Finite Group Databasebeta

← #194168

Subgroups — #194168

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2D4×#117C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic, complemented
4#21313C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6#8976C214yesmaximal, complemented
7#8973C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7891C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117—21no—
13C2×#476—24no—
14C2×#476—21no—
15C2×#476—21no—
16C2×#476C418yescomplemented
17C2×#476C2212yes—
18C2×#476C2212yes—
19C2×#431C2211yescharacteristic
20C22×#106—21no—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#102—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#1988—22no—
37#1987C2214yescomplemented
38#1981—22no—
39#1956C2214yescomplemented
40#1787—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×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#476—28no—
89#476—28no—
90#476—24no—
91#476—24no—
92#476C2×C418yes—
93#476C2×C418yescomplemented
94#431—22no—
95#431—22no—
96C2×#106—28no—
97C2×#106—22no—
98C2×#106—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—
108C2×#103—28no—
109C2×#102—28no—
110C2×#102—22no—
111C2×#102—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—24no—
142#340—24no—
143D4⋊C8—24no—
144D4⋊C8—24no—
145C25—21no—
146C22×Q8—24no—
147C22×Q8—22no—
148C22×Q8—41no—
149C22×Q8—42no—
150C22×Q8—84no—
151C22×D4—24no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—42no—
155C22×D4C22⋊C414yes—
156C22×D4C2×D411yescharacteristic
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—41no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—42no—
177C22×C8—24no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—44no—
182C4⋊Q8—24no—
183C4⋊Q8—24no—
184C4⋊Q8—24no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—42no—
189C4⋊Q8—42no—
190C4⋊Q8—42no—
191C4⋊Q8C22×C414yes—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4C22⋊C414yes—
205C4×D4C22⋊C414yes—
206C4×D4C22×C412yes—
207C4×D4C22×C412yes—
208C4×D4C2×D412yes—
209C4×D4C2×D412yes—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—48no—
214#117—48no—
215#117—44no—
216#117—42no—
217#117—42no—
218#117—44no—
219#117SD1614yes—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C22⋊C414yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D414yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—48no—
254C2×C42—21no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—48no—
261C2×C42C22×C412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D412yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42#4112yes—
267C2×C42#4112yes—
268C2×C42C2411yescharacteristic
269#106—28no—
270#106—28no—
271C4⋊C8—24no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—42no—
276C4⋊C8—42no—
277C4⋊C8C22×C414yes—
278#103—88no—
279#102—28no—
280#102—28no—
281C22⋊C8—42no—
282C4×C8—22no—
283C4×C8—22no—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290C24—21no—
291C24—21no—
292C24—24no—
293C24—44no—
294C24—42no—
295C24—42no—
296C24—82no—
297C24#12011yescharacteristic
298C2×Q8—24no—
299C2×Q8—48no—
300C2×Q8—42no—
301C2×Q8—44no—
302C2×Q8—84no—
303C2×Q8—88no—
304C2×Q8—88no—
305C2×D4—28no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—24no—
312C2×D4—44no—
313C2×D4—48no—
314C2×D4—44no—
315C2×D4C2×C22⋊C412yes—
316C2×D4C2×C22⋊C412yes—
317C2×D4#13714yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—42no—
333C22×C4—48no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4C2×C22⋊C412yes—
358C22×C4C4×D412yes—
359C22×C4C4×D412yes—
360C22×C4#12011yescharacteristic
361C22×C4#12011yescharacteristic
362C22×C4#12011yescharacteristic
363C22×C4#12112yes—
364C22×C4#12112yes—
365C22×C4#12312yes—
366C22×C4C2×SD1612yes—
367C22×C4#13512yes—
368C22×C4AGL(1,ℤ/8)12yes—
369C22×C4#13712yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373M16—84no—
374C2×C8—24no—
375C2×C8—48no—
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4C2×C22⋊C412yes—
401C4⋊C4C2×C22⋊C412yes—
402C4⋊C4C2×SD1614yes—
403C22⋊C4—24no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—84no—
415C42—24no—
416C42—48no—
417C42—42no—
418C42—42no—
419C42—44no—
420C42—42no—
421C42—82no—
422C42C2×C22⋊C412yes—
423C42C4×D412yes—
424C42C4×D412yes—
425C42#12112yes—
426C42#12112yes—
427C42#12312yes—
428C42C23×C411yescharacteristic
429C42C22×D411yescharacteristic
430C42C2×#4111yescharacteristic
431C42C2×#4111yescharacteristic
432C23—22no—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—48no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—48no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—84no—
447C23—88no—
448C23—88no—
449C23#38612yes—
450C23#45012yes—
451C23#45212yes—
452C23C2×#12011yescharacteristic
453Q8—48no—
454Q8—84no—
455Q8—88no—
456D4—24no—
457D4—22no—
458D4—22no—
459D4—22no—
460D4—48no—
461D4—42no—
462D4#41914yes—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—48no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—816no—
499C2×C4—88no—
500C2×C4—84no—
501C2×C4#38612yes—
502C2×C4#38612yes—
503C2×C4#38612yes—
504C2×C4C4×SD1614yes—
505C2×C4#44014yes—
506C2×C4#45012yes—
507C2×C4#45212yes—
508C2×C4#46114yes—
509C2×C4#46314yes—
510C2×C4C22×C22⋊C411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×C4×D411yescharacteristic
513C2×C4C2×#12111yescharacteristic
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12311yescharacteristic
516C2×C4C22×SD1611yescharacteristic
517C2×C4C2×#13511yescharacteristic
518C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
519C2×C4C2×#13712yes—
520C2×C4C2×#13711yescharacteristic
521C2×C4#57512yes—
522C8—44no—
523C8—82no—
524C8—82no—
525C22—24no—
526C22—24no—
527C22—22no—
528C22—22no—
529C22—24no—
530C22—44no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—816no—
538C22—82no—
539C22—82no—
540C22#187714yes—
541C22C2×#38611yescharacteristic
542C22C2×#45011yescharacteristic
543C22C2×#45211yescharacteristic
544C22#300712yes—
545C22#302112yes—
546C4—22no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—84no—
554C4—82no—
555C4—84no—
556C4#187714yes—
557C4C2×#41911yescharacteristic
558C4#289611yescharacteristic
559C4C2×C4×SD1611yescharacteristic
560C4C2×#44011yescharacteristic
561C4C2×#46111yescharacteristic
562C4C2×#46311yescharacteristic
563C4#309512yes—
564C2—24no—
565C2—44no—
566C2—82no—
567C2#1878712yes—
568C2#1879312yes—
569C2#2091111yescharacteristic
570C2C2×#187711yescharacteristic
571C2#2094511yescharacteristic
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. #21804C2semidirect
4. #21313C2semidirect
6. #8976C2semidirect
7. #8973C2semidirect
8. #8045C2semidirect
9. #7891C2semidirect
16. C2×#476C4semidirect
37. #1987C22semidirect
39. #1956C22semidirect
46. #1296C22semidirect
47. #1279C22semidirect
59. C2×C4⋊Q8C2×C4semidirect
93. #476C2×C4semidirect
105. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
D4×#1172
#218042
#213132
C2×#14862
#89762
#89732
#80452
#78912