Finite Group Databasebeta

← #189744

Subgroups — #189744

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2#21740C212yesmaximal
3#8704C212yesmaximal, complemented
4#8703C212yesmaximal, complemented
5#8701C212yesmaximal, complemented
6#8698C212yesmaximal, complemented
7#7915C212yesmaximal, complemented
8#7909C211yesmaximal, characteristic, complemented
9C2×#1280C211yesmaximal, characteristic, complemented
10C22×C4×D4C412yes—
11C22×C4×D4C2211yescharacteristic
12C2×#407C2212yes—
13C22×C22⋊C8C2212yes—
14#2349—22no—
15#2349—22no—
16#2349—22no—
17#2349—22no—
18C2×#389C412yes—
19C2×#385C412yes—
20C2×#385C412yes—
21C2×#382C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C22×#95C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1870—22no—
28#1870—22no—
29#1869—22no—
30#1869—22no—
31#1817C2214yes—
32#1779C2214yes—
33#1280C2212yes—
34#1280C2212yes—
35#1280C2214yes—
36#1280C2212yes—
37#1280C2212yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41C24×C4C2×C412yes—
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C23×C8—22no—
46C2×C4×D4—22no—
47C2×C4×D4—24no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—48no—
54C2×C4×D4D412yes—
55C2×C4×D4D412yes—
56C2×C4×D4D412yes—
57C2×C4×D4D412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C412yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C42C2×C412yes—
68C22×C42C2311yescharacteristic
69#407—24no—
70#407—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8C2314yes—
78C2×C22⋊C8C2314yes—
79#389—24no—
80#389—24no—
81#385—24no—
82#385—24no—
83#385—24no—
84#385—24no—
85#382—22no—
86#382—22no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2311yescharacteristic
101C2×#95C2311yescharacteristic
102C2×#95C2311yescharacteristic
103C2×#95C2311yescharacteristic
104#336—24no—
105#336—24no—
106C25C22×C411yescharacteristic
107C25C22×C411yescharacteristic
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—24no—
111C22×D4—24no—
112C22×D4C22⋊C411yescharacteristic
113C22×D4C22⋊C411yescharacteristic
114C22×D4C22⋊C411yescharacteristic
115C22×D4C22⋊C411yescharacteristic
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—82no—
130C23×C4—82no—
131C23×C4—82no—
132C23×C4—82no—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C22×C412yes—
137C23×C4C22×C412yes—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4#4112yes—
146C23×C4#4114yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C2×M16—44no—
150C2×M16—84no—
151C22×C8—24no—
152C22×C8—24no—
153C22×C8—44no—
154C22×C8—84no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—48no—
164C4×D4—48no—
165C4×D4—48no—
166C4×D4—48no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—44no—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—44no—
208C22⋊C8—44no—
209C22⋊C8—84no—
210C22⋊C8—84no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—42no—
229C24—42no—
230C24C2×C22⋊C411yescharacteristic
231C24C2×C22⋊C411yescharacteristic
232C24C2×C22⋊C411yescharacteristic
233C24C2×C22⋊C411yescharacteristic
234C24#11712yes—
235C24#11712yes—
236C24C23×C411yescharacteristic
237C2×D4—216no—
238C2×D4—216no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4#10014yes—
250C2×D4#10114yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
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—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—82no—
300C22×C4—84no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—84no—
308C22×C4—82no—
309C22×C4—88no—
310C22×C4—88no—
311C22×C4—88no—
312C22×C4—88no—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4#11712yes—
319C22×C4#11712yes—
320C22×C4#12314yes—
321C22×C4#12314yes—
322C22×C4#12314yes—
323C22×C4#12314yes—
324C22×C4#13114yes—
325C22×C4C22×D412yes—
326C22×C4C2×#4112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329M16—88no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—88no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—48no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—44no—
357C42—44no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—21no—
362C23—21no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—42no—
393C23—42no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23#38112yes—
398C23#38112yes—
399C23C2×#10012yes—
400C23C2×#10012yes—
401C23C2×#10112yes—
402C23C2×#10112yes—
403C23C22×C22⋊C411yescharacteristic
404C23C2×#11711yescharacteristic
405C23C2×#11711yescharacteristic
406C23C2×#13112yes—
407C23#56812yes—
408D4—28no—
409D4—28no—
410D4—28no—
411D4—28no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—88no—
457C2×C4—88no—
458C2×C4—88no—
459C2×C4—88no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4#38114yes—
467C2×C4C2×#10012yes—
468C2×C4C2×#10112yes—
469C2×C4C2×#12312yes—
470C2×C4C2×#12312yes—
471C8—84no—
472C8—84no—
473C22—22no—
474C22—22no—
475C22—24no—
476C22—22no—
477C22—22no—
478C22—24no—
479C22—22no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—21no—
484C22—21no—
485C22—24no—
486C22—24no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—44no—
493C22—44no—
494C22—48no—
495C22—44no—
496C22—44no—
497C22—48no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22#182114yes—
507C22C2×#38111yescharacteristic
508C22#288012yes—
509C22#288312yes—
510C22C22×#10011yescharacteristic
511C22C22×#10111yescharacteristic
512C22#292211yescharacteristic
513C22#292311yescharacteristic
514C22#292512yes—
515C4—22no—
516C4—22no—
517C4—24no—
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—82no—
525C4—82no—
526C2—21no—
527C2—21no—
528C2—22no—
529C2—24no—
530C2—42no—
531C2—42no—
532C2—42no—
533C2#1742112yes—
534C2#1742612yes—
535C2#2068411yescharacteristic
536C2#2068911yescharacteristic
537C2C2×#182111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ 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. #8704C2semidirect
4. #8703C2semidirect
5. #8701C2semidirect
6. #8698C2semidirect
7. #7915C2semidirect
8. #7909C2semidirect
9. C2×#1280C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
#217402
#87042
#87032
#87012
#86982
#79152
#79092
C2×#12802