Finite Group Databasebeta

← #480400

Subgroups — #480400

Order 512.

1418 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2328C211yesmaximal, characteristic
2#17362C212yesmaximal, complemented
3#11533C214yesmaximal, complemented
4#11511C212yesmaximal, complemented
5#11505C212yesmaximal, complemented
6#11479C214yesmaximal, complemented
7C22×#127C411yescharacteristic
8C22×C4×D4C411yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#407—22no—
11#2328—22no—
12#2328—22no—
13#2328C412yes—
14#2328C412yes—
15#2328C2212yes—
16#2328C2212yes—
17C2×#390C411yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#386—22no—
20C2×#386—22no—
21C2×#382C2211yescharacteristic
22C2×#376C411yescharacteristic
23#1465—22no—
24#1465C2214yescomplemented
25#1463C2214yes—
26#1462—22no—
27#1459—22no—
28#1459C2214yescomplemented
29#1324C2218yescomplemented
30#1318C2218yescomplemented
31C23×D4—22no—
32C23×D4C2×C411yescharacteristic
33C23×D4C2×C411yescharacteristic
34C24×C4—21no—
35C24×C4D412yes—
36C22×M16—42no—
37C2×#127—22no—
38C2×#127—28no—
39C2×#127C2×C412yes—
40C2×#127C2×C412yes—
41C2×C4×D4—21no—
42C2×C4×D4—21no—
43C2×C4×D4—42no—
44C2×C4×D4—42no—
45C2×C4×D4—42no—
46C2×C4×D4—42no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4C2×C412yes—
49C2×C4×D4C2×C411yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2311yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D412yes—
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#390—22no—
71#390—22no—
72#390C2×C412yes—
73#390C2×C412yes—
74#390C2×C412yes—
75#390C2312yes—
76#386—24no—
77#386—24no—
78#386—24no—
79#386—24no—
80#386—24no—
81#386—24no—
82#386—24no—
83#386—24no—
84#382—22no—
85#382C2×C412yes—
86#382C2312yes—
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—
98C25—21no—
99C25C22⋊C412yes—
100C22×D4—216no—
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—24no—
104C22×D4—24no—
105C22×D4—21no—
106C22×D4—42no—
107C22×D4—41no—
108C22×D4—41no—
109C22×D4—41no—
110C22×D4—41no—
111C22×D4—42no—
112C22×D4—84no—
113C22×D4—84no—
114C22×D4C22⋊C412yes—
115C22×D4C22×C411yescharacteristic
116C22×D4C22×C412yes—
117C22×D4C22×C411yescharacteristic
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—84no—
130C23×C4—84no—
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C2×M16—44no—
140C2×M16—44no—
141C2×M16—88no—
142C22×C8—44no—
143#127—216no—
144#127—24no—
145#127—24no—
146#127—24no—
147#127M1618yescomplemented
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—44no—
157C4×D4—44no—
158C4×D4—44no—
159C4×D4—44no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2411yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—84no—
180C2×C22⋊C4—84no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C2×D412yes—
184C2×C22⋊C4C2×D412yes—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—24no—
189C2×C42—21no—
190C2×C42—42no—
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C412yes—
196C2×C42C22×C412yes—
197C22⋊C8—28no—
198C22⋊C8—44no—
199C22⋊C8—44no—
200C22⋊C8—44no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—24no—
206#95—24no—
207C24—24no—
208C24—22no—
209C24—24no—
210C24—22no—
211C24—22no—
212C24—42no—
213C24—44no—
214C24—44no—
215C24—44no—
216C24—44no—
217C24—42no—
218C24—41no—
219C24—41no—
220C24C2×C22⋊C411yescharacteristic
221C24C2×C22⋊C412yes—
222C24C2×C22⋊C412yes—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—216no—
226C2×D4—216no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—24no—
230C2×D4—42no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—48no—
239C2×D4—48no—
240C2×D4—48no—
241C2×D4—48no—
242C2×D4—42no—
243C2×D4—84no—
244C2×D4—84no—
245C2×D4—84no—
246C2×D4—84no—
247C2×D4—832no—
248C2×D4C2×M1614yes—
249C2×D4C2×M1614yescomplemented
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—84no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—88no—
294C22×C4—88no—
295C22×C4#9912yes—
296C22×C4#9912yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D412yes—
305M16—88no—
306M16—88no—
307C2×C8—48no—
308C2×C8—88no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—84no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C22⋊C4—84no—
324C42—24no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—42no—
331C42—42no—
332C42C2×M1614yescomplemented
333C23—21no—
334C23—21no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—22no—
342C23—22no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—416no—
354C23—416no—
355C23—42no—
356C23—42no—
357C23—44no—
358C23—44no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—88no—
363C23—88no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23#40714yes—
368C23C2×#9911yescharacteristic
369C23C2×#9911yescharacteristic
370C23C22×C22⋊C411yescharacteristic
371D4—44no—
372D4—48no—
373D4—44no—
374D4—44no—
375D4—44no—
376D4—48no—
377D4—816no—
378D4—816no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—88no—
419C2×C4—88no—
420C2×C4—84no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4C2×#9912yes—
424C2×C4C2×#9912yes—
425C2×C4#41312yes—
426C2×C4C22×M1612yes—
427C8—88no—
428C22—24no—
429C22—24no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—24no—
437C22—44no—
438C22—44no—
439C22—48no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—82no—
447C22—88no—
448C22—88no—
449C22—82no—
450C22—84no—
451C22—88no—
452C22—88no—
453C22—82no—
454C22—82no—
455C22#212712yes—
456C22#213412yes—
457C22C2×#40711yescharacteristic
458C22C22×#9911yescharacteristic
459C22C2×#41311yescharacteristic
460C4—22no—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—44no—
465C4—42no—
466C4—42no—
467C4—41no—
468C4—41no—
469C4—82no—
470C4—82no—
471C2—21no—
472C2—21no—
473C2—22no—
474C2—22no—
475C2—44no—
476C2—82no—
477C2—84no—
478C2#1736211yescharacteristic
479C2C2×#212711yescharacteristic
480C2C2×#213411yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#127C4
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
2. #17362C2semidirect
3. #11533C2semidirect
4. #11511C2semidirect
5. #11505C2semidirect
6. #11479C2semidirect
24. #1465C22semidirect
28. #1459C22semidirect
29. #1324C22semidirect
30. #1318C22semidirect
147. #127M16semidirect
249. C2×D4C2×M16semidirect
332. C42C2×M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23282
#173622
#115332
#115112
#115052
#114792