Finite Group Databasebeta

← #189692

Subgroups — #189692

Order 512.

1596 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×#2275C211yesmaximal, characteristic
2#21738C212yesmaximal
3#8701C212yesmaximal, complemented
4#8700C212yesmaximal, complemented
5#8699C212yesmaximal, complemented
6#8698C212yesmaximal, complemented
7#7907C212yesmaximal, complemented
8C2×#1280C212yesmaximal, complemented
9C22×C4×D4C411yescharacteristic
10C22×C4×D4C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×C22⋊C8C2214yes—
13#2275—28no—
14C2×C4×C4⋊C4C411yescharacteristic
15C2×C4×C4⋊C4C411yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C22⋊C4C411yescharacteristic
18C2×C4×C22⋊C4C411yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20C2×C4×C22⋊C4C411yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×#376C2211yescharacteristic
24C22×#95C2211yescharacteristic
25C22×#95C2211yescharacteristic
26#1869—28no—
27#1817C2214yes—
28#1779C2214yes—
29#1280C2214yes—
30#1280C2214yes—
31#1280C2214yes—
32#1280C2214yes—
33C23×D4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8—24no—
41C2×C4×D4—24no—
42C2×C4×D4—24no—
43C2×C4×D4—24no—
44C2×C4×D4—24no—
45C2×C4×D4—24no—
46C2×C4×D4—24no—
47C2×C4×D4—24no—
48C2×C4×D4C818yescomplemented
49C2×C4×D4D418yes—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2311yescharacteristic
68C2×C22⋊C8—28no—
69C2×C22⋊C8—28no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8C2318yes—
74C4×C4⋊C4—24no—
75C4×C4⋊C4—24no—
76C4×C4⋊C4—24no—
77C4×C22⋊C4—24no—
78C4×C22⋊C4—24no—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—24no—
83#376—24no—
84C2×#95—24no—
85C2×#95—24no—
86C2×#95C2×C411yescharacteristic
87C2×#95C2×C411yescharacteristic
88C2×#95C2×C411yescharacteristic
89C2×#95C2×C411yescharacteristic
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C411yescharacteristic
92C2×#95C2311yescharacteristic
93C2×#95C2311yescharacteristic
94C2×#95C2311yescharacteristic
95C2×#95C2311yescharacteristic
96#336—28no—
97C25C22×C411yescharacteristic
98C25C22×C411yescharacteristic
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4C22⋊C414yes—
104C22×D4C2×C814yes—
105C22×D4M1614yes—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—81no—
118C23×C4—81no—
119C23×C4—81no—
120C23×C4—81no—
121C23×C4—81no—
122C23×C4—81no—
123C23×C4—81no—
124C23×C4—81no—
125C23×C4—81no—
126C23×C4—81no—
127C23×C4—81no—
128C23×C4—81no—
129C23×C4C2×C814yes—
130C23×C4C2×C814yescomplemented
131C23×C4M1614yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D414yes—
145C23×C4C2×D414yes—
146C23×C4#4114yes—
147C23×C4#4114yes—
148C23×C4C2411yescharacteristic
149C22×C8—28no—
150C22×C8—48no—
151C22×C8—84no—
152C22×C8—84no—
153C4×D4—232no—
154C4×D4—232no—
155C4×D4—216no—
156C4×D4—216no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C22⋊C414yes—
165C2×C4⋊C4C2×C814yescomplemented
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4C22⋊C414yes—
180C2×C22⋊C4C22⋊C414yes—
181C2×C22⋊C4C2×C814yes—
182C2×C22⋊C4C2×C814yescomplemented
183C2×C22⋊C4M1614yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—81no—
198C2×C42—81no—
199C2×C42—81no—
200C2×C42—81no—
201C2×C42—81no—
202C2×C42—81no—
203C2×C42C2×C814yescomplemented
204C2×C42C2×D414yes—
205C22⋊C8—28no—
206C22⋊C8—28no—
207C22⋊C8—48no—
208C22⋊C8—84no—
209C22⋊C8—84no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—41no—
232C24C2×C22⋊C412yes—
233C24C2×C22⋊C412yes—
