Finite Group Databasebeta

← #193196

Subgroups — #193196

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2366C211yesmaximal, characteristic
2#21803C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic, complemented
4#21170C211yesmaximal, characteristic, complemented
5C2×#1545C211yesmaximal, characteristic
6#8924C214yesmaximal, complemented
7#8921C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C22×#127C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4—21no—
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#428—21no—
16C2×#423C2211yescharacteristic
17#2366—22no—
18#2366—22no—
19#2366—22no—
20#2366C2214yes—
21C2×#390—21no—
22C2×#390—21no—
23C2×#386—22no—
24#2285C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1982—22no—
28#1957C2212yes—
29#1957C2212yes—
30#1949—22no—
31#1948C2212yescomplemented
32#1948C2212yescomplemented
33#1948C2212yescomplemented
34#1947C2212yescomplemented
35#1853—21no—
36#1853—21no—
37#1786—22no—
38C4×#106—22no—
39#1736—21no—
40#1545C418yescomplemented
41C2×#331C2214yes—
42#1279C2214yescomplemented
43#1279C2214yescomplemented
44C23×D4—21no—
45C23×D4—21no—
46C23×D4D412yes—
47C24×C4—22no—
48C22×M16—21no—
49C2×#127—21no—
50C2×#127—21no—
51C2×#127—22no—
52C2×#127—22no—
53C2×#127C2×C414yes—
54C2×#127D412yes—
55C2×#127Q814yes—
56C2×#127C2312yes—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—48no—
62C2×C4×D4—44no—
63C2×C4×D4—44no—
64C2×#117—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#431—22no—
76#431—22no—
77#431—22no—
78#431C2×C414yescomplemented
79#428—44no—
80C2×#107—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—22no—
84#423—22no—
85#423C2×C414yescomplemented
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×#102—24no—
91C2×#102—24no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8C2312yes—
96#390—24no—
97#390—24no—
98#390—24no—
99#390—24no—
100#386—28no—
101#386—28no—
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#382—21no—
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2312yes—
116C4×C4⋊C4C2312yes—
117C2×#95—21no—
118C2×#95—21no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#331C2×C4116yescomplemented
124C25—22no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—44no—
136C22×D4—44no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—82no—
142C22×D4—82no—
143C22×D4C4⋊C418yes—
144C22×D4C2×D414yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—28no—
150C23×C4—48no—
151C23×C4—48no—
152C23×C4—84no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C2×M16—24no—
157C2×M16—42no—
158C2×M16—42no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162#127—24no—
163#127—24no—
164#127—24no—
165#127—24no—
166#127—24no—
167#127—22no—
168#127—22no—
169#127C4⋊C418yes—
170#127C22×C414yes—
171#127C2×Q814yes—
172C4×D4—416no—
173C4×D4—48no—
174C4×D4—48no—
175C4×D4—416no—
176C4×D4—48no—
177C4×D4—48no—
178#117—42no—
179#117—42no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4SD1614yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—84no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—44no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—41no—
215C2×C42C22×C412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42C2411yescharacteristic
223#107—44no—
224#106—44no—
225#106—44no—
226#106—44no—
227C4⋊C8—24no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—42no—
231C4⋊C8C22×C414yescomplemented
232C4⋊C8C22×C414yescomplemented
233#102—84no—
234#102—84no—
235C22⋊C8—24no—
236C22⋊C8—42no—
237C4×C8—24no—
238C4×C8—22no—
239C4×C8—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—41no—
245#95—41no—
246C24—24no—
247C24—28no—
248C24—44no—
249C24—44no—
250C24—44no—
251C24—44no—
252C24—44no—
253C24—44no—
254C24#12011yescharacteristic
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—28no—
261C2×D4—28no—
262C2×D4—28no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—416no—
