Finite Group Databasebeta

← #495897

Subgroups — #495897

Order 512.

663 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33605C211yesmaximal, characteristic
2C2×AGL(1,ℤ/16)C211yesmaximal, characteristic, complemented
3C2×#2110C211yesmaximal, characteristic, complemented
4#21547C211yesmaximal, characteristic, complemented
5#21529C211yesmaximal, characteristic, complemented
6#13249C211yesmaximal, characteristic, complemented
7#13174C211yesmaximal, characteristic, complemented
8#13155C211yesmaximal, characteristic, complemented
9#13153C211yesmaximal, characteristic, complemented
10#13067C211yesmaximal, characteristic, complemented
11#13065C211yesmaximal, characteristic, complemented
12#12831C211yesmaximal, characteristic, complemented
13#12677C211yesmaximal, characteristic, complemented
14#12650C211yesmaximal, characteristic, complemented
15#12647C211yesmaximal, characteristic, complemented
16C2×#509C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C22×M32C2211yescharacteristic
19#3050C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×#444C2211yescharacteristic
22C2×#442C2211yescharacteristic
23#2930—21no—
24C2×#405C2211yescharacteristic
25#2324—21no—
26#2220C412yescomplemented
27#2218C412yescomplemented
28#2217C412yescomplemented
29#2215C412yescomplemented
30AGL(1,ℤ/16)—24no—
31#2159C2211yescharacteristic
32C2×#365C2211yescharacteristic
33C2×#365C2211yescharacteristic
34C2×#361C2211yescharacteristic
35C2×#361C2211yescharacteristic
36C2×#360C2211yescharacteristic
37C2×#360C2211yescharacteristic
38#2140C2211yescharacteristic
39#2139C2211yescharacteristic
40#2110—24no—
41C2×#347C2211yescharacteristic
42C2×#347C2211yescharacteristic
43C2×#347C2211yescharacteristic
44C2×#347C2211yescharacteristic
45#1975C2211yescharacteristic
46#1972C2211yescharacteristic
47#1946C2211yescharacteristic
48#1943C2211yescharacteristic
49#1937—21no—
50#1934—21no—
51#1897—21no—
52#1895—21no—
53#1810C2211yescharacteristic
54#1807C2211yescharacteristic
55#1776C2211yescharacteristic
56#1773C2211yescharacteristic
57#1764—21no—
58C4×#102—21no—
59#1387C2211yescharacteristic
60#1387C2211yescharacteristic
61#1386C2211yescharacteristic
62#1386C2211yescharacteristic
63#1383C2211yescharacteristic
64#1383C2211yescharacteristic
65C2×#135C2311yescharacteristic
66C22×D8C2311yescharacteristic
67C2×#131—21no—
68C22×M16D411yescharacteristic
69C23×C8C2311yescharacteristic
70#525—21no—
71C2×#121—21no—
72C2×C4×D4—21no—
73C2×#117D411yescharacteristic
74#509—22no—
75#509—22no—
76C2×SD32C2×C412yes—
77C2×D16C2×C412yes—
78C2×M32—21no—
79C2×M32—22no—
80C2×M32—22no—
81C2×M32—22no—
82C2×M32—22no—
83C2×M32C2×C412yescomplemented
84C2×M32C2311yescharacteristic
85C2×M32C2311yescharacteristic
86C2×M32C2311yescharacteristic
87C2×M32C2311yescharacteristic
88C22×C16—21no—
89C22×C16C2×C412yescomplemented
90#467C2×C411yescharacteristic, complemented
91#467C2×C411yescharacteristic, complemented
92#466—21no—
93#466—21no—
94#466C2×C412yes—
95#466C2×C412yes—
96#466C2×C411yescharacteristic, complemented
97#466C2×C411yescharacteristic, complemented
98#465—21no—
99#465—21no—
100#444—22no—
101#444—22no—
102#442—22no—
103#442—22no—
104#436—41no—
105#436—41no—
106#435—41no—
107#435—41no—
108C2×#108C2311yescharacteristic
109C2×#108C2311yescharacteristic
110#427C2311yescharacteristic
111C2×#107C2311yescharacteristic
112C2×#106C2311yescharacteristic
113C2×#106C2311yescharacteristic
114#424—21no—
115C2×#104—21no—
