Finite Group Databasebeta

← #480401

Subgroups — #480401

Order 512.

1156 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2#17365C212yesmaximal, complemented
3#11534C214yesmaximal, complemented
4#11512C212yesmaximal, complemented
5#11505C212yesmaximal
6#11480C214yesmaximal, complemented
7C22×#124C411yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9C22×C4×D4C411yescharacteristic
10#2920—22no—
11#2326—22no—
12#2326—22no—
13#2326C412yes—
14#2326C412yes—
15#2326C2212yes—
16#2326C2212yes—
17C2×#391C411yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#388—22no—
20C2×#385—22no—
21C2×#384C2211yescharacteristic
22C2×#376C411yescharacteristic
23#1467—22no—
24#1467C2214yescomplemented
25#1463—22no—
26#1463C2214yes—
27#1461—22no—
28#1461C2214yescomplemented
29#1325C2218yescomplemented
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—42no—
39C2×C4×Q8—42no—
40C2×C4×Q8C2×C411yescharacteristic
41C2×C4×Q8C2×C411yescharacteristic
42C2×C4×Q8D412yes—
43C2×C4×Q8D412yes—
44C2×C4×Q8C2311yescharacteristic
45C2×C4×Q8C2311yescharacteristic
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—42no—
49C2×C4×D4—42no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4C2×C412yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42D412yes—
63#432—24no—
64#432—42no—
65#431—24no—
66#431—42no—
67#423—42no—
68#407—24no—
69C4×M16—42no—
70#391—22no—
71#391C2×C412yes—
72#391C2×C412yes—
73#390—22no—
74#390C2×C412yes—
75#390C2312yes—
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—22no—
93C2×#95—21no—
94#324—24no—
95#324—24no—
96#324—24no—
97#324C2318yes—
98C25—21no—
99C22×Q8—24no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—42no—
103C22×Q8C22⋊C412yes—
104C22×Q8C22×C411yescharacteristic
105C22×Q8C22×C411yescharacteristic
106C22×D4—24no—
107C22×D4—21no—
108C22×D4—41no—
109C22×D4—41no—
110C22×D4—42no—
111C22×D4C22×C412yes—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—84no—
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C2×D412yes—
125C2×M16—44no—
126#124—216no—
127#124—24no—
128#124—24no—
129#124—24no—
130#124M1618yescomplemented
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—42no—
139C4×D4—44no—
140C4×D4—44no—
141C4×D4—44no—
142C4×D4—44no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—84no—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2411yescharacteristic
161C2×C22⋊C4—216no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—84no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—24no—
179C2×C42—21no—
180C2×C42—42no—
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C412yes—
188C2×C42C22×C412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C4⋊C8—44no—
192C4⋊C8—44no—
193C22⋊C8—28no—
194C22⋊C8—44no—
195#97—44no—
196C4×C8—44no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—24no—
202#95—24no—
203C24—24no—
204C24—22no—
205C24—44no—
206C24—42no—
207C24—44no—
208C24—41no—
209C24—41no—
210C24C2×C22⋊C411yescharacteristic
211C2×Q8—22no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—48no—
220C2×Q8—48no—
221C2×Q8C2×M1614yes—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—48no—
228C2×D4—48no—
229C2×D4—42no—
230C2×D4C2×M1614yes—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—84no—
261C22×C4—84no—
262C22×C4—88no—
263C22×C4—88no—
264C22×C4#9912yes—
265C22×C4#9912yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D412yes—
276M16—88no—
277C2×C8—48no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—84no—
288C4⋊C4—84no—
289C22⋊C4—216no—
290C22⋊C4—216no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—84no—
294C22⋊C4—84no—
295C42—24no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—42no—
310C42—42no—
311C42C2×M1614yes—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23—24no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—416no—
321C23—44no—
322C23—42no—
323C23—42no—
324C23—82no—
325C23—88no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23C2×#9911yescharacteristic
330C23C2×#9911yescharacteristic
331C23C22×C22⋊C411yescharacteristic
332Q8—44no—
333Q8—48no—
334Q8—44no—
335D4—44no—
336D4—44no—
337D4—48no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—88no—
377C2×C4—88no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4#40714yes—
382C2×C4C2×#9912yes—
383C2×C4C2×#9912yes—
384C2×C4#41312yes—
385C2×C4C22×M1612yes—
386C8—88no—
387C22—24no—
388C22—24no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—48no—
396C22—44no—
397C22—84no—
398C22—88no—
399C22—88no—
400C22—82no—
401C22—82no—
402C22#212812yes—
403C22#213312yes—
404C22C2×#40711yescharacteristic
405C22C22×#9911yescharacteristic
406C22C2×#41311yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—24no—
412C4—44no—
413C4—42no—
414C4—41no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—82no—
419C4—82no—
420C2—21no—
421C2—21no—
422C2—22no—
423C2—22no—
424C2—84no—
425C2#1736211yescharacteristic
426C2C2×#212811yescharacteristic
427C2C2×#213311yescharacteristic
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×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. #11534C2semidirect
4. #11512C2semidirect
6. #11480C2semidirect
24. #1467C22semidirect
28. #1461C22semidirect
29. #1325C22semidirect
130. #124M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
#173652
#115342
#115122
#115052
#114802