Finite Group Databasebeta

← #194033

Subgroups — #194033

Order 512.

1398 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3039C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1476C211yesmaximal, characteristic
6#8988C214yesmaximal, complemented
7#8986C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, 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
21C2×#419—21no—
22C22×C4×C8C2211yescharacteristic
23#2355—21no—
24#2355C2212yes—
25#2354—21no—
26#2354C2212yes—
27#2352—22no—
28#2352—22no—
29#2314—22no—
30#2311—21no—
31#2285—21no—
32#2285—21no—
33#2285C2211yescharacteristic
34C2×#384C2211yescharacteristic
35#2274—21no—
36#1990—22no—
37#1990—21no—
38#1990—21no—
39#1990C2214yes—
40#1988C2214yes—
41#1823—22no—
42C2×#340—22no—
43#1476—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—22no—
78C2×#117—21no—
79C2×#117D411yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D412yes—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#462—28no—
88#462—24no—
89#462C2×C418yescomplemented
90#461—28no—
91#461—24no—
92#461C2×C418yescomplemented
93#431—22no—
94#431—22no—
95#423—22no—
96#423C2×C414yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100#419—28no—
101C2×#103—22no—
102C2×#103—22no—
103C2×C22⋊C8—21no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2312yes—
109#391—22no—
110#390—22no—
111#388—24no—
112#388—24no—
113#387—24no—
114#387—24no—
115#386—24no—
116#384—21no—
117#384—21no—
118#384D412yes—
119#384D411yescharacteristic
120#384D411yescharacteristic
121#384C2312yes—
122#382—22no—
123#382—22no—
124#382—21no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380—21no—
129#380—22no—
130#380D412yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2312yes—
139C4×C22⋊C4—24no—
140#340—24no—
141#340—24no—
142#340—28no—
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—24no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—42no—
185C22×C8—42no—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8—42no—
193C4⋊Q8—42no—
194C4⋊Q8—42no—
195C4⋊Q8C22×C414yes—
196C4×Q8—22no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8C22⋊C414yes—
200C4×Q8C22×C412yes—
201C4×Q8C2×D412yes—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—22no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4C22⋊C414yes—
210C4×D4C22×C412yes—
211C4×D4C2×D412yes—
212#117—22no—
213#117—24no—
214#117—24no—
215#117—42no—
216#117—42no—
217#117—84no—
218#117—84no—
219#117C22⋊C414yes—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D414yes—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42C22×C412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4112yes—
268C2×C42#4112yes—
269C2×C42C2411yescharacteristic
270C4⋊C8—24no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8C22×C414yes—
275#103—28no—
276#103—28no—
277C22⋊C8—42no—
278C4×C8—22no—
279C4×C8—22no—
280C4×C8—42no—
281C4×C8—42no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—22no—
288C24—21no—
289C24#12011yescharacteristic
290#41—28no—
291#41—28no—
292#41—44no—
293#41—44no—
294C2×Q8—21no—
295C2×Q8—24no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—42no—
301C2×Q8—44no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—48no—
307C2×Q8—48no—
308C2×Q8—82no—
309C2×Q8—82no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×Q8C2×C22⋊C412yes—
313C2×D4—24no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4C2×C22⋊C412yes—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4C2×C22⋊C412yes—
350C22×C4C4×D412yes—
351C22×C4C4×D412yes—
352C22×C4#12011yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4#12312yes—
358C22×C4#13512yes—
359C22×C4#13512yes—
360C22×C4AGL(1,ℤ/8)12yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365Q16—416no—
366SD16—416no—
367M16—48no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—42no—
371C2×C8—48no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—82no—
402C4⋊C4C2×C22⋊C412yes—
403C4⋊C4C2×C22⋊C412yes—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—24no—
408C22⋊C4—44no—
409C22⋊C4—84no—
410C22⋊C4—84no—
411C42—22no—
412C42—22no—
413C42—24no—
414C42—24no—
415C42—42no—
416C42—42no—
417C42—84no—
418C42—84no—
419C42C2×C22⋊C412yes—
420C42C4×D412yes—
421C42C4×D412yes—
422C42#12112yes—
423C42#12112yes—
424C42#12312yes—
425C42C23×C411yescharacteristic
426C42C22×D411yescharacteristic
427C42C2×#4111yescharacteristic
428C42C2×#4111yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—22no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23#38612yes—
441C23#44912yes—
442C23#44912yes—
443C23C2×#12011yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—41no—
447Q8—41no—
448Q8—44no—
449Q8—44no—
450Q8—82no—
451Q8—82no—
452Q8—88no—
453D4—22no—
454D4—22no—
455D4—48no—
456D4—41no—
457D4—41no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—88no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—164no—
493C2×C4#38612yes—
494C2×C4#38612yes—
495C2×C4#38612yes—
496C2×C4#41614yes—
497C2×C4#41814yes—
498C2×C4#44912yes—
499C2×C4#44912yes—
500C2×C4#45512yes—
501C2×C4#45612yes—
502C2×C4#48214yes—
503C2×C4#48214yes—
504C2×C4C22×C22⋊C411yescharacteristic
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×#12111yescharacteristic
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×#12311yescharacteristic
510C2×C4C2×#13511yescharacteristic
511C2×C4C2×#13511yescharacteristic
512C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
513C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
514C2×C4#57712yes—
515C2×C4#57812yes—
516C8—44no—
517C8—44no—
518C8—82no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22#187514yes—
533C22C2×#38611yescharacteristic
534C22C2×#44911yescharacteristic
535C22C2×#44911yescharacteristic
536C22#300911yescharacteristic
537C22#301111yescharacteristic
538C22C2×#45511yescharacteristic
539C22C2×#45611yescharacteristic
540C4—22no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—84no—
547C4—84no—
548C4—82no—
549C4—82no—
550C4#187514yes—
551C4#188214yes—
552C4C2×#41611yescharacteristic
553C4C2×#41811yescharacteristic
554C4#294712yes—
555C4#304612yes—
556C4C2×#48211yescharacteristic
557C4C2×#48211yescharacteristic
558C2—24no—
559C2—82no—
560C2#1863411yescharacteristic
561C2#1863611yescharacteristic
562C2#1865411yescharacteristic
563C2#1865611yescharacteristic
564C2C2×#187511yescharacteristic
565C2#2093111yescharacteristic
566C2C2×#188211yescharacteristic
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×#1990C2semidirect
6. #8988C2semidirect
7. #8986C2semidirect
8. #8043C2semidirect
9. #7897C2semidirect
14. #3039C4semidirect
89. #462C2×C4semidirect
92. #461C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30392
#341882
#218332
C2×#19902
C2×#14762
#89882
#89862
#80432
#78972