Finite Group Databasebeta

← #498376

Subgroups — #498376

Order 512.

627 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33606C211yesmaximal, characteristic
2C2×#2180C211yesmaximal, characteristic, complemented
3C22×#365C211yesmaximal, characteristic, complemented
4#23028C211yesmaximal, characteristic, complemented
5#23019C211yesmaximal, characteristic
6#13250C211yesmaximal, characteristic, complemented
7#13174C211yesmaximal, characteristic, complemented
8#13143C211yesmaximal, characteristic, complemented
9#13141C211yesmaximal, characteristic, complemented
10#13003C211yesmaximal, characteristic, complemented
11#13001C211yesmaximal, characteristic
12#12957C211yesmaximal, characteristic, complemented
13#12955C211yesmaximal, characteristic, complemented
14#12677C211yesmaximal, characteristic, complemented
15#12616C211yesmaximal, characteristic, complemented
16C2×#510C2211yescharacteristic
17C22×M32C2211yescharacteristic
18#3050C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#444C2211yescharacteristic
21C2×#441C2211yescharacteristic
22#2933—21no—
23C22×#108C2211yescharacteristic
24C22×#106C2211yescharacteristic
25#2360—21no—
26#2220C412yescomplemented
27#2219C412yescomplemented
28#2217C412yescomplemented
29#2216C412yescomplemented
30#2180—24no—
31#2159C2211yescharacteristic
32C2×#365—24no—
33C2×#365C2211yescharacteristic
34C2×#365C2211yescharacteristic
35C2×#365C2211yescharacteristic
36C2×#365C2211yescharacteristic
37C2×#362C2211yescharacteristic
38C2×#362C2211yescharacteristic
39C2×#360C2211yescharacteristic
40C2×#360C2211yescharacteristic
41#2141C2211yescharacteristic
42#2139C2211yescharacteristic
43#2112C2211yescharacteristic
44C2×#347C2211yescharacteristic
45C2×#347C2211yescharacteristic
46#1975C2211yescharacteristic
47#1971C2211yescharacteristic
48#1946C2211yescharacteristic
49#1943C2211yescharacteristic
50#1929—21no—
51#1927—21no—
52#1872—21no—
53#1871—21no—
54#1866—21no—
55#1865—21no—
56#1776C2211yescharacteristic
57#1773C2211yescharacteristic
58#1752C2211yescharacteristic
59#1387C2211yescharacteristic
60#1387C2211yescharacteristic
61#1386C2211yescharacteristic
62#1386C2211yescharacteristic
63#1375C2211yescharacteristic
64#1375C2211yescharacteristic
65C2×#135C2311yescharacteristic
66C22×Q16C2311yescharacteristic
67C2×#131—21no—
68C22×M16—21no—
69C23×C8C2311yescharacteristic
70#525—21no—
71C2×C22⋊Q8—21no—
72C2×C4×Q8—21no—
73C22×C4⋊C4—21no—
74#510—22no—
75#510—22no—
76C2×Q32C2×C412yescomplemented
77C2×SD32C2×C412yescomplemented
78C2×M32—22no—
79C2×M32—22no—
80C2×M32C2×C412yes—
81C2×M32C2311yescharacteristic
82C2×M32C2311yescharacteristic
83C2×M32C2311yescharacteristic
84C2×M32C2311yescharacteristic
85C22×C16C2×C412yes—
86#467—21no—
87#467—21no—
88#467C2×C411yescharacteristic
89#467C2×C412yes—
90#467C2×C411yescharacteristic
91#467C2×C412yes—
92#466C2×C411yescharacteristic
93#466C2×C411yescharacteristic
94#465—21no—
95#465—21no—
96#444—22no—
97#444—41no—
98#441—22no—
99#441—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#427C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#106—22no—
112C2×#106—21no—
113C2×#106—41no—
114C2×#106C2311yescharacteristic
115C2×#106C2311yescharacteristic
116#424—21no—
117C2×#104—21no—
118C2×#104—21no—
119#416—21no—
120C2×#103—21no—
121C2×#103—21no—
122C2×#103—21no—
123#408—21no—
124C2×C22⋊C8D411yescharacteristic
125#405—21no—
126C2×#97C2311yescharacteristic
127C2×#97C2311yescharacteristic
128#391—21no—
129#389—21no—
130#389—41no—
131#387—41no—
132#385—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#362—22no—
143#362—41no—
144#360—22no—
145#360—41no—
146#358C2×C412yescomplemented
147#358C2×C412yescomplemented
148#358C2×C412yescomplemented
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×Q8—21no—
162C23×C4—41no—
163C23×C4C2×D411yescharacteristic
164#135—21no—
165#135—21no—
166#135—21no—
167#135—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16C22×C412yes—
174C2×Q16C22×C411yescharacteristic
175C2×SD16—21no—
176C2×D8C22×C411yescharacteristic
177#131—42no—
178#131—81no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—41no—
182C2×M16—41no—
183C2×M16—42no—
184C2×M16—81no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—41no—
190C22×C8—41no—
191C22×C8C22×C411yescharacteristic
192C22×C8C22×C411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8#4111yescharacteristic
197C22×C8#4111yescharacteristic
198C22×C8C2411yescharacteristic
199#123—41no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206#121—21no—
207#121—21no—
208#120—41no—
209C4×Q8—42no—
