Finite Group Databasebeta

← #480277

Subgroups — #480277

Order 512.

1545 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34259C211yesmaximal, characteristic
2#17368C212yesmaximal, complemented
3#11510C212yesmaximal, complemented
4#11508C212yesmaximal, complemented
5#11507C212yesmaximal, complemented
6#11506C212yesmaximal, complemented
7#11432C212yesmaximal, complemented
8C2×#1319C212yesmaximal, complemented
9C22×2−1+4C411yescharacteristic
10C2×#520C2211yescharacteristic
11C2×C4×#41C411yescharacteristic
12#2973—22no—
13Q8×C22⋊C4—22no—
14#2339—22no—
15#2339—22no—
16#2339—22no—
17#2311C411yescharacteristic
18#2311C2211yescharacteristic
19#2311C2211yescharacteristic
20#2310C411yescharacteristic
21#2310C411yescharacteristic
22#2310C2211yescharacteristic
23#2297C411yescharacteristic
24#2297C411yescharacteristic
25#2297C2211yescharacteristic
26C2×#391C411yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29#1467—22no—
30#1465—22no—
31#1464C2214yes—
32#1463C2214yes—
33#1459—22no—
34C2×#324C2214yes—
35#1319C2214yes—
36#1319C2214yes—
37#1319C2214yes—
38#1318C2214yes—
39C2×2−1+4—24no—
40C2×2−1+4—24no—
41C2×2−1+4—44no—
42C2×2−1+4—44no—
43C22×#41—21no—
44C22×#41—21no—
45C22×#41—21no—
46C22×#41C2×C411yescharacteristic
47C22×#41C2×C411yescharacteristic
48C22×#41C2×C411yescharacteristic
49C22×#41C2×C411yescharacteristic
50C23×Q8—21no—
51C23×Q8C2×C411yescharacteristic
52C23×Q8C2×C411yescharacteristic
53C23×Q8C2×C411yescharacteristic
54#520—44no—
55C4×#41—44no—
56C2×C4×Q8—21no—
57C2×C4×Q8D412yes—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4D412yes—
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C2×#117C2311yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76#435—48no—
77#432—24no—
78#407—24no—
79C2×C22⋊C8—24no—
80#391—22no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#391—22no—
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#391C2×C411yescharacteristic
92#391C2×C411yescharacteristic
93#391C2×C411yescharacteristic
94#391C2×C411yescharacteristic
95#391C2311yescharacteristic
96#391C2311yescharacteristic
97#391C2311yescharacteristic
98#390C2×C411yescharacteristic
99#390C2×C411yescharacteristic
100#390C2×C411yescharacteristic
101#390C2311yescharacteristic
102#386—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#381—22no—
109#381—22no—
110#381—22no—
111#380—22no—
112#380—22no—
113#380—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120#324—24no—
121#324—24no—
122#324C2318yes—
123#323—24no—
1242−1+4—432no—
1252−1+4—432no—
126C2×#41—28no—
127C2×#41—28no—
128C2×#41—28no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—24no—
132C2×#41—24no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—42no—
144C2×#41—44no—
145C2×#41—44no—
146C2×#41—44no—
147C2×#41—44no—
148C2×#41—44no—
149C2×#41—44no—
150C2×#41M1614yescomplemented
151C22×Q8—28no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—22no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8C22⋊C411yescharacteristic
169C22×Q8M1614yescomplemented
170C22×Q8C22×C411yescharacteristic
171C22×Q8C22×C411yescharacteristic
172C22×Q8C22×C411yescharacteristic
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C22⋊C411yescharacteristic
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22×C411yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4C22⋊C412yes—
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C2×M16—44no—
196C22×C8—44no—
197C4×Q8—44no—
198C4×Q8—44no—
199C4×D4—28no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205#117—42no—
206#117—42no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C2×D412yes—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42C22⋊C412yes—
241C2×C42C22⋊C412yes—
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D412yes—
247C4⋊C8—44no—
248C22⋊C8—28no—
249C22⋊C8—44no—
250C22⋊C8—44no—
251#97—44no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264C24—21no—
265C24C2×C22⋊C411yescharacteristic
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C411yescharacteristic
268#41—216no—
269#41—216no—
270#41—48no—
271#41—48no—
272#41—48no—
273#41—48no—
274#41—48no—
275#41—48no—
276#41—432no—
277#41—432no—
278#41—432no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—48no—
295C2×Q8—48no—
296C2×Q8—48no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—48no—
304C2×Q8—432no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8C2×M1612yes—
308C2×Q8C2×M1612yes—
309C2×Q8C2×M1612yes—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—21no—
320C2×D4—22no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4#9914yes—
330C2×D4#10014yes—
331C2×D4C2×M1612yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×M1612yes—
389C22×C4C2×M1612yes—
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D412yes—
392M16—164no—
393C2×C8—48no—
394C2×C8—164no—
395C4⋊C4—24no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C42—24no—
416C42—42no—
417C42—42no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—42no—
423C42—42no—
424C42—42no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23#40712yes—
441C23C2×#9912yes—
442C23C2×#9912yes—
443C23C2×#10012yes—
444C23C2×#10012yes—
445C23C22×C22⋊C411yescharacteristic
446Q8—28no—
447Q8—42no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—48no—
454Q8—48no—
455Q8—48no—
456D4—28no—
457D4—28no—
458D4—28no—
459D4—42no—
460D4—42no—
461D4—48no—
462D4—48no—
463D4—48no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
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—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—162no—
519C2×C4#40712yes—
520C2×C4C2×#9912yes—
521C2×C4C2×#10012yes—
522C2×C4C22×M1611yescharacteristic
523C2×C4C22×M1611yescharacteristic
524C8—88no—
525C22—21no—
526C22—22no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—44no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—82no—
544C22—81no—
545C22—81no—
546C22#146214yes—
547C22C2×#40711yescharacteristic
548C22C22×#9911yescharacteristic
549C22C22×#10011yescharacteristic
550C4—21no—
551C4—21no—
552C4—22no—
553C4—22no—
554C4—22no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4—81no—
566C2—21no—
567C2—21no—
568C2—24no—
569C2—42no—
570C2—42no—
571C2—82no—
572C2C2×#146211yescharacteristic
573C2#3336111yescharacteristic
574C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×2−1+4C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17368C2semidirect
3. #11510C2semidirect
4. #11508C2semidirect
5. #11507C2semidirect
6. #11506C2semidirect
7. #11432C2semidirect
8. C2×#1319C2semidirect
150. C2×#41M16semidirect
169. C22×Q8M16semidirect

Maximal subgroups

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

subgroupindex
#342592
#173682
#115102
#115082
#115072
#115062
#114322
C2×#13192