Finite Group Databasebeta

← #201551

Subgroups — #201551

Order 512.

1408 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic, complemented
2C2×#2536C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic, complemented
4#17448C212yesmaximal
5#9490C214yesmaximal, complemented
6#9486C214yesmaximal, complemented
7#7959C212yesmaximal, complemented
8C22×C22⋊Q8—21no—
9C22×C4×Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#2984—24no—
12#2983—22no—
13#2983C2212yescomplemented
14#2982C2212yescomplemented
15Q8×M16—24no—
16#2920—22no—
17#2920C2212yes—
18#2920C2212yes—
19#2536—21no—
20#2536—21no—
21#2536—22no—
22#2536C414yes—
23#2536C2212yes—
24#2536C2212yes—
25C2×#398C412yes—
26C2×#394C412yes—
27C2×#388—22no—
28C2×#382C2211yescharacteristic
29C2×#380—21no—
30#1821—21no—
31#1801C2212yescomplemented
32#1801C2212yescomplemented
33#1784—21no—
34C4×#101—21no—
35C4×#101—21no—
36#1490—24no—
37#1486C2218yes—
38#1478—24no—
39#1286C2218yescomplemented
40C23×Q8D411yescharacteristic
41C23×D4C2×C411yescharacteristic
42C24×C4—21no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—48no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D412yes—
55C2×C4×Q8C2311yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2312yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C42D412yes—
74C22×C42C2311yescharacteristic
75#446—216no—
76#446—28no—
77C8×Q8—28no—
78#432—24no—
79#432—24no—
80#432—22no—
81#432—22no—
82#432—22no—
83#432—22no—
84#431—24no—
85#431—24no—
86#431—22no—
87#431—22no—
88#431D414yes—
89#431C2314yes—
90#423—24no—
91#423—22no—
92#423C2314yes—
93C2×#101—22no—
94C2×#101—22no—
95#407—22no—
96#407—22no—
97C4×M16—24no—
98C4×M16—22no—
99C4×M16D414yes—
100#398—22no—
101#398—22no—
102#398C2×C414yes—
103#398C2×C414yes—
104#394—22no—
105#394—22no—
106#394C2×C414yes—
107#394C2×C414yes—
108#388—24no—
109#388—24no—
110#388—24no—
111#388—24no—
112#382—21no—
113#382—21no—
114#382C2×C412yes—
115#382C2312yes—
116#380—24no—
117#380—22no—
118#380—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95C2×C412yes—
122#343—22no—
123#343—22no—
124D4⋊C8—216no—
125C25—21no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—42no—
130C22×Q8C22⋊C412yes—
131C22×Q8C2×D411yescharacteristic
132C22×Q8C2×D411yescharacteristic
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—44no—
137C22×D4C22×C412yes—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—24no—
152C2×M16—24no—
153C22⋊Q8—48no—
154C22⋊Q8—416no—
155C22⋊Q8—48no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—24no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—42no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42C22⋊C412yes—
214C2×C42C22⋊C412yes—
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42C2411yescharacteristic
225C4⋊C8—28no—
226C4⋊C8—28no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8—24no—
231#101—44no—
232#101—44no—
233C22⋊C8—24no—
234C22⋊C8—24no—
235C22⋊C8—42no—
236C22⋊C8—42no—
237#97—28no—
238#97—24no—
239#97—42no—
240C4×C8—28no—
241C4×C8—42no—
242C4×C8C2×D418yes—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95C22×C414yes—
249C24—24no—
250C24—22no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24—41no—
257C24—41no—
258C24C2×C22⋊C411yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—42no—
267C2×Q8—48no—
268C2×Q8—48no—
269C2×Q8AGL(1,ℤ/8)14yes—
270C2×D4—28no—
271C2×D4—24no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4—42no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—88no—
304C22×C4—88no—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C4×D412yes—
313C22×C4C4×D412yes—
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12112yes—
317C22×C4#12112yes—
318C22×C4#12311yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323M16—48no—
324M16—44no—
325M16—44no—
326C2×C8—28no—
327C2×C8—44no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—44no—
337C4⋊C4—48no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—48no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—48no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42C4×D412yes—
358C42C4×D412yes—
359C42#12012yes—
360C42#12112yes—
361C42#12112yes—
362C42#12312yes—
363C42C22×D412yes—
364C42C2×#4112yes—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—22no—
369C23—24no—
370C23—42no—
371C23—44no—
372C23—48no—
373C23—48no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—44no—
379C23—42no—
380C23—82no—
381C23—82no—
382C23—84no—
383C23—84no—
384C23—82no—
385C23—82no—
386C23#38611yescharacteristic
387C23#38611yescharacteristic
388C23C22×C22⋊C411yescharacteristic
389Q8—28no—
390Q8—48no—
391Q8—44no—
392D4—28no—
393D4—44no—
394D4—48no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—88no—
428C2×C4—82no—
429C2×C4—88no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—88no—
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4#41814yes—
438C2×C4C2×C4×D412yes—
439C2×C4C2×#12011yescharacteristic
440C2×C4C2×#12112yes—
441C2×C4C2×#12311yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)12yes—
443C2×C4#57512yes—
444C8—48no—
445C8—84no—
446C22—24no—
447C22—24no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—82no—
458C22—82no—
459C22—88no—
460C22—88no—
461C22—82no—
462C22—82no—
463C22#189112yes—
464C22#191212yes—
465C22C2×#38611yescharacteristic
466C22C2×#41811yescharacteristic
467C22#289611yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—24no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—84no—
480C4—82no—
481C4#305514yes—
482C2—21no—
483C2—21no—
484C2—22no—
485C2—22no—
486C2—84no—
487C2C2×#189111yescharacteristic
488C2#2097311yescharacteristic
489C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60248C2semidirect
3. #20727C2semidirect
5. #9490C2semidirect
6. #9486C2semidirect
7. #7959C2semidirect
13. #2983C22semidirect
14. #2982C22semidirect
31. #1801C22semidirect
32. #1801C22semidirect
39. #1286C22semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#25362
#207272
#174482
#94902
#94862
#79592