Finite Group Databasebeta

← #196871

Subgroups — #196871

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic, complemented
3#20682C211yesmaximal, characteristic, complemented
4C2×#1507C212yesmaximal
5#9149C214yesmaximal, complemented
6#9148C214yesmaximal, complemented
7#8019C212yesmaximal, complemented
8C22×#123—21no—
9C22×#121C2211yescharacteristic
10C22×#117—21no—
11C23×C42C2211yescharacteristic
12C2×#407C2212yes—
13#2854—22no—
14#2854—21no—
15#2854—21no—
16#2854C414yescomplemented
17#2854C2212yes—
18#2854C2212yes—
19C2×#390C412yescomplemented
20C2×#389C412yescomplemented
21C2×#388—22no—
22C2×#386—21no—
23C2×#380C2211yescharacteristic
24C4×#117—28no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yescomplemented
27C2×C4×C22⋊C4—21no—
28C2×C4×C22⋊C4C2212yes—
29C2×#376C2212yescomplemented
30C2×C43—21no—
31#1817—22no—
32#1817C2212yescomplemented
33#1817C2212yescomplemented
34#1728—22no—
35#1507—28no—
36C2×#327C2218yes—
37#1294C2218yescomplemented
38C23×D4—21no—
39C23×D4C2×C411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C2×#123—24no—
45C2×#123—22no—
46C2×#123—22no—
47C2×#123—48no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—24no—
51C2×#121—24no—
52C2×#121C2×C412yes—
53C2×#121C2312yes—
54C2×#117—216no—
55C2×#117—28no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C42—24no—
66C22×C42—22no—
67C22×C42—24no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42C2×C412yes—
71C22×C42C2×C412yes—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D412yes—
75C22×C42D412yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#407—24no—
79#407—24no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8C2314yes—
82#390—22no—
83#390—22no—
84#390C2×C414yes—
85#390C2×C414yes—
86#389—22no—
87#389—22no—
88#389C2×C414yes—
89#389C2×C414yes—
90#388—24no—
91#388—24no—
92#388—24no—
93#388—24no—
94#386—24no—
95#386—22no—
96#386—22no—
97#380—21no—
98#380—21no—
99#380C2×C412yes—
100#380C2312yes—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—28no—
103C4×C22⋊C4—28no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4D418yes—
106#376—28no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95C2314yes—
111C43—28no—
112#336—24no—
113#336—24no—
114#327—216no—
115C25—21no—
116C25C2×D411yescharacteristic
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—24no—
121C22×D4—44no—
122C22×D4—44no—
123C22×D4—44no—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4C22×C412yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—28no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—44no—
137C23×C4—42no—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D414yes—
152C23×C4#4114yes—
153C23×C4C2411yescharacteristic
154C2×M16—48no—
155C2×M16—44no—
156C22×C8—24no—
157#123—48no—
158#123—416no—
159#123—48no—
160#121—28no—
161#121—24no—
162#121—24no—
163#121—42no—
164#121—44no—
165#121—42no—
166#117—264no—
167C2×C4⋊C4—216no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—48no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C414yes—
178C2×C22⋊C4—216no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—48no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—44no—
192C2×C42—28no—
193C2×C42—216no—
194C2×C42—28no—
195C2×C42—28no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—28no—
199C2×C42—28no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—28no—
207C2×C42—28no—
208C2×C42—44no—
209C2×C42—48no—
210C2×C42—44no—
211C2×C42C22⋊C412yes—
212C2×C42C22⋊C412yes—
213C2×C42C22⋊C412yes—
214C2×C42C22⋊C412yes—
215C2×C42C22×C412yes—
216C2×C42C22×C412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C22⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—44no—
222#95—28no—
223#95—22no—
224#95—24no—
225#95—24no—
226#95—22no—
227#95—24no—
228#95C2×D418yes—
