Finite Group Databasebeta

← #6484971

Subgroups — #6484971

Order 512.

1856 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#33095C214yesmaximal, complemented
4#29907C214yesmaximal, complemented
5#26318C214yesmaximal, complemented
6#26278C214yesmaximal, complemented
7#25058C214yesmaximal, complemented
8#24992C214yesmaximal, complemented
9#21981C211yesmaximal, characteristic, complemented
10#21859C211yesmaximal, characteristic
11#21818C211yesmaximal, characteristic, complemented
12#21386C211yesmaximal, characteristic
13#21359C211yesmaximal, characteristic, complemented
14C2×#536—21no—
15C2×#532C2211yescharacteristic
16C22×C4×Q8—21no—
17C22×C4×Q8C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2854C2214yes—
21#2618—22no—
22#2548C2212yes—
23#2546C2212yes—
24#2545—22no—
25#2538C2212yes—
26#2538C2212yes—
27#2536—22no—
28Q8×C4⋊C4—22no—
29Q8×C4⋊C4—22no—
30Q8×C4⋊C4C2214yes—
31Q8×C4⋊C4C2214yes—
32#2311—21no—
33C2×#389—21no—
34C2×#389—22no—
35#2285C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#384C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40#2100C2218yes—
41#2094C2218yes—
42#2084C2218yes—
43#2071C2218yes—
44#1990C2212yes—
45#1988C2212yes—
46#1984C2214yes—
47#1984C2214yes—
48#1982C2212yes—
49#1957C2212yes—
50#1948C2212yes—
51#1948C2212yes—
52#1948C2214yes—
53#1926C2218yes—
54#1926C2218yes—
55#1919C2218yes—
56#1919C2218yes—
57#1874C2212yes—
58#1874C2214yes—
59#1874C2212yes—
60#1853C2212yes—
61#1853C2214yes—
62#1853C2212yes—
63#1844C2212yes—
64#1834C2212yes—
65#1817C2214yes—
66#1802C2214yes—
67C4×#106C2214yes—
68C4×#102C2214yes—
69C22×#41—21no—
70C23×Q8D411yescharacteristic
71#536—48no—
72#532—24no—
73#532—24no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C2311yescharacteristic
76C2×#127C2311yescharacteristic
77C2×#124—22no—
78C2×#124—21no—
79C2×#123—24no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×#121—21no—
83C2×#121C2312yes—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—48no—
89C2×C4×Q8D412yes—
90C2×C4×Q8D412yes—
91C2×C4×Q8C2312yes—
92C2×C4×Q8C2312yes—
93C2×C4×D4—21no—
94C2×C4×D4C2312yes—
95C2×#117—21no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4D412yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C42—22no—
106C22×C42—21no—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109#431—24no—
110#431—24no—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×#106C2318yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×#102C2314yes—
117C2×#102C2318yes—
118C2×#102C2314yes—
119C2×C22⋊C8C2312yes—
120C2×C22⋊C8C2312yes—
121C2×C4×C8C2312yes—
122C2×C4×C8C2312yes—
123#398—24no—
124#396—24no—
125#396—24no—
126#394—24no—
127#391—22no—
128#390—22no—
129#390C2314yes—
130#390C2314yes—
131#389—24no—
132#389—24no—
133#389—28no—
134#389—28no—
135#389C2314yes—
136#389C2314yes—
137#388—24no—
138#388—24no—
139#386—24no—
140#384—24no—
141#384—21no—
142#384—21no—
143#384C2312yes—
144#384C2318yes—
145#384C2311yescharacteristic
146#384C2312yes—
147#384C2311yescharacteristic
148#382—21no—
149#380—21no—
150#380—22no—
151#380—22no—
152#380C2312yes—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4C2318yes—
158C4×C4⋊C4C2312yes—
159C4×C4⋊C4C2312yes—
160C4×C4⋊C4C2312yes—
161C2×#95—22no—
162C2×#95—21no—
163#340C2318yes—
164#340C2314yes—
165#340C2318yes—
166#340C2314yes—
167#340C2314yes—
168#340C2314yes—
169#336C2318yes—
170#336C2318yes—
171C2×#41—44no—
172C2×#41—44no—
173C2×#41—44no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—21no—
177C22×Q8SD1618yes—
178C22×Q8C2×D412yes—
179C22×Q8C2×D412yes—
180C22×D4—22no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4C2412yes—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2411yescharacteristic
193C22×C8C2414yes—
194C22×C8C2414yes—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—48no—
198#127—22no—
199#127—22no—
200#124—48no—
201#124—44no—
202#123—416no—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205#121—24no—
206#121—24no—