116C2×#104—21no—
117#416—21no—
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—21no—
121#408—21no—
122#407—21no—
123#405—22no—
124#405—22no—
125#405—21no—
126C2×#97C2311yescharacteristic
127C2×#97C2311yescharacteristic
128C2×C4×C8D411yescharacteristic
129C2×C4×C8D411yescharacteristic
130#390—21no—
131#390—41no—
132#389—21no—
133#388—41no—
134#386—41no—
135#385—41no—
136#380—21no—
137C4×C4⋊C4—21no—
138#367C2×C412yescomplemented
139#366C2×C412yescomplemented
140#365—22no—
141#365—22no—
142#361—22no—
143#361—22no—
144#360—22no—
145#360—22no—
146#358C2×C412yescomplemented
147#357C2×C412yes—
148#357C2×C412yes—
149#357C2×C412yescomplemented
150#347—22no—
151#347—22no—
152#347—22no—
153#347D414yescomplemented
154#341—21no—
155#340—21no—
156#340—21no—
157#340—21no—
158#339—21no—
159#339—21no—
160C2×#41—21no—
161C22×D4—21no—
162C22×D4—41no—
163C23×C4—41no—
164C23×C4C2×D411yescharacteristic
165#135—21no—
166#135—21no—
167#135—21no—
168#135—21no—
169C2×Q16C22×C411yescharacteristic
170C2×SD16—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8C22×C412yes—
177C2×D8C22×C411yescharacteristic
178#131—42no—
179#131—42no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—22no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—41no—
186C2×M16—41no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—41no—
192C22×C8C22×C411yescharacteristic
193C22×C8C22×C411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8#4111yescharacteristic
198C22×C8#4111yescharacteristic
199C22×C8C2411yescharacteristic
200#123—41no—
201C22⋊Q8—21no—
202C22⋊Q8—21no—
203#121—22no—
204#121—22no—
205#121—21no—
206#121—21no—
207#121—41no—
208#121—41no—
209#120—41no—
210C4×D4—42no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—42no—
214#117—22no—
215#117—21no—
216#117—41no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C42—41no—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231SD32—24no—
232D16—24no—
233M32—24no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—24no—
239M32C2×D412yes—
240M32C2×D412yes—
241M32#4112yes—
242M32#4112yes—
243C2×C16—24no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—24no—
247C2×C16C22×C412yes—
248C2×C16C22×C412yes—
249#108—21no—
250#108—21no—
251#108—21no—
252#108—21no—
253#107—21no—
254#107C22×C411yescharacteristic
255#107C22×C411yescharacteristic
256#107C22×C412yes—
257#106—21no—
258#106—21no—
259#106—21no—
260#106—21no—
261#106—21no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264#104—42no—
265#104—42no—
266#103—21no—
267#103—21no—
268#102—22no—
269#102—22no—
270#102—21no—
271#102—21no—
272#102—42no—
273#102—42no—
274C22⋊C8—41no—
275C22⋊C8—41no—
276C22⋊C8—41no—
277C22⋊C8—41no—
278#97—21no—
279#97—21no—
280#97—41no—
281#97C2×D412yes—
282#97#4112yes—
283C4×C8—22no—
284C4×C8—22no—
285C4×C8—41no—
286#95—41no—
287#95—41no—
288#95—41no—
289C24—41no—
290C24AGL(1,ℤ/8)11yescharacteristic
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295C2×Q8—21no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—161no—
309C2×D4—161no—
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—82no—
321C22×C4—161no—
322C22×C4—161no—
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4AGL(1,ℤ/8)11yescharacteristic
326C22×C4AGL(1,ℤ/8)11yescharacteristic
327C22×C4AGL(1,ℤ/8)11yescharacteristic
328C22×C4C22×D411yescharacteristic
329Q16—21no—
330Q16—21no—
331SD16—41no—
332SD16—41no—
333D8—21no—
334D8—22no—
335D8—22no—
336D8—21no—
337D8—41no—
338D8—41no—
