Finite Group Databasebeta

← #196285

Subgroups — #196285

Order 512.

1433 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic, complemented
2C4×#535C211yesmaximal, characteristic
3#17448C212yesmaximal, complemented
4#17366C211yesmaximal, characteristic, complemented
5#9106C214yesmaximal, complemented
6#9101C214yesmaximal, complemented
7#7856C212yesmaximal, complemented
8C2×#535C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#2990—24no—
12#2983C2212yescomplemented
13#2973—22no—
14#2972C2212yescomplemented
15#2970—24no—
16#2920C2212yes—
17#2920C2212yes—
18C2×#432—22no—
19#2328C2212yes—
20#2326C2212yes—
21#2324—22no—
22#2321—22no—
23#2317C411yescharacteristic
24#2311C411yescharacteristic
25#2309C411yescharacteristic
26C4×#127C412yes—
27C4×#124C412yes—
28C4×#121—22no—
29C4×#120—22no—
30C4×#117C2211yescharacteristic
31#1490—24no—
32#1486—24no—
33#1478C2218yes—
34#1465—22no—
35#1464—22no—
36#1461C2212yescomplemented
37#1460C2212yescomplemented
38#1274C2218yescomplemented
39C22×#41C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C24×C4—21no—
42#535—28no—
43C2×#127C2×C412yes—
44C2×#124C2×C412yes—
45C2×#121—22no—
46C2×#120—22no—
47C4×#41—22no—
48C4×#41—22no—
49C4×#41—22no—
50C4×#41—22no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2311yescharacteristic
53C2×C4×D4—24no—
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×C4×D4C2312yes—
65C2×C4×D4C2311yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C2×#117D412yes—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C42D412yes—
73C22×C42C2311yescharacteristic
74#446—28no—
75#435—216no—
76C8×D4—28no—
77#432—24no—
78#432—24no—
79#432—22no—
80#432—24no—
81#432—22no—
82#432—22no—
83#432—22no—
84#431—22no—
85#431D414yescomplemented
86#431C2314yes—
87#431C2314yes—
88#423—22no—
89#423—22no—
90C2×C4⋊C8—24no—
91#407—22no—
92#407—22no—
93C2×C22⋊C8—24no—
94C4×M16—22no—
95C4×M16D414yescomplemented
96C2×#97—24no—
97#391C2×C412yes—
98#390—24no—
99#390—22no—
100#390C2×C412yes—
101#390C2×C412yes—
102#390C2×C412yes—
103#389—22no—
104#388—24no—
105#388—24no—
106#386—22no—
107#386—24no—
108#386—24no—
109#385—24no—
110#385—22no—
111#384C2×C412yes—
112#382C2×C412yes—
113#380—22no—
114#379—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—24no—
120#376C2312yes—
121C43C2×C412yes—
122D4⋊C8—216no—
123#324—24no—
124#323—24no—
125C25—21no—
126C2×#41—24no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—42no—
130C22×Q8C22×C411yescharacteristic
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—42no—
137C22×D4C22⋊C412yes—
138C22×D4C22×C412yes—
139C22×D4C22×C411yescharacteristic
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—84no—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—24no—
153C2×M16—42no—
154C22×C8—28no—
155#127—24no—
156#127M1618yescomplemented
157#124—24no—
158#124—24no—
159#121—216no—
160#120—216no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4—24no—
175C4×D4—22no—
176C4×D4—24no—
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—88no—
184#117—22no—
185#117—22no—
186#117—24no—
187#117—24no—
188#117—24no—
189#117—42no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C2×D412yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C2×D412yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C22⋊C412yes—
221C2×C42C22⋊C412yes—
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42C2411yescharacteristic
233C4⋊C8—28no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C4⋊C8—42no—
238C22⋊C8—28no—
239C22⋊C8—24no—
240C22⋊C8—24no—
241C22⋊C8—24no—
242C22⋊C8—42no—
243#97—28no—
244#97—24no—
245#97—42no—
246C4×C8—24no—
247C4×C8C2×D418yes—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252C24—24no—
253C24—22no—
254C24—42no—
255C24—42no—
256C24—82no—
257C24—82no—
258C24C2×C22⋊C411yescharacteristic
259C24C2×C22⋊C412yes—
260#41—48no—
261#41—48no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—216no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—48no—
275C2×D4—48no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—84no—
281C2×D4—84no—
282C2×D4C2×M1614yes—
283C2×D4C2×M1614yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—48no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—88no—
305C22×C4—81no—
306C22×C4—82no—
307C22×C4—84no—
308C22×C4—82no—
309C22×C4—88no—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C4×D412yes—
317C22×C4C4×D412yes—
318C22×C4#12012yes—
319C22×C4#12112yes—
320C22×C4#12112yes—
321C22×C4#12312yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325M16—84no—
326M16—84no—
327C2×C8—28no—
328C2×C8—44no—
329C2×C8—88no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—84no—
340C22⋊C4—24no—
341C22⋊C4—216no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C22⋊C4—84no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—24no—
360C42—44no—
361C42—82no—
362C42C4×D412yes—
363C42C4×D412yes—
364C42#12011yescharacteristic
365C42#12011yescharacteristic
366C42#12112yes—
367C42#12112yes—
368C42#12311yescharacteristic
369C42#12311yescharacteristic
370C42C2×M1614yes—
371C42C22×D412yes—
372C42C2×#4112yes—
373C23—21no—
374C23—21no—
375C23—22no—
376C23—22no—
377C23—24no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—44no—
383C23—44no—
384C23—48no—
385C23—42no—
386C23—42no—
387C23—84no—
388C23—84no—
389C23—88no—
390C23—88no—
391C23#38612yes—
392C23#40714yes—
393C23C22×C22⋊C411yescharacteristic
394Q8—44no—
395D4—28no—
396D4—28no—
397D4—44no—
398D4—44no—
399D4—48no—
400D4—44no—
401D4—816no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—88no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4#38611yescharacteristic
440C2×C4#38611yescharacteristic
441C2×C4#38611yescharacteristic
442C2×C4#38612yes—
443C2×C4#38611yescharacteristic
444C2×C4C2×C4×D412yes—
445C2×C4C2×#12011yescharacteristic
446C2×C4C2×#12112yes—
447C2×C4C2×#12311yescharacteristic
448C2×C4C22×M1612yes—
449C2×C4#56812yes—
450C8—48no—
451C8—84no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—24no—
458C22—44no—
459C22—42no—
460C22—42no—
461C22—48no—
462C22—42no—
463C22—42no—
464C22—44no—
465C22—82no—
466C22—84no—
467C22—816no—
468C22—82no—
469C22—82no—
470C22C2×#38611yescharacteristic
471C22C2×#40711yescharacteristic
472C22#287911yescharacteristic
473C4—22no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—22no—
478C4—21no—
479C4—21no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—82no—
485C4—82no—
486C4#191012yes—
487C4D4×M1614yes—
488C2—21no—
489C2—21no—
490C2—42no—
491C2—44no—
492C2—84no—
493C2#2089811yescharacteristic
494C2C2×#191011yescharacteristic
495C2#2101211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#535C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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. #60282C2semidirect
3. #17448C2semidirect
4. #17366C2semidirect
5. #9106C2semidirect
6. #9101C2semidirect
7. #7856C2semidirect
12. #2983C22semidirect
14. #2972C22semidirect
36. #1461C22semidirect
37. #1460C22semidirect
38. #1274C22semidirect
85. #431D4semidirect
95. C4×M16D4semidirect
156. #127M16semidirect

Maximal subgroups

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

subgroupindex
#602822
C4×#5352
#174482
#173662
#91062
#91012
#78562