Finite Group Databasebeta

← #229552

Subgroups — #229552

Order 512.

538 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#19127C211yesmaximal, characteristic
2#18089C211yesmaximal, characteristic, complemented
3#18016C211yesmaximal, characteristic, complemented
4#18015C211yesmaximal, characteristic
5#17836C211yesmaximal, characteristic, complemented
6#9851C211yesmaximal, characteristic, complemented
7#9636C211yesmaximal, characteristic, complemented
8#9586C211yesmaximal, characteristic, complemented
9#9585C211yesmaximal, characteristic, complemented
10#8971C211yesmaximal, characteristic, complemented
11#8969C211yesmaximal, characteristic
12#8936C211yesmaximal, characteristic, complemented
13#8934C211yesmaximal, characteristic, complemented
14#8268C211yesmaximal, characteristic, complemented
15#8192C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3107C2211yescharacteristic
18#3106C2211yescharacteristic
19#2990C2211yescharacteristic
20#2990C2211yescharacteristic
21#2986C2211yescharacteristic
22#2979C2211yescharacteristic
23C8×#41C2211yescharacteristic
24#1994C2211yescharacteristic
25#1962C2211yescharacteristic
26C4×#104C2211yescharacteristic
27C4×#97C2211yescharacteristic
28#1684C411yescharacteristic, complemented
29#1683C411yescharacteristic, complemented
30#1682C411yescharacteristic, complemented
31#1681C411yescharacteristic, complemented
32#1672C411yescharacteristic, complemented
33#1671C411yescharacteristic, complemented
34#1666C411yescharacteristic, complemented
35#1665C411yescharacteristic, complemented
36#1592—21no—
37#1591—21no—
38#1590—21no—
39#1587—21no—
40#1586—21no—
41#1585—21no—
42#1585—21no—
43#1583—21no—
44#1582—21no—
45SD16⋊C8—21no—
46C8×SD16—21no—
47C8×D8—21no—
48#1574C2211yescharacteristic
49#1574C2211yescharacteristic
50#1574C2211yescharacteristic
51#1574C2211yescharacteristic
52#1573—21no—
53#1573—21no—
54#1568—21no—
55#1567C2211yescharacteristic
56#1564C2211yescharacteristic
57#1564C2211yescharacteristic
58C2×#335—21no—
59#1544C2211yescharacteristic
60#1543C2211yescharacteristic
61#1493C2211yescharacteristic
62#1492C2211yescharacteristic
63#1485C2211yescharacteristic
64#1484C2211yescharacteristic
65#1476C2211yescharacteristic
66#1451C2211yescharacteristic
67#1450C2211yescharacteristic
68#1450C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1278C2211yescharacteristic
72#1276C2211yescharacteristic
73#1276C2211yescharacteristic
74#1276C2211yescharacteristic
75#533C2311yescharacteristic
76#532C2311yescharacteristic
77C4×#41C2311yescharacteristic
78#498C2×C411yescharacteristic
79#492C2×C411yescharacteristic
80#487C2×C411yescharacteristic
81#486C2×C411yescharacteristic
82#485—21no—
83#482—21no—
84#477C2×C411yescharacteristic
85#476C2×C411yescharacteristic
86#475C2×C411yescharacteristic
87#474C2×C411yescharacteristic
88#465—21no—
89#465—21no—
90#462C2×C411yescharacteristic
91#461C2×C411yescharacteristic
92#460C2×C411yescharacteristic
93#459C2×C411yescharacteristic
94#452—21no—
95#449—21no—
96#446—21no—
97#446—21no—
98#446—21no—
99#446—21no—
100C8×Q8—21no—
101C8×Q8—21no—
102C8×Q8C2×C411yescharacteristic
103C8×Q8C2×C411yescharacteristic
104C4×Q16—21no—
105C4×SD16—21no—
106C4×SD16—21no—
107C4×SD16—21no—
108C4×SD16—21no—
109C4×D8—21no—
110#436—21no—
111#436—21no—
112#435—41no—
113#435—41no—
114C8×D4—21no—
115C8×D4—21no—
116C8×D4—21no—
117C8×D4—21no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4C2×C411yescharacteristic
121C8×D4C2×C411yescharacteristic
122#433—41no—
123#432—21no—
124#432—21no—
125#431—21no—
126#431C2311yescharacteristic
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134C2×C4⋊C8—21no—
135C2×C4⋊C8—21no—
136C2×#104—21no—
137C2×#104—41no—
138#408—41no—
139C4×M16C2311yescharacteristic
140C2×#97—21no—
141C2×#97—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C43C2311yescharacteristic
146#339—21no—
147#339—41no—
148#335—21no—
149#335—21no—
150#335—21no—
151#335—21no—
152#335—21no—
153#335—21no—
154#335C2×C411yescharacteristic
155#335C2×C411yescharacteristic
156#335D411yescharacteristic
157#335D411yescharacteristic
158#334—21no—
159#334—21no—
160#334—21no—
161#334—21no—
162#334C2×C411yescharacteristic
163#334C2×C411yescharacteristic
164#334D411yescharacteristic
165#334D411yescharacteristic
166#333C2×C411yescharacteristic
167#333C2×C411yescharacteristic
168#332C2×C411yescharacteristic
169#332C2×C411yescharacteristic
170#332C2×C411yescharacteristic
171#332C2×C411yescharacteristic
172#331C2×C411yescharacteristic
173#331C2×C411yescharacteristic
174Q8⋊C8—21no—
175Q8⋊C8—21no—
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8D411yescharacteristic
179Q8⋊C8D411yescharacteristic
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8D411yescharacteristic
185D4⋊C8D411yescharacteristic
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190C82—21no—
191C82—21no—
192C2×#41—21no—
193C2×#41#4111yescharacteristic
194C2×Q16—21no—
195C2×SD16—21no—
196C2×SD16—21no—
197C2×SD16—21no—
