16C22semidirect
29. C2×C4⋊C16C22semidirect
31. C2×#358C22semidirect
32. C2×#358C22semidirect
44. #1969C22semidirect
45. #1969C22semidirect
48. #1846C22semidirect
49. #1845C22semidirect
53. #1380C22semidirect
54. #1380C22semidirect
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×#21872
C2×#21402
#279272
C22⋊C4×D82
#134822
#134802
#134492
#130622
#130142
#129132
#128682
#128132
#128112