Finite Group Databasebeta

← #498510

Subgroups — #498510

Order 512.

608 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33605C211yesmaximal, characteristic
2#23031C211yesmaximal, characteristic, complemented
3#23019C211yesmaximal, characteristic, complemented
4#13155C211yesmaximal, characteristic, complemented
5#13153C211yesmaximal, characteristic, complemented
6#13119C211yesmaximal, characteristic, complemented
7#13118C211yesmaximal, characteristic, complemented
8#13040C211yesmaximal, characteristic, complemented
9#13037C211yesmaximal, characteristic, complemented
10#13004C211yesmaximal, characteristic, complemented
11#13002C211yesmaximal, characteristic, complemented
12#12650C211yesmaximal, characteristic, complemented
13#12647C211yesmaximal, characteristic, complemented
14#12616C211yesmaximal, characteristic, complemented
15#12563C211yesmaximal, 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
30#2159C2211yescharacteristic
31C2×#361C2211yescharacteristic
32C2×#361C2211yescharacteristic
33C2×#360C2211yescharacteristic
34C2×#360C2211yescharacteristic
35#2140C2211yescharacteristic
36#2139C2211yescharacteristic
37#2112C2211yescharacteristic
38C2×#347C2211yescharacteristic
39C2×#347C2211yescharacteristic
40#1975C2211yescharacteristic
41#1972C2211yescharacteristic
42#1946C2211yescharacteristic
43#1943C2211yescharacteristic
44#1929—21no—
45#1925—21no—
46#1875—21no—
47#1871—21no—
48#1866—21no—
49#1862—21no—
50#1776C2211yescharacteristic
51#1773C2211yescharacteristic
52#1752C2211yescharacteristic
53#1387C2211yescharacteristic
54#1387C2211yescharacteristic
55#1386C2211yescharacteristic
56#1386C2211yescharacteristic
57#1383C2211yescharacteristic
58#1383C2211yescharacteristic
59#1383C2211yescharacteristic
60#1383C2211yescharacteristic
61#1375C2211yescharacteristic
62#1375C2211yescharacteristic
63C2×#135C2311yescharacteristic
64C22×D8C2311yescharacteristic
65C2×#131—21no—
66C22×M16—21no—
67C23×C8C2311yescharacteristic
68#525—21no—
69C2×#121—21no—
70C2×C4×D4—21no—
71C2×#117—21no—
72#509—41no—
73#509—41no—
74C2×SD32C2×C412yes—
75C2×D16C2×C412yes—
76C2×M32—41no—
77C2×M32—41no—
78C2×M32C2×C412yescomplemented
79C2×M32C2311yescharacteristic
80C2×M32C2311yescharacteristic
81C2×M32C2311yescharacteristic
82C2×M32C2311yescharacteristic
83C22×C16C2×C412yescomplemented
84#467C2×C411yescharacteristic, complemented
85#467C2×C411yescharacteristic, complemented
86#466—21no—
87#466—21no—
88#466C2×C412yes—
89#466C2×C412yes—
90#466C2×C411yescharacteristic, complemented
91#466C2×C411yescharacteristic, complemented
92#465—21no—
93#465—21no—
94#444—22no—
95#444—41no—
96#442—22no—
97#442—41no—
98#436—41no—
99#435—41no—
100#435—41no—
101C8×D4—41no—
102C2×#108—22no—
103C2×#108—21no—
104C2×#108—41no—
105C2×#108C2311yescharacteristic
106C2×#108C2311yescharacteristic
107#427—22no—
108#427—41no—
109#427C2311yescharacteristic
110C2×#107—21no—
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—
122C2×C22⋊C8D411yescharacteristic
123#405—21no—
124C2×#97C2311yescharacteristic
125C2×#97C2311yescharacteristic
126#390—21no—
127#390—41no—
128#389—21no—
129#388—41no—
130#386—41no—
131C4×C4⋊C4—21no—
132C4×C22⋊C4—41no—
133C4×C22⋊C4D411yescharacteristic
134#367C2×C412yescomplemented
135#366C2×C412yescomplemented
136#361—22no—
137#361—41no—
138#360—22no—
139#360—41no—
140#358C2×C412yescomplemented
141#357C2×C412yes—
142#357C2×C412yes—
143#357C2×C412yescomplemented
144C22⋊C16—21no—
145C22⋊C16—21no—
146#347—41no—
147#347—41no—
148#340—21no—
149#340—21no—
150#339—21no—
151#339—21no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154C2×#41—21no—
155C22×D4—21no—
156C22×D4—41no—
157C23×C4—41no—
158C23×C4C2×D411yescharacteristic
159#135—21no—
160#135—21no—
161#135—21no—
162#135—21no—
163C2×Q16C22×C411yescharacteristic
164C2×SD16—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8C22×C412yes—
171C2×D8C22×C411yescharacteristic
172#131—42no—
173#131—81no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—41no—
177C2×M16—41no—
178C2×M16—42no—
179C2×M16—81no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—41no—
185C22×C8—41no—
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8#4111yescharacteristic
192C22×C8#4111yescharacteristic
193C22×C8C2411yescharacteristic
194#123—41no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197#121—22no—
198#121—22no—
199#121—21no—
200#121—21no—
201#121—41no—
202#121—41no—
203#120—41no—
204C4×D4—42no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—81no—
208#117—21no—
209#117—42no—
210#117—81no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225SD32—42no—
226D16—42no—
227M32—21no—
228M32—21no—
229M32—21no—