207#121—44no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—416no—
215C4×Q8—416no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—44no—
219#117—44no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4—84no—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—28no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—48no—
269C2×C42—44no—
270C2×C42C2×D414yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4114yes—
274C2×C42#4114yes—
275C2×C42C2414yes—
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278#106—28no—
279#106—48no—
280#106—44no—
281C4⋊C8—24no—
282C4⋊C8—24no—
283#102—28no—
284#102—44no—
285#102—48no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C4×C8—24no—
289C4×C8—24no—
290#95—28no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—24no—
297C24—22no—
298C24C22×D411yescharacteristic
299#41—816no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—232no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8C2×SD1618yes—
309C2×Q8C2×SD1618yes—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—81no—
322C2×D4—81no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4#12314yes—
351C22×C4#12314yes—
352C22×C4C2×SD1618yes—
353C22×C4C2×SD1614yes—
354C22×C4AGL(1,ℤ/8)14yes—
355C22×C4C22×D414yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D414yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4C2511yescharacteristic
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C4⋊C4—24no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—416no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—81no—
395C4⋊C4—81no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—88no—
399C4⋊C42−1+414yes—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—84no—
406C22⋊C4—84no—
407C42—28no—
408C42—24no—
409C42—24no—
410C42—416no—
411C42—44no—
412C42—44no—
413C42—44no—
414C42#12314yes—
415C42#12314yes—
416C42C2×#4112yes—
417C42C2×#4112yes—
418C42C2×#4112yes—
419C42C2×#4112yes—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23C2×#12312yes—
432C23C22×SD1612yes—
433C23C2×AGL(1,ℤ/8)12yes—
434C23C23×D411yescharacteristic
435Q8—216no—
436Q8—216no—
437Q8—84no—
438D4—44no—
439D4—44no—
440D4—84no—
441D4—88no—
442D4—84no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—216no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—88no—
479C2×C4—84no—
480C2×C4—88no—
481C2×C4#481116yes—
482C2×C4C2×#12312yes—
483C2×C4C2×#12312yes—
484C2×C4C2×#12312yes—
485C2×C4C2×#12312yes—
486C2×C4C2×#12312yes—
487C2×C4C22×SD1618yes—
488C2×C4C22×SD1612yes—
489C2×C4C22×SD1612yes—
490C2×C4C2×AGL(1,ℤ/8)12yes—
491C2×C4C2×AGL(1,ℤ/8)12yes—
492C2×C4C22×#4111yescharacteristic
493C2×C4C22×#4111yescharacteristic
494C2×C4C2×2−1+412yes—
495C2×C4#58512yes—
496C8—44no—
497C8—44no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22C2×#48114yes—
512C22C22×#12311yescharacteristic
513C22#348611yescharacteristic
514C22#348711yescharacteristic
515C22C23×SD1611yescharacteristic
516C22C22×AGL(1,ℤ/8)11yescharacteristic
517C22#358511yescharacteristic
518C22#358611yescharacteristic
519C4—28no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—44no—
524C4—44no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—84no—
530C4C2×#48114yes—
531C4C2×#48114yes—
532C4#346112yes—
533C4#350712yes—
534C2—24no—
535C2—162no—
536C2#3518211yescharacteristic
537C2C22×#48111yescharacteristic
538C2#6051711yescharacteristic
539C2#6182011yescharacteristic
540C2#6182711yescharacteristic
541C2#6188311yescharacteristic
542C2#6189011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
3. #33095C2semidirect
4. #29907C2semidirect
5. #26318C2semidirect
6. #26278C2semidirect
7. #25058C2semidirect
8. #24992C2semidirect
9. #21981C2semidirect
11. #21818C2semidirect
13. #21359C2semidirect

Maximal subgroups

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

subgroupindex
#351782
C2×Q8×C4⋊C42
#330952
#299072
#263182
#262782
#250582
#249922
#219812
#218592
#218182
#213862
#213592