Finite Group Databasebeta

← #194175

Subgroups — #194175

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#461C211yesmaximal, characteristic
2D4×#117C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic, complemented
4#21313C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6#8988C214yesmaximal, complemented
7#8985C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7891C211yesmaximal, characteristic, complemented
10C23×SD16—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C2×#461—24no—
15C2×#461—21no—
16C2×#461—21no—
17C2×#461C418yescomplemented
18C2×#461C2212yes—
19C2×#461C2212yes—
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#103—21no—
24#2354C2212yes—
25#2352—22no—
26D4×C4⋊C4C2212yes—
27#2344—21no—
28#2342—22no—
29D4×C22⋊C4—21no—
30#2293—21no—
31#2290—22no—
32#2285C2211yescharacteristic
33#2280—21no—
34C42×D4—21no—
35C4×#117C2211yescharacteristic
36#1990C2214yescomplemented
37#1988C2214yescomplemented
38#1987—22no—
39#1987—22no—
40#1824—22no—
41#1736—21no—
42#1736—21no—
43#1486—24no—
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1296C2214yescomplemented
47#1279C2214yescomplemented
48C23×Q8—21no—
49C23×D4D411yescharacteristic
50C24×C4—21no—
51C22×SD16—28no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—48no—
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×C414yescomplemented
64C2×C4⋊Q8C2312yes—
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C414yes—
71C2×C4×D4D414yes—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—21no—
79C2×#117—22no—
80C2×#117—44no—
81C2×#117—44no—
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4D411yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C42—22no—
91C22×C42C2311yescharacteristic
92#461—28no—
93#461—28no—
94#461—24no—
95#461—24no—
96#461C2×C418yes—
97#461C2×C418yescomplemented
98#431—22no—
99#431—22no—
100#423—22no—
101#423—22no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8C2×C414yescomplemented
107C2×C4⋊C8C2312yes—
108C2×C4⋊C8C2312yes—
109C2×#103—28no—
110C2×#103—22no—
111C2×#103—22no—
112C2×C22⋊C8—21no—
113C2×C4×C8C2312yes—
114#390—22no—
115#389—22no—
116#388—24no—
117#388—24no—
118#387—24no—
119#386—24no—
120#385—24no—
121#384C2312yes—
122#382—21no—
123#381—22no—
124#380—21no—
125#380—22no—
126#380—22no—
127#380D412yes—
128#379—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4D412yes—
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139#376—21no—
140C43—21no—
141#340—28no—
142#340—24no—
143#340—24no—
144D4⋊C8—24no—
145D4⋊C8—24no—
146C25—21no—
147C22×Q8—24no—
148C22×Q8—22no—
149C22×Q8—44no—
150C22×Q8—44no—
151C22×Q8—41no—
152C22×Q8—42no—
153C22×D4—24no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—42no—
157C22×D4C22⋊C414yes—
158C22×D4C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—24no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—41no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×SD16—28no—
177C2×SD16—28no—
178C2×SD16—432no—
179C2×SD16—48no—
180C2×SD16—48no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—42no—
184C22×C8—24no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—44no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—24no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—42no—
196C4⋊Q8—42no—
197C4⋊Q8—42no—
198C4⋊Q8C22×C414yes—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4C22⋊C414yes—
212C4×D4C22⋊C414yes—
213C4×D4C22×C412yes—
214C4×D4C22×C412yes—
215C4×D4C2×D412yes—
216C4×D4C2×D412yes—
217#117—22no—
218#117—22no—
219#117—22no—
220#117—48no—
221#117—48no—
222#117—44no—
223#117—42no—
224#117—42no—
225#117—44no—
226#117SD1614yescomplemented
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C22⋊C414yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D414yes—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—48no—
261C2×C42—21no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—48no—
268C2×C42C22×C412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4112yes—
274C2×C42#4112yes—
275C2×C42C2411yescharacteristic
276C4⋊C8—24no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—42no—
281C4⋊C8—42no—
282C4⋊C8C22×C414yes—
283#103—28no—
284#103—28no—
285C22⋊C8—42no—
286C4×C8—22no—
