Finite Group Databasebeta

← #215508

Subgroups — #215508

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2#21906C211yesmaximal, characteristic, complemented
3#21902C211yesmaximal, characteristic, complemented
4#9654C212yesmaximal, complemented
5#9652C212yesmaximal, complemented
6#9651C212yesmaximal, complemented
7#9650C212yesmaximal, complemented
8#8301C212yesmaximal
9#8299C212yesmaximal, complemented
10C22×#123C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13#2322—24no—
14#2322—24no—
15C2×#389C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#388C2211yescharacteristic
18C2×#388C2211yescharacteristic
19C2×#387C411yescharacteristic
20C2×#387C411yescharacteristic
21C2×#386C411yescharacteristic
22C2×#385C411yescharacteristic
23C2×#385C411yescharacteristic
24C2×#385C411yescharacteristic
25C2×#381C411yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1924—24no—
28#1923—24no—
29#1797C2212yes—
30#1797C2212yes—
31#1779C2212yes—
32#1779C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#324C2214yes—
37C2×#324C2214yes—
38C2×#323C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2311yescharacteristic
44C2×#123—24no—
45C2×#123—24no—
46C2×#123—24no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—44no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—44no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2314yes—
80C2×C22⋊C8C2314yes—
81C2×#97—22no—
82C2×C4×C8—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#387—24no—
92#387—24no—
93#386—24no—
94#385—24no—
95#385—24no—
96#385—24no—
97#381—24no—
98C4×C22⋊C4D414yes—
99C4×C22⋊C4D414yes—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110#336—22no—
111#336—22no—
112#336—22no—
113#336—22no—
114#324—44no—
115#324—44no—
116#324—44no—
117#323—44no—
118C25C22×C411yescharacteristic
119C25C22×C411yescharacteristic
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—42no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—81no—
136C23×C4—81no—
137C23×C4—81no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4—81no—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
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×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2411yescharacteristic
156C22×C8—24no—
157C22×C8—24no—
158#123—232no—
159C4×D4—48no—
160C4×D4—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4—81no—
184C2×C4⋊C4—81no—
185C2×C4⋊C4M1614yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—81no—
202C2×C22⋊C4—81no—
203C2×C22⋊C4C22⋊C414yes—
204C2×C22⋊C4C22⋊C414yes—
205C2×C22⋊C4M1614yes—
206C2×C42—22no—
207C2×C42—42no—
208C2×C42—81no—
209C2×C42—81no—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C4⋊C8—82no—
215C4⋊C8—82no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—82no—
221C22⋊C8—82no—
222C22⋊C8—82no—
223C22⋊C8—82no—
224#97—82no—
225C4×C8—82no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95C22⋊C414yes—
235#95C22⋊C414yes—
236C24—22no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—22no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24C2×C22⋊C412yes—
252C24C2×M1612yes—
253C24#13114yes—
254C24C23×C411yescharacteristic
255C2×D4—28no—
256C2×D4—48no—
257C2×D4—48no—
258C2×D4—44no—
259C2×D4—44no—
260C22×C4—28no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
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—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×M1612yes—
331C22×C4C2×M1612yes—
332C22×C4C2×M1612yes—
333C22×C4C2×M1612yes—
334C22×C4C2×M1612yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4H(5,𝔽2)14yes—
338C2×C8—44no—
339C2×C8—44no—
340C2×C8—44no—
341C2×C8—44no—
342C2×C8—44no—
343C2×C8—44no—
344C4⋊C4—28no—
345C4⋊C4—28no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C22⋊C4—28no—
357C22⋊C4—28no—
358C22⋊C4—28no—
359C22⋊C4—28no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—44no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—24no—
382C23—22no—
383C23—24no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—42no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23#40814yes—
408C23C22×C22⋊C411yescharacteristic
409C23#51812yes—
410C23C22×M1611yescharacteristic
411C23C22×M1611yescharacteristic
412C23C2×#13112yes—
413D4—416no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—416no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—88no—
471C2×C4—88no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4#40714yes—
477C2×C4#53512yes—
478C2×C4#53512yes—
479C8—84no—
480C8—84no—
481C22—22no—
482C22—22no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—48no—
493C22—48no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—84no—
502C22—84no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22#230911yescharacteristic
508C22C2×#40711yescharacteristic
509C22C2×#40811yescharacteristic
510C22#288311yescharacteristic
511C22#288311yescharacteristic
512C22#288911yescharacteristic
513C22#288911yescharacteristic
514C22#297412yes—
515C22#297412yes—
516C4—22no—
517C4—22no—
518C4—24no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C2—22no—
531C2—22no—
532C2—44no—
533C2—82no—
534C2#1736811yescharacteristic
535C2#1737011yescharacteristic
536C2#1738711yescharacteristic
537C2#1738711yescharacteristic
538C2#2178811yescharacteristic
539C2#2179511yescharacteristic
540C2#2180211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
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. #9654C2semidirect
5. #9652C2semidirect
6. #9651C2semidirect
7. #9650C2semidirect
9. #8299C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
#219062
#219022
#96542
#96522
#96512
#96502
#83012
#82992