Finite Group Databasebeta

← #480402

Subgroups — #480402

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2#17362C212yesmaximal
3#11535C214yesmaximal, complemented
4#11512C212yesmaximal, complemented
5#11506C212yesmaximal, complemented
6#11481C214yesmaximal, complemented
7C22×#124C411yescharacteristic
8C22×C4×Q8C411yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#407—22no—
11#2326—22no—
12#2326—22no—
13#2326C412yes—
14#2326C412yes—
15#2326C2212yes—
16#2326C2212yes—
17C2×#391C2211yescharacteristic
18C2×#390C411yescharacteristic
19C2×#388—22no—
20C2×#385—22no—
21C2×#384C2211yescharacteristic
22C2×#376C411yescharacteristic
23#1465—22no—
24#1465C2214yes—
25#1463C2214yes—
26#1462—22no—
27#1459—22no—
28#1459C2214yes—
29#1325C2218yescomplemented
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—42no—
42C2×C4×Q8—42no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C2×C4×D4—42no—
46C2×C4×D4—42no—
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4D412yes—
50C2×C4×D4D412yes—
51C2×C4×D4C2311yescharacteristic
52C2×C4×D4C2311yescharacteristic
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D412yes—
60C22×C42—21no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64#407—24no—
65#407—24no—
66#407—42no—
67#407—42no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—42no—
70#391—22no—
71#391C2×C412yes—
72#391C2312yes—
73#390—22no—
74#390C2×C412yes—
75#390C2×C412yes—
76#388—24no—
77#388—24no—
78#388—24no—
79#388—24no—
80#385—24no—
81#385—24no—
82#385—24no—
83#385—24no—
84#384—22no—
85#384C2×C412yes—
86#384C2312yes—
87#376—21no—
88#376—21no—
89#376C2×C412yes—
90#376C2×C412yes—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94#324—24no—
95#324C2318yes—
96#323—24no—
97#323—24no—
98C25C22⋊C412yes—
99C22×Q8—24no—
100C22×Q8—21no—
101C22×Q8—42no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8C22×C412yes—
105C22×D4—24no—
106C22×D4—42no—
107C22×D4—41no—
108C22×D4—41no—
109C22×D4C22⋊C412yes—
110C22×D4C22×C411yescharacteristic
111C22×D4C22×C411yescharacteristic
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—84no—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C412yes—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C2×M16—44no—
130C2×M16—44no—
131C2×M16—88no—
132C22×C8—44no—
133#124—216no—
134#124—24no—
135#124—24no—
136#124—24no—
137#124M1618yescomplemented
138C4×Q8—44no—
139C4×Q8—44no—
140C4×Q8—44no—
141C4×Q8—44no—
142C4×D4—42no—
143C4×D4—42no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—42no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—84no—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2411yescharacteristic
166C2×C22⋊C4—216no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—84no—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C2×D412yes—
177C2×C22⋊C4C2×D412yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194C22⋊C8—28no—
195C22⋊C8—44no—
196C22⋊C8—44no—
197C22⋊C8—44no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—24no—
203#95—24no—
204C24—24no—
205C24—22no—
206C24—22no—
207C24—42no—
208C24—44no—
209C24—44no—
210C24C2×C22⋊C411yescharacteristic
211C24C2×C22⋊C412yes—
212C24C2×C22⋊C412yes—
213C2×Q8—24no—
214C2×Q8—22no—
215C2×Q8—42no—
216C2×Q8—48no—
217C2×Q8—48no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8C2×M1614yescomplemented
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—24no—
225C2×D4—42no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—48no—
231C2×D4—48no—
232C2×D4C2×M1614yes—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—48no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—84no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4#9912yes—
278C22×C4#9912yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D412yes—
287M16—88no—
288M16—88no—
289C2×C8—48no—
290C2×C8—88no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C22⋊C4—216no—
301C22⋊C4—216no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—84no—
307C22⋊C4—84no—
308C42—24no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—42no—
315C42—44no—
316C42—44no—
317C42—42no—
318C42C2×M1614yescomplemented
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—44no—
329C23—44no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—416no—
335C23—44no—
336C23—44no—
337C23—82no—
338C23—82no—
339C23—88no—
340C23#40714yes—
341C23C2×#9911yescharacteristic
342C23C2×#9911yescharacteristic
343C23C22×C22⋊C411yescharacteristic
344Q8—44no—
345Q8—44no—
346Q8—48no—
347D4—44no—
348D4—48no—
349D4—44no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—48no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4C2×#9912yes—
398C2×C4C2×#9912yes—
399C2×C4#41312yes—
400C2×C4C22×M1612yes—
401C8—88no—
402C22—24no—
403C22—24no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—24no—
411C22—44no—
412C22—44no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—82no—
419C22—88no—
420C22—88no—
421C22—82no—
422C22#212712yes—
423C22#213512yes—
424C22C2×#40711yescharacteristic
425C22C22×#9911yescharacteristic
426C22C2×#41311yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—41no—
435C4—41no—
436C4—44no—
437C4—82no—
438C4—82no—
439C2—21no—
440C2—21no—
441C2—22no—
442C2—22no—
443C2—44no—
444C2—82no—
445C2#1736211yescharacteristic
446C2C2×#212711yescharacteristic
447C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #11535C2semidirect
4. #11512C2semidirect
5. #11506C2semidirect
6. #11481C2semidirect
29. #1325C22semidirect
30. #1319C22semidirect
137. #124M16semidirect
221. C2×Q8C2×M16semidirect
318. C42C2×M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
#173622
#115352
#115122
#115062
#114812