287C4×C8—22no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294C24—21no—
295C24—21no—
296C24—24no—
297C24—44no—
298C24—42no—
299C24—42no—
300C24—82no—
301C24#12011yescharacteristic
302C2×Q8—24no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—42no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—84no—
310C2×Q8—88no—
311C2×D4—28no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—24no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—44no—
321C2×D4C2×C22⋊C412yes—
322C2×D4C2×C22⋊C412yes—
323C2×D4AGL(1,ℤ/8)14yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—42no—
339C22×C4—48no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4C2×C22⋊C412yes—
364C22×C4C4×D412yes—
365C22×C4C4×D412yes—
366C22×C4#12011yescharacteristic
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4#12312yes—
372C22×C4C2×SD1612yes—
373C22×C4#13512yes—
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4#13712yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379SD16—416no—
380SD16—416no—
381M16—84no—
382C2×C8—24no—
383C2×C8—48no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—44no—
399C4⋊C4—48no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4C2×C22⋊C412yes—
409C4⋊C4C2×C22⋊C412yes—
410C4⋊C4C2×SD1614yescomplemented
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—44no—
421C22⋊C4—42no—
422C22⋊C4—84no—
423C42—24no—
424C42—48no—
425C42—42no—
426C42—42no—
427C42—44no—
428C42—42no—
429C42—82no—
430C42C2×C22⋊C412yes—
431C42C4×D412yes—
432C42C4×D412yes—
433C42#12112yes—
434C42#12112yes—
435C42#12312yes—
436C42C23×C411yescharacteristic
437C42C22×D411yescharacteristic
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C23—22no—
441C23—22no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—48no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—48no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—84no—
455C23—88no—
456C23—88no—
457C23#38612yes—
458C23#45012yes—
459C23#45212yes—
460C23C2×#12011yescharacteristic
461Q8—44no—
462Q8—44no—
463Q8—88no—
464Q8—84no—
465D4—24no—
466D4—22no—
467D4—22no—
468D4—22no—
469D4—48no—
470D4—42no—
471D4#41814yes—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—816no—
508C2×C4—88no—
509C2×C4—84no—
510C2×C4#38612yes—
511C2×C4#38612yes—
512C2×C4#38612yes—
513C2×C4C4×SD1614yes—
514C2×C4#44014yes—
515C2×C4#45012yes—
516C2×C4#45212yes—
517C2×C4#46114yes—
518C2×C4#46314yes—
519C2×C4C22×C22⋊C411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C2×C4×D411yescharacteristic
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×#12111yescharacteristic
524C2×C4C2×#12311yescharacteristic
525C2×C4C22×SD1611yescharacteristic
526C2×C4C2×#13511yescharacteristic
527C2×C4C2×AGL(1,ℤ/8)12yes—
528C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
529C2×C4C2×#13711yescharacteristic
530C2×C4#57512yes—
531C8—44no—
532C8—82no—
533C8—82no—
534C22—24no—
535C22—24no—
536C22—22no—
537C22—22no—
538C22—24no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—816no—
547C22—82no—
548C22—82no—
549C22#187714yes—
550C22C2×#38611yescharacteristic
551C22C2×#45011yescharacteristic
552C22C2×#45211yescharacteristic
553C22#300612yes—
554C22#302212yes—
555C4—22no—
556C4—21no—
557C4—21no—
558C4—22no—
559C4—44no—
560C4—42no—
561C4—42no—
562C4—84no—
563C4—82no—
564C4—84no—
565C4#187714yes—
566C4C2×#41811yescharacteristic
567C4#289611yescharacteristic
568C4C2×C4×SD1611yescharacteristic
569C4C2×#44011yescharacteristic
570C4C2×#46111yescharacteristic
571C4C2×#46311yescharacteristic
572C4#309412yes—
573C2—24no—
574C2—44no—
575C2—82no—
576C2#1879912yes—
577C2#1880512yes—
578C2#2091211yescharacteristic
579C2C2×#187711yescharacteristic
580C2#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. #21834C2semidirect
4. #21313C2semidirect
6. #8988C2semidirect
7. #8985C2semidirect
8. #8045C2semidirect
9. #7891C2semidirect
17. C2×#461C4semidirect
36. #1990C22semidirect
37. #1988C22semidirect
46. #1296C22semidirect
47. #1279C22semidirect
63. C2×C4⋊Q8C2×C4semidirect
97. #461C2×C4semidirect
106. C2×C4⋊C8C2×C4semidirect
226. #117SD16semidirect
410. C4⋊C4C2×SD16semidirect

Maximal subgroups

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

subgroupindex
C22×#4612
D4×#1172
#218342
#213132
C2×#14862
#89882
#89852
#80452
#78912