Finite Group Databasebeta

← #194283

Subgroups — #194283

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3208C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#21803C211yesmaximal, characteristic
4C2×#1982C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic, complemented
6#8979C214yesmaximal, complemented
7#8978C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×#127C2211yescharacteristic
12#3208—24no—
13#3208C418yescomplemented
14C2×#492—21no—
15C2×#486—21no—
16C2×#460C2214yes—
17C2×#450—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102—21no—
22C22×#102—21no—
23#2370—22no—
24#2369—21no—
25#2365—21no—
26#2355—22no—
27#2355C2212yescomplemented
28Q8×C4⋊C4C2212yescomplemented
29#2329—22no—
30#2317—21no—
31C4×C22⋊Q8—22no—
32#2285C2211yescharacteristic
33#2274—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1982—22no—
36#1982—22no—
37#1982C2214yes—
38#1957C2212yes—
39#1957C2212yes—
40#1786—22no—
41#1481—24no—
42C2×Q8⋊C8C2214yescomplemented
43#1296C2212yescomplemented
44#1296C2212yescomplemented
45#1279C2214yescomplemented
46C23×D4—21no—
47C23×D4—21no—
48C23×D4—21no—
49C22×SD16—24no—
50C23×C8—21no—
51#533—22no—
52#533—22no—
53#533C2×C414yes—
54#533D414yes—
55C2×C4⋊Q8D411yescharacteristic
56C2×C4⋊Q8D411yescharacteristic
57C2×#127—21no—
58C2×#127—22no—
59C2×#127—21no—
60C2×#127—21no—
61C2×#127—22no—
62C2×#127—22no—
63C2×#127C2×C414yes—
64C2×#127C2312yes—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8D412yes—
68C2×C4×Q8—22no—
69C2×C4×Q8—21no—
70C2×C4×Q8C2312yes—
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#492—28no—
78#486—28no—
79#460—28no—
80#460C2×C4116yescomplemented
81#450—216no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431C2×C414yes—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—44no—
98C2×#102—44no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2312yes—
103#391—24no—
104#391—22no—
105#389—22no—
106#389—22no—
107#389—24no—
108#389—22no—
109#387—24no—
110#387—24no—
111#387—24no—
112#385—24no—
113#384—22no—
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#382—22no—
117#382—21no—
118#380—21no—
119#380D412yes—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4C2312yes—
132C4×C22⋊C4—24no—
133C2×#95—21no—
134Q8⋊C8—28no—
135C25—41no—
136C22×Q8—21no—
137C22×Q8C2×D412yes—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—41no—
145C22×D4—41no—
146C22×D4—41no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—82no—
155C22×D4—82no—
156C22×D4—82no—
157C22×D4—82no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×SD16—216no—
164C2×SD16—84no—
165C2×SD16—84no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—42no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8C22⋊C414yes—
175C4⋊Q8C22⋊C414yes—
176#127—24no—
177#127—24no—
178#127—24no—
179#127—22no—
180#127—22no—
181#127—22no—
182#127—42no—
183#127—42no—
184#127—42no—
185#127C22×C414yes—
186#125—24no—
187#125—24no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C22⋊Q8C22⋊C418yes—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—84no—
197C4×Q8C22×C414yes—
198C4×Q8C2×D414yes—
199#117—24no—
200#117—24no—
201#117—42no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4SD1614yescomplemented
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—82no—
234C2×C42C22×C412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8C22×C414yes—
248#102—216no—
249#102—48no—
250#102—48no—
251#102—84no—
252#102—84no—
253C22⋊C8—24no—
254C22⋊C8—42no—
255C4×C8—24no—
256C4×C8—22no—
257C4×C8—22no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266C24—42no—
267C24—44no—
268C24—82no—
269C24—82no—
270C24—82no—
271C24—82no—
272C24—82no—
273C24—82no—
274C24#12011yescharacteristic
