Finite Group Databasebeta

← #194036

Subgroups — #194036

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#21885C211yesmaximal, characteristic, complemented
4#21400C211yesmaximal, characteristic, complemented
5C22×D4⋊C8C211yesmaximal, characteristic
6#8981C214yesmaximal, complemented
7#8973C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C2×#518—21no—
12C22×C4×D4C2211yescharacteristic
13#3077—24no—
14#3077C418yescomplemented
15C2×#478—22no—
16C2×#476C2212yes—
17C2×#474C2212yes—
18C2×#428—21no—
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#418—21no—
22C22×C4×C8C2211yescharacteristic
23#2349—22no—
24#2349—22no—
25D4×C4⋊C4—22no—
26D4×C4⋊C4C2214yes—
27#2316—21no—
28#2315—22no—
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31C2×#380—22no—
32#1990C2212yescomplemented
33#1987C2212yescomplemented
34#1981C2212yescomplemented
35#1956C2212yescomplemented
36#1892—22no—
37#1892—22no—
38#1803—22no—
39C4×#103—22no—
40C2×D4⋊C8—24no—
41C2×D4⋊C8C2214yes—
42#1296C2214yescomplemented
43#1279C2214yescomplemented
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C22×M16—21no—
47C23×C8—21no—
48C23×C8—21no—
49#533—22no—
50#533—22no—
51#533—22no—
52#533C2×C414yescomplemented
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58#518—44no—
59#518—44no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C2×C4×D4D414yes—
70C2×C4×D4D412yes—
71C2×C4×D4C2312yes—
72C2×#117—21no—
73C2×#117—21no—
74C2×#117—21no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4D411yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—24no—
79C22×C42C2311yescharacteristic
80#478—216no—
81#476—24no—
82#476C2×C418yescomplemented
83#474—24no—
84#474C2×C418yescomplemented
85#428—28no—
86C2×#107—22no—
87C2×#106—22no—
88#423—22no—
89#423C2×C414yescomplemented
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96#418—28no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#102—22no—
100C2×#102—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8—21no—
104C2×C4×C8C2312yes—
105#389—24no—
106#389—24no—
107#385—28no—
108#385—28no—
109#384D412yes—
110#384C2312yes—
111#382—21no—
112#382—21no—
113#380—24no—
114#380—24no—
115#380—21no—
116#380—21no—
117#380—24no—
118#380D412yes—
119#380D412yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4C2312yes—
124C2×#95—24no—
125#340—24no—
126#340—24no—
127D4⋊C8—28no—
128C25—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C22⋊C414yes—
136C22×D4SD1614yes—
137C22×D4Q1614yes—
138C22×D4C2×D412yes—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—28no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—44no—
157C22×C8—42no—
158C22×C8—42no—
159C22×C8—82no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8C22×C412yes—
167C4⋊Q8C22×C412yes—
168#125—24no—
169#125—42no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—48no—
186C4×D4—416no—
187C4×D4—48no—
188C4×D4C22⋊C418yes—
189C4×D4C22×C414yes—
190C4×D4C2×D414yes—
191#117—24no—
192#117—42no—
193#117—42no—
194#117—42no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—84no—
211C2×C4⋊C4—84no—
212C2×C4⋊C4C22⋊C414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D414yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—416no—
224C2×C22⋊C4—41no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—41no—
228C2×C42C22×C412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
236#107—28no—
237#106—28no—
238C4⋊C8—24no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—42no—
243C4⋊C8C22×C414yes—
244#103—48no—
245#103—84no—
246#102—28no—
247#102—28no—
248C4×C8—22no—
249C4×C8—22no—
250C4×C8—42no—
251#95—28no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256C24—22no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—82no—
261C24—82no—
262C24#12011yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—44no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—28no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—28no—
