Finite Group Databasebeta

← #192159

Subgroups — #192159

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34185C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic, complemented
3#21938C211yesmaximal, characteristic, complemented
4C2×#1459C211yesmaximal, characteristic
5#13960C211yesmaximal, characteristic
6#8858C214yesmaximal, complemented
7#8855C214yesmaximal, complemented
8C2×#1294C211yesmaximal, characteristic, complemented
9C2×#1294C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2353—22no—
14#2353—22no—
15#2352—22no—
16#2352—22no—
17#2313C2212yes—
18#2312C2212yes—
19#2308—21no—
20C2×#386—21no—
21#2283—21no—
22#2280—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#2273—21no—
27C22×#95C2211yescharacteristic
28#1853—22no—
29#1853C2214yes—
30#1852—22no—
31#1852C2214yes—
32#1808—22no—
33#1807—22no—
34#1459C418yescomplemented
35C2×#324C2212yes—
36C2×#323C2212yes—
37#1441C2212yes—
38#1439C2212yes—
39#1434—21no—
40C22⋊#95—21no—
41C22⋊#95—21no—
42#1294C2214yes—
43#1294C2214yes—
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—21no—
49C23×C8—21no—
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C414yes—
57C2×C4×D4Q814yes—
58C2×#117—22no—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4C2311yescharacteristic
69C22×C42D411yescharacteristic
70C2×#107—24no—
71C2×#106—24no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8C2×C414yes—
77C2×C22⋊C8C2×C414yes—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80#388—24no—
81#388—24no—
82#387—24no—
83#387—24no—
84#386—24no—
85#386—24no—
86#386—44no—
87#383—22no—
88#382—22no—
89#381—22no—
90#381—22no—
91#380—22no—
92#380—21no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380D412yes—
101#380D412yes—
102#380C2312yes—
103#380C2312yes—
104#379—22no—
105C4×C4⋊C4—22no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—24no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111#376—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95C2312yes—
122#340—24no—
123#340—24no—
124#324C2×C418yescomplemented
125#323C2×C418yescomplemented
126C25C2×D411yescharacteristic
127C25C2×D411yescharacteristic
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C4⋊C414yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—44no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4—82no—
150C23×C4C4⋊C414yes—
151C23×C4C22×C412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×Q812yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—24no—
167C22×C8—24no—
168C22×C8—82no—
169C22×C8—82no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178#117—84no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2×Q812yes—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4C4⋊C414yes—
219C2×C22⋊C4C22×C412yes—
220C2×C22⋊C4C2×Q812yes—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42C4⋊C414yes—
227#107—84no—
228#106—84no—
229C22⋊C8—24no—
230C22⋊C8—24no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8C22×C414yes—
234C22⋊C8C22×C414yes—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—44no—
246#95—44no—
247#95—41no—
248#95—41no—
249#95—42no—
250#95—82no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24C2×C4⋊C412yes—
262C24C2×C4⋊C412yes—
263C24#12011yescharacteristic
264C24#12011yescharacteristic
265C24#12312yes—
266C24#13512yes—
267C24#13512yes—
268C24AGL(1,ℤ/8)12yes—
269C24#13712yes—
270C24C22×D411yescharacteristic
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—28no—
274C2×D4—28no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—44no—
280C2×D4—44no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—48no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4C2×C4⋊C412yes—
337C22×C4C2×C4⋊C412yes—
338C22×C4C2×C4⋊C412yes—
339C22×C4C2×C4⋊C412yes—
340C22×C4C4×D412yes—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4C4×D412yes—
344C22×C4#12011yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4C22⋊Q812yes—
347C22×C4C22⋊Q812yes—
348C22×C4C22⋊Q812yes—
349C22×C4C22⋊Q812yes—
350C22×C4#12312yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—44no—
360C2×C8—44no—
361C2×C8—84no—
362C2×C8—84no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C22⋊C4—88no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C42—24no—
391C42—24no—
392C42—84no—
393C42—84no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—42no—
409C23—44no—
410C23—44no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—44no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—81no—
422C23—81no—
423C23—82no—
424C23—82no—
425C23#38512yes—
426C23#38512yes—
427C23#42714yes—
428C23#42814yes—
429C23#44912yes—
430C23#44912yes—
431C23#45212yes—
432C23#45212yes—
433C23#48214yes—
434C23#48514yes—
435C23C22×C4⋊C411yescharacteristic
436C23C2×#12011yescharacteristic
437C23C2×#12311yescharacteristic
438C23C2×#13511yescharacteristic
439C23C2×#13511yescharacteristic
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×#13711yescharacteristic
442D4—48no—
443D4—48no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—48no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—84no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—816no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—164no—
501C2×C4—164no—
502C2×C4#38512yes—
503C2×C4#38512yes—
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×C22⋊Q811yescharacteristic
508C8—44no—
509C8—44no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—22no—
514C22—21no—
515C22—21no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—24no—
521C22—24no—
522C22—22no—
523C22—24no—
524C22—22no—
525C22—24no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22#185414yes—
540C22#185414yes—
541C22C2×#38511yescharacteristic
542C22C2×#42711yescharacteristic
543C22C2×#42811yescharacteristic
544C22C2×#44911yescharacteristic
545C22C2×#45211yescharacteristic
546C22#301812yes—
547C22#302612yes—
548C22C2×#48211yescharacteristic
549C22C2×#48511yescharacteristic
550C4—21no—
551C4—21no—
552C4—22no—
553C4—22no—
554C4—21no—
555C4—21no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—84no—
560C4—84no—
561C4—84no—
562C4—162no—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—24no—
568C2—24no—
569C2—82no—
570C2#1838412yes—
571C2#1838512yes—
572C2C2×#185411yescharacteristic
573C2#2085511yescharacteristic
574C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21940C2semidirect
3. #21938C2semidirect
6. #8858C2semidirect
7. #8855C2semidirect
8. C2×#1294C2semidirect
9. C2×#1294C2semidirect
34. #1459C4semidirect
124. #324C2×C4semidirect
125. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341852
#219402
#219382
C2×#14592
#139602
#88582
#88552
C2×#12942