Finite Group Databasebeta

← #497033

Subgroups — #497033

Order 512.

634 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
2#21546C211yesmaximal, characteristic, complemented
3#21526C211yesmaximal, characteristic, complemented
4#13143C211yesmaximal, characteristic, complemented
5#13142C211yesmaximal, characteristic, complemented
6#13119C211yesmaximal, characteristic, complemented
7#13118C211yesmaximal, characteristic, complemented
8#13067C211yesmaximal, characteristic, complemented
9#13065C211yesmaximal, characteristic, complemented
10#13040C211yesmaximal, characteristic, complemented
11#13037C211yesmaximal, characteristic, complemented
12#12956C211yesmaximal, characteristic, complemented
13#12955C211yesmaximal, characteristic, complemented
14#12831C211yesmaximal, characteristic, complemented
15C2×#1375C211yesmaximal, characteristic, complemented
16C2×#509C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C22×M32C2211yescharacteristic
19#3050C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×#444C2211yescharacteristic
22C2×#442C2211yescharacteristic
23#2929—21no—
24C22×#97C2211yescharacteristic
25C4×#121—21no—
26#2220C412yescomplemented
27#2218C412yescomplemented
28#2217C412yescomplemented
29#2215C412yescomplemented
30#2159C2211yescharacteristic
31C2×#365C2211yescharacteristic
32C2×#365C2211yescharacteristic
33C2×#361C2211yescharacteristic
34C2×#361C2211yescharacteristic
35C2×#360C2211yescharacteristic
36C2×#360C2211yescharacteristic
37#2140C2211yescharacteristic
38#2139C2211yescharacteristic
39#1975C2211yescharacteristic
40#1972C2211yescharacteristic
41#1946C2211yescharacteristic
42#1943C2211yescharacteristic
43#1938—21no—
44#1936—21no—
45#1896—21no—
46#1893—21no—
47#1810C2211yescharacteristic
48#1807C2211yescharacteristic
49#1776C2211yescharacteristic
50#1773C2211yescharacteristic
51#1765—21no—
52#1762—21no—
53#1387C2211yescharacteristic
54#1387C2211yescharacteristic
55#1386C2211yescharacteristic
56#1386C2211yescharacteristic
57#1383C2211yescharacteristic
58#1383C2211yescharacteristic
59#1375C2211yescharacteristic
60#1375C2211yescharacteristic
61#1375C2211yescharacteristic
62#1375C2211yescharacteristic
63C2×#135C2311yescharacteristic
64C22×D8C2311yescharacteristic
65C2×#131—21no—
66C23×C8D411yescharacteristic
67C23×C8C2311yescharacteristic
68#525—21no—
69C2×#121—21no—
70C2×C4×D4—21no—
71C2×C4×D4—41no—
72C22×C42D411yescharacteristic
73#509—41no—
74#509—41no—
75C2×SD32C2×C412yescomplemented
76C2×D16C2×C412yescomplemented
77C2×M32—21no—
78C2×M32—41no—
79C2×M32—41no—
80C2×M32—41no—
81C2×M32—41no—
82C2×M32C2×C412yescomplemented
83C2×M32C2311yescharacteristic
84C2×M32C2311yescharacteristic
85C2×M32C2311yescharacteristic
86C2×M32C2311yescharacteristic
87C22×C16—21no—
88C22×C16C2×C412yescomplemented
89#467C2×C411yescharacteristic, complemented
90#467C2×C411yescharacteristic, complemented
91#466—21no—
92#466—21no—
93#466C2×C412yes—
94#466C2×C412yes—
95#466C2×C411yescharacteristic, complemented
96#466C2×C411yescharacteristic, complemented
97#465—21no—
98#465—21no—
99#444—41no—
100#444—41no—
101#442—41no—
102#442—41no—
103#435—41no—
104#435—41no—
105C8×D4—41no—
106C8×D4—41no—
107C2×#108C2311yescharacteristic
108C2×#108C2311yescharacteristic
109#427C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#106C2311yescharacteristic
112C2×#106C2311yescharacteristic
113#424—21no—
114C2×#104—21no—
115C2×#104—21no—
116#416—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102—21no—
120#408—21no—
121#407—21no—
122#405—21no—
123C2×#97—41no—
124C2×#97—41no—
125C2×#97D411yescharacteristic
126C2×#97D411yescharacteristic
127C2×#97C2311yescharacteristic
128C2×#97C2311yescharacteristic
129#390—21no—
130#389—21no—
131#388—41no—
132#386—41no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C22⋊C4—41no—
136#367C2×C412yescomplemented
137#366C2×C412yescomplemented
138#365—41no—
139#365—41no—
140#361—41no—
141#361—41no—
142#360—41no—
143#360—41no—
144#358C2×C412yescomplemented
145#357C2×C412yescomplemented
146#357C2×C412yescomplemented
147#357C2×C412yescomplemented
148#341—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#339—21no—
153#339—21no—
154C2×#41—21no—
155C22×D4—21no—
156C22×D4—41no—
157C23×C4—21no—
158C23×C4—41no—
159C23×C4C2×D411yescharacteristic
160#135—21no—
161#135—21no—
162#135—21no—
163#135—21no—
164C2×Q16C22×C411yescharacteristic
165C2×SD16—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8C22×C412yes—
172C2×D8C22×C411yescharacteristic
173#131—42no—
174#131—42no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—41no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—41no—
186C22×C8—41no—
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8#4111yescharacteristic
193C22×C8#4111yescharacteristic
194C22×C8C2411yescharacteristic
195#123—41no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198#121—22no—
199#121—22no—
200#121—21no—
201#121—21no—
202#121—41no—
203#121—41no—
204#120—41no—
