Finite Group Databasebeta

← #480390

Subgroups — #480390

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2#17362C212yesmaximal
3#11531C214yesmaximal, complemented
4#11511C212yesmaximal, complemented
5#11506C212yesmaximal, complemented
6#11477C214yesmaximal, complemented
7C22×#124C411yescharacteristic
8C22×C4×Q8C411yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#407—22no—
11C4×#124—22no—
12C4×#124—22no—
13C4×#124C412yes—
14C4×#124C412yes—
15C4×#124C2212yes—
16C4×#124C2212yes—
17C2×#391C2211yescharacteristic
18C2×#390C411yescharacteristic
19C2×#388—22no—
20C2×#382C2211yescharacteristic
21C2×C4×C22⋊C4—22no—
22C2×C43C411yescharacteristic
23#1465—22no—
24#1465C2214yes—
25#1463C2214yes—
26#1462—22no—
27#1459—22no—
28#1459C2214yes—
29#1324C2218yescomplemented
30#1319C2218yescomplemented
31C23×Q8C2×C411yescharacteristic
32C23×D4C2×C411yescharacteristic
33C24×C4D412yes—
34C22×M16—42no—
35C2×#124—22no—
36C2×#124—28no—
37C2×#124C2×C412yes—
38C2×#124C2×C412yes—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—41no—
42C2×C4×Q8—41no—
43C2×C4×Q8—41no—
44C2×C4×Q8—41no—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8C2×C412yes—
47C2×C4×D4—41no—
48C2×C4×D4—41no—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C2×C4×D4D412yes—
55C2×C4×D4C2311yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42—21no—
64C22×C42—21no—
65C22×C42—21no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69#407—24no—
70#407—24no—
71#407—42no—
72#407—42no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—42no—
75#391—22no—
76#391C2×C412yes—
77#391C2312yes—
78#390—22no—
79#390C2×C412yes—
80#390C2×C412yes—
81#388—24no—
82#388—24no—
83#388—24no—
84#388—24no—
85#382—22no—
86#382C2×C412yes—
87#382C2312yes—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—24no—
92C2×#95—22no—
93C2×#95—22no—
94C43—21no—
95C43—21no—
96C43C2×C412yes—
97C43C2×C412yes—
98#324—24no—
99#324C2318yes—
100#323—24no—
101#323—24no—
102C25C22⋊C412yes—
103C22×Q8—24no—
104C22×Q8—21no—
105C22×Q8—42no—
106C22×Q8—41no—
107C22×Q8—41no—
108C22×Q8C22×C412yes—
109C22×D4—24no—
110C22×D4—42no—
111C22×D4—41no—
112C22×D4—41no—
113C22×D4C22⋊C412yes—
114C22×D4C22×C411yescharacteristic
115C22×D4C22×C411yescharacteristic
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—48no—
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C2×M16—44no—
134C2×M16—44no—
135C2×M16—88no—
136C22×C8—44no—
137#124—216no—
138#124—24no—
139#124—24no—
140#124—24no—
141#124M1618yescomplemented
142C4×Q8—42no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—42no—
158C4×D4—41no—
159C4×D4—41no—
160C4×D4—41no—
161C4×D4—41no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2411yescharacteristic
175C2×C22⋊C4—216no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—48no—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C2×D412yes—
186C2×C22⋊C4C2×D412yes—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—24no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—42no—
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C412yes—
212C2×C42C22×C412yes—
213C22⋊C8—28no—
214C22⋊C8—44no—
215C22⋊C8—44no—
216C22⋊C8—44no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221C24—24no—
222C24—22no—
223C24—22no—
224C24—42no—
225C24—44no—
226C24—44no—
227C24C2×C22⋊C411yescharacteristic
228C24C2×C22⋊C412yes—
229C24C2×C22⋊C412yes—
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—42no—
233C2×Q8—48no—
234C2×Q8—48no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8C2×M1614yescomplemented
239C2×D4—24no—
240C2×D4—22no—
241C2×D4—24no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—48no—
248C2×D4—48no—
249C2×D4C2×M1614yes—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—48no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4#9911yescharacteristic
295C22×C4#9911yescharacteristic
296C22×C4#9911yescharacteristic
297C22×C4#9911yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D412yes—
306M16—88no—
307M16—88no—
308C2×C8—48no—
309C2×C8—88no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C22⋊C4—216no—
318C22⋊C4—216no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—48no—
324C22⋊C4—48no—
325C42—24no—
326C42—22no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—42no—
336C42—44no—
337C42—44no—
338C42—44no—
339C42—44no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42C2×M1614yescomplemented
346C23—21no—
347C23—21no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—44no—
356C23—44no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—416no—
362C23—44no—
363C23—44no—
364C23—82no—
365C23—82no—
366C23—88no—
367C23#40714yes—
368C23C2×#9911yescharacteristic
369C23C2×#9911yescharacteristic
370C23C22×C22⋊C411yescharacteristic
371Q8—44no—
372Q8—44no—
373Q8—48no—
374D4—44no—
375D4—48no—
376D4—44no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4#35414yes—
426C2×C4C2×#9911yescharacteristic
427C2×C4C2×#9911yescharacteristic
428C2×C4C2×#9911yescharacteristic
429C2×C4C2×#9911yescharacteristic
430C2×C4#41312yes—
431C2×C4C22×M1612yes—
432C8—88no—
433C22—24no—
434C22—24no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—24no—
442C22—44no—
443C22—44no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—82no—
450C22—88no—
451C22—88no—
452C22—82no—
453C22C2×#35412yes—
454C22#213212yes—
455C22C2×#40711yescharacteristic
456C22C22×#9911yescharacteristic
457C22C2×#41311yescharacteristic
458C4—21no—
459C4—21no—
460C4—22no—
461C4—22no—
462C4—44no—
463C4—44no—
464C4—42no—
465C4—42no—
466C4—41no—
467C4—41no—
468C4—44no—
469C4—82no—
470C4C2×#35412yes—
471C4C2×#35412yes—
472C2—21no—
473C2—21no—
474C2—22no—
475C2—22no—
476C2—44no—
477C2—82no—
478C2#1736211yescharacteristic
479C2C22×#35411yescharacteristic
480C2C2×#213211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11531C2semidirect
4. #11511C2semidirect
5. #11506C2semidirect
6. #11477C2semidirect
29. #1324C22semidirect
30. #1319C22semidirect
141. #124M16semidirect
238. C2×Q8C2×M16semidirect
345. C42C2×M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
#173622
#115312
#115112
#115062
#114772