Finite Group Databasebeta

← #480389

Subgroups — #480389

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2#17365C212yesmaximal, complemented
3#11530C214yesmaximal, complemented
4#11511C212yesmaximal, complemented
5#11505C212yesmaximal
6#11476C214yesmaximal, complemented
7C22×#124C411yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9C22×C4×D4C411yescharacteristic
10#2920—22no—
11C4×#124—22no—
12C4×#124—22no—
13C4×#124C412yes—
14C4×#124C412yes—
15C4×#124C2212yes—
16C4×#124C2212yes—
17C2×#391C411yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#388—22no—
20C2×#382C2211yescharacteristic
21C2×C4×C22⋊C4—22no—
22C2×C43C411yescharacteristic
23#1467—22no—
24#1467C2214yescomplemented
25#1463—22no—
26#1463C2214yes—
27#1461—22no—
28#1461C2214yescomplemented
29#1324C2218yescomplemented
30#1318C2218yes—
31C23×Q8C2×C411yescharacteristic
32C23×D4C2×C411yescharacteristic
33C24×C4—21no—
34C2×#124—22no—
35C2×#124—28no—
36C2×#124C2×C412yes—
37C2×#124C2×C412yes—
38C2×C4×Q8—41no—
39C2×C4×Q8—41no—
40C2×C4×Q8—41no—
41C2×C4×Q8—41no—
42C2×C4×Q8C2×C411yescharacteristic
43C2×C4×Q8C2×C411yescharacteristic
44C2×C4×Q8D412yes—
45C2×C4×Q8D412yes—
46C2×C4×Q8C2311yescharacteristic
47C2×C4×Q8C2311yescharacteristic
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×C4×D4—41no—
53C2×C4×D4—41no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C412yes—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D412yes—
68#432—24no—
69#432—42no—
70#431—24no—
71#431—42no—
72#423—42no—
73#407—24no—
74C4×M16—42no—
75#391—22no—
76#391C2×C412yes—
77#391C2×C412yes—
78#390—22no—
79#390C2×C412yes—
80#390C2312yes—
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#324—24no—
100#324—24no—
101#324C2318yes—
102C25—21no—
103C22×Q8—24no—
104C22×Q8—41no—
105C22×Q8—41no—
106C22×Q8—42no—
107C22×Q8C22⋊C412yes—
108C22×Q8C22×C411yescharacteristic
109C22×Q8C22×C411yescharacteristic
110C22×D4—24no—
111C22×D4—21no—
112C22×D4—41no—
113C22×D4—41no—
114C22×D4—42no—
115C22×D4C22×C412yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—48no—
122C23×C4—41no—
123C23×C4—41no—
124C23×C4—41no—
125C23×C4—41no—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D412yes—
131C2×M16—44no—
132#124—216no—
133#124—24no—
134#124—24no—
135#124—24no—
136#124M1618yescomplemented
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2411yescharacteristic
172C2×C22⋊C4—216no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—24no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—44no—
198C2×C42—44no—
199C2×C42—42no—
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C412yes—
203C2×C42C22⋊C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C4⋊C8—44no—
211C4⋊C8—44no—
212C22⋊C8—28no—
213C22⋊C8—44no—
214#97—44no—
215C4×C8—44no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220C24—24no—
221C24—22no—
222C24—44no—
223C24—44no—
224C24—42no—
225C24—41no—
226C24—41no—
227C24C2×C22⋊C411yescharacteristic
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8C2×M1614yes—
239C2×D4—24no—
240C2×D4—22no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—48no—
245C2×D4—48no—
246C2×D4—42no—
247C2×D4C2×M1614yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—84no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4—84no—
285C22×C4#9911yescharacteristic
286C22×C4#9911yescharacteristic
287C22×C4#9911yescharacteristic
288C22×C4#9911yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D412yes—
299M16—88no—
300C2×C8—48no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C22⋊C4—216no—
311C22⋊C4—216no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C42—24no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—22no—
331C42—21no—
332C42—22no—
333C42—21no—
334C42—42no—
335C42—44no—
336C42—44no—
337C42—44no—
338C42—44no—
339C42—42no—
340C42C2×M1614yes—
341C23—21no—
342C23—21no—
343C23—22no—
344C23—22no—
345C23—24no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23—44no—
350C23—416no—
351C23—42no—
352C23—42no—
353C23—82no—
354C23—88no—
355C23—82no—
356C23—82no—
357C23—82no—
358C23C2×#9911yescharacteristic
359C23C2×#9911yescharacteristic
360C23C22×C22⋊C411yescharacteristic
361Q8—44no—
362Q8—48no—
363Q8—44no—
364D4—44no—
365D4—44no—
366D4—48no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4#35514yes—
414C2×C4#40714yes—
415C2×C4C2×#9911yescharacteristic
416C2×C4C2×#9911yescharacteristic
417C2×C4C2×#9911yescharacteristic
418C2×C4C2×#9911yescharacteristic
419C2×C4#41312yes—
420C2×C4C22×M1612yes—
421C8—88no—
422C22—24no—
423C22—24no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—48no—
431C22—44no—
432C22—84no—
433C22—88no—
434C22—88no—
435C22—82no—
436C22—82no—
437C22#212612yes—
438C22C2×#35512yes—
439C22C2×#40711yescharacteristic
440C22C22×#9911yescharacteristic
441C22C2×#41311yescharacteristic
442C4—21no—
443C4—21no—
444C4—22no—
445C4—22no—
446C4—24no—
447C4—44no—
448C4—44no—
449C4—42no—
450C4—41no—
451C4—41no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4C2×#35512yes—
456C4C2×#35512yes—
457C2—21no—
458C2—21no—
459C2—22no—
460C2—22no—
461C2—84no—
462C2#1736211yescharacteristic
463C2C2×#212611yescharacteristic
464C2C22×#35511yescharacteristic
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×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17365C2semidirect
3. #11530C2semidirect
4. #11511C2semidirect
6. #11476C2semidirect
24. #1467C22semidirect
28. #1461C22semidirect
29. #1324C22semidirect
136. #124M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
#173652
#115302
#115112
#115052
#114762