Finite Group Databasebeta

← #482389

Subgroups — #482389

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic, complemented
3#17365C211yesmaximal, characteristic, complemented
4#11882C214yesmaximal, complemented
5#11802C214yesmaximal, complemented
6#11781C214yesmaximal, complemented
7C22×#127C411yescharacteristic
8C22×#124—21no—
9C22×C4×Q8C2211yescharacteristic
10#2920—21no—
11#2920—21no—
12#2548—21no—
13#2548—21no—
14#2548—21no—
15#2548—21no—
16#2548C412yes—
17#2548C412yes—
18#2548C2212yes—
19#2548C2212yes—
20C2×#394C412yes—
21C2×#394C2212yes—
22C2×#390—21no—
23C2×#390—21no—
24C2×#390—21no—
25C2×C4×C4⋊C4C411yescharacteristic
26#1821C2212yes—
27#1801—21no—
28#1801C2212yescomplemented
29#1784—21no—
30C4×#101—21no—
31C4×#101C2212yescomplemented
32#1467—21no—
33#1467C2212yescomplemented
34#1463—21no—
35#1463C2212yes—
36#1461—21no—
37#1461C2212yescomplemented
38#1345C22116yescomplemented
39C23×Q8D411yescharacteristic
40C23×D4—21no—
41C23×D4—21no—
42C23×D4C2×C412yes—
43C2×#127—21no—
44C2×#127—21no—
45C2×#127—41no—
46C2×#127—41no—
47C2×#127—41no—
48C2×#127—41no—
49C2×#127C2×C412yes—
50C2×#127C2×C412yes—
51C2×#124—22no—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—48no—
56C2×C4×Q8—41no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—21no—
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80#432—22no—
81#432—41no—
82#432—41no—
83#432—41no—
84#431—22no—
85#431—41no—
86#431—41no—
87#431—41no—
88#423—22no—
89#423—41no—
90C2×#101—22no—
91C2×#101—22no—
92#407—22no—
93#407—22no—
94C4×M16—22no—
95C4×M16—41no—
96#394—22no—
97#394—22no—
98#394—22no—
99#394—22no—
100#394C2×C414yes—
101#394C2×C414yes—
102#394C2×C414yes—
103#394C2314yes—
104#390—22no—
105#390—22no—
106#390—22no—
107#390—22no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390—22no—
112#390—22no—
113#390—22no—
114#390—22no—
115#390—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C2×#95C2×C412yes—
121#343—22no—
122#343C2314yes—
123#324—22no—
124#324—22no—
125#324—22no—
126#324C2314yes—
127C25—22no—
128C22×Q8—24no—
129C22×Q8—41no—
130C22×Q8—41no—
131C22×Q8—42no—
132C22×Q8C22⋊C411yescharacteristic
133C22×Q8C22⋊C411yescharacteristic
134C22×Q8C2×D411yescharacteristic
135C22×Q8C2×D411yescharacteristic
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—44no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—41no—
150C22×D4—41no—
151C22×D4—41no—
152C22×D4—41no—
153C22×D4—84no—
154C22×D4C22×C414yes—
155C23×C4—22no—
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—24no—
164C2×M16—42no—
165#127—42no—
166#127—42no—
167#127—42no—
168#127—42no—
169#127—42no—
170#127—42no—
171#127—42no—
172#127—42no—
173#124—48no—
174#124—48no—
175#124—48no—
176#124—48no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2411yescharacteristic
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—48no—
224C2×C22⋊C4—84no—
225C2×C22⋊C4—84no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—44no—
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257#101—44no—
258#101—44no—
259C22⋊C8—24no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263#97—42no—
264#97—42no—
265C4×C8—42no—
266C4×C8—42no—
267#95—22no—
268#95C22×C414yes—
269C24—24no—
270C24—44no—
271C24—44no—
272C24—44no—
273C24—44no—
274C24—44no—
275C24—44no—
276C24—42no—
277C24—42no—
278C24C2×C22⋊C411yescharacteristic
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8AGL(1,ℤ/8)14yes—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—88no—
310C2×D4—88no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—88no—
316C2×D4—88no—
317C2×D4—84no—
318C2×D4—84no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4#9911yescharacteristic
346C22×C4#9911yescharacteristic
347C22×C4#9911yescharacteristic
348C22×C4#9911yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363M16—44no—
364M16—44no—
365M16—84no—
366C2×C8—44no—
367C2×C8—44no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C22⋊C4—84no—
389C22⋊C4—84no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—44no—
415C42—44no—
416C23—22no—
417C23—21no—
418C23—22no—
419C23—21no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—84no—
428C23—84no—
429C23—84no—
430C23—84no—
431C23—88no—
432C23—88no—
433C23—88no—
434C23—88no—
435C23—84no—
436C23—84no—
437C23C2×#9911yescharacteristic
438C23C2×#9911yescharacteristic
439C23C22×C22⋊C411yescharacteristic
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—44no—
444D4—84no—
445D4—84no—
446D4—88no—
447D4—88no—
448D4—88no—
449D4—88no—
450D4—84no—
451D4—84no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—88no—
499C2×C4#35514yes—
500C2×C4C2×#9911yescharacteristic
501C2×C4C2×#9911yescharacteristic
502C2×C4C2×#9911yescharacteristic
503C2×C4C2×#9911yescharacteristic
504C2×C4#41812yes—
505C2×C4#41812yes—
506C2×C4#42112yes—
507C2×C4C2×AGL(1,ℤ/8)12yes—
508C8—84no—
509C8—84no—
510C22—24no—
511C22—24no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—84no—
519C22—84no—
520C22—84no—
521C22—84no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—168no—
526C22—168no—
527C22#212212yes—
528C22C2×#35512yes—
529C22C22×#9911yescharacteristic
530C22C2×#41811yescharacteristic
531C22C2×#42111yescharacteristic
532C4—21no—
533C4—21no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—84no—
547C4C2×#35512yes—
548C4C2×#35512yes—
549C2—21no—
550C2—21no—
551C2—22no—
552C2—22no—
553C2—164no—
554C2#1757711yescharacteristic
555C2C2×#212211yescharacteristic
556C2C22×#35511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#127C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #17365C2semidirect
4. #11882C2semidirect
5. #11802C2semidirect
6. #11781C2semidirect
28. #1801C22semidirect
31. C4×#101C22semidirect
33. #1467C22semidirect
37. #1461C22semidirect
38. #1345C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
#207272
#173652
#118822
#118022
#117812