Finite Group Databasebeta

← #6629631

Subgroups — #6629631

Order 512.

2497 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3#33100C214yesmaximal, complemented
4#33099C212yesmaximal, complemented
5#33098C212yesmaximal, complemented
6#33096C214yesmaximal, complemented
7#33095C212yesmaximal, complemented
8#33094C212yesmaximal, complemented
9#21059C214yesmaximal, complemented
10#21033C214yesmaximal, complemented
11#20909C212yesmaximal
12#20844C212yesmaximal, complemented
13#20682C211yesmaximal, characteristic, complemented
14C2×#525C2211yescharacteristic
15C23×C4⋊C4C2212yes—
16C23×C42—21no—
17C23×C42C2211yescharacteristic
18C2×#428C2214yes—
19C2×#417C2214yes—
20C2×#407C2212yes—
21#2855C2211yescharacteristic
22#2855C2211yescharacteristic
23#2855C2212yes—
24#2854C2211yescharacteristic
25#2854C2211yescharacteristic
26#2854C2212yes—
27#2853—22no—
28#2849—21no—
29#2849—21no—
30C4×#123—22no—
31C4×#123—21no—
32C4×#123—21no—
33#2297C2212yes—
34#2296C2212yes—
35#2280C2211yescharacteristic
36C2×C4×C4⋊C4—22no—
37C2×C4×C4⋊C4—22no—
38C2×C4×C4⋊C4—21no—
39C2×C4×C4⋊C4—21no—
40C2×C4×C4⋊C4C2214yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2214yes—
44C22×#95—21no—
45#1928C2218yescomplemented
46#1927C2218yescomplemented
47#1926C2218yescomplemented
48#1925C2218yescomplemented
49#1921C2218yescomplemented
50#1920C2218yescomplemented
51#1919C2218yescomplemented
52#1918C2218yescomplemented
53#1874C2214yes—
54#1874C2214yes—
55#1872C2214yes—
56#1872C2214yes—
57#1853C2214yes—
58#1853C2214yes—
59#1852C2214yes—
60#1852C2214yes—
61#1817C2212yes—
62#1817C2212yes—
63#1817C2214yes—
64#1736C2214yes—
65#1736C2214yes—
66#1728C2214yes—
67C22×#41C2311yescharacteristic
68C24×C4—22no—
69C24×C4D411yescharacteristic
70C24×C4D411yescharacteristic
71C24×C4C2311yescharacteristic
72C22×M16C2314yes—
73#525—28no—
74C2×#123—22no—
75C2×#123—21no—
76C2×#123—21no—
77C2×#123—21no—
78C2×C22⋊Q8C2311yescharacteristic
79C2×C22⋊Q8C2311yescharacteristic
80C2×#121C2311yescharacteristic
81C2×#121C2311yescharacteristic
82C2×#120—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×#117C2312yes—
86C2×#117C2311yescharacteristic
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—216no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C42—28no—
95C22×C42—24no—
96C22×C42—24no—
97C22×C42—22no—
98C22×C42—22no—
99C22×C42—48no—
100C22×C42D412yes—
101C22×C42D412yes—
102C22×C42D411yescharacteristic
103C22×C42D411yescharacteristic
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108#428—216no—
109C2×#107C2318yes—
110C2×#106C2318yes—
111#417—216no—
112C2×#103C2318yes—
113C2×#102C2318yes—
114#407—28no—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117#391C2312yes—
118#391C2312yes—
119#390C2312yes—
120#390C2312yes—
121#389C2312yes—
122#389C2312yes—
123#389C2312yes—
124#389C2312yes—
125#388—22no—
126#388—22no—
127#388—22no—
128#388—22no—
129#387—22no—
130#387—22no—
131#387—22no—
132#387—22no—
133#386—22no—
134#386—22no—
135#386—22no—
136#386—21no—
137#386—21no—
138#385—22no—
139#385—22no—
140#385—22no—
141#385—21no—
142#385—21no—
143#381—22no—
144#381—21no—
145#381—21no—
146#381—21no—
147#380C2311yescharacteristic
148#380C2312yes—
149#380C2311yescharacteristic
150#379—21no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4C2318yes—
160C4×C4⋊C4C2318yes—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—22no—
164C4×C22⋊C4—22no—
165C2×#95—24no—
166C2×#95—24no—
167C2×#95—22no—
168C2×#95—22no—
169#340C2318yes—
170#340C2318yes—
171#340C2318yes—
172#340C2318yes—
173#336C2318yes—
174#336C2318yes—
175C25C2×D411yescharacteristic
176C2×#41—24no—
177C2×#41—28no—
178C22×Q8C2411yescharacteristic
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C2411yescharacteristic
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—216no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—21no—
193C23×C4—44no—
194C23×C4—48no—
195C23×C4—48no—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4#4118yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C2×M16—28no—
212C2×M16—28no—
213C2×M16—28no—
214C22×C8C2418yes—
215#123—48no—
216#123—44no—
217#123—44no—
218#123—44no—
219C22⋊Q8—24no—
220C22⋊Q8—24no—
221#121—24no—
222#121—24no—
223#120—44no—