205C4×D4—42no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—42no—
209C4×D4—84no—
210#117—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—41no—
223C2×C42—41no—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226SD32—42no—
227D16—42no—
228M32—21no—
229M32—21no—
230M32—21no—
231M32—21no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32—21no—
236M32—42no—
237M32—42no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—42no—
241C2×C16—42no—
242C2×C16C22×C412yes—
243C2×C16C22×C412yes—
244#108—21no—
245#108—21no—
246#108—21no—
247#108—21no—
248#107—21no—
249#107C22×C411yescharacteristic
250#107C22×C411yescharacteristic
251#107C22×C412yes—
252#106—21no—
253#106—21no—
254#106—21no—
255#106—21no—
256#106—21no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259#104—42no—
260#104—42no—
261#103—21no—
262#103—21no—
263#102—22no—
264#102—22no—
265#102—21no—
266#102—21no—
267#102—42no—
268#102—42no—
269C22⋊C8—41no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272C22⋊C8—41no—
273#97—21no—
274#97—21no—
275#97—21no—
276#97—21no—
277#97—41no—
278#97—41no—
279#97—41no—
280C4×C8—41no—
281#95—41no—
282#95—41no—
283#95—41no—
284C24—41no—
285C24AGL(1,ℤ/8)11yescharacteristic
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290C2×Q8—21no—
291C2×D4—22no—
292C2×D4—21no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—82no—
304C2×D4—82no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—82no—
318C22×C4—81no—
319C22×C4—82no—
320C22×C4—82no—
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—41no—
338M16—41no—
339M16—41no—
340M16—41no—
341M16—82no—
342C2×C8—21no—
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—42no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—82no—
362C2×C8C4×D411yescharacteristic
363C2×C8C4×D411yescharacteristic
364C2×C8#13512yes—
365C2×C8C23×C411yescharacteristic
366C2×C8C2×#4111yescharacteristic
367C2×C8H(5,𝔽2)11yescharacteristic
368C2×C8H(5,𝔽2)11yescharacteristic
369C2×C8H(5,𝔽2)11yescharacteristic
370C2×C82−1+411yescharacteristic
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—82no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—42no—
397C42—41no—
398C42—82no—
399C42#13512yes—
400C16—22no—
401C16—22no—
402C16—41no—
403C16—41no—
404C23—21no—
405C23—22no—
406C23—21no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23#44211yescharacteristic
417C23#44211yescharacteristic
418C23C2×AGL(1,ℤ/8)11yescharacteristic
419Q8—41no—
420Q8—41no—
421D4—42no—
422D4—41no—
423D4—42no—
424D4—41no—
425D4—82no—
426D4—84no—
427D4—81no—
428D4—82no—
429D4—84no—
430D4—81no—
431D4—81no—
432D4—81no—
433C2×C4—21no—
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—22no—
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—84no—
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—81no—
464C2×C4—164no—
465C2×C4#44211yescharacteristic
466C2×C4#44211yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4#53411yescharacteristic
469C2×C4#53711yescharacteristic
470C2×C4C2×#13511yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
473C8—21no—
474C8—21no—
475C8—22no—
476C8—41no—
477C8—81no—
478C8—81no—
479C8#51811yescharacteristic
480C8#52011yescharacteristic
481C8#53811yescharacteristic
482C8#53911yescharacteristic
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—42no—
491C22—42no—
492C22—81no—
493C22—82no—
494C22—82no—
495C22—81no—
496C22—81no—
497C22—82no—
498C22—161no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22AGL(1,ℤ/16)12yes—
503C22C2×#44211yescharacteristic
504C22#318811yescharacteristic
505C22#324711yescharacteristic
506C4—21no—
507C4—21no—
508C4—42no—
509C4—81no—
510C4—81no—
511C4—82no—
512C4—81no—
513C4—81no—
514C4—161no—
515C4—161no—
516C4#232511yescharacteristic
517C4C4×#13511yescharacteristic
518C4#318811yescharacteristic
519C4#324711yescharacteristic
520C2—21no—
521C2—21no—
522C2—22no—
523C2—41no—
524C2—161no—
525C2—162no—
526C2#2218411yescharacteristic
527C2C2×AGL(1,ℤ/16)11yescharacteristic
528C2#3346811yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#33605C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21546C2semidirect
3. #21526C2semidirect
4. #13143C2semidirect
5. #13142C2semidirect
6. #13119C2semidirect
7. #13118C2semidirect
8. #13067C2semidirect
9. #13065C2semidirect
10. #13040C2semidirect
11. #13037C2semidirect
12. #12956C2semidirect
13. #12955C2semidirect
14. #12831C2semidirect
15. C2×#1375C2semidirect
26. #2220C4semidirect
27. #2218C4semidirect
28. #2217C4semidirect
29. #2215C4semidirect
75. C2×SD32C2×C4semidirect
76. C2×D16C2×C4semidirect
82. C2×M32C2×C4semidirect
88. C22×C16C2×C4semidirect
89. #467C2×C4semidirect
90. #467C2×C4semidirect
95. #466C2×C4semidirect
96. #466C2×C4semidirect
136. #367C2×C4semidirect
137. #366C2×C4semidirect
144. #358C2×C4semidirect
145. #357C2×C4semidirect
146. #357C2×C4semidirect
147. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
#336052
#215462
#215262
#131432
#131422
#131192
#131182
#130672
#130652
#130402
#130372
#129562
#129552
#128312
C2×#13752