278C2×D4—28no—
279C2×D4#103116yescomplemented
280C2×D4C2×C22⋊C414yes—
281C2×D4C2×SD1614yes—
282C2×D4C2×Q1614yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—416no—
298C22×C4—88no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—82no—
303C22×C4—88no—
304C22×C4—88no—
305C22×C4C2×C22⋊C412yes—
306C22×C4C4×D412yes—
307C22×C4C4×D412yes—
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12312yes—
314C22×C4C2×SD1612yes—
315C22×C4C2×Q1612yes—
316C22×C4AGL(1,ℤ/8)12yes—
317C22×C4#13712yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321M16—48no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—42no—
325C2×C8—48no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—44no—
331C2×C8—84no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—44no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—82no—
353C4⋊C4—88no—
354C4⋊C4C2×C22⋊C414yes—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—28no—
358C22⋊C4—416no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C42—42no—
363C42—42no—
364C42—44no—
365C42—44no—
366C42C2×C22⋊C412yes—
367C42C4×D412yes—
368C42C4×D412yes—
369C42#12112yes—
370C42#12112yes—
371C42#12312yes—
372C42C23×C411yescharacteristic
373C42C22×D411yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C23—22no—
377C23—22no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—216no—
382C23—44no—
383C23—44no—
384C23—84no—
385C23—84no—
386C23—88no—
387C23—88no—
388C23#38612yes—
389C23#45012yes—
390C23#45112yes—
391C23C2×#12011yescharacteristic
392Q8—44no—
393Q8—44no—
394Q8—82no—
395Q8—81no—
396Q8—81no—
397D4—24no—
398D4—24no—
399D4—24no—
400D4—216no—
401D4C2×#10318yes—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—48no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—88no—
424C2×C4—84no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—1616no—
428C2×C4#38612yes—
429C2×C4#38612yes—
430C2×C4#38612yes—
431C2×C4C2×#10314yes—
432C2×C4#41714yes—
433C2×C4#45012yes—
434C2×C4#45112yes—
435C2×C4#48314yes—
436C2×C4#48414yes—
437C2×C4C22×C22⋊C411yescharacteristic
438C2×C4C2×C4×D411yescharacteristic
439C2×C4C2×C4×D411yescharacteristic
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×#12111yescharacteristic
442C2×C4C2×#12311yescharacteristic
443C2×C4C22×SD1611yescharacteristic
444C2×C4C22×Q1611yescharacteristic
445C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4#57612yes—
448C2×C4#57712yes—
449C8—44no—
450C8—44no—
451C8—82no—
452C22—22no—
453C22—22no—
454C22—24no—
455C22—24no—
456C22—28no—
457C22—44no—
458C22—44no—
459C22—48no—
460C22—48no—
461C22—816no—
462C22#187214yes—
463C22C2×#38611yescharacteristic
464C22C2×#45011yescharacteristic
465C22C2×#45111yescharacteristic
466C22#301212yes—
467C22#301812yes—
468C4—22no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—44no—
473C4—82no—
474C4—82no—
475C4—84no—
476C4—84no—
477C4#187214yes—
478C4C22×#10311yescharacteristic
479C4C2×#41711yescharacteristic
480C4#294612yes—
481C4#304111yescharacteristic
482C4#304211yescharacteristic
483C4C2×#48311yescharacteristic
484C4C2×#48411yescharacteristic
485C2—24no—
486C2—48no—
487C2#1866312yes—
488C2#1866512yes—
489C2C2×#187211yescharacteristic
490C2#2092911yescharacteristic
491C2#2094211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #21885C2semidirect
4. #21400C2semidirect
6. #8981C2semidirect
7. #8973C2semidirect
8. #8045C2semidirect
9. #7897C2semidirect
14. #3077C4semidirect
32. #1990C22semidirect
33. #1987C22semidirect
34. #1981C22semidirect
35. #1956C22semidirect
42. #1296C22semidirect
43. #1279C22semidirect
52. #533C2×C4semidirect
82. #476C2×C4semidirect
84. #474C2×C4semidirect
89. #423C2×C4semidirect
279. C2×D4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
#341812
#218852
#214002
C22×D4⋊C82
#89812
#89732
#80452
#78972