339M16—42no—
340M16—42no—
341M16—41no—
342M16—41no—
343M16—41no—
344M16—41no—
345M16—82no—
346C2×C8—21no—
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—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—82no—
364C2×C8C4×D411yescharacteristic
365C2×C8C4×D411yescharacteristic
366C2×C8#12112yes—
367C2×C8C23×C411yescharacteristic
368C2×C8C22×D411yescharacteristic
369C2×C8C2×#4111yescharacteristic
370C2×C8C2×#4111yescharacteristic
371C2×C8C2×#4111yescharacteristic
372C2×C8C2×#4111yescharacteristic
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—161no—
385C22⋊C4—42no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—82no—
394C22⋊C4—161no—
395C22⋊C4—161no—
396C22⋊C4—161no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—41no—
401C42—161no—
402C42#12112yes—
403C16—22no—
404C16—22no—
405C16—22no—
406C16C4×D414yes—
407C23—21no—
408C23—21no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23#44211yescharacteristic
420C23#44211yescharacteristic
421C23C2×AGL(1,ℤ/8)11yescharacteristic
422Q8—41no—
423Q8—41no—
424D4—42no—
425D4—41no—
426D4—42no—
427D4—41no—
428D4—81no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—161no—
433D4—161no—
434D4—162no—
435D4—162no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
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—164no—
467C2×C4#44211yescharacteristic
468C2×C4#44211yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4#53211yescharacteristic
472C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
473C2×C4#57511yescharacteristic
474C2×C4#57511yescharacteristic
475C8—22no—
476C8—21no—
477C8—21no—
478C8—41no—
479C8—81no—
480C8—81no—
481C8C2×C4×D411yescharacteristic
482C8C4×#4111yescharacteristic
483C8#52511yescharacteristic
484C8#52911yescharacteristic
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—82no—
497C22—81no—
498C22—82no—
499C22—82no—
500C22—81no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22C2×#44211yescharacteristic
506C22#305511yescharacteristic
507C22#318011yescharacteristic
508C4—21no—
509C4—21no—
510C4—42no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—82no—
515C4—81no—
516C4—161no—
517C4—161no—
518C4C4×#12111yescharacteristic
519C4#294311yescharacteristic
520C4#305511yescharacteristic
521C4#318011yescharacteristic
522C2—21no—
523C2—21no—
524C2—41no—
525C2—41no—
526C2—161no—
527C2—162no—
528C2#2152611yescharacteristic
529C2#3346911yescharacteristic
530C2#3347111yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#33605C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×AGL(1,ℤ/16)C2semidirect
3. C2×#2110C2semidirect
4. #21547C2semidirect
5. #21529C2semidirect
6. #13249C2semidirect
7. #13174C2semidirect
8. #13155C2semidirect
9. #13153C2semidirect
10. #13067C2semidirect
11. #13065C2semidirect
12. #12831C2semidirect
13. #12677C2semidirect
14. #12650C2semidirect
15. #12647C2semidirect
26. #2220C4semidirect
27. #2218C4semidirect
28. #2217C4semidirect
29. #2215C4semidirect
83. C2×M32C2×C4semidirect
89. C22×C16C2×C4semidirect
90. #467C2×C4semidirect
91. #467C2×C4semidirect
96. #466C2×C4semidirect
97. #466C2×C4semidirect
138. #367C2×C4semidirect
139. #366C2×C4semidirect
146. #358C2×C4semidirect
149. #357C2×C4semidirect
153. #347D4semidirect

Maximal subgroups

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

subgroupindex
#336052
C2×AGL(1,ℤ/16)2
C2×#21102
#215472
#215292
#132492
#131742
#131552
#131532
#130672
#130652
#128312
#126772
#126502
#126472