Finite Group Databasebeta

← #196833

Subgroups — #196833

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#20714C211yesmaximal, characteristic, complemented
4C2×#1517C212yesmaximal, complemented
5#9157C214yesmaximal, complemented
6#9150C214yesmaximal, complemented
7#8026C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#431C2212yes—
12#2857C412yescomplemented
13#2856C412yescomplemented
14#2854—21no—
15#2854C2212yes—
16#2850—22no—
17#2847—21no—
18#2847C2212yes—
19C4×#121—22no—
20C4×#121C414yescomplemented
21#2297—21no—
22#2285C2211yescharacteristic
23#2283—28no—
24C2×#383C2212yes—
25C2×#382C2212yes—
26C2×#381—21no—
27C2×#380—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C22⋊C4C2212yes—
30C2×#376—21no—
31#1817C2212yes—
32#1797—22no—
33#1779C2212yes—
34C4×C22⋊C8—22no—
35#1517—28no—
36C2×#327C2218yes—
37#1294C2218yes—
38C22×#41—21no—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42#532—24no—
43#532—24no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#127C2×C411yescharacteristic
46C2×#124—21no—
47#525—48no—
48C2×#123—22no—
49C2×C22⋊Q8—22no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121—21no—
53C2×#121C2×C414yes—
54C2×#121C2312yes—
55C2×#120—22no—
56C2×C4×D4—24no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C412yes—
59C2×#117—21no—
60C22×C4⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C42—24no—
63C22×C42—24no—
64C22×C42—22no—
65C22×C42C2×C412yes—
66C22×C42C2×C412yes—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D412yes—
70C22×C42D412yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#431—24no—
74#431—24no—
75C2×C4⋊C8—22no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8C2314yes—
78C2×C4×C8—22no—
79#391—22no—
80#390—22no—
81#390—22no—
82#390C2×C414yes—
83#389—22no—
84#389C2×C414yes—
85#388—24no—
86#388—24no—
87#386—24no—
88#386—22no—
89#386—24no—
90#386—22no—
91#384C2×C412yes—
92#383—28no—
93#382—21no—
94#382—24no—
95#382D418yescomplemented
96#381—28no—
97#380—28no—
98#380—22no—
99#380—21no—
100#380C2×C414yes—
101#380C2312yes—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C412yes—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—28no—
107#376—28no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95C2314yes—
112#336—24no—
113#327—216no—
114C25C2×D411yescharacteristic
115C2×#41—42no—
116C2×#41—42no—
117C2×#41—44no—
118C2×#41—44no—
119C22×Q8—21no—
120C22×D4—24no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4C22×C412yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—24no—
150C4⋊Q8—24no—
151#127—24no—
152#124—44no—
153#123—48no—
154C22⋊Q8—48no—
155#121—24no—
156#121—24no—
157#121—48no—
158#121—48no—
159#121—42no—
160#121—42no—
161#121—48no—
162#120—48no—
163C4×D4—28no—
164C4×D4—416no—
165C4×D4—44no—
166#117—44no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C414yes—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—48no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—28no—
192C2×C42—28no—
193C2×C42—28no—
194C2×C42—28no—
195C2×C42—28no—
196C2×C42—22no—
197C2×C42—48no—
198C2×C42C22⋊C412yes—
199C2×C42C22⋊C412yes—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D414yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4114yes—
210C4⋊C8—44no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—48no—
214C4×C8—44no—
215#95—28no—
216#95—28no—
217#95—24no—
218#95—24no—
219#95—22no—
220#95—28no—
221#95—24no—
222#95—22no—
223#95C2×D418yes—
224C24—21no—
225C24—21no—
226C24—24no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—42no—
231C24C2×C22⋊C412yes—
232C24C22×D411yescharacteristic
233#41—88no—
234#41—88no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×D4—24no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—48no—
242C2×D4—48no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—48no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—28no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4#12012yes—
306C22×C4#12012yes—
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4#13514yes—
310C22×C4AGL(1,ℤ/8)14yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D412yes—
314C22×C4C2×#4112yes—
315C2×C8—48no—
316C2×C8—44no—
317C4⋊C4—24no—
318C4⋊C4—44no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—416no—
324C22⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—416no—
327C22⋊C4—48no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—48no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—416no—
339C42—44no—
340C42C4×D414yes—
341C42C4×D414yes—
342C42#12114yes—
343C42#12114yes—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—24no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—48no—
355C23—48no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—44no—
361C23—42no—
362C23—42no—
363C23—44no—
364C23—42no—
365C23#42114yes—
366C23C22×C22⋊C411yescharacteristic
367C23C2×#12011yescharacteristic
368C23C2×#12311yescharacteristic
369C23C2×#13512yes—
370C23C2×AGL(1,ℤ/8)12yes—
371Q8—84no—
372D4—84no—
373D4—84no—
374D4—816no—
375D4—84no—
376D4—84no—
377D4—84no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—88no—
416C2×C4—88no—
417C2×C4—88no—
418C2×C4#38612yes—
419C2×C4#38612yes—
420C2×C4#38612yes—
421C2×C4#38612yes—
422C2×C4#41014yes—
423C2×C4#41612yes—
424C2×C4#41612yes—
425C2×C4#41812yes—
426C2×C4#41812yes—
427C2×C4#44918yes—
428C2×C4#48218yes—
429C2×C4C2×C4×D412yes—
430C2×C4C2×#12112yes—
431C8—84no—
432C22—24no—
433C22—24no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—48no—
442C22—48no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—84no—
448C22—82no—
449C22—84no—
450C22—82no—
451C22—82no—
452C22C2×#38611yescharacteristic
453C22C2×#41012yes—
454C22C2×#41611yescharacteristic
455C22C2×#41811yescharacteristic
456C22C2×#42112yes—
457C22C2×#44912yes—
458C22C2×#48212yes—
459C4—22no—
460C4—22no—
461C4—22no—
462C4—24no—
463C4—22no—
464C4—42no—
465C4—42no—
466C4—88no—
467C4—82no—
468C4—84no—
469C4#187514yes—
470C4#187514yes—
471C2—22no—
472C2—22no—
473C2—44no—
474C2—162no—
475C2#1757312yes—
476C2#1758312yes—
477C2C2×#187511yescharacteristic
478C2#2097911yescharacteristic
479C2#2098311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #20714C2semidirect
4. C2×#1517C2semidirect
5. #9157C2semidirect
6. #9150C2semidirect
7. #8026C2semidirect
12. #2857C4semidirect
13. #2856C4semidirect
20. C4×#121C4semidirect
95. #382D4semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#22832
#207142
C2×#15172
#91572
#91502
#80262