Finite Group Databasebeta

← #238128

Subgroups — #238128

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2#21859C211yesmaximal, characteristic
3#21386C211yesmaximal, characteristic
4#21258C211yesmaximal, characteristic
5#10353C214yesmaximal, complemented
6#10351C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic, complemented
8#7901C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#417—21no—
15C2×#417—21no—
16#2571—22no—
17#2571—22no—
18#2571—22no—
19#2571C2214yes—
20C2×#400—22no—
21C2×#389—21no—
22C2×#389—21no—
23C2×#384C2211yescharacteristic
24C2×#382—21no—
25C2×#382—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990C2212yes—
29#1988C2212yes—
30#1983—22no—
31#1982C2212yes—
32#1982C2212yes—
33#1981C2212yes—
34#1957C2212yes—
35#1874—22no—
36#1874—21no—
37#1874—21no—
38#1802—22no—
39C4×#102—22no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×#41—21no—
44C22×M16—21no—
45#532—22no—
46#532—22no—
47#532—22no—
48#532Q814yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×#127C2311yescharacteristic
51C2×#124—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64C22×C42—21no—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#431—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71#423—22no—
72#423—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77#417—44no—
78#417—44no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—24no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2312yes—
90#400—28no—
91#400—28no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—24no—
96#384—21no—
97#384—21no—
98#384D412yes—
99#384C2312yes—
100#382—24no—
101#382—24no—
102#382—24no—
103#382—24no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4C2312yes—
111C4×C4⋊C4C2312yes—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117#340—24no—
118#340—22no—
119#340—22no—
120C2×#41—24no—
121C2×#41—42no—
122C2×#41—42no—
123C2×#41—42no—
124C2×#41—42no—
125C2×#41—42no—
126C22×Q8—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—41no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×M16—42no—
141C2×M16—42no—
142C2×M16—42no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8C2×Q812yes—
150#127—21no—
151#127—21no—
152#127—21no—
153#127C2×Q812yes—
154#124—24no—
155#124—42no—
156#121—24no—
157#121—24no—
158#121—42no—
159#121—42no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—42no—
163C4×D4—42no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—84no—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D414yes—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—44no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×Q812yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42C2411yescharacteristic
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—42no—
214C4⋊C8—42no—
215#103—44no—
216#103—44no—
217#102—44no—
218#102—44no—
219#102—44no—
220#102—44no—
221#102—84no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—42no—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234C24—41no—
235C24#12011yescharacteristic
236#41—48no—
237#41—48no—
238#41—84no—
239#41—84no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—82no—
262C2×D4—82no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4C22⋊Q812yes—
295C22×C4C22⋊Q812yes—
296C22×C4C22⋊Q812yes—
297C22×C4#12412yes—
298C22×C4#12412yes—
299C22×C4#12412yes—
300C22×C4#13512yes—
301C22×C4#13512yes—
302C22×C4AGL(1,ℤ/8)12yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307M16—84no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—82no—
340C4⋊C4—88no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C42—44no—
344C42—42no—
345C42—42no—
346C42—44no—
347C42—44no—
348C42C22⋊Q812yes—
349C42C22⋊Q812yes—
350C42C22⋊Q812yes—
351C42#12412yes—
352C42#12412yes—
353C42#12412yes—
354C42C22×Q811yescharacteristic
355C42C2×#4111yescharacteristic
356C42C2×#4111yescharacteristic
357C42C2×#4111yescharacteristic
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23#39312yes—
370C23#44912yes—
371C23#44912yes—
372C23C2×#12011yescharacteristic
373Q8—42no—
374Q8—42no—
375Q8—81no—
376Q8—81no—
377D4—44no—
378D4—42no—
379D4—42no—
380D4—44no—
381D4—82no—
382D4—88no—
383D4—81no—
384D4—81no—
385D4—82no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—88no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—88no—
420C2×C4#39312yes—
421C2×C4#39312yes—
422C2×C4#39312yes—
423C2×C4#44912yes—
424C2×C4#44912yes—
425C2×C4#47814yes—
426C2×C4#47814yes—
427C2×C4#48814yes—
428C2×C4#49014yes—
429C2×C4C2×C22⋊Q811yescharacteristic
430C2×C4C2×C22⋊Q811yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×#12411yescharacteristic
433C2×C4C2×#12411yescharacteristic
434C2×C4C2×#12411yescharacteristic
435C2×C4C2×#13511yescharacteristic
436C2×C4C2×#13511yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4#57512yes—
440C2×C4#57512yes—
441C8—82no—
442C8—82no—
443C8—82no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22#202514yes—
458C22C2×#39311yescharacteristic
459C22C2×#44911yescharacteristic
460C22C2×#44911yescharacteristic
461C22#300511yescharacteristic
462C22#300511yescharacteristic
463C22#301911yescharacteristic
464C22#302011yescharacteristic
465C4—22no—
466C4—21no—
467C4—21no—
468C4—22no—
469C4—84no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4#202514yes—
476C4C2×#47811yescharacteristic
477C4C2×#47811yescharacteristic
478C4#308012yes—
479C4C2×#48811yescharacteristic
480C4C2×#49011yescharacteristic
481C4#314212yes—
482C2—24no—
483C2—162no—
484C2#1875011yescharacteristic
485C2#1876211yescharacteristic
486C2#1877211yescharacteristic
487C2#1878011yescharacteristic
488C2C2×#202511yescharacteristic
489C2#2345311yescharacteristic
490C2#2346211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2571C2semidirect
5. #10353C2semidirect
6. #10351C2semidirect
7. #8043C2semidirect
8. #7901C2semidirect
9. #7892C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
#218592
#213862
#212582
#103532
#103512
#80432
#79012
#78922