Finite Group Databasebeta

← #480263

Subgroups — #480263

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2C2×#1463C212yesmaximal, complemented
3#11508C214yesmaximal, complemented
4#11506C212yesmaximal, complemented
5#11505C212yesmaximal, complemented
6#11432C214yesmaximal, complemented
7C23×#41C411yescharacteristic
8C22×#117C411yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407—22no—
11#2311—22no—
12#2311—22no—
13#2311C412yes—
14#2311C412yes—
15#2311C2212yes—
16#2311C2212yes—
17C2×#391C411yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#390C411yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#386—22no—
22C2×#380—22no—
23#1463—22no—
24#1463—22no—
25#1463C2214yes—
26#1463C2214yescomplemented
27C2×#324—22no—
28C2×#324C2214yescomplemented
29#1319C2218yescomplemented
30#1318C2218yescomplemented
31C22×#41—28no—
32C22×#41—21no—
33C22×#41—21no—
34C22×#41—44no—
35C22×#41—42no—
36C22×#41—42no—
37C22×#41C2×C412yes—
38C22×#41C2×C412yes—
39C23×Q8C2×C411yescharacteristic
40C23×D4—21no—
41C23×D4C2×C411yescharacteristic
42C24×C4—21no—
43C24×C4C2×C411yescharacteristic
44C24×C4D412yes—
45C22×M16—42no—
46C2×#117—21no—
47C2×#117—21no—
48C2×#117—44no—
49C2×#117C2×C412yes—
50C2×#117C2×C412yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—42no—
53C22×C4⋊C4—42no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68#407—24no—
69#407—24no—
70#407—42no—
71#407—42no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—42no—
74#391—22no—
75#391—22no—
76#391C2×C412yes—
77#391C2×C412yes—
78#391C2×C412yes—
79#391C2312yes—
80#390—22no—
81#390—22no—
82#390C2×C412yes—
83#390C2×C412yes—
84#390C2×C412yes—
85#390C2312yes—
86#386—24no—
87#386—24no—
88#386—24no—
89#386—24no—
90#380—24no—
91#380—24no—
92#380—24no—
93#380—24no—
94C2×#95—22no—
95C2×#95—22no—
96#324—24no—
97#324—24no—
98#324—24no—
99#324C2318yes—
100C25—21no—
101C25C22⋊C411yescharacteristic
102C2×#41—28no—
103C2×#41—28no—
104C2×#41—24no—
105C2×#41—24no—
106C2×#41—24no—
107C2×#41—432no—
108C2×#41—432no—
109C2×#41—44no—
110C2×#41—44no—
111C2×#41—44no—
112C2×#41—44no—
113C2×#41—44no—
114C2×#41—44no—
115C2×#41—44no—
116C2×#41—44no—
117C2×#41M1618yescomplemented
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—42no—
121C22×Q8—41no—
122C22×Q8—41no—
123C22×Q8C22×C412yes—
124C22×D4—28no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—24no—
128C22×D4—21no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—42no—
132C22×D4—44no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4C22×C412yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—28no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—44no—
153C23×C4—84no—
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C412yes—
158C23×C4C22⋊C412yes—
159C23×C4C22⋊C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C2×M16—44no—
167C2×M16—44no—
168C2×M16—88no—
169C22×C8—44no—
170#117—44no—
171#117—44no—
172#117—44no—
173#117—44no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2411yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—84no—
209C2×C22⋊C4—84no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C22⋊C8—28no—
217C22⋊C8—44no—
218C22⋊C8—44no—
219C22⋊C8—44no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224C24—22no—
225C24—22no—
226C24—24no—
227C24—22no—
228C24—42no—
229C24—42no—
230C24—44no—
231C24—42no—
232C24—44no—
233C24—41no—
234C24—41no—
235C24#9912yes—
236C24#9912yes—
237C24C2×C22⋊C411yescharacteristic
238C24C2×C22⋊C411yescharacteristic
239C24C2×C22⋊C411yescharacteristic
240#41—416no—
241#41—416no—
242#41—432no—
243#41—432no—
244#41—416no—
245#41—416no—
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—48no—
249C2×Q8—48no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8C2×M1614yescomplemented
255C2×D4—24no—
256C2×D4—28no—
257C2×D4—28no—
258C2×D4—22no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—48no—
262C2×D4—48no—
263C2×D4—42no—
264C2×D4—416no—
265C2×D4—416no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—42no—
271C2×D4C2×M1614yescomplemented
272C22×C4—216no—
273C22×C4—216no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—28no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—28no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—416no—
311C22×C4—416no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—48no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4#10014yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×M1614yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D412yes—
345M16—88no—
346M16—88no—
347C2×C8—48no—
348C2×C8—88no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—84no—
364C22⋊C4—84no—
365C22⋊C4—84no—
366C22⋊C4—84no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C23—21no—
372C23—21no—
373C23—22no—
374C23—22no—
375C23—21no—
376C23—21no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—48no—
389C23—48no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—416no—
394C23—44no—
395C23—42no—
396C23—42no—
397C23—88no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23#40712yes—
403C23C2×#9912yes—
404C23C2×#9911yescharacteristic
405C23C2×#9911yescharacteristic
406C23C2×#9912yes—
407C23C2×#10012yes—
408C23C22×C22⋊C411yescharacteristic
409Q8—44no—
410Q8—44no—
411Q8—48no—
412D4—44no—
413D4—44no—
414D4—48no—
415D4—416no—
416D4—416no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—28no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—416no—
439C2×C4—416no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4#40712yes—
469C2×C4C2×#10012yes—
470C2×C4C2×#10012yes—
471C2×C4C22×M1612yes—
472C8—88no—
473C22—24no—
474C22—24no—
475C22—21no—
476C22—22no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—22no—
481C22—21no—
482C22—21no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—42no—
487C22—42no—
488C22—48no—
489C22—48no—
490C22—42no—
491C22—42no—
492C22—48no—
493C22—44no—
494C22—84no—
495C22—88no—
496C22—88no—
497C22—82no—
498C22—82no—
499C22#146214yes—
500C22#212712yes—
501C22#213312yes—
502C22C2×#40711yescharacteristic
503C22C22×#9911yescharacteristic
504C22C22×#10011yescharacteristic
505C4—22no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—44no—
514C4—82no—
515C4—82no—
516C2—21no—
517C2—21no—
518C2—22no—
519C2—22no—
520C2—24no—
521C2—42no—
522C2—84no—
523C2C2×#146211yescharacteristic
524C2C2×#212711yescharacteristic
525C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1463C2semidirect
3. #11508C2semidirect
4. #11506C2semidirect
5. #11505C2semidirect
6. #11432C2semidirect
26. #1463C22semidirect
28. C2×#324C22semidirect
29. #1319C22semidirect
30. #1318C22semidirect
117. C2×#41M16semidirect
254. C2×Q8C2×M16semidirect
271. C2×D4C2×M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
C2×#14632
#115082
#115062
#115052
#114322