Finite Group Databasebeta

← #480112

Subgroups — #480112

Order 512.

1211 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2#17349C212yesmaximal, complemented
3#11480C214yesmaximal, complemented
4#11475C214yesmaximal, complemented
5#11450C212yesmaximal, complemented
6#11433C212yesmaximal, complemented
7C2×#535C411yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C22×C4×D4C411yescharacteristic
10C2×#431—22no—
11#2328C412yes—
12#2326C2212yes—
13#2324—22no—
14#2321—22no—
15#2317C2211yescharacteristic
16#2311C411yescharacteristic
17#2309C2211yescharacteristic
18C4×#127C2212yes—
19C4×#124C412yes—
20C4×#121—22no—
21C4×#120—22no—
22C4×#117C411yescharacteristic
23#1461C2214yescomplemented
24#1460—22no—
25#1459—22no—
26#1459C2214yescomplemented
27#1458—22no—
28#1458C2214yes—
29#1325C2218yescomplemented
30#1318C2218yes—
31C22×#41C2×C411yescharacteristic
32C23×D4C2×C411yescharacteristic
33C24×C4—21no—
34#535—24no—
35#535C818yescomplemented
36C2×#127C2×C412yes—
37C2×#124C2×C412yes—
38C2×#121—22no—
39C2×#120—22no—
40C4×#41—42no—
41C4×#41—42no—
42C2×C4×Q8C2×C411yescharacteristic
43C2×C4×Q8C2311yescharacteristic
44C2×C4×D4—24no—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—42no—
48C2×C4×D4—42no—
49C2×C4×D4C2×C412yes—
50C2×C4×D4C2×C412yes—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4D412yes—
53C2×C4×D4D412yes—
54C2×C4×D4C2311yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C2×#117D412yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C22⋊C4—21no—
61C22×C42C2×C411yescharacteristic
62C22×C42D412yes—
63#431—24no—
64#431—24no—
65#431—42no—
66#431—42no—
67C2×C4⋊C8—42no—
68C2×C22⋊C8—24no—
69C2×C4×C8—42no—
70#391C2×C412yes—
71#390—24no—
72#390—22no—
73#390C2×C412yes—
74#390C2×C412yes—
75#390C2312yes—
76#389—22no—
77#388—24no—
78#388—24no—
79#386—22no—
80#386—24no—
81#386—24no—
82#385—24no—
83#385—22no—
84#384C2312yes—
85#382C2×C412yes—
86#380—22no—
87#379—22no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—24no—
93#376C2×C412yes—
94C43C2×C412yes—
95#324—24no—
96#324C2318yes—
97#323—24no—
98#323—24no—
99C25—21no—
100C2×#41—42no—
101C2×#41—42no—
102C2×#41C2×C814yes—
103C2×#41M1614yes—
104C22×Q8C22×C411yescharacteristic
105C22×D4—24no—
106C22×D4—21no—
107C22×D4—22no—
108C22×D4—42no—
109C22×D4—42no—
110C22×D4C22⋊C412yes—
111C22×D4C2×C814yescomplemented
112C22×D4C22×C412yes—
113C22×D4C22×C411yescharacteristic
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⋊C412yes—
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D412yes—
126C22×C8—44no—
127#127—24no—
128#127C2×C818yes—
129#124—24no—
130#124C2×C818yescomplemented
131#121—216no—
132#120—216no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—44no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—44no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—88no—
150#117—22no—
151#117—42no—
152#117—42no—
153#117—42no—
154#117—42no—
155#117C2×C814yescomplemented
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2411yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C2×D412yes—
177C2×C22⋊C4C2×D412yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C412yes—
192C2×C42C22×C412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C4⋊C8—44no—
196C4⋊C8—44no—
197C22⋊C8—28no—
198C22⋊C8—44no—
199C4×C8—44no—
200C4×C8—44no—
201#95—22no—
202#95—22no—
203#95—24no—
204#95—24no—
205C24—22no—
206C24—24no—
207C24—42no—
208C24—42no—
209C24—82no—
210C24—82no—
211C24C2×C22⋊C412yes—
212C24C2×C22⋊C411yescharacteristic
213#41—48no—
214#41—48no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8C22×C812yes—
218C2×Q8C2×M1612yes—
219C2×D4—216no—
220C2×D4—22no—
221C2×D4—24no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—48no—
229C2×D4—48no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—84no—
233C2×D4—84no—
234C2×D4C22⋊C818yes—
235C2×D4C22×C814yes—
236C2×D4C22×C812yes—
237C2×D4C2×M1612yes—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—48no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—82no—
261C22×C4—84no—
262C22×C4—82no—
263C22×C4—81no—
264C22×C4—88no—
265C22×C4—88no—
266C22×C4C22⋊C818yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C22×C812yes—
275C22×C4C2×M1612yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D412yes—
278C2×C8—48no—
279C2×C8—88no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—84no—
289C22⋊C4—24no—
290C22⋊C4—216no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—44no—
295C22⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—84no—
298C22⋊C4—84no—
299C22⋊C4—84no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—21no—
308C42—22no—
309C42—22no—
310C42—21no—
311C42—42no—
312C42—44no—
313C42—82no—
314C42#9912yes—
315C42#9912yes—
316C42C22×C814yes—
317C23—21no—
318C23—22no—
319C23—21no—
320C23—22no—
321C23—24no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—44no—
327C23—44no—
328C23—48no—
329C23—42no—
330C23—42no—
331C23—88no—
332C23—84no—
333C23—88no—
334C23—84no—
335C23C2×C22⋊C812yes—
336C23C2×C22⋊C814yes—
337C23C22×C22⋊C411yescharacteristic
338Q8—42no—
339Q8—42no—
340D4—48no—
341D4—42no—
342D4—42no—
343D4—44no—
344D4—44no—
345D4—48no—
346D4—816no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—84no—
381C2×C4—88no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4C2×C22⋊C814yes—
385C2×C4C2×C22⋊C812yes—
386C2×C4C2×#9912yes—
387C2×C4C2×#9911yescharacteristic
388C2×C4C2×#9912yes—
389C2×C4C2×#9911yescharacteristic
390C2×C4#41312yes—
391C2×C4C23×C811yescharacteristic
392C2×C4C22×M1611yescharacteristic
393C8—88no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—24no—
398C22—44no—
399C22—42no—
400C22—42no—
401C22—48no—
402C22—41no—
403C22—42no—
404C22—42no—
405C22—41no—
406C22—44no—
407C22—82no—
408C22—84no—
409C22—816no—
410C22—82no—
411C22—82no—
412C22C22×C22⋊C811yescharacteristic
413C22C22×#9911yescharacteristic
414C22C2×#41311yescharacteristic
415C4—21no—
416C4—22no—
417C4—21no—
418C4—21no—
419C4—24no—
420C4—44no—
421C4—42no—
422C4—41no—
423C4—42no—
424C4—41no—
425C4—82no—
426C4—82no—
427C4#145914yes—
428C4#212812yes—
429C2—21no—
430C2—21no—
431C2—42no—
432C2—44no—
433C2—84no—
434C2C2×#145911yescharacteristic
435C2C2×#212811yescharacteristic
436C2#3336911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC2×C4C23×C8
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC2×#535C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17349C2semidirect
3. #11480C2semidirect
4. #11475C2semidirect
5. #11450C2semidirect
6. #11433C2semidirect
23. #1461C22semidirect
26. #1459C22semidirect
29. #1325C22semidirect
35. #535C8semidirect
111. C22×D4C2×C8semidirect
130. #124C2×C8semidirect
155. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
#173492
#114802
#114752
#114502
#114332