Finite Group Databasebeta

← #190578

Subgroups — #190578

Order 512.

1176 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2333C211yesmaximal, characteristic
2#21906C211yesmaximal, characteristic, complemented
3#21902C211yesmaximal, characteristic, complemented
4#8702C212yesmaximal, complemented
5#8559C212yesmaximal, complemented
6#8557C212yesmaximal, complemented
7#8307C212yesmaximal
8#8155C212yesmaximal, complemented
9#8155C212yesmaximal, complemented
10C22×#121C2211yescharacteristic
11#2333—24no—
12#2333—24no—
13C2×#390C411yescharacteristic
14C2×#390C411yescharacteristic
15C2×#388C2211yescharacteristic
16C2×#388C2211yescharacteristic
17C2×#387C411yescharacteristic
18C2×#387C411yescharacteristic
19C2×#386C411yescharacteristic
20C2×#386C411yescharacteristic
21C2×#386C411yescharacteristic
22C2×#386C411yescharacteristic
23C2×#381C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1924—24no—
28#1923—24no—
29#1797C2212yes—
30#1797C2212yes—
31#1779C2212yes—
32#1779C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#323C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C23×D4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C2×#121—22no—
46C2×#121—22no—
47C2×#121—24no—
48C2×#121—24no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊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×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2311yescharacteristic
63C22×C22⋊C4C2311yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8C2314yes—
75C2×C22⋊C8C2314yes—
76C2×#97—22no—
77C2×C4×C8—22no—
78#390—24no—
79#390—24no—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#387—24no—
85#387—24no—
86#386—24no—
87#386—24no—
88#386—24no—
89#386—24no—
90#381—22no—
91#381—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4D414yes—
97C4×C22⋊C4D414yes—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2311yescharacteristic
107C2×#95C2311yescharacteristic
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#323—28no—
113#323—28no—
114C25C22×C411yescharacteristic
115C25C22×C411yescharacteristic
116C25C22×C411yescharacteristic
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—81no—
127C22×D4—81no—
128C22×D4—81no—
129C22×D4—81no—
130C22×D4—81no—
131C22×D4—81no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—42no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4—81no—
141C23×C4—81no—
142C23×C4—81no—
143C23×C4—81no—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2411yescharacteristic
158C22×C8—24no—
159C22×C8—24no—
160#121—216no—
161#121—216no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—81no—
168C2×C4⋊C4—81no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—81no—
188C2×C22⋊C4—81no—
189C2×C22⋊C4—81no—
190C2×C22⋊C4—81no—
191C2×C22⋊C4—81no—
192C2×C22⋊C4—81no—
193C2×C22⋊C4C22⋊C414yes—
194C2×C22⋊C4C22⋊C414yes—
195C2×C22⋊C4#4114yes—
196C2×C22⋊C4#4114yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—81no—
200C2×C42—81no—
201C2×C42—81no—
202C2×C42—81no—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C4⋊C8—82no—
208C4⋊C8—82no—
209C22⋊C8—24no—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—82no—
214C22⋊C8—82no—
215C22⋊C8—82no—
216C22⋊C8—82no—
217#97—82no—
218C4×C8—82no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95C22⋊C414yes—
230#95C22⋊C414yes—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24C2×C22⋊C412yes—
256C24#11712yes—
257C24#11712yes—
258C24C23×C411yescharacteristic
259C2×D4—28no—
260C2×D4—28no—
261C2×D4—28no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—84no—
269C2×D4—84no—
270C2×D4—84no—
271C2×D4—84no—
272C2×D4—84no—
273C2×D4—84no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
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—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—44no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—44no—
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—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4#11712yes—
337C22×C4#11712yes—
338C22×C4#13114yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C2×C8—44no—
344C2×C8—44no—
345C2×C8—44no—
346C2×C8—44no—
347C2×C8—44no—
348C2×C8—44no—
349C4⋊C4—28no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—28no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—82no—
371C42—82no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—24no—
386C23—24no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23C22×C22⋊C411yescharacteristic
428C23C2×#11711yescharacteristic
429C23C2×#11711yescharacteristic
430C23C2×#13112yes—
431C23#56812yes—
432D4—88no—
433D4—88no—
434D4—88no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
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—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
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—42no—
468C2×C4—42no—
469C2×C4—41no—
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—84no—
482C2×C4—84no—
483C2×C4—88no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—88no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#40814yes—
492C2×C4#52512yes—
493C2×C4#52512yes—
494C8—84no—
495C8—84no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—84no—
518C22—84no—
519C22—84no—
520C22—84no—
521C22—84no—
522C22—84no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22#228011yescharacteristic
532C22C2×#40811yescharacteristic
533C22#287911yescharacteristic
534C22#288311yescharacteristic
535C22#288311yescharacteristic
536C22#288811yescharacteristic
537C22#288811yescharacteristic
538C22#292512yes—
539C22#292512yes—
540C4—24no—
541C4—22no—
542C4—22no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C2—22no—
554C2—22no—
555C2—82no—
556C2—82no—
557C2—82no—
558C2#1738611yescharacteristic
559C2#1738911yescharacteristic
560C2#1740511yescharacteristic
561C2#1740811yescharacteristic
562C2#2072111yescharacteristic
563C2#2072411yescharacteristic
564C2#2072911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21906C2semidirect
3. #21902C2semidirect
4. #8702C2semidirect
5. #8559C2semidirect
6. #8557C2semidirect
8. #8155C2semidirect
9. #8155C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23332
#219062
#219022
#87022
#85592
#85572
#83072
#81552