Finite Group Databasebeta

← #224785

Subgroups — #224785

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#559C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic, complemented
3C4×#446C211yesmaximal, characteristic, complemented
4#9676C212yesmaximal, complemented
5#9674C211yesmaximal, characteristic
6#9646C212yesmaximal, complemented
7#9638C211yesmaximal, characteristic
8#8564C211yesmaximal, characteristic, complemented
9#8563C212yesmaximal, complemented
10#8172C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#559C411yescharacteristic
13C2×#446—21no—
14C2×#446—21no—
15#2374—22no—
16#2367—21no—
17#2367C412yes—
18#2366C411yescharacteristic
19#2357—21no—
20#2357C412yes—
21#2355C411yescharacteristic
22#2355C411yescharacteristic
23C4×#127C2211yescharacteristic
24C4×#124—21no—
25C4×#124C2212yes—
26C4×#123—22no—
27C42×Q8C2211yescharacteristic
28C42×Q8C2211yescharacteristic
29C42×D4—21no—
30C42×D4C2212yes—
31#1940—22no—
32#1939—21no—
33#1842C2212yes—
34#1842C2211yescharacteristic
35#1771—22no—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×#97C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1279C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53#559—24no—
54#559—24no—
55C2×#127C2×C411yescharacteristic
56C2×#124—21no—
57C2×#124C2×C412yes—
58C2×#123—22no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C412yes—
70C2×C4×D4C2×C412yes—
71C2×C4×D4C2312yes—
72C22×C42—22no—
73#446—28no—
74#446—44no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2311yescharacteristic
84C2×#97—22no—
85C2×#97—22no—
86C2×#97—22no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2312yes—
92C2×C4×C8C2311yescharacteristic
93#391—21no—
94#391C2×C412yes—
95#390—21no—
96#390—21no—
97#390—21no—
98#390C2×C412yes—
99#390C2×C412yes—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#386—22no—
109#385—22no—
110#384C2×C411yescharacteristic
111#382—21no—
112#382C2×C412yes—
113#381—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C43—21no—
138C43C2311yescharacteristic
139C43C2312yes—
140#336—22no—
141#336—21no—
142#336—22no—
143#336—21no—
144Q8⋊C8—22no—
145Q8⋊C8—22no—
146Q8⋊C8—42no—
147D4⋊C8—24no—
148D4⋊C8—24no—
149D4⋊C8—42no—
150D4⋊C8—42no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22×C411yescharacteristic
153C22×D4—21no—
154C22×D4—21no—
155C22×D4C22×C412yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165#127—22no—
166#127M1614yes—
167#124—28no—
168#124—24no—
169#124—24no—
170#123—216no—
171C4×Q8—24no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—41no—
183C4×Q8M1614yes—
184C4×D4—28no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—42no—
193C4×D4—44no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—81no—
197C4×D4—81no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—82no—
231C2×C42—82no—
232C2×C42C22×C412yes—
233C2×C42C22×C412yes—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2411yescharacteristic
245C4⋊C8—24no—
246C4⋊C8—28no—
247C4⋊C8—22no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—22no—
251C4⋊C8—21no—
252C4⋊C8—44no—
253C4⋊C8—42no—
254C4⋊C8—82no—
255C4⋊C8—81no—
256C4⋊C8#4112yes—
257#97—28no—
258#97—44no—
259#97—44no—
260C4×C8—24no—
261C4×C8—22no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—22no—
265C4×C8—21no—
266C4×C8—42no—
267C4×C8—42no—
268C4×C8#4112yes—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280#95C22⋊C412yes—
281C24—22no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8C2×M1612yes—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—81no—
301C2×D4—81no—
302C2×D4C2×M1614yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—28no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—22no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—81no—
366C4⋊C4#10414yes—
367C4⋊C4C2×M1612yes—
368C4⋊C4#13112yes—
369C22⋊C4—28no—
370C22⋊C4—28no—
371C22⋊C4—28no—
372C22⋊C4—44no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—21no—
383C42—22no—
384C42—21no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—21no—
389C42—22no—
390C42—21no—
391C42—24no—
392C42—22no—
393C42—24no—
394C42—41no—
395C42—42no—
396C42—42no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—42no—
402C42—81no—
403C42—82no—
404C42—82no—
405C42C2×M1612yes—
406C42C2×M1612yes—
407C42AGL(1,ℤ/8)12yes—
408C42#13712yes—
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C42H(5,𝔽2)11yescharacteristic
412C422−1+411yescharacteristic
413C23—24no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23C22×C22⋊C411yescharacteristic
421Q8—22no—
422Q8—22no—
423Q8—41no—
424Q8—41no—
425Q8—44no—
426D4—24no—
427D4—24no—
428D4—42no—
429D4—42no—
430D4—48no—
431D4—84no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—88no—
468C2×C4—88no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#40712yes—
474C2×C4#40712yes—
475C2×C4#41712yes—
476C2×C4#41712yes—
477C2×C4C2×#10414yes—
478C2×C4C2×#10412yes—
479C2×C4C4×#4111yescharacteristic
480C2×C4#52011yescharacteristic
481C2×C4#54811yescharacteristic
482C2×C4#54811yescharacteristic
483C2×C4#55011yescharacteristic
484C2×C4#55011yescharacteristic
485C2×C4C22×M1611yescharacteristic
486C2×C4C22×M1611yescharacteristic
487C2×C4C2×#13111yescharacteristic
488C2×C4#56811yescharacteristic
489C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
490C2×C4C2×#13711yescharacteristic
491C2×C4#57611yescharacteristic
492C2×C4#57711yescharacteristic
493C8—24no—
494C8—42no—
495C8—44no—
496C8—42no—
497C8—81no—
498C8—82no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22#234111yescharacteristic
507C22C2×#40711yescharacteristic
508C22#288011yescharacteristic
509C22C2×#41711yescharacteristic
510C22#289711yescharacteristic
511C22C22×#10411yescharacteristic
512C22#290111yescharacteristic
513C4—21no—
514C4—21no—
515C4—22no—
516C4—22no—
517C4—24no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—41no—
524C4—42no—
525C4—82no—
526C4—81no—
527C4#148914yes—
528C4#296511yescharacteristic
529C4#297011yescharacteristic
530C4#298911yescharacteristic
531C4#299011yescharacteristic
532C4#337311yescharacteristic
533C4#337511yescharacteristic
534C4#339911yescharacteristic
535C4#340011yescharacteristic
536C2—84no—
537C2C2×#148911yescharacteristic
538C2#1745311yescharacteristic
539C2#1751511yescharacteristic
540C2#1751711yescharacteristic
541C2#2253211yescharacteristic
542C2#2254611yescharacteristic
543C2#2255011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22722C2semidirect
3. C4×#446C2semidirect
4. #9676C2semidirect
6. #9646C2semidirect
8. #8564C2semidirect
9. #8563C2semidirect
10. #8172C2semidirect
11. C4×Q8⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5592
#227222
C4×#4462
#96762
#96742
#96462
#96382
#85642
#85632
#81722
C4×Q8⋊C82