Finite Group Databasebeta

← #197496

Subgroups — #197496

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34634C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#17575C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9200C214yesmaximal, complemented
6#9199C214yesmaximal
7#7981C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C22×#117—21no—
10C22×#117C2211yescharacteristic
11C2×#431C2212yes—
12#2629—22no—
13#2628C412yescomplemented
14#2628C412yescomplemented
15#2627—21no—
16#2627C2212yes—
17#2626—21no—
18#2626C2212yes—
19#2396—22no—
20#2396C414yescomplemented
21#2317—24no—
22#2316—21no—
23#2316C2212yes—
24#2315—21no—
25#2315C2212yes—
26#2296—21no—
27#2296—24no—
28#2285—21no—
29#2285C2211yescharacteristic
30#2285C2212yes—
31#2280—21no—
32#1520C2218yes—
33#1518—24no—
34#1503—24no—
35#1458—22no—
36#1458—22no—
37C2×#324C2212yes—
38C2×#323C2212yes—
39#1289C2218yes—
40C24×C4D411yescharacteristic
41#533—22no—
42#533—22no—
43#533—22no—
44#533—22no—
45C2×C4⋊Q8C2×C411yescharacteristic, complemented
46C2×C4⋊Q8C2×C411yescharacteristic, complemented
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2312yes—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C412yescomplemented
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—21no—
57C2×#117—21no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—44no—
63C2×#117—44no—
64C2×#117C2×C412yes—
65C2×#117C2×C412yes—
66C2×#117D412yes—
67C2×#117D412yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4C2311yescharacteristic
74C22×C42—22no—
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77#431—22no—
78#431—22no—
79#431—22no—
80#431—22no—
81C2×C4⋊C8—22no—
82C2×C22⋊C8C2314yes—
83C2×C4×C8—22no—
84#400—22no—
85#400C2×C414yescomplemented
86#399—22no—
87#399C2×C414yes—
88#398—24no—
89#398—24no—
90#396—24no—
91#396—24no—
92#395—22no—
93#395C2×C414yescomplemented
94#393—22no—
95#393C2×C414yes—
96#389—28no—
97#389—22no—
98#389—22no—
99#389—28no—
100#385—28no—
101#385—22no—
102#385—22no—
103#385—28no—
104#384—22no—
105#384—22no—
106#384—21no—
107#382—22no—
108#382C2×C412yescomplemented
109#382D414yes—
110#381—22no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—21no—
117#380C2312yes—
118#380C2314yes—
119#379—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2×C412yescomplemented
126C4×C4⋊C4D414yes—
127C4×C22⋊C4—24no—
128#328—216no—
129#324—24no—
130#323—24no—
131C25C2×D411yescharacteristic
132C22×Q8—21no—
133C22×Q8C22×C412yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4SD1614yes—
144C23×C4Q1614yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—24no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156#125—42no—
157#125—42no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—42no—
161C22⋊Q8—42no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—42no—
165C4×Q8—42no—
166#117—24no—
167#117—24no—
168#117—24no—
169#117—24no—
170#117—24no—
171#117—24no—
172#117—24no—
173#117—24no—
174#117—48no—
175#117—44no—
176#117—48no—
177#117—41no—
178#117—41no—
179#117—41no—
180#117—41no—
181#117—44no—
182#117—44no—
183#117—42no—
184#117—44no—
185#117—42no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22×C412yes—
227C2×C22⋊C4C2×D412yes—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—48no—
237C2×C42C22⋊C412yes—
238C2×C42C22⋊C412yes—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4112yes—
246C2×C42#4112yes—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C22⋊C8—24no—
250C22⋊C8—24no—
251C4×C8—42no—
252C4×C8—42no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95C22×C414yes—
262#95C2×D418yes—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—81no—
268C24—81no—
269C24C2×C22⋊C412yes—
270C24C2×SD1612yes—
271C24C2×Q1612yes—
272C24C22×D411yescharacteristic
273C2×Q8—24no—
274C2×Q8—42no—
275C2×Q8—82no—
276C2×Q8—82no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—28no—
303C22×C4—48no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4#10318yes—
311C22×C4#10318yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4#12012yes—
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12312yes—
323C22×C4#12311yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4C2×SD1612yes—
326C22×C4C2×SD1612yes—
327C22×C4C2×Q1612yes—
328C22×C4C2×Q1612yes—
329C22×C4AGL(1,ℤ/8)12yes—
330C22×C4#13712yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D412yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4112yes—
335C2×C8—44no—
336C2×C8—84no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—48no—
343C4⋊C4—44no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—48no—
352C4⋊C4—82no—
353C4⋊C4—84no—
354C4⋊C4—82no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—48no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—84no—
372C22⋊C4—84no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—48no—
378C42—42no—
379C42—42no—
380C42—44no—
381C42—48no—
382C42C4×D412yes—
383C42C4×D412yes—
384C42C4×D412yes—
385C42C4×D412yes—
386C42#12112yes—
387C42#12112yes—
388C42#12112yes—
389C42#12112yes—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—81no—
403C23—81no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23C2×#10314yes—
409C23C22×C22⋊C411yescharacteristic
410C23C2×#12011yescharacteristic
411C23C2×#12311yescharacteristic
412C23C22×SD1611yescharacteristic
413C23C22×Q1611yescharacteristic
414C23C2×AGL(1,ℤ/8)11yescharacteristic
415C23C2×#13711yescharacteristic
416Q8—84no—
417Q8—84no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—816no—
455C2×C4#38612yes—
456C2×C4#38612yes—
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C2×C4C2×#10314yes—
462C2×C4C2×#10312yes—
463C2×C4C2×#10312yes—
464C2×C4#41712yes—
465C2×C4#41712yes—
466C2×C4#45012yes—
467C2×C4#45012yes—
468C2×C4#45112yes—
469C2×C4#45112yes—
470C2×C4#48312yes—
471C2×C4#48312yes—
472C2×C4#48412yes—
473C2×C4#48412yes—
474C2×C4C2×C4×D412yes—
475C2×C4C2×#12112yes—
476C8—84no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—24no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—42no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22C2×#38611yescharacteristic
497C22C22×#10311yescharacteristic
498C22C2×#41711yescharacteristic
499C22C2×#45011yescharacteristic
500C22C2×#45111yescharacteristic
501C22C2×#48311yescharacteristic
502C22C2×#48411yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—24no—
508C4—42no—
509C4—42no—
510C4—84no—
511C4—88no—
512C4—82no—
513C4—82no—
514C4#187212yes—
515C4#187212yes—
516C4#187212yes—
517C4#187212yes—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—21no—
522C2—21no—
523C2—21no—
524C2—44no—
525C2C2×#187211yescharacteristic
526C2#2101211yescharacteristic
527C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17575C2semidirect
5. #9200C2semidirect
7. #7981C2semidirect
13. #2628C4semidirect
14. #2628C4semidirect
20. #2396C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
46. C2×C4⋊Q8C2×C4semidirect
51. C2×C4×Q8C2×C4semidirect
85. #400C2×C4semidirect
93. #395C2×C4semidirect
108. #382C2×C4semidirect
125. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#346342
#340522
#175752
C2×#14582
#92002
#91992
#79812