Finite Group Databasebeta

← #480396

Subgroups — #480396

Order 512.

1167 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2#17363C212yesmaximal, complemented
3#11534C214yesmaximal, complemented
4#11516C212yesmaximal, complemented
5#11509C212yesmaximal, complemented
6#11475C214yesmaximal, complemented
7C2×#535C411yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C22×C4×D4C411yescharacteristic
10C2×#432—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#1467C2214yescomplemented
24#1466—22no—
25#1464—22no—
26#1464C2214yes—
27#1459—22no—
28#1459C2214yescomplemented
29#1325C2218yescomplemented
30#1318C2218yes—
31C22×#41C2×C411yescharacteristic
32C23×D4C2×C411yescharacteristic
33C24×C4—21no—
34#535—28no—
35C2×#127C2×C412yes—
36C2×#124C2×C412yes—
37C2×#121—22no—
38C2×#120—22no—
39C4×#41—42no—
40C4×#41—42no—
41C2×C4×Q8C2×C411yescharacteristic
42C2×C4×Q8C2311yescharacteristic
43C2×C4×D4—24no—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—42no—
47C2×C4×D4—42no—
48C2×C4×D4C2×C412yes—
49C2×C4×D4C2×C412yes—
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2311yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117D412yes—
57C2×#117C2311yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4—21no—
60C22×C42C2×C411yescharacteristic
61C22×C42D412yes—
62#432—24no—
63#432—24no—
64#432—42no—
65#432—42no—
66C2×C4⋊C8—42no—
67C2×C22⋊C8—24no—
68C2×#97—42no—
69#391C2×C412yes—
70#390—24no—
71#390—22no—
72#390C2×C412yes—
73#390C2×C412yes—
74#390C2312yes—
75#389—22no—
76#388—24no—
77#388—24no—
78#386—22no—
79#386—24no—
80#386—24no—
81#385—24no—
82#385—22no—
83#384C2312yes—
84#382C2×C412yes—
85#380—22no—
86#379—22no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—24no—
92#376C2×C412yes—
93C43C2×C412yes—
94#324—24no—
95#324C2318yes—
96#323—24no—
97#323—24no—
98C25—21no—
99C2×#41—24no—
100C2×#41—42no—
101C2×#41—42no—
102C22×Q8C22×C411yescharacteristic
103C22×D4—24no—
104C22×D4—24no—
105C22×D4—21no—
106C22×D4—42no—
107C22×D4—42no—
108C22×D4C22⋊C412yes—
109C22×D4C22×C412yes—
110C22×D4C22×C411yescharacteristic
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—84no—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C412yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D412yes—
123C22×C8—44no—
124#127—24no—
125#127M1618yes—
126#124—24no—
127#124—24no—
128#121—216no—
129#120—216no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—42no—
133C4×D4—42no—
134C4×D4—42no—
135C4×D4—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—44no—
141C4×D4—42no—
142C4×D4—42no—
143C4×D4—44no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—88no—
147#117—24no—
148#117—42no—
149#117—42no—
150#117—42no—
151#117—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2411yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4C22⋊C412yes—
172C2×C22⋊C4C2×D412yes—
173C2×C22⋊C4C2×D412yes—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C411yescharacteristic
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—
196#97—44no—
197#95—22no—
198#95—22no—
199#95—24no—
200#95—24no—
201C24—24no—
202C24—22no—
203C24—42no—
204C24—42no—
205C24—82no—
206C24—82no—
207C24C2×C22⋊C412yes—
208C24C2×C22⋊C411yescharacteristic
209#41—416no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—216no—
217C2×D4—22no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—416no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—84no—
227C2×D4—84no—
228C2×D4C2×M1614yes—
229C2×D4C2×M1614yes—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—48no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—82no—
255C22×C4—84no—
256C22×C4—82no—
257C22×C4—81no—
258C22×C4—88no—
259C22×C4—88no—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D412yes—
269C2×C8—48no—
270C2×C8—88no—
271C4⋊C4—24no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—84no—
280C22⋊C4—24no—
281C22⋊C4—216no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—84no—
289C22⋊C4—84no—
290C22⋊C4—84no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—21no—
299C42—22no—
300C42—22no—
301C42—21no—
302C42—42no—
303C42—44no—
304C42—82no—
305C42#9912yes—
306C42#9912yes—
307C42C2×M1614yes—
308C23—21no—
309C23—21no—
310C23—22no—
311C23—22no—
312C23—24no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—44no—
319C23—48no—
320C23—42no—
321C23—42no—
322C23—88no—
323C23—84no—
324C23—88no—
325C23—84no—
326C23#40714yes—
327C23#41312yes—
328C23C22×C22⋊C411yescharacteristic
329Q8—44no—
330D4—44no—
331D4—44no—
332D4—48no—
333D4—44no—
334D4—48no—
335D4—816no—
336C2×C4—28no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—88no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4C2×#9912yes—
376C2×C4C2×#9912yes—
377C2×C4C2×#9911yescharacteristic
378C2×C4C2×#9911yescharacteristic
379C2×C4C22×M1612yes—
380C8—88no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—24no—
385C22—44no—
386C22—42no—
387C22—42no—
388C22—48no—
389C22—41no—
390C22—42no—
391C22—42no—
392C22—41no—
393C22—44no—
394C22—82no—
395C22—84no—
396C22—816no—
397C22—82no—
398C22—82no—
399C22C2×#40711yescharacteristic
400C22C22×#9911yescharacteristic
401C22C2×#41311yescharacteristic
402C4—21no—
403C4—22no—
404C4—22no—
405C4—22no—
406C4—44no—
407C4—42no—
408C4—41no—
409C4—42no—
410C4—41no—
411C4—42no—
412C4—82no—
413C4—82no—
414C4#212812yes—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—44no—
419C2—84no—
420C2#1736211yescharacteristic
421C2C2×#212811yescharacteristic
422C2#3336911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC2×#535C4
LCSG ▷ C22×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. #17363C2semidirect
3. #11534C2semidirect
4. #11516C2semidirect
5. #11509C2semidirect
6. #11475C2semidirect
23. #1467C22semidirect
28. #1459C22semidirect
29. #1325C22semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
#173632
#115342
#115162
#115092
#114752