210C4×Q8—81no—
211C4×D4—41no—
212C4×D4—41no—
213#117—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—81no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229Q32—24no—
230SD32—24no—
231M32—21no—
232M32—21no—
233M32—24no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32C2×D412yes—
239M32#4112yes—
240C2×C16—24no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16C22×C412yes—
244C2×C16C22×C412yes—
245#108—21no—
246#108—21no—
247#108—21no—
248#108—42no—
249#108C2×D412yes—
250#107—21no—
251#107C22×C412yes—
252#107C22×C411yescharacteristic
253#107C22×C411yescharacteristic
254#106—21no—
255#106—21no—
256#106—21no—
257#106—21no—
258#106—42no—
259#106C2×D412yes—
260C4⋊C8—41no—
261C4⋊C8—41no—
262#104—42no—
263#104—81no—
264#103—22no—
265#103—22no—
266#103—21no—
267#103—21no—
268#103—42no—
269#103—81no—
270#102—21no—
271#102—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—41no—
275C22⋊C8—41no—
276#97—21no—
277#97—21no—
278#97—21no—
279#97—21no—
280#97—41no—
281C4×C8—41no—
282#95—21no—
283#95—41no—
284#95—41no—
285#95—41no—
286#95—41no—
287C24AGL(1,ℤ/8)11yescharacteristic
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—161no—
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4AGL(1,ℤ/8)11yescharacteristic
320C22×C4AGL(1,ℤ/8)11yescharacteristic
321C22×C4AGL(1,ℤ/8)11yescharacteristic
322C22×C4C22×D411yescharacteristic
323Q16—22no—
324Q16—21no—
325Q16—22no—
326Q16—21no—
327Q16—41no—
328Q16—41no—
329SD16—41no—
330SD16—41no—
331D8—21no—
332D8—21no—
333M16—22no—
334M16—41no—
335M16—41no—
336M16—41no—
337M16—82no—
338M16—82no—
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—21no—
354C2×C8—21no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—82no—
358C2×C8—82no—
359C2×C8C4×D411yescharacteristic
360C2×C8C4×D411yescharacteristic
361C2×C8C23×C411yescharacteristic
362C2×C8C22×D411yescharacteristic
363C2×C8C2×#4111yescharacteristic
364C2×C8C2×#4111yescharacteristic
365C2×C8H(5,𝔽2)11yescharacteristic
366C2×C82−1+411yescharacteristic
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—161no—
382C22⋊C4—42no—
383C22⋊C4—41no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—41no—
395C42—82no—
396C42—161no—
397C16—22no—
398C16—41no—
399C16—41no—
400C16C4×D414yes—
401C23—21no—
402C23—22no—
403C23—21no—
404C23—41no—
405C23—41no—
406C23#44211yescharacteristic
407C23#44211yescharacteristic
408C23C2×AGL(1,ℤ/8)11yescharacteristic
409Q8—41no—
410Q8—42no—
411Q8—42no—
412Q8—41no—
413Q8—81no—
414Q8—81no—
415D4—41no—
416D4—41no—
417D4—81no—
418D4—82no—
419D4—81no—
420D4—161no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—164no—
455C2×C4#44211yescharacteristic
456C2×C4#44211yescharacteristic
457C2×C4C2×C4×D411yescharacteristic
458C2×C4D4211yescharacteristic
459C2×C4#54711yescharacteristic
460C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57711yescharacteristic
463C8—21no—
464C8—21no—
465C8—41no—
466C8—41no—
467C8—42no—
468C8—81no—
469C8C2×C4×D411yescharacteristic
470C8#52011yescharacteristic
471C8#54811yescharacteristic
472C8#55311yescharacteristic
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22#218012yescomplemented
486C22C2×#44211yescharacteristic
487C22#326511yescharacteristic
488C22#334511yescharacteristic
489C4—21no—
490C4—21no—
491C4—81no—
492C4—81no—
493C4—82no—
494C4—81no—
495C4—82no—
496C4—81no—
497C4—81no—
498C4—82no—
499C4—161no—
500C4#236111yescharacteristic
501C4#295211yescharacteristic
502C4#326511yescharacteristic
503C4#334511yescharacteristic
504C2—21no—
505C2—21no—
506C2—22no—
507C2—41no—
508C2—161no—
509C2#2302911yescharacteristic
510C2C2×#218011yescharacteristic
511C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#3050C22
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×#2180C2semidirect
3. C22×#365C2semidirect
4. #23028C2semidirect
6. #13250C2semidirect
7. #13174C2semidirect
8. #13143C2semidirect
9. #13141C2semidirect
10. #13003C2semidirect
12. #12957C2semidirect
13. #12955C2semidirect
14. #12677C2semidirect
15. #12616C2semidirect
26. #2220C4semidirect
27. #2219C4semidirect
28. #2217C4semidirect
29. #2216C4semidirect
76. C2×Q32C2×C4semidirect
77. C2×SD32C2×C4semidirect
136. #367C2×C4semidirect
137. #366C2×C4semidirect
141. #365D4semidirect
146. #358C2×C4semidirect
147. #358C2×C4semidirect
148. #358C2×C4semidirect
149. #357C2×C4semidirect
485. C22#2180semidirect

Maximal subgroups

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

subgroupindex
#336062
C2×#21802
C22×#3652
#230282
#230192
#132502
#131742
#131432
#131412
#130032
#130012
#129572
#129552
#126772
#126162