Finite Group Databasebeta

← #194174

Subgroups — #194174

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2D4×#117C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4#21259C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6#8984C214yesmaximal, complemented
7#8981C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7891C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117—21no—
13C2×#474—24no—
14C2×#474—21no—
15C2×#474—21no—
16C2×#474C418yescomplemented
17C2×#474C2212yes—
18C2×#474C2212yes—
19C2×#431C2211yescharacteristic
20C22×#107—21no—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#419—21no—
24C2×#419—21no—
25C22×#102—21no—
26#2354C2212yes—
27#2352—22no—
28D4×C4⋊C4C2212yes—
29#2344—21no—
30#2342—22no—
31D4×C22⋊C4—21no—
32#2293—21no—
33#2290—22no—
34#2285C2211yescharacteristic
35#2280—21no—
36C42×D4—21no—
37C4×#117C2211yescharacteristic
38#1990C2214yescomplemented
39#1988—22no—
40#1981—22no—
41#1981C2214yescomplemented
42#1823—22no—
43#1486—24no—
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1296C2214yescomplemented
47#1279C2214yescomplemented
48C23×Q8—21no—
49C23×D4D411yescharacteristic
50C24×C4—21no—
51C22×M16—21no—
52C23×C8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yescomplemented
60C2×C4⋊Q8C2312yes—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4C2×C414yes—
67C2×C4×D4D414yes—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—21no—
75C2×#117—22no—
76C2×#117—44no—
77C2×#117—44no—
78C2×#117D411yescharacteristic
79C2×#117D411yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4D411yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C42—22no—
87C22×C42C2311yescharacteristic
88#474—28no—
89#474—28no—
90#474—24no—
91#474—24no—
92#474C2×C418yes—
93#474C2×C418yescomplemented
94#431—22no—
95#431—22no—
96C2×#107—28no—
97C2×#107—22no—
98C2×#107—22no—
99#423—22no—
100#423—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2×C414yescomplemented
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2312yes—
108#419—44no—
109#419—44no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#102—28no—
113C2×#102—22no—
114C2×#102—22no—
115C2×C22⋊C8—21no—
116C2×C4×C8C2312yes—
117#390—22no—
118#389—22no—
119#388—24no—
120#388—24no—
121#387—24no—
122#386—24no—
123#385—24no—
124#384C2312yes—
125#382—21no—
126#381—22no—
127#380—21no—
128#380—22no—
129#380—22no—
130#380D412yes—
131#379—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4D412yes—
135C4×C4⋊C4D412yes—
136C4×C4⋊C4C2312yes—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#376—21no—
143C43—21no—
144#340—28no—
145D4⋊C8—24no—
146D4⋊C8—24no—
147C25—21no—
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—41no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×D4—24no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—42no—
160C22×D4C22⋊C414yes—
161C22×D4C2×D411yescharacteristic
162C22×D4C2×D411yescharacteristic
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—24no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—41no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—42no—
182C22×C8—24no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—44no—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—42no—
194C4⋊Q8—42no—
195C4⋊Q8—42no—
196C4⋊Q8C22×C414yes—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—44no—
208C4×D4—44no—
209C4×D4C22⋊C414yes—
210C4×D4C22⋊C414yes—
211C4×D4C22×C412yes—
212C4×D4C22×C412yes—
213C4×D4C2×D412yes—
214C4×D4C2×D412yes—
215#117—22no—
216#117—22no—
217#117—22no—
218#117—48no—
219#117—48no—
220#117—44no—
221#117—42no—
222#117—42no—
223#117—44no—
224#117D814yes—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C22⋊C414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D414yes—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—48no—
259C2×C42—21no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—48no—
266C2×C42C22×C412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4112yes—
272C2×C42#4112yes—
273C2×C42C2411yescharacteristic
274#107—28no—
275#107—28no—
276C4⋊C8—24no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—42no—
281C4⋊C8—42no—
282C4⋊C8C22×C414yes—
283#103—48no—
284#103—48no—
285#102—28no—
286#102—28no—
287C22⋊C8—42no—
288C4×C8—22no—
289C4×C8—22no—
