Finite Group Databasebeta

← #503518

Subgroups — #503518

Order 512.

823 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic, complemented
2C2×#2243C211yesmaximal, characteristic, complemented
3C2×#2242C211yesmaximal, characteristic, complemented
4#33182C211yesmaximal, characteristic
5#13366C211yesmaximal, characteristic, complemented
6#13365C211yesmaximal, characteristic, complemented
7#13364C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic, complemented
9#13284C211yesmaximal, characteristic, complemented
10#13282C211yesmaximal, characteristic, complemented
11#12994C211yesmaximal, characteristic, complemented
12#12993C211yesmaximal, characteristic, complemented
13#12992C211yesmaximal, characteristic
14#12991C211yesmaximal, characteristic
15#12600C211yesmaximal, characteristic, complemented
16C2×#544—21no—
17#3164—24no—
18#3164—24no—
19C2×#501—21no—
20C2×#501—21no—
21C2×#500C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#499C2211yescharacteristic
24C2×#499C2211yescharacteristic
25C2×#495C2211yescharacteristic
26C2×#468—21no—
27C2×#467C2211yescharacteristic
28C2×#466C2211yescharacteristic
29C2×#428—21no—
30C2×#427C2211yescharacteristic
31C22×#107C2211yescharacteristic
32C2×#405C2211yescharacteristic
33#2620—21no—
34#2243—24no—
35#2242—24no—
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#357C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×C22⋊C16C2211yescharacteristic
45C2×C22⋊C16C2211yescharacteristic
46C2×#346C2211yescharacteristic
47C2×C4×C16C2211yescharacteristic
48#1973—21no—
49#1970C2211yescharacteristic
50#1969C2211yescharacteristic
51#1959—21no—
52#1845C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1374C2211yescharacteristic
62#1374C2211yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×SD16—21no—
65C22×D8C2311yescharacteristic
66C22×M16D411yescharacteristic
67C23×C8C2311yescharacteristic
68#544—28no—
69#544—44no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—41no—
73C2×#125—21no—
74C2×#125—21no—
75C2×#125—41no—
76C2×#124—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—41no—
79C2×C22⋊Q8—41no—
80C2×#121—21no—
81C2×#117—21no—
82C2×#117D411yescharacteristic
83C22×C4⋊C4—21no—
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88#501—28no—
89#501—28no—
90#500—22no—
91#500—22no—
92#500—22no—
93#500D414yescomplemented
94#499—22no—
95#499—22no—
96#499—22no—
97#499D414yescomplemented
98#495—22no—
99#495—22no—
100#468—44no—
101#467—22no—
102#467—22no—
103#466—22no—
104#466—22no—
105#428—28no—
106#427—22no—
107#427—22no—
108C2×#107—21no—
109C2×#107—21no—
110C2×#107—21no—
111C2×#107—22no—
112C2×#107—22no—
113C2×#107—21no—
114C2×#107C2311yescharacteristic
115C2×#107C2311yescharacteristic
116C2×#107C2311yescharacteristic
117C2×#107C2311yescharacteristic
118C2×#107C2311yescharacteristic
119C2×#107C2311yescharacteristic
120C2×#106—21no—
121C2×#106—21no—
122C2×#106—21no—
123C2×#103—21no—
124C2×#103—21no—
125C2×#102—21no—
126C2×#102—21no—
127#405—22no—
128#405—22no—
129C2×#97D411yescharacteristic
130C2×#97D411yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#396—41no—
134#396—41no—
135#394—41no—
136#394—41no—
137#391—21no—
138#390—21no—
139#384—21no—
140#358—22no—
141#358—22no—
142#358—22no—
143#358—22no—
144#357—22no—
145#357—22no—
146#357—22no—
147#357—22no—
148C22⋊C16—22no—
149C22⋊C16—22no—
150#346—22no—
151C4×C16—22no—
152C22×Q8—21no—
153C22×Q8—41no—
154C22×Q8—41no—
155C22×D4—21no—
156C22×D4—41no—
157C23×C4—41no—
158C23×C4C2×D411yescharacteristic
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×SD16—42no—
166C2×SD16—42no—
167C2×SD16—42no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16SD1614yes—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8#4112yes—
185C22×C8#4112yes—
186C22×C8C2411yescharacteristic
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189C4⋊Q8—42no—
190C4⋊Q8—48no—
191C4⋊Q8—42no—
192#125—24no—
193#125—24no—
194#125—42no—
195#125—48no—
196#125—42no—
197#124—42no—
198#124—42no—
199C22⋊Q8—48no—
200C22⋊Q8—48no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203#121—42no—
204#121—42no—
205#117—22no—
206#117—22no—
207#117—24no—
208#117—42no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4D811yescharacteristic
228C2×C4⋊C4D811yescharacteristic
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4D811yescharacteristic
