Finite Group Databasebeta

← #201404

Subgroups — #201404

Order 512.

1489 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic
2#35010C211yesmaximal, characteristic
3#21759C212yesmaximal, complemented
4C2×#1799C211yesmaximal, characteristic
5#9469C214yesmaximal, complemented
6#9467C214yesmaximal, complemented
7C2×#1289C212yesmaximal, complemented
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10#2972—24no—
11#2972—22no—
12#2972C2214yes—
13C2×C8×D4—24no—
14C2×#431—21no—
15C22×#99—21no—
16C22×C22⋊C8—21no—
17C22×C22⋊C8C2212yes—
18#2855—21no—
19#2848—21no—
20#2627C412yescomplemented
21#2626C412yescomplemented
22#2579—22no—
23#2564C414yescomplemented
24#2487C2212yes—
25#2417—22no—
26#2409C2212yes—
27#2315—21no—
28#2308—21no—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31#1892—24no—
32#1872—24no—
33#1824C2214yescomplemented
34#1799—22no—
35#1799C2214yes—
36#1787C2214yescomplemented
37C2×#342—21no—
38#1289C2218yescomplemented
39C23×Q8C2311yescharacteristic
40C23×D4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C23×C8D414yes—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8C2×C412yes—
53C2×C22⋊Q8C2×C412yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4D412yes—
60C2×C4×D4D412yes—
61C2×C4×D4C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42D411yescharacteristic
73C8×D4—216no—
74C8×D4—216no—
75#431—24no—
76#431—24no—
77C2×C4⋊C8—22no—
78C2×#103—28no—
79C2×#99—24no—
80C2×#99—44no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8D414yes—
89C2×C22⋊C8C2314yes—
90C2×C4×C8—22no—
91#400C2×C414yescomplemented
92#399C2×C414yescomplemented
93#398—24no—
94#396—24no—
95#395C2×C414yescomplemented
96#393C2×C414yescomplemented
97#391—22no—
98#389—22no—
99#387—24no—
100#386—24no—
101#386—22no—
102#386—22no—
103#385—24no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#381—22no—
109#380—21no—
110#380—21no—
111#380—22no—
112#380—24no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380C2×C412yes—
117#380C2314yes—
118#380C2312yes—
119#379—22no—
120C2×#95—22no—
121C2×#95C2×C412yes—
122#342—24no—
123#340—28no—
124C25C22⋊C411yescharacteristic
125C25C2×D411yescharacteristic
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—82no—
130C22×Q8C22×C412yes—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—24no—
134C22×D4C22⋊C412yes—
135C22×D4SD1614yes—
136C22×D4Q1614yes—
137C22×D4C2×D412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—42no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163C22×C8—24no—
164C22×C8—28no—
165C22×C8—24no—
166C22×C8—84no—
167C22×C8—84no—
168C22×C8C2×D418yes—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C22⋊Q8—41no—
174C22⋊Q8—42no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—41no—
178C22⋊Q8—42no—
179C22⋊Q8—84no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—84no—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
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—24no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22×C412yes—
228C2×C22⋊C4C2×D412yes—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42C22⋊C412yes—
234C2×C42C2×D412yes—
235C4⋊C8—28no—
236#103—88no—
237#99—48no—
238C22⋊C8—28no—
239C22⋊C8—24no—
240C22⋊C8—24no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243C4×C8—28no—
244#95—22no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—42no—
251#95C22×C414yes—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—81no—
256C24—81no—
257C24—81no—
258C24—81no—
259C24#10314yescomplemented
260C24C2×C22⋊C412yes—
261C24C2×C22⋊C411yescharacteristic
262C24C2×C22⋊C411yescharacteristic
263C24#12112yes—
264C24#12112yes—
