Finite Group Databasebeta

← #196898

Subgroups — #196898

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C22×#383C211yesmaximal, characteristic, complemented
3#20715C211yesmaximal, characteristic, complemented
4C2×#1514C212yesmaximal, complemented
5#9157C214yesmaximal, complemented
6#9147C214yesmaximal, complemented
7#8018C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#432C2212yes—
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
23C2×#383—21no—
24C2×#383—21no—
25C2×#383—28no—
26C2×#383C2212yes—
27C2×#383C2212yes—
28C22×#95—21no—
29C22×#95—21no—
30C22×#95C2212yes—
31#1817C2212yes—
32#1797—22no—
33#1779C2212yes—
34#1749—22no—
35#1514—28no—
36C2×#327C2218yes—
37#1294C2218yes—
38C22×#41—21no—
39C24×C4—22no—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43#532—24no—
44#532—24no—
45C2×C4⋊Q8C2×C411yescharacteristic, complemented
46C2×#127C2×C411yescharacteristic, complemented
47C2×#124—21no—
48#525—48no—
49C2×#123—22no—
50C2×C22⋊Q8—22no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—21no—
54C2×#121C2×C414yes—
55C2×#121C2312yes—
56C2×#120—22no—
57C2×C4×D4—24no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yescomplemented
60C2×#117—21no—
61C22×C42—24no—
62C22×C42—24no—
63C22×C42C2×C412yes—
64C22×C42C2×C412yes—
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D412yes—
68C22×C42D412yes—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#432—24no—
72#432—24no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8C2314yes—
76C2×#97—22no—
77#391—22no—
78#390—22no—
79#390—22no—
80#390C2×C414yescomplemented
81#389—22no—
82#389C2×C414yescomplemented
83#388—24no—
84#388—24no—
85#386—24no—
86#386—22no—
87#386—24no—
88#386—22no—
89#384C2×C412yescomplemented
90#383—28no—
91#383—28no—
92#383—24no—
93#383—28no—
94#383D418yescomplemented
95#382—21no—
96#380—22no—
97#380—21no—
98#380C2×C414yes—
99#380C2312yes—
100C4×C4⋊C4—21no—
101C4×C4⋊C4C2×C412yescomplemented
102C4×C22⋊C4—24no—
103C2×#95—28no—
104C2×#95—28no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—28no—
111C2×#95C2314yes—
112#336—24no—
113#327—216no—
114C25C2×D411yescharacteristic
115C2×#41—44no—
116C2×#41—44no—
117C2×#41—42no—
118C2×#41—42no—
119C22×Q8—21no—
120C22×D4—24no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4C22×C412yes—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—28no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—42no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4—48no—
139C23×C4—48no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C22×C412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—24no—
155C4⋊Q8—24no—
156#127—24no—
157#124—44no—
158#123—48no—
159C22⋊Q8—48no—
160#121—24no—
161#121—24no—
162#121—48no—
163#121—48no—
164#121—48no—
165#121—42no—
166#121—42no—
167#120—48no—
168C4×D4—28no—
169C4×D4—416no—
170C4×D4—44no—
171#117—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C414yes—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
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—22no—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22⋊C412yes—
199C2×C42C22⋊C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D414yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42#4114yes—
208C4⋊C8—44no—
209C22⋊C8—24no—
210C22⋊C8—24no—
211C22⋊C8—48no—
212#97—44no—
213#95—28no—
214#95—28no—
215#95—28no—
216#95—24no—
217#95—24no—
218#95—22no—
219#95—28no—
220#95—28no—
221#95—24no—
222#95—22no—
223#95C2×D418yes—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—24no—
228C24—24no—
229C24—22no—
230C24C2×C22⋊C412yes—
231C24AGL(1,ℤ/8)14yes—
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—42no—
241C2×D4—44no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—48no—
248C2×D4—48no—
249C2×D4—48no—
250C2×D4—42no—
251C2×D4—42no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
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—22no—
276C22×C4—28no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—416no—
296C22×C4—416no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—416no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—816no—
304C22×C4—816no—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4#12012yes—
313C22×C4#12012yes—
314C22×C4#12312yes—
315C22×C4#12312yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D412yes—
319C22×C4C2×#4112yes—
320C2×C8—48no—
321C2×C8—44no—
322C4⋊C4—24no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—48no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C22⋊C4—48no—
329C22⋊C4—48no—
330C22⋊C4—48no—
331C22⋊C4—44no—
332C22⋊C4—48no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—44no—
342C42C4×D414yes—
343C42C4×D414yes—
344C42#12114yes—
345C42#12114yes—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—216no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—24no—
358C23—216no—
359C23—22no—
360C23—44no—
361C23—42no—
362C23—42no—
363C23—44no—
364C23—42no—
365C23#41014yes—
366C23#41814yes—
367C23#42114yes—
368C23C22×C22⋊C411yescharacteristic
369C23C2×#12011yescharacteristic
370C23C2×#12311yescharacteristic
371C23C2×AGL(1,ℤ/8)12yes—
372C23#57512yes—
373Q8—84no—
374D4—84no—
375D4—84no—
376D4—84no—
377D4—84no—
378D4—84no—
379D4—816no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—42no—
424C2×C4—88no—
425C2×C4—832no—
426C2×C4—88no—
427C2×C4—88no—
428C2×C4#38612yes—
429C2×C4#38612yes—
430C2×C4#38612yes—
431C2×C4#38612yes—
432C2×C4C2×C4×D412yes—
433C2×C4C2×#12112yes—
434C8—84no—
435C22—216no—
436C22—216no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—21no—
444C22—21no—
445C22—24no—
446C22—21no—
447C22—21no—
448C22—24no—
449C22—84no—
450C22—82no—
451C22—84no—
452C22—82no—
453C22—82no—
454C22#151418yescomplemented
455C22C2×#38611yescharacteristic
456C22C2×#41012yes—
457C22C2×#41811yescharacteristic
458C22#289611yescharacteristic
459C22C2×#42112yes—
460C22#300612yes—
461C22#309412yes—
462C4—22no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—88no—
472C4—84no—
473C4—82no—
474C2—22no—
475C2—22no—
476C2—28no—
477C2—162no—
478C2C2×#151412yes—
479C2#1759312yes—
480C2#2091211yescharacteristic
481C2#2097911yescharacteristic
482C2#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. C22×#383C2semidirect
3. #20715C2semidirect
4. C2×#1514C2semidirect
5. #9157C2semidirect
6. #9147C2semidirect
7. #8018C2semidirect
12. #2857C4semidirect
13. #2856C4semidirect
20. C4×#121C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
46. C2×#127C2×C4semidirect
59. C2×C4×D4C2×C4semidirect
80. #390C2×C4semidirect
82. #389C2×C4semidirect
89. #384C2×C4semidirect
94. #383D4semidirect
101. C4×C4⋊C4C2×C4semidirect
454. C22#1514semidirect

Maximal subgroups

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

subgroupindex
#366242
C22×#3832
#207152
C2×#15142
#91572
#91472
#80182