Finite Group Databasebeta

← #511980

Subgroups — #511980

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#135C211yesmaximal, characteristic
2C2×#2229C211yesmaximal, characteristic, complemented
3C2×#2228C211yesmaximal, characteristic, complemented
4#33208C211yesmaximal, characteristic, complemented
5#12938C211yesmaximal, characteristic, complemented
6#12937C211yesmaximal, characteristic, complemented
7#12936C211yesmaximal, characteristic, complemented
8#12935C211yesmaximal, characteristic, complemented
9#12921C211yesmaximal, characteristic, complemented
10#12921C211yesmaximal, characteristic, complemented
11#12814C211yesmaximal, characteristic, complemented
12#12813C211yesmaximal, characteristic, complemented
13#12783C211yesmaximal, characteristic, complemented
14#12733C211yesmaximal, characteristic
15#12733C211yesmaximal, characteristic
16C22×#135C2211yescharacteristic
17C2×C4×#41—21no—
18C2×#499C2211yescharacteristic
19C4×#135—24no—
20C4×#135—24no—
21C2×C4×Q16C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×SD16—21no—
24C2×C4×SD16—21no—
25C2×C4×D8C2211yescharacteristic
26C2×C4×D8C2211yescharacteristic
27C2×#427C2211yescharacteristic
28C2×#416—21no—
29C22×C4×C8C2211yescharacteristic
30#2851—21no—
31#2229—24no—
32#2228—24no—
33C2×#367C2211yescharacteristic
34C2×#367C2211yescharacteristic
35C2×#366C2211yescharacteristic
36C2×#366C2211yescharacteristic
37C2×C4⋊C16C2211yescharacteristic
38C2×C4⋊C16C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#358C2211yescharacteristic
42C2×#358C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×#357C2211yescharacteristic
45C2×#357C2211yescharacteristic
46C2×#357C2211yescharacteristic
47C2×C22⋊C16C2211yescharacteristic
48C2×C22⋊C16C2211yescharacteristic
49#1991—21no—
50#1942C2211yescharacteristic
51#1942C2211yescharacteristic
52#1941—21no—
53#1846C2211yescharacteristic
54#1808C2211yescharacteristic
55#1808C2211yescharacteristic
56#1807—21no—
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1374C2211yescharacteristic
62#1374C2211yescharacteristic
63C22×#41—21no—
64C2×#135—22no—
65C2×#135—22no—
66C2×#135—22no—
67C2×#135—22no—
68C2×#135—22no—
69C2×#135Q814yes—
70C22×Q16C2311yescharacteristic
71C22×SD16—21no—
72C22×D8C2311yescharacteristic
73C23×C8D411yescharacteristic
74C23×C8C2311yescharacteristic
75C2×#125—21no—
76C4×#41—28no—
77C4×#41—44no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—41no—
83C2×C4×D4—41no—
84C2×#117—21no—
85C2×#117—41no—
86C22×C42—41no—
87C22×C42D411yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92#499—22no—
93#499—22no—
94C4×Q16—22no—
95C4×Q16—22no—
96C4×Q16—22no—
97C4×Q16D414yes—
98C4×SD16—28no—
99C4×SD16—28no—
100C4×D8—22no—
101C4×D8—22no—
102C4×D8—22no—
103C4×D8D414yes—
104#427—22no—
105#427—22no—
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#107C2311yescharacteristic
112C2×#106—21no—
113C2×#106—21no—
114C2×#106—21no—
115#416—28no—
116C2×#103—21no—
117C2×#103—21no—
118C2×#102—21no—
119C2×#102—21no—
120C2×C4×C8—22no—
121C2×C4×C8—22no—
122C2×C4×C8D411yescharacteristic
123C2×C4×C8D411yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#389—21no—
127#389—21no—
128#387—41no—
129#387—41no—
130#385—41no—
131#385—41no—
132#382—21no—
133#380—21no—
134#380—21no—
135#367—22no—
136#367—22no—
137#366—22no—
138#366—22no—
139C4⋊C16—22no—
140C4⋊C16—22no—
141#358—22no—
142#358—22no—
143#358—22no—
144#358—22no—
145#357—22no—
146#357—22no—
147#357—22no—
148#357—22no—
149C22⋊C16—22no—
150C22⋊C16—22no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155C2×#41—24no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—42no—
159C2×#41—42no—
160C2×#41—42no—
161C22×Q8—21no—
162C22×D4—21no—
163C22×D4—41no—
164C23×C4—41no—
165C23×C4—41no—
166C23×C4—41no—
167C23×C4D811yescharacteristic
168C23×C4D811yescharacteristic
169C23×C4C2×D411yescharacteristic
170#135—28no—
171#135—28no—
172#135—28no—
173#135—28no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×Q16—21no—
177C2×Q16C2×D412yes—
178C2×Q16C2×Q812yes—
179C2×Q16#4112yes—
180C2×SD16—24no—
181C2×SD16—24no—
182C2×SD16—24no—
183C2×D8—21no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8C2×D412yes—
187C2×D8C2×Q812yes—
188C2×D8#4112yes—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—41no—
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×Q812yes—
200C22×C8#4112yes—
201C22×C8C2411yescharacteristic
202#125—42no—