266C2×D4—416no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—44no—
275C2×D4—48no—
276C2×D4—48no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—82no—
282C2×D4—82no—
283C2×D4—88no—
284C2×D4—88no—
285C2×D4C2×C4⋊C418yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—416no—
297C22×C4—416no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—88no—
311C22×C4—832no—
312C22×C4C2×C4⋊C412yes—
313C22×C4C4×D412yes—
314C22×C4C4×D412yes—
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4C22⋊Q812yes—
319C22×C4C22⋊Q812yes—
320C22×C4#12312yes—
321C22×C4C2×SD1612yes—
322C22×C4#13512yes—
323C22×C4AGL(1,ℤ/8)12yes—
324C22×C4AGL(1,ℤ/8)12yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328M16—48no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4C2×SD1614yes—
357C4⋊C4AGL(1,ℤ/8)14yes—
358C22⋊C4—416no—
359C22⋊C4—416no—
360C22⋊C4—88no—
361C22⋊C4—82no—
362C42—44no—
363C42—42no—
364C42—42no—
365C42—44no—
366C42—44no—
367C42—82no—
368C42C2×C4⋊C412yes—
369C42C4×D412yes—
370C42C4×D412yes—
371C42C22⋊Q812yes—
372C42C22⋊Q812yes—
373C42#12312yes—
374C42C23×C411yescharacteristic
375C42C22×Q811yescharacteristic
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—28no—
383C23—416no—
384C23—416no—
385C23—416no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—88no—
393C23—88no—
394C23—88no—
395C23—88no—
396C23#38512yes—
397C23#44912yes—
398C23#45012yes—
399C23C2×#12011yescharacteristic
400D4—48no—
401D4—416no—
402D4—44no—
403D4—48no—
404D4—416no—
405D4—44no—
406D4—44no—
407D4—48no—
408D4—44no—
409D4—84no—
410D4—88no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—832no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4#38512yes—
445C2×C4#38512yes—
446C2×C4#38512yes—
447C2×C4C4×SD1614yes—
448C2×C4#44214yes—
449C2×C4#44912yes—
450C2×C4#45012yes—
451C2×C4#47614yes—
452C2×C4#47814yes—
453C2×C4C22×C4⋊C411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C22⋊Q811yescharacteristic
457C2×C4C2×C22⋊Q811yescharacteristic
458C2×C4C2×#12311yescharacteristic
459C2×C4C22×SD1611yescharacteristic
460C2×C4C2×#13511yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4C2×AGL(1,ℤ/8)12yes—
463C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
464C2×C4#57512yes—
465C8—44no—
466C8—44no—
467C8—82no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—48no—
475C22—48no—
476C22—48no—
477C22—832no—
478C22—84no—
479C22—84no—
480C22—84no—
481C22—84no—
482C22#186414yes—
483C22C2×#38511yescharacteristic
484C22C2×#45011yescharacteristic
485C22C2×#44911yescharacteristic
486C22#300311yescharacteristic
487C22#300511yescharacteristic
488C22#301911yescharacteristic
489C22#301911yescharacteristic
490C4—21no—
491C4—21no—
492C4—22no—
493C4—22no—
494C4—44no—
495C4—44no—
496C4—82no—
497C4—82no—
498C4—161no—
499C4—161no—
500C4#186414yes—
501C4C2×C4×SD1611yescharacteristic
502C4C2×#44211yescharacteristic
503C4C4×AGL(1,ℤ/8)12yes—
504C4C2×#47611yescharacteristic
505C4C2×#47811yescharacteristic
506C4#307712yes—
507C2—24no—
508C2—88no—
509C2#1874711yescharacteristic
510C2#1875011yescharacteristic
511C2#1876711yescharacteristic
512C2#1877011yescharacteristic
513C2#2085611yescharacteristic
514C2C2×#186411yescharacteristic
515C2#2087311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ 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
3. #21359C2semidirect
4. #21170C2semidirect
6. #8924C2semidirect
7. #8921C2semidirect
8. #7901C2semidirect
9. #7892C2semidirect
31. #1948C22semidirect
32. #1948C22semidirect
33. #1948C22semidirect
34. #1947C22semidirect
40. #1545C4semidirect
42. #1279C22semidirect
43. #1279C22semidirect
78. #431C2×C4semidirect
85. #423C2×C4semidirect
123. #331C2×C4semidirect
231. C4⋊C8C22×C4semidirect
232. C4⋊C8C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23662
#218032
#213592
#211702
C2×#15452
#89242
#89212
#79012
#78922