234C24#11712yes—
235C24#11712yes—
236C24C22×C812yes—
237C24C22×C812yes—
238C24C2×M1612yes—
239C24C2×M1612yes—
240C24C2×M1612yes—
241C24C2×M1612yes—
242C24C23×C411yescharacteristic
243C2×D4—28no—
244C2×D4—28no—
245C2×D4—28no—
246C2×D4—28no—
247C2×D4—28no—
248C2×D4—28no—
249C2×D4C22⋊C8116yes—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—84no—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4#11712yes—
332C22×C4#11712yes—
333C22×C4#12318yes—
334C22×C4#12318yes—
335C22×C4C22×C812yes—
336C22×C4C22×C812yes—
337C22×C4C22×C812yes—
338C22×C4C22×C812yes—
339C22×C4C22×C812yes—
340C22×C4C2×M1612yes—
341C22×C4C2×M1612yes—
342C22×C4C2×M1612yes—
343C22×C4C2×M1612yes—
344C22×C4C2×M1612yes—
345C22×C4C22×D412yes—
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C2×C8—48no—
349C2×C8—48no—
350C2×C8—48no—
351C2×C8—816no—
352C4⋊C4—28no—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C22⋊C4—28no—
356C22⋊C4—28no—
357C22⋊C4—28no—
358C22⋊C4—28no—
359C22⋊C4—28no—
360C22⋊C4—28no—
361C42—28no—
362C42—28no—
363C42—28no—
364C42—82no—
365C42—82no—
366C42—82no—
367C42—82no—
368C42—82no—
369C42—82no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—22no—
374C23—21no—
375C23—22no—
376C23—21no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23#38114yes—
413C23C2×C22⋊C814yes—
414C23C2×C22⋊C814yes—
415C23#40714yes—
416C23#43114yes—
417C23#43114yes—
418C23#43214yes—
419C23#43214yes—
420C23C22×C22⋊C411yescharacteristic
421C23C2×#11711yescharacteristic
422C23C2×#11711yescharacteristic
423C23C23×C811yescharacteristic
424C23C22×M1611yescharacteristic
425C23C22×M1611yescharacteristic
426C23C22×M1611yescharacteristic
427D4—232no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
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—48no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
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—88no—
495C2×C4—88no—
496C2×C4—88no—
497C2×C4—88no—
498C2×C4—88no—
499C2×C4#38114yes—
500C2×C4C2×C22⋊C814yes—
501C2×C4C2×#12314yes—
502C8—88no—
503C22—21no—
504C22—21no—
505C22—24no—
506C22—21no—
507C22—21no—
508C22—24no—
509C22—21no—
510C22—21no—
511C22—24no—
512C22—21no—
513C22—21no—
514C22—24no—
515C22—21no—
516C22—21no—
517C22—24no—
518C22—21no—
519C22—21no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22#181718yes—
537C22C2×#38111yescharacteristic
538C22C22×C22⋊C811yescharacteristic
539C22C2×#40711yescharacteristic
540C22#288311yescharacteristic
541C22#288311yescharacteristic
542C22#288811yescharacteristic
543C22#288911yescharacteristic
544C22C2×#43111yescharacteristic
545C22C2×#43111yescharacteristic
546C22C2×#43211yescharacteristic
547C22C2×#43211yescharacteristic
548C4—24no—
549C4—24no—
550C4—24no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—28no—
566C2—44no—
567C2#1735011yescharacteristic
568C2#1735111yescharacteristic
569C2#1736711yescharacteristic
570C2#1736811yescharacteristic
571C2C2×#181711yescharacteristic
572C2#2068911yescharacteristic
573C2#2069511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8701C2semidirect
4. #8700C2semidirect
5. #8699C2semidirect
6. #8698C2semidirect
7. #7907C2semidirect
8. C2×#1280C2semidirect
48. C2×C4×D4C8semidirect
130. C23×C4C2×C8semidirect
165. C2×C4⋊C4C2×C8semidirect
182. C2×C22⋊C4C2×C8semidirect
203. C2×C42C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#22752
#217382
#87012
#87002
#86992
#86982
#79072
C2×#12802