230M32—21no—
231M32—21no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32—42no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—42no—
239C2×C16C22×C412yes—
240C2×C16C22×C412yes—
241#108—21no—
242#108—21no—
243#108—21no—
244#108—42no—
245#108#4112yes—
246#107—21no—
247#107—42no—
248#107C22×C411yescharacteristic
249#107C22×C411yescharacteristic
250#107C22×C412yes—
251#106—21no—
252#106—21no—
253#106—21no—
254#106—21no—
255#106#4112yes—
256C4⋊C8—41no—
257C4⋊C8—41no—
258#104—42no—
259#104—81no—
260#103—21no—
261#103—21no—
262#102—22no—
263#102—22no—
264#102—21no—
265#102—21no—
266#102—42no—
267#102—81no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272#97—21no—
273#97—21no—
274#97—21no—
275#97—21no—
276#97—41no—
277C4×C8—41no—
278#95—21no—
279#95—41no—
280#95—41no—
281C24—41no—
282C24AGL(1,ℤ/8)11yescharacteristic
283#41—41no—
284#41—41no—
285#41—41no—
286#41—41no—
287C2×Q8—21no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—161no—
301C2×D4—161no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—161no—
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4AGL(1,ℤ/8)11yescharacteristic
318C22×C4AGL(1,ℤ/8)11yescharacteristic
319C22×C4AGL(1,ℤ/8)11yescharacteristic
320C22×C4C22×D411yescharacteristic
321Q16—21no—
322Q16—21no—
323SD16—41no—
324SD16—41no—
325D8—22no—
326D8—21no—
327D8—21no—
328D8—22no—
329D8—41no—
330D8—41no—
331M16—22no—
332M16—41no—
333M16—41no—
334M16—41no—
335M16—82no—
336M16—82no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
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—41no—
354C2×C8—41no—
355C2×C8—82no—
356C2×C8—82no—
357C2×C8C4×D411yescharacteristic
358C2×C8C4×D411yescharacteristic
359C2×C8C23×C411yescharacteristic
360C2×C8C2×#4111yescharacteristic
361C2×C8C2×#4111yescharacteristic
362C2×C8C2×#4111yescharacteristic
363C2×C8H(5,𝔽2)11yescharacteristic
364C2×C8H(5,𝔽2)11yescharacteristic
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—161no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—81no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—161no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—41no—
393C42—82no—
394C42—161no—
395C16—22no—
396C16—22no—
397C16—41no—
398C16—41no—
399C23—21no—
400C23—21no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23#44211yescharacteristic
412C23#44211yescharacteristic
413C23C2×AGL(1,ℤ/8)11yescharacteristic
414Q8—41no—
415Q8—41no—
416D4—42no—
417D4—41no—
418D4—42no—
419D4—41no—
420D4—82no—
421D4—81no—
422D4—81no—
423D4—81no—
424D4—81no—
425D4—161no—
426D4—162no—
427D4—162no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
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—164no—
459C2×C4#44211yescharacteristic
460C2×C4#44211yescharacteristic
461C2×C4C2×C4×D411yescharacteristic
462C2×C4#54611yescharacteristic
463C2×C4#54611yescharacteristic
464C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
465C2×C4#57611yescharacteristic
466C2×C4#57811yescharacteristic
467C8—21no—
468C8—21no—
469C8—41no—
470C8—41no—
471C8—42no—
472C8—81no—
473C8C4×#4111yescharacteristic
474C8#51811yescharacteristic
475C8#55311yescharacteristic
476C8#55311yescharacteristic
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—82no—
489C22—81no—
490C22—82no—
491C22—82no—
492C22—81no—
493C22—161no—
494C22—161no—
495C22—161no—
496C22—161no—
497C22C2×#44211yescharacteristic
498C22#330811yescharacteristic
499C22#331811yescharacteristic
500C4—21no—
501C4—21no—
502C4—81no—
503C4—81no—
504C4—82no—
505C4—81no—
506C4—81no—
507C4—82no—
508C4—81no—
509C4—161no—
510C4#236411yescharacteristic
511C4#295111yescharacteristic
512C4#330811yescharacteristic
513C4#331811yescharacteristic
514C2—21no—
515C2—21no—
516C2—41no—
517C2—41no—
518C2—161no—
519C2—162no—
520C2#2313211yescharacteristic
521C2#3346811yescharacteristic
522C2#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. #23031C2semidirect
3. #23019C2semidirect
4. #13155C2semidirect
5. #13153C2semidirect
6. #13119C2semidirect
7. #13118C2semidirect
8. #13040C2semidirect
9. #13037C2semidirect
10. #13004C2semidirect
11. #13002C2semidirect
12. #12650C2semidirect
13. #12647C2semidirect
14. #12616C2semidirect
15. #12563C2semidirect
26. #2220C4semidirect
27. #2218C4semidirect
28. #2217C4semidirect
29. #2215C4semidirect
78. C2×M32C2×C4semidirect
83. C22×C16C2×C4semidirect
84. #467C2×C4semidirect
85. #467C2×C4semidirect
90. #466C2×C4semidirect
91. #466C2×C4semidirect
134. #367C2×C4semidirect
135. #366C2×C4semidirect
140. #358C2×C4semidirect
143. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
#336052
#230312
#230192
#131552
#131532
#131192
#131182
#130402
#130372
#130042
#130022
#126502
#126472
#126162
#125632