229C24—22no—
230C24—22no—
231C24—24no—
232C24—22no—
233C24—42no—
234C24—42no—
235C24—44no—
236C24—42no—
237C24—42no—
238C24—44no—
239C24C2×C22⋊C411yescharacteristic
240C24C2×C22⋊C411yescharacteristic
241C24#12012yes—
242C24#12312yes—
243C24C22×D411yescharacteristic
244C2×D4—24no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—88no—
256C2×D4—88no—
257C2×D4—88no—
258C22×C4—216no—
259C22×C4—216no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
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—24no—
277C22×C4—216no—
278C22×C4—21no—
279C22×C4—28no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4#10418yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C4×D414yes—
315C22×C4C4×D414yes—
316C22×C4#12012yes—
317C22×C4#12114yes—
318C22×C4#12114yes—
319C22×C4#12312yes—
320C22×C4AGL(1,ℤ/8)14yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D412yes—
324C22×C4C2×#4112yes—
325M16—88no—
326C2×C8—48no—
327C2×C8—44no—
328C4⋊C4—232no—
329C4⋊C4—24no—
330C4⋊C4—44no—
331C4⋊C4—48no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—48no—
335C22⋊C4—232no—
336C22⋊C4—48no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C42—232no—
345C42—24no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—24no—
350C42—24no—
351C42—24no—
352C42—24no—
353C42—416no—
354C42—416no—
355C23—22no—
356C23—22no—
357C23—21no—
358C23—21no—
359C23—22no—
360C23—24no—
361C23—22no—
362C23—21no—
363C23—21no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—42no—
369C23—42no—
370C23—48no—
371C23—48no—
372C23—44no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—42no—
377C23—84no—
378C23—84no—
379C23—88no—
380C23—88no—
381C23—84no—
382C23#38612yes—
383C23#38612yes—
384C23C2×#10414yes—
385C23C22×C22⋊C411yescharacteristic
386C23C2×#12011yescharacteristic
387C23C2×#12311yescharacteristic
388C23C2×AGL(1,ℤ/8)12yes—
389C23#57512yes—
390D4—84no—
391D4—88no—
392D4—84no—
393D4—84no—
394D4—84no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—216no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—216no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—24no—
416C2×C4—216no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—416no—
428C2×C4—416no—
429C2×C4—42no—
430C2×C4—48no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—88no—
439C2×C4—84no—
440C2×C4#38612yes—
441C2×C4#38612yes—
442C2×C4#41014yes—
443C2×C4#41814yes—
444C2×C4C2×#10414yes—
445C2×C4C2×#10414yes—
446C2×C4C2×C4×D412yes—
447C2×C4C2×#12112yes—
448C8—84no—
449C22—24no—
450C22—24no—
451C22—24no—
452C22—24no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—21no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—48no—
463C22—48no—
464C22—42no—
465C22—42no—
466C22—84no—
467C22—84no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—168no—
472C22#189614yes—
473C22C2×#38611yescharacteristic
474C22C2×#41012yes—
475C22C2×#41811yescharacteristic
476C22#289611yescharacteristic
477C22C22×#10412yes—
478C22#300612yes—
479C22#309412yes—
480C4—24no—
481C4—24no—
482C4—24no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—416no—
487C4—42no—
488C4—84no—
489C4—82no—
490C4#189914yes—
491C2—22no—
492C2—24no—
493C2—22no—
494C2—42no—
495C2—162no—
496C2#1753712yes—
497C2#1754412yes—
498C2#2091211yescharacteristic
499C2C2×#189611yescharacteristic
500C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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×C4×#117C2semidirect
3. #20682C2semidirect
5. #9149C2semidirect
6. #9148C2semidirect
7. #8019C2semidirect
16. #2854C4semidirect
19. C2×#390C4semidirect
20. C2×#389C4semidirect
26. C2×C4×C4⋊C4C22semidirect
29. C2×#376C22semidirect
32. #1817C22semidirect
33. #1817C22semidirect
37. #1294C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×C4×#1172
#206822
C2×#15072
#91492
#91482
#80192