Finite Group Databasebeta

← #244207

Subgroups — #244207

Order 512.

868 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#500C211yesmaximal, characteristic, complemented
2C4×#498C211yesmaximal, characteristic, complemented
3C4×#492C211yesmaximal, characteristic
4C4×#492C211yesmaximal, characteristic
5C4×#487C211yesmaximal, characteristic, complemented
6C4×#486C211yesmaximal, characteristic
7#10685C211yesmaximal, characteristic, complemented
8#10684C211yesmaximal, characteristic, complemented
9#10681C211yesmaximal, characteristic, complemented
10#10672C211yesmaximal, characteristic
11#10315C211yesmaximal, characteristic
12#10314C211yesmaximal, characteristic, complemented
13#10310C211yesmaximal, characteristic
14#10308C211yesmaximal, characteristic
15#8131C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#498C2211yescharacteristic
18C2×#492C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×C4×SD16—22no—
23C2×C4×SD16—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×#127C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1968—22no—
35#1968—22no—
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1956C2211yescharacteristic
40#1956C2211yescharacteristic
41#1956C2211yescharacteristic
42#1942—22no—
43#1941—22no—
44#1934—22no—
45#1933—22no—
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52C4×#107—22no—
53C4×#106—22no—
54C4×#103—22no—
55C4×#102—22no—
56C42×C8C2211yescharacteristic
57C42×C8C2211yescharacteristic
58C42×C8C2211yescharacteristic
59#1273C2211yescharacteristic
60#1273C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×SD16—22no—
64C22×SD16—22no—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75#500—22no—
76#500—22no—
77#498—22no—
78#498—22no—
79#492—22no—
80#492—22no—
81#492—22no—
82#492—22no—
83#487—22no—
84#487—22no—
85#486—22no—
86#486—22no—
87C4×SD16—48no—
88C4×SD16—48no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×C4×C8D412yes—
104C2×C4×C8D412yes—
105C2×C4×C8D412yes—
106C2×C4×C8D412yes—
107C2×C4×C8D412yes—
108C2×C4×C8D412yes—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391—21no—
116#391—21no—
117#391—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C43C2311yescharacteristic
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C23×C4—41no—
153C2×SD16—28no—
154C2×SD16—28no—
155C2×SD16—44no—
156C2×SD16—44no—
157C2×SD16—44no—
158C2×SD16—44no—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8#4112yes—
175C4⋊Q8#4112yes—
176C4⋊Q8#4112yes—
177#127—21no—
178#127—21no—
179#127—21no—
180#127#4112yes—
181C4×Q8—44no—
182C4×Q8—44no—
183C4×Q8—44no—
184C4×D4—44no—
185C4×D4—44no—
186C4×D4—44no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C42—41no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2411yescharacteristic
221#107—28no—
222#107—44no—
223#106—28no—
224#106—44no—
225#106—44no—
226#106—44no—
227#103—28no—
228#103—44no—
229#103—44no—
230#103—44no—
231#102—28no—
232#102—44no—
233#102—44no—
234#102—44no—
235C4×C8—24no—
236C4×C8—24no—
237C4×C8—24no—
238C4×C8—24no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—42no—
252C4×C8SD1618yescomplemented
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259C24—41no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4#12711yescharacteristic
295C22×C4#12711yescharacteristic
296C22×C4#12711yescharacteristic
297C22×C4#12711yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301SD16—416no—
302SD16—416no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8C2×SD1614yes—
318C2×C8C2×SD1614yes—
319C2×C8#13514yes—
320C2×C8#13514yes—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—82no—
355C42—82no—
356C42—82no—
357C42C2×SD1614yes—
358C42AGL(1,ℤ/8)14yes—
359C42#13714yes—
360C42C2×#4111yescharacteristic
361C42C2×#4111yescharacteristic
362C42C2×#4111yescharacteristic
363C42C2×#4111yescharacteristic
364C23—42no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23C2×#12711yescharacteristic
372Q8—44no—
373Q8—44no—
374Q8—44no—
375Q8—82no—
376Q8—82no—
377Q8—82no—
378D4—44no—
379D4—44no—
380D4—44no—
381D4—82no—
382D4—82no—
383D4—82no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—164no—
412C2×C4#49212yes—
413C2×C4#49212yes—
414C2×C4#49612yes—
415C2×C4#49612yes—
416C2×C4#49712yes—
417C2×C4#49712yes—
418C2×C4#52511yescharacteristic
419C2×C4#52511yescharacteristic
420C2×C4#52511yescharacteristic
421C2×C4#52511yescharacteristic
422C2×C4#52511yescharacteristic
423C2×C4#52511yescharacteristic
424C2×C4C22×SD1612yes—
425C2×C4C2×#13512yes—
426C2×C4C2×#13512yes—
427C2×C4C2×AGL(1,ℤ/8)12yes—
428C2×C4C2×#13712yes—
429C2×C4#57512yes—
430C8—24no—
431C8—24no—
432C8—24no—
433C8—24no—
434C8—42no—
435C8—42no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22#239711yescharacteristic
444C22C2×#49211yescharacteristic
445C22#314711yescharacteristic
446C22#314711yescharacteristic
447C22C2×#49611yescharacteristic
448C22C2×#49711yescharacteristic
449C22#315011yescharacteristic
450C4—22no—
451C4—22no—
452C4—22no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4#171414yes—
461C4#310112yes—
462C4#310312yes—
463C4#310512yes—
464C4#310612yes—
465C4#310712yes—
466C4#310812yes—
467C2—162no—
468C2#1959611yescharacteristic
469C2C2×#171411yescharacteristic
470C2#1961911yescharacteristic
471C2#1962011yescharacteristic
472C2#2379611yescharacteristic
473C2#2379711yescharacteristic
474C2#2379911yescharacteristic
subgroupG/H
centerC22×C4#127
commutator, Frattini, cosocleC2×C42C24
socleC23C2×#127
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#500C2semidirect
2. C4×#498C2semidirect
5. C4×#487C2semidirect
7. #10685C2semidirect
8. #10684C2semidirect
9. #10681C2semidirect
12. #10314C2semidirect
15. #8131C2semidirect
252. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
C4×#5002
C4×#4982
C4×#4922
C4×#4872
C4×#4862
#106852
#106842
#106812
#106722
#103152
#103142
#103102
#103082
#81312