224C4×D4—44no—
225C4×D4—44no—
226#117—24no—
227#117—28no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4#41116yes—
244C2×C4⋊C4C2411yescharacteristic
245C2×C4⋊C4C2411yescharacteristic
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2418yes—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C42—28no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—28no—
266C2×C42—28no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—28no—
273C2×C42—28no—
274C2×C42—48no—
275C2×C42—416no—
276C2×C42—416no—
277C2×C42—48no—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2412yes—
285C2×C42C2412yes—
286#107—216no—
287#106—216no—
288#103—216no—
289#102—216no—
290C22⋊C8—28no—
291C22⋊C8—28no—
292#95—28no—
293#95—28no—
294#95—21no—
295#95—22no—
296#95—21no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—21no—
310C24#12314yes—
311C24C22×D411yescharacteristic
312C24C22×D411yescharacteristic
313C24C22×D411yescharacteristic
314#41—416no—
315C2×Q8—24no—
316C2×Q8—22no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—44no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—44no—
326C22×C4—232no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—232no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—216no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—464no—
365C22×C4—416no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4#12314yes—
373C22×C4#12314yes—
374C22×C4#12314yes—
375C22×C4AGL(1,ℤ/8)14yes—
376C22×C4#13714yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4114yes—
385C22×C4C2×#4114yes—
386C22×C4C2×#4114yes—
387C22×C4C2×#4118yes—
388C22×C4C2511yescharacteristic
389M16—416no—
390C2×C8—28no—
391C2×C8—28no—
392C2×C8—28no—
393C4⋊C4—216no—
394C4⋊C4—216no—
395C4⋊C4—216no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4C2×#41116yes—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C42—24no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—416no—
430C42—42no—
431C42—44no—
432C42—416no—
433C23—216no—
434C23—216no—
435C23—28no—
436C23—28no—
437C23—28no—
438C23—22no—
439C23—22no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—22no—
445C23—22no—
446C23—41no—
447C23—42no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23C2×#12312yes—
452C23C2×#12312yes—
453C23C2×#12312yes—
454C23#52514yes—
455C23C2×AGL(1,ℤ/8)12yes—
456C23C2×#13712yes—
457C23#57514yes—
458C23C23×D411yescharacteristic
459Q8—44no—
460D4—44no—
461D4—84no—
462D4—84no—
463D4—84no—
464C2×C4—216no—
465C2×C4—216no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—216no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—432no—
490C2×C4—432no—
491C2×C4—432no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—84no—
509C2×C4C2×#12312yes—
510C2×C4C2×#12312yes—
511C2×C4C2×#12312yes—
512C2×C4#52514yes—
513C2×C4#529116yes—
514C2×C4C2×AGL(1,ℤ/8)14yes—
515C2×C4C2×#13714yes—
516C2×C4C22×#4112yes—
517C2×C4C22×#4114yes—
518C8—48no—
519C22—232no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—82no—
536C22—82no—
537C22#285314yes—
538C22#309214yes—
539C22#309314yes—
540C22C22×#12311yescharacteristic
541C22C2×#52512yes—
542C22C22×AGL(1,ℤ/8)11yescharacteristic
543C22C22×#13711yescharacteristic
544C22C2×#57512yes—
545C4—22no—
546C4—22no—
547C4—24no—
548C4—24no—
549C4—42no—
550C4—42no—
551C4—416no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—82no—
556C4—82no—
557C4#285314yes—
558C4C2×#52914yes—
559C2—24no—
560C2—24no—
561C2—24no—
562C2—161no—
563C2C2×#285311yescharacteristic
564C2C2×#309211yescharacteristic
565C2C2×#309311yescharacteristic
566C2#6055812yes—
567C2#6056012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C4⋊C4C2semidirect
3. #33100C2semidirect
4. #33099C2semidirect
5. #33098C2semidirect
6. #33096C2semidirect
7. #33095C2semidirect
8. #33094C2semidirect
9. #21059C2semidirect
10. #21033C2semidirect
12. #20844C2semidirect
13. #20682C2semidirect
45. #1928C22semidirect
46. #1927C22semidirect
47. #1926C22semidirect
48. #1925C22semidirect
49. #1921C22semidirect
50. #1920C22semidirect
51. #1919C22semidirect
52. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#366262
C22×C4×C4⋊C42
#331002
#330992
#330982
#330962
#330952
#330942
#210592
#210332
#209092
#208442
#206822