Finite Group Databasebeta

← #229558

Subgroups — #229558

Order 512.

524 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19161C211yesmaximal, characteristic
2#18090C211yesmaximal, characteristic, complemented
3#18019C211yesmaximal, characteristic
4#18016C211yesmaximal, characteristic
5#17839C211yesmaximal, characteristic
6#9845C211yesmaximal, characteristic, complemented
7#9636C211yesmaximal, characteristic, complemented
8#9585C211yesmaximal, characteristic, complemented
9#9585C211yesmaximal, characteristic, complemented
10#8972C211yesmaximal, characteristic
11#8971C211yesmaximal, characteristic
12#8936C211yesmaximal, characteristic, complemented
13#8936C211yesmaximal, characteristic, complemented
14#8266C211yesmaximal, characteristic, complemented
15#8194C211yesmaximal, characteristic
16#3170C2211yescharacteristic
17#3107C2211yescharacteristic
18#3107C2211yescharacteristic
19#2991C2211yescharacteristic
20#2990C2211yescharacteristic
21#2986C2211yescharacteristic
22#2986C2211yescharacteristic
23#2965C2211yescharacteristic
24#1994C2211yescharacteristic
25#1994C2211yescharacteristic
26C4×#104C2211yescharacteristic
27C42×C8C2211yescharacteristic
28#1700C411yescharacteristic, complemented
29#1700C411yescharacteristic, complemented
30#1699C411yescharacteristic, complemented
31#1699C411yescharacteristic, complemented
32#1694C411yescharacteristic, complemented
33#1693C411yescharacteristic, complemented
34#1690C411yescharacteristic, complemented
35#1689C411yescharacteristic, complemented
36#1593—21no—
37#1591—21no—
38#1590—21no—
39#1588—21no—
40#1587—21no—
41#1587—21no—
42#1585—21no—
43#1584—21no—
44#1584—21no—
45#1579—21no—
46C8×Q16—21no—
47C8×SD16—21no—
48#1574C2211yescharacteristic
49#1574C2211yescharacteristic
50#1574C2211yescharacteristic
51#1574C2211yescharacteristic
52#1572—21no—
53#1571—21no—
54#1569—21no—
55#1567C2211yescharacteristic
56#1565—21no—
57#1564C2211yescharacteristic
58#1564C2211yescharacteristic
59#1544C2211yescharacteristic
60#1544C2211yescharacteristic
61#1493C2211yescharacteristic
62#1493C2211yescharacteristic
63#1485C2211yescharacteristic
64#1485C2211yescharacteristic
65#1476C2211yescharacteristic
66#1451C2211yescharacteristic
67#1450C2211yescharacteristic
68#1450C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1278C2211yescharacteristic
72#1276C2211yescharacteristic
73#1276C2211yescharacteristic
74#1276C2211yescharacteristic
75#533C2311yescharacteristic
76#533C2311yescharacteristic
77C4×#41C2311yescharacteristic
78#500C2×C411yescharacteristic
79#494C2×C411yescharacteristic
80#487C2×C411yescharacteristic
81#487C2×C411yescharacteristic
82#485—21no—
83#485—21no—
84#475C2×C411yescharacteristic
85#475C2×C411yescharacteristic
86#474C2×C411yescharacteristic
87#474C2×C411yescharacteristic
88#467—21no—
89#467—21no—
90#462C2×C411yescharacteristic
91#462C2×C411yescharacteristic
92#461C2×C411yescharacteristic
93#461C2×C411yescharacteristic
94#452—21no—
95#452—21no—
96#446—21no—
97#446—21no—
98#446—21no—
99#446—21no—
100#446—21no—
101#446C2×C411yescharacteristic
102#446C2×C411yescharacteristic
103C8×Q8—21no—
104C8×Q8—21no—
105C8×Q8—21no—
106C8×Q8—21no—
107C8×Q8—21no—
108C4×Q16—21no—
109C4×Q16—21no—
110C4×Q16—21no—
111C4×Q16—21no—
112C4×SD16—21no—
113C4×SD16—21no—
114#436—21no—
115#436—41no—
116#436C2×C411yescharacteristic
117#436C2×C411yescharacteristic
118#435—21no—
119#435—21no—
120#435—41no—
121C8×D4—21no—
122#433—41no—
123#433—41no—
124#432—21no—
125#431—21no—
126#431—21no—
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#431C2311yescharacteristic
133#423—21no—
134#423C2311yescharacteristic
135#423C2311yescharacteristic
136C2×C4⋊C8—21no—
137C2×#104—21no—
138C4×M16—21no—
139C4×M16C2311yescharacteristic
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C43C2311yescharacteristic
145#339—21no—
146#339—41no—
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#332C2×C411yescharacteristic
167#332C2×C411yescharacteristic
168#332C2×C411yescharacteristic
169#332C2×C411yescharacteristic
170#332C2×C411yescharacteristic
171#332C2×C411yescharacteristic
172#332C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174Q8⋊C8—21no—
175Q8⋊C8—21no—
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8—21no—
180Q8⋊C8—21no—
181Q8⋊C8—21no—
182Q8⋊C8D411yescharacteristic
183Q8⋊C8D411yescharacteristic
184D4⋊C8D411yescharacteristic
185D4⋊C8D411yescharacteristic
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190C82—21no—