265C24C2×SD1612yes—
266C24C2×Q1612yes—
267C24#13514yes—
268C24C22×D411yescharacteristic
269C2×Q8—24no—
270C2×Q8—44no—
271C2×Q8—48no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—82no—
275C2×Q8—84no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—28no—
281C2×D4—28no—
282C2×D4—28no—
283C2×D4—28no—
284C2×D4#10318yescomplemented
285C2×D4C2×SD1614yes—
286C2×D4C2×Q1614yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—82no—
313C22×C4—81no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4—84no—
330C22×C4#10314yescomplemented
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C4×D412yes—
337C22×C4C4×D412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4#12011yescharacteristic
341C22×C4#12012yes—
342C22×C4#12011yescharacteristic
343C22×C4#12112yes—
344C22×C4#12112yes—
345C22×C4#12311yescharacteristic
346C22×C4#12311yescharacteristic
347C22×C4#12312yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4112yes—
352C2×C8—28no—
353C2×C8—48no—
354C2×C8—48no—
355C2×C8—84no—
356C2×C8—88no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—84no—
368C4⋊C4—82no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—84no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C23—24no—
390C23—22no—
391C23—22no—
392C23—24no—
393C23—41no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—81no—
408C23—81no—
409C23—82no—
410C23—82no—
411C23—84no—
412C23—84no—
413C23—84no—
414C23#38612yes—
415C23#38611yescharacteristic
416C23#38611yescharacteristic
417C23C2×#10314yes—
418C23C2×#10312yes—
419C23C2×#10312yes—
420C23#41612yes—
421C23#41612yes—
422C23#46514yes—
423C23#46514yes—
424C23#46714yes—
425C23#46714yes—
426C23C22×C22⋊C411yescharacteristic
427C23C2×#12112yes—
428C23C22×SD1611yescharacteristic
429C23C22×Q1611yescharacteristic
430C23C2×#13512yes—
431Q8—48no—
432Q8—82no—
433Q8—82no—
434D4—216no—
435D4—28no—
436D4—28no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—164no—
484C2×C4—168no—
485C2×C4#38611yescharacteristic
486C2×C4#38611yescharacteristic
487C2×C4#38612yes—
488C2×C4C2×#10314yes—
489C2×C4C2×C4×D412yes—
490C2×C4C2×#12011yescharacteristic
491C2×C4C2×#12311yescharacteristic
492C8—48no—
493C22—21no—
494C22—21no—
495C22—22no—
496C22—21no—
497C22—22no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—21no—
502C22—22no—
503C22—24no—
504C22—42no—
505C22—44no—
506C22—42no—
507C22—42no—
508C22—41no—
509C22—41no—
510C22—42no—
511C22—42no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—48no—
517C22—42no—
518C22—42no—
519C22—82no—
520C22—84no—
521C22#189212yes—
522C22#189212yes—
523C22#189214yes—
524C22C2×#38611yescharacteristic
525C22C22×#10311yescharacteristic
526C22C2×#41611yescharacteristic
527C22C2×#46512yes—
528C22C2×#46712yes—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—22no—
533C4—22no—
534C4—24no—
535C4—42no—
536C4—42no—
537C4—84no—
538C4—82no—
539C4—82no—
540C4—164no—
541C4#302612yes—
542C2—21no—
543C2—22no—
544C2—21no—
545C2—22no—
546C2—22no—
547C2—24no—
548C2—22no—
549C2—44no—
550C2#2096011yescharacteristic
551C2C2×#189211yescharacteristic
552C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21759C2semidirect
5. #9469C2semidirect
6. #9467C2semidirect
7. C2×#1289C2semidirect
20. #2627C4semidirect
21. #2626C4semidirect
23. #2564C4semidirect
33. #1824C22semidirect
36. #1787C22semidirect
38. #1289C22semidirect
91. #400C2×C4semidirect
92. #399C2×C4semidirect
95. #395C2×C4semidirect
96. #393C2×C4semidirect
259. C24#103semidirect
284. C2×D4#103semidirect
330. C22×C4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#350102
#217592
C2×#17992
#94692
#94672
C2×#12892