290#95—24no—
291#95—24no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296C24—21no—
297C24—21no—
298C24—24no—
299C24—44no—
300C24—42no—
301C24—42no—
302C24—82no—
303C24#12011yescharacteristic
304C2×Q8—24no—
305C2×Q8—42no—
306C2×Q8—44no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—48no—
312C2×Q8—48no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×Q8—84no—
316C2×Q8—84no—
317C2×D4—28no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—44no—
325C2×D4—48no—
326C2×D4—44no—
327C2×D4C2×C22⋊C412yes—
328C2×D4C2×C22⋊C412yes—
329C2×D4#13714yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—42no—
345C22×C4—48no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—84no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4C2×C22⋊C412yes—
371C22×C4C4×D412yes—
372C22×C4C4×D412yes—
373C22×C4#12011yescharacteristic
374C22×C4#12011yescharacteristic
375C22×C4#12011yescharacteristic
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12312yes—
379C22×C4C2×D812yes—
380C22×C4#13512yes—
381C22×C4AGL(1,ℤ/8)12yes—
382C22×C4AGL(1,ℤ/8)12yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386M16—84no—
387C2×C8—24no—
388C2×C8—48no—
389C2×C8—44no—
390C2×C8—44no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—44no—
404C4⋊C4—48no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4C2×C22⋊C412yes—
414C4⋊C4C2×C22⋊C412yes—
415C4⋊C4C2×D814yes—
416C22⋊C4—24no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—84no—
428C42—24no—
429C42—48no—
430C42—42no—
431C42—42no—
432C42—44no—
433C42—42no—
434C42—82no—
435C42C2×C22⋊C412yes—
436C42C4×D412yes—
437C42C4×D412yes—
438C42#12112yes—
439C42#12112yes—
440C42#12312yes—
441C42C23×C411yescharacteristic
442C42C22×D411yescharacteristic
443C42C2×#4111yescharacteristic
444C42C2×#4111yescharacteristic
445C23—22no—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—48no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—48no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—84no—
460C23—88no—
461C23—88no—
462C23#38612yes—
463C23#44712yes—
464C23#44912yes—
465C23C2×#12011yescharacteristic
466Q8—44no—
467Q8—44no—
468Q8—88no—
469Q8—82no—
470Q8—82no—
471D4—24no—
472D4—22no—
473D4—22no—
474D4—22no—
475D4—48no—
476D4—42no—
477D4#41914yescomplemented
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—48no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—816no—
514C2×C4—88no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#38612yes—
518C2×C4#38612yes—
519C2×C4#38612yes—
520C2×C4C4×D814yes—
521C2×C4#44214yes—
522C2×C4#44712yes—
523C2×C4#44912yes—
524C2×C4#45914yes—
525C2×C4#46314yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12311yescharacteristic
532C2×C4C22×D811yescharacteristic
533C2×C4C2×#13511yescharacteristic
534C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
535C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
536C2×C4C2×#13712yes—
537C2×C4#57512yes—
538C8—44no—
539C8—82no—
540C8—82no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—24no—
546C22—44no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—816no—
554C22—82no—
555C22—82no—
556C22#187914yes—
557C22C2×#38611yescharacteristic
558C22C2×#44711yescharacteristic
559C22C2×#44911yescharacteristic
560C22#300712yes—
561C22#302212yes—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—22no—
566C4—44no—
567C4—42no—
568C4—42no—
569C4—84no—
570C4—82no—
571C4—84no—
572C4#187914yes—
573C4C2×#41911yescharacteristic
574C4#289611yescharacteristic
575C4C2×C4×D811yescharacteristic
576C4C2×#44211yescharacteristic
577C4C2×#45911yescharacteristic
578C4C2×#46311yescharacteristic
579C4#309512yes—
580C2—24no—
581C2—44no—
582C2—82no—
583C2#1879812yes—
584C2#1880412yes—
585C2#2091111yescharacteristic
586C2C2×#187911yescharacteristic
587C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #21259C2semidirect
6. #8984C2semidirect
7. #8981C2semidirect
8. #8045C2semidirect
9. #7891C2semidirect
16. C2×#474C4semidirect
38. #1990C22semidirect
41. #1981C22semidirect
46. #1296C22semidirect
47. #1279C22semidirect
59. C2×C4⋊Q8C2×C4semidirect
93. #474C2×C4semidirect
105. C2×C4⋊C8C2×C4semidirect
477. D4#419semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
D4×#1172
#218332
#212592
C2×#14862
#89842
#89812
#80452
#78912