Finite Group Databasebeta

← #194151

Subgroups — #194151

Order 512.

1386 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3039C211yesmaximal, characteristic
2#34121C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4C2×#1988C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic
6#8988C214yesmaximal, complemented
7#8986C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C22×#137—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13#3039—24no—
14#3039C418yescomplemented
15C2×#462—21no—
16C2×#462C2212yes—
17C2×#461—21no—
18C2×#461C2212yes—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#419—21no—
23#2357—21no—
24#2356—21no—
25#2355C2212yes—
26#2354C2212yes—
27#2353—22no—
28#2352—22no—
29#2298—22no—
30#2297—21no—
31#2285C2211yescharacteristic
32#2283—21no—
33C4×#117—21no—
34C4×#117—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1990C2214yes—
37#1988—21no—
38#1988—21no—
39#1988—22no—
40#1988C2214yes—
41#1823—22no—
42C2×#340—22no—
43#1477—24no—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C2×#137—28no—
51C2×#137—44no—
52C2×#137—44no—
53C22×Q16—22no—
54C22×SD16—22no—
55C22×M16—21no—
56C23×C8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2×C414yes—
64C2×C4⋊Q8C2312yes—
65C4×#41—22no—
66C4×#41—22no—
67C4×#41C2×C414yes—
68C4×#41D414yes—
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76C2×#117—22no—
77C2×#117—21no—
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D412yes—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#462—28no—
86#462—24no—
87#462C2×C418yescomplemented
88#461—28no—
89#461—24no—
90#461C2×C418yescomplemented
91#431—22no—
92#431—22no—
93#423—22no—
94#423C2×C414yes—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102#419—28no—
103C2×#103—22no—
104C2×#103—22no—
105C2×C22⋊C8—21no—
106C2×C4×C8C2312yes—
107#391—22no—
108#390—22no—
109#388—24no—
110#388—24no—
111#387—24no—
112#387—24no—
113#386—24no—
114#384C2312yes—
115#383—22no—
116#382—22no—
117#382—21no—
118#381—22no—
119#380—21no—
120#380—22no—
121#380D412yes—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D412yes—
131C4×C4⋊C4C2312yes—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136#376—22no—
137#376—22no—
138C2×#95—21no—
139C43—22no—
140#340—28no—
141#340—24no—
142#340—24no—
143Q8⋊C8—24no—
144D4⋊C8—24no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41C22⋊C414yes—
151C22×Q8—24no—
152C22×Q8—22no—
153C22×Q8—41no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8C2×D411yescharacteristic
160C22×D4—21no—
161C22×D4C2×D411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—41no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170#137—432no—
171C2×Q16—28no—
172C2×Q16—44no—
173C2×Q16—44no—
174C2×SD16—28no—
175C2×SD16—44no—
176C2×SD16—44no—
177C2×M16—24no—
178C2×M16—42no—
179C2×M16—42no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—44no—
184C22×C8—42no—
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—42no—
192C4⋊Q8—42no—
193C4⋊Q8—42no—
194C4⋊Q8C22×C414yes—
195C4×Q8—22no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8C22⋊C414yes—
199C4×Q8C22×C412yes—
200C4×Q8C2×D412yes—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4C22⋊C414yes—
209C4×D4C22×C412yes—
210C4×D4C2×D412yes—
211#117—22no—
212#117—24no—
213#117—24no—
214#117—44no—
215#117—42no—
216#117—42no—
217#117—44no—
218#117C22⋊C414yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C22×C412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42#4112yes—
264C2×C42#4112yes—
265C2×C42C2411yescharacteristic
266C4⋊C8—24no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—42no—
271C4⋊C8—42no—
272C4⋊C8C22×C414yes—
273#103—28no—
274#103—28no—
275C22⋊C8—42no—
276C4×C8—22no—
277C4×C8—22no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—22no—
286#95—22no—
287C24—21no—