237C2×C22⋊C4D811yescharacteristic
238C2×C42—41no—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
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—21no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—21no—
253#107—24no—
254#107—24no—
255#107—24no—
256#107—21no—
257#107—21no—
258#107—21no—
259#107—21no—
260#107—21no—
261#107—21no—
262#107—21no—
263#107—21no—
264#107—42no—
265#107C2×D412yes—
266#107C2×D412yes—
267#107C2×D412yes—
268#107C2×D412yes—
269#106—24no—
270#106—24no—
271#106—24no—
272#103—42no—
273#103—42no—
274#102—42no—
275#102—42no—
276#97—22no—
277#97—22no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8C2×D412yes—
281C4×C8C2×D412yes—
282#95—41no—
283#95—41no—
284C24—41no—
285C24C2×D811yescharacteristic
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—161no—
295C2×Q8—161no—
296C2×Q8—161no—
297C2×Q8—161no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—82no—
305C2×D4—161no—
306C2×D4—161no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—44no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—161no—
320C22×C4—161no—
321C22×C4#12412yes—
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4C2×D811yescharacteristic
325C22×C4C2×D811yescharacteristic
326C22×C4C2×D811yescharacteristic
327C22×C4C2×SD1612yes—
328C22×C4C22×D411yescharacteristic
329Q16—41no—
330Q16—41no—
331Q16—41no—
332Q16—41no—
333SD16—84no—
334D8—41no—
335D8—41no—
336D8—41no—
337D8—41no—
338M16—28no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8#12012yes—
348C2×C8#12412yes—
349C2×C8C2×SD1612yes—
350C2×C8C2×SD1612yes—
351C2×C8#13712yes—
352C2×C8C22×D411yescharacteristic
353C2×C8C22×D411yescharacteristic
354C2×C8C2×#4111yescharacteristic
355C2×C8C2×#4111yescharacteristic
356C2×C82−1+411yescharacteristic
357C2×C82−1+411yescharacteristic
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—82no—
380C4⋊C4—161no—
381C4⋊C4—161no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—44no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—161no—
390C22⋊C4—161no—
391C42—21no—
392C42—21no—
393C42—44no—
394C42#12012yes—
395C42AGL(1,ℤ/8)12yes—
396C16—41no—
397C16—41no—
398C16—41no—
399C16—41no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23#48612yes—
414C23C22×D811yescharacteristic
415Q8—81no—
416Q8—81no—
417Q8—81no—
418Q8—81no—
419Q8—164no—
420Q8—164no—
421D4—81no—
422D4—81no—
423D4—81no—
424D4—81no—
425D4—164no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—164no—
464C2×C4#44712yes—
465C2×C4#44712yes—
466C2×C4#48612yes—
467C2×C4C2×#12011yescharacteristic
468C2×C4C2×#12411yescharacteristic
469C2×C4#53611yescharacteristic
470C2×C4C22×SD1611yescharacteristic
471C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C8—22no—
474C8—22no—
475C8#44814yes—
476C8#54811yescharacteristic
477C8#54811yescharacteristic
478C8#54811yescharacteristic
479C8#54811yescharacteristic
480C22—22no—
481C22—22no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—161no—
488C22—161no—
489C22—161no—
490C22—161no—
491C22—161no—
492C22—161no—
493C22—161no—
494C22C2×#44711yescharacteristic
495C22C2×#48611yescharacteristic
496C22#322611yescharacteristic
497C22#341511yescharacteristic
498C22#341611yescharacteristic
499C22#341611yescharacteristic
500C22#341711yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—41no—
505C4—41no—
506C4—82no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4—161no—
511C4—161no—
512C4—161no—
513C4#243611yescharacteristic
514C4C2×#44811yescharacteristic
515C4#314111yescharacteristic
516C4#322711yescharacteristic
517C2—42no—
518C2—321no—
519C2#2590311yescharacteristic
520C2#3352611yescharacteristic
521C2#3352711yescharacteristic
522C2#3370211yescharacteristic
523C2#3370311yescharacteristic
524C2#3380911yescharacteristic
525C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3164C2semidirect
2. C2×#2243C2semidirect
3. C2×#2242C2semidirect
5. #13366C2semidirect
6. #13365C2semidirect
7. #13364C2semidirect
8. #13363C2semidirect
9. #13284C2semidirect
10. #13282C2semidirect
11. #12994C2semidirect
12. #12993C2semidirect
15. #12600C2semidirect
93. #500D4semidirect
97. #499D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C2×#22432
C2×#22422
#331822
#133662
#133652
#133642
#133632
#132842
#132822
#129942
#129932
#129922
#129912
#126002