191C82—21no—
192C2×#41#4111yescharacteristic
193C2×Q16—21no—
194C2×Q16—21no—
195C2×Q16—21no—
196C2×Q16—21no—
197C2×SD16—21no—
198C2×SD16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—41no—
205C22×C8—21no—
206C22×C8—41no—
207C22×C8#4111yescharacteristic
208C22×C8#4111yescharacteristic
209C4⋊Q8C22×C411yescharacteristic
210C4⋊Q8C22×C411yescharacteristic
211C4⋊Q8C22×C411yescharacteristic
212C4⋊Q8C22×C411yescharacteristic
213#125—21no—
214#125—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8C22×C411yescharacteristic
224C4×Q8C2×D411yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4C22×C411yescharacteristic
228C4×D4C2×D411yescharacteristic
229#117—21no—
230#117—21no—
231#117—21no—
232#117#4111yescharacteristic
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42C2411yescharacteristic
238#107—21no—
239#107—21no—
240#107—21no—
241#107—21no—
242#106—21no—
243#106—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—41no—
251C4⋊C8—41no—
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#103—21no—
270#103—21no—
271#103—21no—
272#103—21no—
273#103—21no—
274#103—21no—
275#103—21no—
276#103—21no—
277#103—21no—
278#103—21no—
279#102—21no—
280#102—21no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291#97—21no—
292#97—21no—
293#97—21no—
294#97—41no—
295#97—41no—
296#97—41no—
297C4×C8—24no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—41no—
307C4×C8—41no—
308C4×C8—41no—
309C4×C8—42no—
310C4×C8C22×C411yescharacteristic
311C4×C8C22×C411yescharacteristic
312C4×C8C2×D411yescharacteristic
313C4×C8C2×D411yescharacteristic
314C4×C8C2×D411yescharacteristic
315C4×C8C2×D411yescharacteristic
316#41—41no—
317#41—41no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8C4×D411yescharacteristic
323C2×D4—21no—
324C2×D4C4×D411yescharacteristic
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)11yescharacteristic
332Q16—42no—
333Q16—42no—
334Q16—81no—
335Q16—81no—
336SD16—42no—
337SD16—81no—
338M16—42no—
339M16—41no—
340M16—41no—
341M16—81no—
342M16—81no—
343M16—81no—
344M16—161no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
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—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—161no—
367C2×C8C4×D411yescharacteristic
368C2×C8C4×D411yescharacteristic
369C2×C8C4×D411yescharacteristic
370C2×C8C4×D411yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4C4×D411yescharacteristic
386C4⋊C4C4×D411yescharacteristic
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—21no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42C23×C411yescharacteristic
404C42C22×D411yescharacteristic
405C42C22×D411yescharacteristic
406C422−1+411yescharacteristic
407C23—21no—
408C23#55311yescharacteristic
409Q8—21no—
410Q8—21no—
411Q8—41no—
412Q8—41no—
413Q8—41no—
414Q8—41no—
415Q8—41no—
416Q8—41no—
417Q8—41no—
418Q8—41no—
419D4—21no—
420D4—21no—
421D4—81no—
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—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4#52011yescharacteristic
446C2×C4D4211yescharacteristic
447C2×C4#54711yescharacteristic
448C2×C4#54811yescharacteristic
449C8—22no—
450C8—22no—
451C8—21no—
452C8—21no—
453C8—21no—
454C8—41no—
455C8—41no—
456C8—41no—
457C8—41no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8#44312yes—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22#236111yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—44no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4C2×#44311yescharacteristic
489C4#295511yescharacteristic
490C4#295611yescharacteristic
491C4#295611yescharacteristic
492C4#329011yescharacteristic
493C4#329311yescharacteristic
494C2—41no—
495C2—81no—
496C2#2301511yescharacteristic
497C2#2302111yescharacteristic
498C2#2304511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radicalC2×#41#41
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. #18090C2semidirect
6. #9845C2semidirect
7. #9636C2semidirect
8. #9585C2semidirect
9. #9585C2semidirect
12. #8936C2semidirect
13. #8936C2semidirect
14. #8266C2semidirect
28. #1700C4semidirect
29. #1700C4semidirect
30. #1699C4semidirect
31. #1699C4semidirect
32. #1694C4semidirect
33. #1693C4semidirect
34. #1690C4semidirect
35. #1689C4semidirect

Maximal subgroups

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

subgroupindex
#191612
#180902
#180192
#180162
#178392
#98452
#96362
#95852
#89722
#89712
#89362
#82662
#81942