288C24#12011yescharacteristic
289#41—28no—
290#41—28no—
291#41—44no—
292#41—44no—
293C2×Q8—21no—
294C2×Q8—24no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×Q8—44no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8—82no—
308C2×Q8—82no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8C2×C22⋊C412yes—
312C2×D4—24no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4C2×C22⋊C412yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4C2×C22⋊C412yes—
347C22×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4#12011yescharacteristic
350C22×C4#12011yescharacteristic
351C22×C4#12011yescharacteristic
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4#12312yes—
355C22×C4#13512yes—
356C22×C4#13512yes—
357C22×C4AGL(1,ℤ/8)12yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362Q16—416no—
363SD16—416no—
364M16—48no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—44no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—44no—
372C2×C8—84no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4C2×C22⋊C412yes—
394C4⋊C4C2×C22⋊C412yes—
395C4⋊C4#13714yescomplemented
396C22⋊C4—24no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—24no—
400C22⋊C4—48no—
401C22⋊C4—44no—
402C22⋊C4—84no—
403C42—22no—
404C42—22no—
405C42—24no—
406C42—24no—
407C42—48no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—44no—
413C42—84no—
414C42C2×C22⋊C412yes—
415C42C4×D412yes—
416C42C4×D412yes—
417C42#12112yes—
418C42#12112yes—
419C42#12312yes—
420C42C23×C411yescharacteristic
421C42C22×D411yescharacteristic
422C42C2×#4111yescharacteristic
423C42C2×#4111yescharacteristic
424C23—22no—
425C23—22no—
426C23—22no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23#38612yes—
432C23#44912yes—
433C23#44912yes—
434C23C2×#12011yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—41no—
438Q8—41no—
439Q8—44no—
440Q8—44no—
441Q8—82no—
442Q8—82no—
443Q8—88no—
444D4—22no—
445D4—22no—
446D4—48no—
447D4—41no—
448D4—41no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—84no—
479C2×C4—88no—
480C2×C4—88no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4#38612yes—
484C2×C4#38612yes—
485C2×C4#38612yes—
486C2×C4#41614yes—
487C2×C4#41814yes—
488C2×C4#44912yes—
489C2×C4#44912yes—
490C2×C4#48214yes—
491C2×C4#48214yes—
492C2×C4C22×C22⋊C411yescharacteristic
493C2×C4C2×C4×D411yescharacteristic
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C2×C4C2×#13511yescharacteristic
499C2×C4C2×#13511yescharacteristic
500C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
501C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
502C2×C4C2×#13712yes—
503C2×C4#57512yes—
504C8—44no—
505C8—82no—
506C8—82no—
507C22—24no—
508C22—24no—
509C22—22no—
510C22—22no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22#187514yes—
517C22C2×#38611yescharacteristic
518C22C2×#44911yescharacteristic
519C22C2×#44911yescharacteristic
520C22#300712yes—
521C22#302212yes—
522C4—22no—
523C4—21no—
524C4—22no—
525C4—21no—
526C4—44no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—84no—
531C4—82no—
532C4—84no—
533C4#187514yes—
534C4C2×#41611yescharacteristic
535C4C2×#41811yescharacteristic
536C4C4×#13712yes—
537C4#303911yescharacteristic
538C4#304011yescharacteristic
539C4C2×#48211yescharacteristic
540C4C2×#48211yescharacteristic
541C2—24no—
542C2—82no—
543C2#1880212yes—
544C2#1880412yes—
545C2C2×#187511yescharacteristic
546C2#2092711yescharacteristic
547C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
3. #21833C2semidirect
4. C2×#1988C2semidirect
6. #8988C2semidirect
7. #8986C2semidirect
8. #8045C2semidirect
9. #7901C2semidirect
14. #3039C4semidirect
87. #462C2×C4semidirect
90. #461C2×C4semidirect
395. C4⋊C4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#30392
#341212
#218332
C2×#19882
C2×#14772
#89882
#89862
#80452
#79012