Finite Group Databasebeta

← #194044

Subgroups — #194044

Order 512.

1342 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#21885C211yesmaximal, characteristic, complemented
4#21400C211yesmaximal, characteristic, complemented
5C22×D4⋊C8C211yesmaximal, characteristic
6#8984C214yesmaximal, complemented
7#8976C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3077—24no—
13#3077C418yescomplemented
14C2×#478—22no—
15C2×#476C2212yes—
16C2×#474C2212yes—
17C2×#428—21no—
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#418—21no—
21C22×C4×C8C2211yescharacteristic
22#2357—22no—
23#2357—22no—
24#2354—22no—
25#2354C2214yes—
26#2312—22no—
27#2309—21no—
28#2285C2211yescharacteristic
29#2285C2211yescharacteristic
30#2280—22no—
31#2274—21no—
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×C414yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C414yes—
65C2×C4×D4D414yes—
66C2×C4×D4D412yes—
67C2×C4×D4C2312yes—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117—24no—
71C2×#117—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C42C2311yescharacteristic
75#478—216no—
76#476—24no—
77#476C2×C418yes—
78#474—24no—
79#474C2×C418yes—
80#428—28no—
81C2×#107—22no—
82C2×#106—22no—
83#423—22no—
84#423C2×C414yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91#418—28no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#102—22no—
95C2×#102—22no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8C2312yes—
100#390—24no—
101#390—24no—
102#388—28no—
103#388—28no—
104#384D412yes—
105#384C2312yes—
106#382—21no—
107#382—21no—
108#381—24no—
109#380—21no—
110#380—21no—
111#380—24no—
112#380D412yes—
113#380D412yes—
114#379—24no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4C2312yes—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—24no—
123#340—24no—
124#340—24no—
125D4⋊C8—28no—
126C25—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C22⋊C414yes—
136C22×D4C2×D412yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—44no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—82no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8C22×C412yes—
164C4⋊Q8C22×C412yes—
165#125—24no—
166#125—42no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4C22⋊C418yes—
183C4×D4C22×C414yes—
184C4×D4C2×D414yes—
185#117—24no—
186#117—42no—
187#117—42no—
188#117—42no—
189#117—42no—
190#117—42no—
191#117—84no—
192#117—84no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D414yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—41no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—28no—
221C2×C42—41no—
222C2×C42C22×C412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230#107—28no—
231#106—28no—
232C4⋊C8—24no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—42no—
237C4⋊C8C22×C414yes—
238#103—48no—
239#103—84no—
240#102—28no—
241#102—28no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—42no—
245#95—28no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250C24—22no—
251C24—24no—
252C24—42no—
253C24—42no—
254C24—82no—
255C24—82no—
256C24#12011yescharacteristic
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—44no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—28no—
271C2×D4—28no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4C2×C22⋊C414yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—28no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—88no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—82no—
296C22×C4C2×C22⋊C412yes—
297C22×C4C4×D412yes—
298C22×C4C4×D412yes—
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4#12312yes—
305C22×C4#13512yes—
306C22×C4#13512yes—
307C22×C4#13712yes—
308C22×C4#13712yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312M16—48no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—42no—
316C2×C8—48no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—44no—
322C2×C8—84no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—44no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—82no—
344C4⋊C4C2×C22⋊C414yes—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—28no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—88no—
352C22⋊C4—88no—
353C42—42no—
354C42—42no—
355C42—44no—
356C42—44no—
357C42—88no—
358C42C2×C22⋊C412yes—
359C42C4×D412yes—
360C42C4×D412yes—
361C42#12112yes—
362C42#12112yes—
363C42#12312yes—
364C42C23×C411yescharacteristic
365C42C22×D411yescharacteristic
366C42C2×#4111yescharacteristic
367C42C2×#4111yescharacteristic
368C23—22no—
369C23—22no—
370C23—24no—
371C23—48no—
372C23—42no—
373C23—42no—
374C23—44no—
375C23—44no—
376C23—84no—
377C23—84no—
378C23—88no—
379C23—88no—
380C23#38612yes—
381C23#45212yes—
382C23#45212yes—
383C23C2×#12011yescharacteristic
384Q8—44no—
385Q8—44no—
386Q8—82no—
387Q8—81no—
388Q8—81no—
389D4—24no—
390D4—24no—
391D4—42no—
392D4—42no—
393D4—48no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—88no—
416C2×C4—84no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4#38612yes—
420C2×C4#38612yes—
421C2×C4#38612yes—
422C2×C4#41614yes—
423C2×C4#41914yes—
424C2×C4#45212yes—
425C2×C4#45212yes—
426C2×C4#48514yes—
427C2×C4#48514yes—
428C2×C4C22×C22⋊C411yescharacteristic
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12311yescharacteristic
434C2×C4C2×#13511yescharacteristic
435C2×C4C2×#13511yescharacteristic
436C2×C4C2×#13711yescharacteristic
437C2×C4C2×#13711yescharacteristic
438C2×C4#57612yes—
439C2×C4#57712yes—
440C8—44no—
441C8—44no—
442C8—82no—
443C22—22no—
444C22—22no—
445C22—24no—
446C22—24no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—816no—
451C22—84no—
452C22—84no—
453C22#187314yes—
454C22C2×#38611yescharacteristic
455C22C2×#45211yescharacteristic
456C22C2×#45211yescharacteristic
457C22#301212yes—
458C22#301812yes—
459C4—22no—
460C4—21no—
461C4—21no—
462C4—22no—
463C4—44no—
464C4—82no—
465C4—82no—
466C4—84no—
467C4—84no—
468C4#187314yes—
469C4C2×#41611yescharacteristic
470C4C2×#41911yescharacteristic
471C4#294612yes—
472C4#304111yescharacteristic
473C4#304211yescharacteristic
474C4C2×#48511yescharacteristic
475C4C2×#48511yescharacteristic
476C2—24no—
477C2—84no—
478C2#1867912yes—
479C2#1868112yes—
480C2C2×#187311yescharacteristic
481C2#2092911yescharacteristic
482C2#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. #8984C2semidirect
7. #8976C2semidirect
8. #8045C2semidirect
9. #7897C2semidirect
13. #3077C4semidirect
32. #1990C22semidirect
33. #1987C22semidirect
34. #1981C22semidirect
35. #1956C22semidirect
42. #1296C22semidirect
43. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
#341862
#218852
#214002
C22×D4⋊C82
#89842
#89762
#80452
#78972