Finite Group Databasebeta

← #498377

Subgroups — #498377

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 530 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×#2155C211yesmaximal, characteristic, complemented
4#23031C211yesmaximal, characteristic, complemented
5#23019C211yesmaximal, characteristic, complemented
6#13249C211yesmaximal, characteristic, complemented
7#13174C211yesmaximal, characteristic, complemented
8#13143C211yesmaximal, characteristic, complemented
9#13142C211yesmaximal, characteristic, complemented
10#13004C211yesmaximal, characteristic, complemented
11#13002C211yesmaximal, characteristic, complemented
12#12956C211yesmaximal, characteristic, complemented
13#12955C211yesmaximal, characteristic, complemented
14#12677C211yesmaximal, characteristic, complemented
15#12616C211yesmaximal, characteristic, complemented
16C2×#509C2211yescharacteristic
17C22×M32C2211yescharacteristic
18#3050C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×#444C2211yescharacteristic
21C2×#442C2211yescharacteristic
22#2933—21no—
23C22×#108C2211yescharacteristic
24C2×#427C2211yescharacteristic
25#2363—21no—
26#2220C412yescomplemented
27#2218C412yescomplemented
28#2217C412yescomplemented
29#2215C412yescomplemented
30AGL(1,ℤ/16)—24no—
31#2159C2211yescharacteristic
32#2155—24no—
33C2×#365C2211yescharacteristic
34C2×#365C2211yescharacteristic
35C2×#365C2211yescharacteristic
36C2×#365C2211yescharacteristic
37C2×#361C2211yescharacteristic
38C2×#361C2211yescharacteristic
39C2×#360C2211yescharacteristic
40C2×#360C2211yescharacteristic
41#2140C2211yescharacteristic
42#2139C2211yescharacteristic
43#2112C2211yescharacteristic
44C2×#347C2211yescharacteristic
45C2×#347C2211yescharacteristic
46#1975C2211yescharacteristic
47#1972C2211yescharacteristic
48#1946C2211yescharacteristic
49#1943C2211yescharacteristic
50#1929—21no—
51#1925—21no—
52#1875—21no—
53#1871—21no—
54#1866—21no—
55#1862—21no—
56#1776C2211yescharacteristic
57#1773C2211yescharacteristic
58#1752C2211yescharacteristic
59#1387C2211yescharacteristic
60#1387C2211yescharacteristic
61#1386C2211yescharacteristic
62#1386C2211yescharacteristic
63#1375C2211yescharacteristic
64#1375C2211yescharacteristic
65C2×#135C2311yescharacteristic
66C22×D8C2311yescharacteristic
67C2×#131—21no—
68C22×M16—21no—
69C23×C8C2311yescharacteristic
70#525—21no—
71C2×#121—21no—
72C2×C4×D4—21no—
73C2×#117—21no—
74#509—22no—
75#509—22no—
76C2×SD32C2×C412yes—
77C2×D16C2×C412yes—
78C2×M32—22no—
79C2×M32—22no—
80C2×M32C2×C412yescomplemented
81C2×M32C2311yescharacteristic
82C2×M32C2311yescharacteristic
83C2×M32C2311yescharacteristic
84C2×M32C2311yescharacteristic
85C22×C16C2×C412yescomplemented
86#467C2×C411yescharacteristic, complemented
87#467C2×C411yescharacteristic, complemented
88#466—21no—
89#466—21no—
90#466C2×C412yes—
91#466C2×C412yes—
92#466C2×C411yescharacteristic, complemented
93#466C2×C411yescharacteristic, complemented
94#465—21no—
95#465—21no—
96#444—22no—
97#444—41no—
98#442—22no—
99#442—41no—
100#436—41no—
101#435—41no—
102#435—41no—
103C8×D4—41no—
104C2×#108—22no—
105C2×#108—21no—
106C2×#108—41no—
107C2×#108C2311yescharacteristic
108C2×#108C2311yescharacteristic
109#427—22no—
110#427—41no—
111#427C2311yescharacteristic
112C2×#107—21no—
113C2×#107C2311yescharacteristic
114C2×#106C2311yescharacteristic
115C2×#106C2311yescharacteristic
116#424—21no—
117C2×#104—21no—
118C2×#104—21no—
119#416—21no—
120C2×#102—21no—
121C2×#102—21no—
122C2×#102—21no—
123#408—21no—
124C2×C22⋊C8D411yescharacteristic
125#405—21no—
126C2×#97C2311yescharacteristic
127C2×#97C2311yescharacteristic
128#390—21no—
129#390—41no—
130#389—21no—
131#388—41no—
132#386—41no—
133C4×C4⋊C4—21no—
134C4×C22⋊C4—41no—
135C4×C22⋊C4D411yescharacteristic
136#367C2×C412yescomplemented
137#366C2×C412yescomplemented
138#365—22no—
139#365—22no—
140#365—22no—
141#365D414yescomplemented
142#361—22no—
143#361—41no—
144#360—22no—
145#360—41no—
146#358C2×C412yescomplemented
147#357C2×C412yes—
148#357C2×C412yes—
149#357C2×C412yescomplemented
150C22⋊C16—21no—
151C22⋊C16—21no—
152#347—22no—
153#347—22no—
154#340—21no—
155#340—21no—
156#339—21no—
157#339—21no—
158#336D411yescharacteristic
159#336D411yescharacteristic
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—81no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—41no—
183C2×M16—41no—
184C2×M16—42no—
185C2×M16—81no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—41no—
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—81no—
214#117—21no—
215#117—42no—
216#117—81no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—41no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231SD32—24no—
232D16—24no—
233M32—21no—
234M32—21no—
235M32—24no—
236M32—21no—
