Finite Group Databasebeta

← #207461

Subgroups — #207461

Order 512.

1062 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
1#36631C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic, complemented
3C4×#431C211yesmaximal, characteristic, complemented
4#9864C212yesmaximal, complemented
5#9674C212yesmaximal, complemented
6#8940C214yesmaximal, complemented
7#8183C214yesmaximal, complemented
8C2×#529—21no—
9C2×#431C2212yes—
10#2861—21no—
11#2860—21no—
12#2860C412yes—
13#2859C411yescharacteristic
14#2858C411yescharacteristic
15#2857C411yescharacteristic
16#2856C411yescharacteristic
17#2854—21no—
18#2854C412yes—
19#2850—21no—
20C4×C4⋊Q8C2211yescharacteristic
21C4×#127C2211yescharacteristic
22C4×#124—21no—
23C4×#121—21no—
24C4×#121C2212yes—
25#2293—21no—
26#2293C2212yes—
27C2×C43C2211yescharacteristic
28#1839—21no—
29#1831—21no—
30#1817—21no—
31#1797—21no—
32C4×C4⋊C8C2212yes—
33C4×C4⋊C8C2212yes—
34C4×C4⋊C8C2212yes—
35C4×C22⋊C8—21no—
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yes—
38#1744—21no—
39#1744C2212yes—
40C42×C8—21no—
41C2×D4⋊C8C2218yes—
42#1279C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45#529—48no—
46C2×#126—22no—
47C2×#125—21no—
48C2×#124—21no—
49C2×#124—21no—
50C2×#123—22no—
51C2×C22⋊Q8—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121C2×C412yes—
55C2×C4×Q8—21no—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2312yes—
60C2×#117—22no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#431—28no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8C2314yes—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76C2×C4×C8C2314yes—
77#391—21no—
78#391—21no—
79#390—21no—
80#390—21no—
81#390—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390C2×C412yes—
86#390C2×C412yes—
87#390C2×C412yes—
88#390C2×C412yes—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#386—22no—
106#386—22no—
107#384C2×C411yescharacteristic
108#384C2×C411yescharacteristic
109#382—21no—
110#380—21no—
111#380C2×C412yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2312yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130D4⋊C8—216no—
131C22×Q8—21no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4C22×C412yes—
136C23×C4—22no—
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8—24no—
145C22×C8—24no—
146C4⋊Q8—24no—
147#127—24no—
148#126—48no—
149#125—44no—
150#124—44no—
151#124—44no—
152#123—48no—
153C22⋊Q8—44no—
154#121—28no—
155#121—44no—
156#121—44no—
157C4×Q8—44no—
158C4×D4—28no—
159C4×D4—44no—
160C4×D4—48no—
161C4×D4—44no—
162#117—48no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2411yescharacteristic
222C4⋊C8—24no—
223C4⋊C8—44no—
224C4⋊C8#4118yes—
225C22⋊C8—28no—
226C22⋊C8—44no—
227C4×C8—28no—
228C4×C8—28no—
229C4×C8—44no—
230C4×C8—44no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237C24—22no—
238C24C2×C22⋊C411yescharacteristic
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
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—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4#13114yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C2×C8—28no—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—44no—
300C2×C8—44no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—44no—
349C42#13114yes—
350C42#13514yes—
351C42AGL(1,ℤ/8)14yes—
352C42C2×#4114yes—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23#40812yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—84no—
369D4—48no—
370D4—84no—
371D4—88no—
372D4—84no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—28no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—88no—
416C2×C4#40812yes—
417C2×C4#40812yes—
418C2×C4#40812yes—
419C2×C4#41612yes—
420C2×C4#41612yes—
421C2×C4#41812yes—
422C2×C4#41812yes—
423C2×C4#42112yes—
424C2×C4#42112yes—
425C2×C4C4×#4112yes—
426C2×C4#52512yes—
427C2×C4#53212yes—
428C2×C4C2×#13112yes—
429C2×C4C2×#13112yes—
430C2×C4C2×#13512yes—
431C2×C4C2×AGL(1,ℤ/8)12yes—
432C8—48no—
433C8—84no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22#229711yescharacteristic
449C22C2×#40811yescharacteristic
450C22C2×#40811yescharacteristic
451C22C2×#41611yescharacteristic
452C22C2×#41811yescharacteristic
453C22C2×#42111yescharacteristic
454C22C2×#42111yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—22no—
461C4—22no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4#148214yes—
470C4#296614yes—
471C4#310614yes—
472C2—42no—
473C2—162no—
474C2C2×#148211yescharacteristic
475C2#1745811yescharacteristic
476C2#1746311yescharacteristic
477C2#1746311yescharacteristic
478C2#2137511yescharacteristic
479C2#2138311yescharacteristic
480C2#2139011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2856C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20777C2semidirect
3. C4×#431C2semidirect
4. #9864C2semidirect
5. #9674C2semidirect
6. #8940C2semidirect
7. #8183C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#207772
C4×#4312
#98642
#96742
#89402
#81832