198C2×SD16—21no—
199C2×D8—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—41no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—41no—
208C22×C8—41no—
209C22×C8—41no—
210C22×C8#4111yescharacteristic
211C22×C8#4111yescharacteristic
212C4⋊Q8C22×C411yescharacteristic
213C4⋊Q8C22×C411yescharacteristic
214C4⋊Q8C22×C411yescharacteristic
215#127C22×C411yescharacteristic
216#125—21no—
217#124—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220#121—21no—
221#121—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8C22×C411yescharacteristic
225C4×Q8C2×D411yescharacteristic
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4C22×C411yescharacteristic
231C4×D4C2×D411yescharacteristic
232#117—21no—
233#117—21no—
234#117#4111yescharacteristic
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C2411yescharacteristic
240#107—21no—
241#107—21no—
242#106—21no—
243#106—21no—
244#106—21no—
245#106—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8C22×C411yescharacteristic
257C4⋊C8C22×C411yescharacteristic
258C4⋊C8C22×C411yescharacteristic
259C4⋊C8C22×C411yescharacteristic
260C4⋊C8C22×C411yescharacteristic
261C4⋊C8C22×C411yescharacteristic
262C4⋊C8C2×D411yescharacteristic
263C4⋊C8C2×D411yescharacteristic
264C4⋊C8C2×D411yescharacteristic
265C4⋊C8C2×D411yescharacteristic
266C4⋊C8C2×D411yescharacteristic
267C4⋊C8C2×D411yescharacteristic
268#104—42no—
269#104—82no—
270#103—21no—
271#103—21no—
272#103—21no—
273#103—21no—
274#103—21no—
275#103—21no—
276#102—21no—
277#102—21no—
278#102—21no—
279#102—21no—
280#102—21no—
281#102—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—41no—
291C22⋊C8—41no—
292C22⋊C8—41no—
293C22⋊C8—41no—
294#97—24no—
295#97—21no—
296#97—41no—
297#97—41no—
298#97—41no—
299#97—42no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—41no—
311C4×C8C22×C411yescharacteristic
312C4×C8C22×C411yescharacteristic
313C4×C8C2×D411yescharacteristic
314C4×C8C2×D411yescharacteristic
315C4×C8C2×D411yescharacteristic
316C4×C8C2×D411yescharacteristic
317#41—22no—
318#41—41no—
319#41—81no—
320#41—81no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8C4×D411yescharacteristic
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—41no—
330C2×D4C4×D411yescharacteristic
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339Q16—81no—
340SD16—42no—
341SD16—42no—
342SD16—81no—
343SD16—81no—
344D8—42no—
345M16—41no—
346M16—41no—
347M16—81no—
348M16—81no—
349M16—161no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—81no—
373C2×C8—161no—
374C2×C8—161no—
375C2×C8—161no—
376C2×C8C4×D411yescharacteristic
377C2×C8C4×D411yescharacteristic
378C2×C8C4×D411yescharacteristic
379C2×C8C4×D411yescharacteristic
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4C4×D411yescharacteristic
391C4⋊C4C4×D411yescharacteristic
392C22⋊C4—21no—
393C22⋊C4—21no—
394C22⋊C4—21no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—22no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42C23×C411yescharacteristic
409C42C22×D411yescharacteristic
410C42C22×D411yescharacteristic
411C422−1+411yescharacteristic
412C23—21no—
413C23—41no—
414C23#55311yescharacteristic
415Q8—21no—
416Q8—21no—
417Q8—41no—
418Q8—41no—
419Q8—41no—
420Q8—41no—
421Q8—81no—
422D4—21no—
423D4—42no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—81no—
429D4—161no—
430D4#44412yes—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4#52011yescharacteristic
456C2×C4D4211yescharacteristic
457C2×C4#54711yescharacteristic
458C2×C4#54811yescharacteristic
459C8—22no—
460C8—22no—
461C8—21no—
462C8—21no—
463C8—21no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—81no—
477C8—81no—
478C8—81no—
479C8#44312yes—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—22no—
484C22—41no—
485C22—41no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22#236111yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—82no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4C2×#44311yescharacteristic
501C4C2×#44411yescharacteristic
502C4#295611yescharacteristic
503C4#295711yescharacteristic
504C4#328811yescharacteristic
505C4#328911yescharacteristic
506C2—41no—
507C2—42no—
508C2—161no—
509C2#2301511yescharacteristic
510C2#2301911yescharacteristic
511C2#2304911yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#3106C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18089C2semidirect
3. #18016C2semidirect
5. #17836C2semidirect
6. #9851C2semidirect
7. #9636C2semidirect
8. #9586C2semidirect
9. #9585C2semidirect
10. #8971C2semidirect
12. #8936C2semidirect
13. #8934C2semidirect
14. #8268C2semidirect
15. #8192C2semidirect
28. #1684C4semidirect
29. #1683C4semidirect
30. #1682C4semidirect
31. #1681C4semidirect
32. #1672C4semidirect
33. #1671C4semidirect
34. #1666C4semidirect
35. #1665C4semidirect

Maximal subgroups

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

subgroupindex
#191272
#180892
#180162
#180152
#178362
#98512
#96362
#95862
#95852
#89712
#89692
#89362
#89342
#82682
#81922