237M32—21no—
238M32—21no—
239M32—21no—
240M32C2×D412yes—
241M32#4112yes—
242C2×C16—24no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16C22×C412yes—
246C2×C16C22×C412yes—
247#108—21no—
248#108—21no—
249#108—21no—
250#108—42no—
251#108C2×D412yes—
252#107—21no—
253#107—42no—
254#107C22×C411yescharacteristic
255#107C22×C411yescharacteristic
256#107C22×C412yes—
257#106—21no—
258#106—21no—
259#106—21no—
260#106—21no—
261#106C2×D412yes—
262C4⋊C8—41no—
263C4⋊C8—41no—
264#104—42no—
265#104—81no—
266#103—21no—
267#103—21no—
268#102—22no—
269#102—22no—
270#102—21no—
271#102—21no—
272#102—42no—
273#102—81no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276C22⋊C8—41no—
277C22⋊C8—41no—
278#97—21no—
279#97—21no—
280#97—21no—
281#97—21no—
282#97—41no—
283C4×C8—41no—
284#95—21no—
285#95—41no—
286#95—41no—
287C24—41no—
288C24AGL(1,ℤ/8)11yescharacteristic
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—161no—
307C2×D4—161no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—161no—
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4AGL(1,ℤ/8)11yescharacteristic
324C22×C4AGL(1,ℤ/8)11yescharacteristic
325C22×C4AGL(1,ℤ/8)11yescharacteristic
326C22×C4C22×D411yescharacteristic
327Q16—21no—
328Q16—21no—
329SD16—41no—
330SD16—41no—
331D8—21no—
332D8—22no—
333D8—22no—
334D8—21no—
335D8—41no—
336D8—41no—
337M16—22no—
338M16—41no—
339M16—41no—
340M16—41no—
341M16—82no—
342M16—82no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
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—41no—
360C2×C8—41no—
361C2×C8—82no—
362C2×C8—82no—
363C2×C8C4×D411yescharacteristic
364C2×C8C4×D411yescharacteristic
365C2×C8C23×C411yescharacteristic
366C2×C8C22×D411yescharacteristic
367C2×C8C2×#4111yescharacteristic
368C2×C8C2×#4111yescharacteristic
369C2×C8H(5,𝔽2)11yescharacteristic
370C2×C82−1+411yescharacteristic
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—161no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—81no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—161no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—41no—
399C42—82no—
400C42—161no—
401C16—22no—
402C16—41no—
403C16—41no—
404C16C4×D414yes—
405C23—21no—
406C23—21no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23#44211yescharacteristic
418C23#44211yescharacteristic
419C23C2×AGL(1,ℤ/8)11yescharacteristic
420Q8—41no—
421Q8—41no—
422D4—42no—
423D4—41no—
424D4—42no—
425D4—41no—
426D4—82no—
427D4—81no—
428D4—81no—
429D4—81no—
430D4—81no—
431D4—161no—
432D4—162no—
433D4—162no—
434C2×C4—21no—
435C2×C4—21no—
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—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
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—81no—
464C2×C4—164no—
465C2×C4#44211yescharacteristic
466C2×C4#44211yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4D4211yescharacteristic
469C2×C4#54711yescharacteristic
470C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57711yescharacteristic
473C8—21no—
474C8—21no—
475C8—41no—
476C8—41no—
477C8—42no—
478C8—81no—
479C8C2×C4×D411yescharacteristic
480C8#52011yescharacteristic
481C8#54811yescharacteristic
482C8#55311yescharacteristic
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—82no—
495C22—82no—
496C22—81no—
497C22—81no—
498C22—82no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22C2×#44211yescharacteristic
504C22#326511yescharacteristic
505C22#334511yescharacteristic
506C4—21no—
507C4—21no—
508C4—81no—
509C4—81no—
510C4—82no—
511C4—81no—
512C4—82no—
513C4—81no—
514C4—81no—
515C4—161no—
516C4#236111yescharacteristic
517C4#295211yescharacteristic
518C4#326511yescharacteristic
519C4#334511yescharacteristic
520C2—21no—
521C2—21no—
522C2—41no—
523C2—41no—
524C2—161no—
525C2—162no—
526C2#2302911yescharacteristic
527C2#3346811yescharacteristic
528C2#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 ◁ #1752 ◁ 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×#2155C2semidirect
4. #23031C2semidirect
5. #23019C2semidirect
6. #13249C2semidirect
7. #13174C2semidirect
8. #13143C2semidirect
9. #13142C2semidirect
10. #13004C2semidirect
11. #13002C2semidirect
12. #12956C2semidirect
13. #12955C2semidirect
14. #12677C2semidirect
15. #12616C2semidirect
26. #2220C4semidirect
27. #2218C4semidirect
28. #2217C4semidirect
29. #2215C4semidirect
80. C2×M32C2×C4semidirect
85. C22×C16C2×C4semidirect
86. #467C2×C4semidirect
87. #467C2×C4semidirect
92. #466C2×C4semidirect
93. #466C2×C4semidirect
136. #367C2×C4semidirect
137. #366C2×C4semidirect
141. #365D4semidirect
146. #358C2×C4semidirect
149. #357C2×C4semidirect

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×#21552
#230312
#230192
#132492
#131742
#131432
#131422
#130042
#130022
#129562
#129552
#126772
#126162