Finite Group Databasebeta

← #195565

Subgroups — #195565

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#21385C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6#9065C214yesmaximal, complemented
7#9059C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8—21no—
13#3229—24no—
14#3229C418yescomplemented
15C2×#498C2212yes—
16C2×#487C2212yes—
17C2×#483—22no—
18C2×#475—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2365—22no—
23#2365—22no—
24#2365—22no—
25#2365C2214yes—
26C2×#391—21no—
27C2×#389—21no—
28C2×#387—22no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1989—22no—
33#1988C2212yes—
34#1987C2212yes—
35#1987C2212yes—
36#1981C2212yes—
37#1872—21no—
38#1872—21no—
39C4×#103—22no—
40#1736—21no—
41#1736—21no—
42#1546—24no—
43C2×#332C2214yes—
44#1279C2214yes—
45#1279C2214yes—
46C23×Q8—21no—
47C22×M16—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8D412yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2312yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—44no—
70C2×C4×Q8—44no—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D412yes—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C42—22no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83#498—24no—
84#498C2×C418yescomplemented
85#487—24no—
86#487C2×C418yescomplemented
87#483—216no—
88#475—216no—
89#431—22no—
90#431—22no—
91#431—22no—
92#431C2×C414yes—
93C2×#106—24no—
94#423—22no—
95#423—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100C2×#103—24no—
101C2×#103—22no—
102C2×#103—22no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2312yes—
107#391—24no—
108#391—24no—
109#389—24no—
110#389—24no—
111#387—28no—
112#387—28no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4C2312yes—
122C4×C4⋊C4C2312yes—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—21no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#332—28no—
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—24no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—42no—
153C2×M16—42no—
154C2×M16—42no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8C22⋊C418yes—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C22×C412yes—
174C4⋊Q8C22×C414yes—
175C4⋊Q8C2×D414yes—
176#125—24no—
177#125—42no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—48no—
185C4×Q8—416no—
186C4×Q8—48no—
187C4×Q8—42no—
188C4×Q8—42no—
189#117—24no—
190#117—42no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4C22⋊C418yes—
215C2×C4⋊C4SD1614yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C22⋊C4—41no—
222C2×C42—28no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—48no—
228C2×C42—44no—
229C2×C42—41no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C22×C412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42C2411yescharacteristic
240#106—216no—
241C4⋊C8—24no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—42no—
247#103—216no—
248#103—44no—
249#103—44no—
250C22⋊C8—24no—
251C22⋊C8—42no—
252C4×C8—22no—
253C4×C8—42no—
254C4×C8C22×C414yes—
255#95—28no—
256#95—28no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261C24#12011yescharacteristic
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—28no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—48no—
276C2×Q8—48no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—42no—
282C2×Q8—42no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4C2×C22⋊C412yes—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12112yes—
316C22×C4#12112yes—
317C22×C4#12312yes—
318C22×C4C2×SD1612yes—
319C22×C4#13512yes—
320C22×C4AGL(1,ℤ/8)12yes—
321C22×C4#13712yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325M16—84no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—28no—
342C4⋊C4—28no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—416no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4C2×C22⋊C418yes—
362C4⋊C4C2×SD1614yes—
363C4⋊C4#13714yes—
364C22⋊C4—44no—
365C42—44no—
366C42—416no—
367C42—42no—
368C42—42no—
369C42—44no—
370C42—44no—
371C42—44no—
372C42C2×C22⋊C412yes—
373C42C4×D412yes—
374C42C4×D412yes—
375C42#12112yes—
376C42#12112yes—
377C42#12312yes—
378C42C23×C411yescharacteristic
379C42C22×D411yescharacteristic
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23#38612yes—
387C23#45012yes—
388C23#45212yes—
389C23C2×#12011yescharacteristic
390Q8—44no—
391Q8—416no—
392Q8—44no—
393Q8—44no—
394Q8—44no—
395Q8—44no—
396Q8—44no—
397Q8—82no—
398Q8—82no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—28no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—416no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4#38612yes—
438C2×C4C4×SD1614yes—
439C2×C4#44014yes—
440C2×C4#45012yes—
441C2×C4#45212yes—
442C2×C4#46114yes—
443C2×C4#46314yes—
444C2×C4C22×C22⋊C411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×#12111yescharacteristic
448C2×C4C2×#12111yescharacteristic
449C2×C4C2×#12311yescharacteristic
450C2×C4C22×SD1611yescharacteristic
451C2×C4C2×#13511yescharacteristic
452C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
453C2×C4C2×#13711yescharacteristic
454C2×C4C2×#13712yes—
455C2×C4#57512yes—
456C8—44no—
457C8—82no—
458C8—82no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22#187714yes—
466C22C2×#38611yescharacteristic
467C22C2×#45011yescharacteristic
468C22C2×#45211yescharacteristic
469C22#300411yescharacteristic
470C22#300511yescharacteristic
471C22#302011yescharacteristic
472C22#302011yescharacteristic
473C4—22no—
474C4—21no—
475C4—21no—
476C4—22no—
477C4—44no—
478C4—44no—
479C4—48no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4#187714yes—
485C4C2×C4×SD1611yescharacteristic
486C4C2×#44011yescharacteristic
487C4C4×#13712yes—
488C4C2×#46111yescharacteristic
489C4C2×#46311yescharacteristic
490C4#303712yes—
491C2—24no—
492C2#1875711yescharacteristic
493C2#1876111yescharacteristic
494C2#1877711yescharacteristic
495C2#1878111yescharacteristic
496C2C2×#187711yescharacteristic
497C2#2092611yescharacteristic
498C2#2095411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #9065C2semidirect
7. #9059C2semidirect
14. #3229C4semidirect
84. #498C2×C4semidirect
86. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
C2×#23652
#213852
#213122
C2×#15462
#90652
#90592
#79012
#78922