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—81no—
280C2×Q8—81no—
281C2×Q8C2×C22⋊C414yes—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—88no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—82no—
306C2×D4—88no—
307C2×D4—88no—
308C2×D4—88no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4—84no—
312C2×D4—84no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4C2×C22⋊C412yes—
330C22×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12112yes—
336C22×C4#12112yes—
337C22×C4#12312yes—
338C22×C4C2×SD1612yes—
339C22×C4#13512yes—
340C22×C4AGL(1,ℤ/8)12yes—
341C22×C4#13712yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345SD16—816no—
346C2×C8—24no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—84no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—28no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4C2×C22⋊C414yes—
379C4⋊C4C2×SD1614yes—
380C22⋊C4—28no—
381C22⋊C4—44no—
382C42—24no—
383C42—24no—
384C42—44no—
385C42—44no—
386C42—82no—
387C42—84no—
388C42—84no—
389C42C2×C22⋊C412yes—
390C42C4×D412yes—
391C42C4×D412yes—
392C42#12112yes—
393C42#12112yes—
394C42#12312yes—
395C42C23×C411yescharacteristic
396C42C22×D411yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—44no—
404C23—88no—
405C23—88no—
406C23—88no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—164no—
414C23—164no—
415C23—164no—
416C23—164no—
417C23#38612yes—
418C23#44812yes—
419C23#44912yes—
420C23C2×#12011yescharacteristic
421Q8—28no—
422Q8—44no—
423Q8—84no—
424D4—48no—
425D4—48no—
426D4—44no—
427D4—44no—
428D4—84no—
429D4—88no—
430D4—88no—
431D4—84no—
432D4—88no—
433D4—82no—
434D4—82no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4#38612yes—
466C2×C4#38612yes—
467C2×C4#38612yes—
468C2×C4C4×SD1614yes—
469C2×C4#44014yes—
470C2×C4#44812yes—
471C2×C4#44912yes—
472C2×C4#46014yes—
473C2×C4#46414yes—
474C2×C4C22×C22⋊C411yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C2×#12111yescharacteristic
478C2×C4C2×#12111yescharacteristic
479C2×C4C2×#12311yescharacteristic
480C2×C4C22×SD1611yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
483C2×C4C2×#13711yescharacteristic
484C2×C4#57612yes—
485C2×C4#57712yes—
486C8—44no—
487C8—82no—
488C8—82no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—84no—
496C22—84no—
497C22—84no—
498C22—1616no—
499C22—162no—
500C22—162no—
501C22—162no—
502C22—162no—
503C22#188014yes—
504C22C2×#38611yescharacteristic
505C22C2×#44811yescharacteristic
506C22C2×#44911yescharacteristic
507C22#300811yescharacteristic
508C22#301011yescharacteristic
509C22#302111yescharacteristic
510C22#302111yescharacteristic
511C4—22no—
512C4—21no—
513C4—21no—
514C4—22no—
515C4—44no—
516C4—44no—
517C4—44no—
518C4—82no—
519C4—82no—
520C4—82no—
521C4#188014yes—
522C4#289712yes—
523C4C2×C4×SD1611yescharacteristic
524C4C2×#44011yescharacteristic
525C4C2×#46011yescharacteristic
526C4C2×#46411yescharacteristic
527C4#309612yes—
528C2—24no—
529C2—164no—
530C2#1889411yescharacteristic
531C2#1889611yescharacteristic
532C2#1891411yescharacteristic
533C2#1891611yescharacteristic
534C2#2091411yescharacteristic
535C2C2×#188011yescharacteristic
536C2#2094511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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
2. #34213C2semidirect
5. C2×#1481C2semidirect
6. #8979C2semidirect
7. #8978C2semidirect
8. #8039C2semidirect
9. #7892C2semidirect
13. #3208C4semidirect
27. #2355C22semidirect
28. Q8×C4⋊C4C22semidirect
42. C2×Q8⋊C8C22semidirect
43. #1296C22semidirect
44. #1296C22semidirect
45. #1279C22semidirect
80. #460C2×C4semidirect
219. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32082
#342132
#218032
C2×#19822
C2×#14812
#89792
#89782
#80392
#78922