Finite Group Databasebeta

← #189817

Subgroups — #189817

Order 512.

1332 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2283C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4#8714C214yesmaximal, complemented
5#8713C212yesmaximal
6#8712C212yesmaximal
7#8021C214yesmaximal, complemented
8C23×C42C2211yescharacteristic
9C2×#432C2212yes—
10#2283—21no—
11#2283—21no—
12#2283—21no—
13#2283—21no—
14#2283C412yescomplemented
15#2283C412yescomplemented
16#2283C2212yes—
17#2283C2212yes—
18C2×#383—21no—
19C2×#382C411yescharacteristic, complemented
20C2×#382C411yescharacteristic, complemented
21C2×#381—21no—
22C2×#380—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4C411yescharacteristic, complemented
26C2×C4×C4⋊C4C411yescharacteristic, complemented
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4—21no—
30C2×#376—21no—
31#1817C2212yes—
32#1797—21no—
33#1797—21no—
34#1779C2212yes—
35#1749—21no—
36#1749—21no—
37C4⋊#95—21no—
38#1434C2212yes—
39C22⋊#95C2212yes—
40C4×#95—21no—
41#1294C2218yes—
42#1294C2218yes—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C42—28no—
55C22×C42—21no—
56C22×C42—21no—
57C22×C42—22no—
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#432—28no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8C2314yes—
75C2×#97—22no—
76#383—22no—
77#383—22no—
78#383—22no—
79#383—22no—
80#382—21no—
81#382—21no—
82#382—21no—
83#382—21no—
84#382C2×C412yes—
85#382C2×C412yes—
86#382C2×C412yes—
87#382C2×C412yes—
88#381—22no—
89#381—22no—
90#381—22no—
91#381—22no—
92#380—28no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380C2×C412yes—
98#380C2×C412yes—
99#380C2312yes—
100#380C2312yes—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—28no—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—44no—
115#376—22no—
116#376—22no—
117#376—22no—
118#376—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95C2314yes—
131#336—22no—
132#336—22no—
133C25C2×D411yescharacteristic
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—42no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—44no—
209C2×C42—44no—
210C2×C42—44no—
211C2×C42—44no—
212C2×C42—44no—
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42#4118yes—
230C4⋊C8—44no—
231C22⋊C8—28no—
232C22⋊C8—44no—
233C22⋊C8—44no—
234#97—44no—
235#95—28no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—28no—
253#95—44no—
254#95—44no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—41no—
260C24—41no—
261C24C2×C22⋊C411yescharacteristic
262C24C2×C22⋊C411yescharacteristic
263C24AGL(1,ℤ/8)12yes—
264C24#13712yes—
265C24C22×D411yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—28no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—416no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—416no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4#10414yes—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4#12314yes—
357C22×C4#12314yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4114yes—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—44no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—416no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C22⋊C4—416no—
374C22⋊C4—44no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—44no—
392C42—44no—
393C42—44no—
394C42—44no—
395C42—88no—
396C42#11714yes—
397C42#11714yes—
398C42#12314yes—
399C42#12314yes—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—24no—
409C23—24no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—24no—
416C23—24no—
417C23—21no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23#41012yes—
425C23#41712yes—
426C23#41712yes—
427C23C2×#10412yes—
428C23C22×C22⋊C411yescharacteristic
429C23C2×#12312yes—
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23C2×#13711yescharacteristic
432C23#57512yes—
433C2×C4—28no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—48no—
482C2×C4—42no—
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—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4#38112yes—
503C2×C4#38112yes—
504C2×C4#38112yes—
505C2×C4#38112yes—
506C2×C4#41012yes—
507C2×C4C2×#10412yes—
508C2×C4C2×#10412yes—
509C2×C4#42112yes—
510C2×C4C2×#11712yes—
511C2×C4C2×#12312yes—
512C8—84no—
513C22—21no—
514C22—21no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—21no—
520C22—22no—
521C22—21no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—48no—
531C22—48no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22#150914yes—
537C22C2×#38111yescharacteristic
538C22C2×#41011yescharacteristic
539C22C2×#41011yescharacteristic
540C22C2×#41711yescharacteristic
541C22#289611yescharacteristic
542C22C22×#10411yescharacteristic
543C22C2×#42111yescharacteristic
544C22#309412yes—
545C22#309512yes—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—22no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—88no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4#182614yes—
563C2—22no—
564C2—22no—
565C2—24no—
566C2—42no—
567C2C2×#150911yescharacteristic
568C2#1753711yescharacteristic
569C2#1758011yescharacteristic
570C2#1758111yescharacteristic
571C2#2068811yescharacteristic
572C2#2070211yescharacteristic
573C2C2×#182611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8714C2semidirect
7. #8021C2semidirect
14. #2283C4semidirect
15. #2283C4semidirect
19. C2×#382C4semidirect
20. C2×#382C4semidirect
25. C2×C4×C4⋊C4C4semidirect
26. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22832
#207152
C4×#3802
#87142
#87132
#87122
#80212