203#125—42no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—42no—
211C4×D4—48no—
212C4×D4—48no—
213C4×D4—42no—
214#117—24no—
215#117—48no—
216#117—42no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—48no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42D811yescharacteristic
241C2×C42D811yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
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—
253C2×C16—21no—
254C2×C16—21no—
255C2×C16—21no—
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—21no—
265#107—21no—
266#107C2×D412yes—
267#107C2×D412yes—
268#106—24no—
269#106—42no—
270#106—42no—
271#103—24no—
272#103—24no—
273#102—24no—
274#102—24no—
275C4×C8—22no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—22no—
279C4×C8C2×D412yes—
280C4×C8C2×D412yes—
281#95—41no—
282#95—41no—
283#95—41no—
284#95—41no—
285#95—41no—
286#95—41no—
287C24—41no—
288C24C2×D811yescharacteristic
289#41—216no—
290#41—48no—
291#41—48no—
292#41—48no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—41no—
322C22×C4—44no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—44no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4C22⋊Q812yes—
335C22×C4C2×D811yescharacteristic
336C22×C4C2×D811yescharacteristic
337C22×C4C2×D811yescharacteristic
338C22×C4C2×D811yescharacteristic
339C22×C4C2×D811yescharacteristic
340C22×C4#13712yes—
341C22×C4C22×D411yescharacteristic
342Q16—22no—
343Q16—22no—
344Q16—22no—
345Q16C22⋊Q814yes—
346SD16—216no—
347D8—22no—
348D8—22no—
349D8—22no—
350D8C22⋊Q814yes—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—44no—
363C2×C8#12012yes—
364C2×C8C22⋊Q812yes—
365C2×C8C22×D411yescharacteristic
366C2×C8C22×D411yescharacteristic
367C2×C8C22×Q811yescharacteristic
368C2×C8C2×#4111yescharacteristic
369C2×C8C2×#4111yescharacteristic
370C2×C8C2×#4111yescharacteristic
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—44no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—44no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C42—21no—
395C42—21no—
396C42—22no—
397C42—22no—
398C42—44no—
399C42—44no—
400C42—44no—
401C42#12012yes—
402C42AGL(1,ℤ/8)12yes—
403C16—41no—
404C16—41no—
405C16—41no—
406C16—41no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23#47412yes—
421C23C22×D811yescharacteristic
422Q8—24no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426D4—24no—
427D4—416no—
428D4—42no—
429D4—42no—
430D4—42no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—416no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—88no—
455C2×C4—88no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
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#44712yes—
473C2×C4#44712yes—
474C2×C4#47412yes—
475C2×C4#50812yes—
476C2×C4C2×#12011yescharacteristic
477C2×C4C2×C22⋊Q811yescharacteristic
478C2×C4#52511yescharacteristic
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4C2×#13711yescharacteristic
481C2×C4#57511yescharacteristic
482C8—22no—
483C8—22no—
484C8—41no—
485C8C2×C22⋊Q811yescharacteristic
486C8C2×C22⋊Q811yescharacteristic
487C8#52511yescharacteristic
488C8#52511yescharacteristic
489C22—22no—
490C22—22no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—44no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22C2×#44711yescharacteristic
504C22C2×#47411yescharacteristic
505C22#310011yescharacteristic
506C22C2×#50811yescharacteristic
507C22#341411yescharacteristic
508C22#341411yescharacteristic
509C22#341411yescharacteristic
510C4—22no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—44no—
515C4—41no—
516C4—82no—
517C4—82no—
518C4—161no—
519C4—161no—
520C4—161no—
521C4—161no—
522C4#284811yescharacteristic
523C4#300711yescharacteristic
524C4#307711yescharacteristic
525C4#311011yescharacteristic
526C2—42no—
527C2—84no—
528C2#3295311yescharacteristic
529C2#3351611yescharacteristic
530C2#3351911yescharacteristic
531C2#3362611yescharacteristic
532C2#3363011yescharacteristic
533C2#3366211yescharacteristic
534C2#3367111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2229C2semidirect
3. C2×#2228C2semidirect
4. #33208C2semidirect
5. #12938C2semidirect
6. #12937C2semidirect
7. #12936C2semidirect
8. #12935C2semidirect
9. #12921C2semidirect
10. #12921C2semidirect
11. #12814C2semidirect
12. #12813C2semidirect
13. #12783C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1352
C2×#22292
C2×#22282
#332082
#129382
#129372
#129362
#129352
#129212
#128142
#128132
#127832
#127332