Finite Group Databasebeta

← #204219

Subgroups — #204219

Order 512.

1201 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2663C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic, complemented
3#20524C211yesmaximal, characteristic, complemented
4C2×#1472C212yesmaximal, complemented
5C2×#1468C212yesmaximal
6#9654C212yesmaximal, complemented
7#9652C212yesmaximal, complemented
8#9651C212yesmaximal, complemented
9#9650C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12C2×#407C2212yes—
13C22×C22⋊C8C2212yes—
14#2663—24no—
15#2663—24no—
16C2×#400C411yescharacteristic
17C2×#399C411yescharacteristic
18C2×#397C411yescharacteristic
19C2×#397C411yescharacteristic
20C2×#395C411yescharacteristic
21C2×#394C411yescharacteristic
22C2×#394C411yescharacteristic
23C2×#393C411yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388C2211yescharacteristic
27C2×#388C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1850—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1753—24no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1472—28no—
36#1468—28no—
37C2×#324C2214yes—
38C2×#324C2214yes—
39C2×#324C2214yes—
40C2×#323C2214yes—
41C23×Q8C2×C411yescharacteristic
42C23×D4C2×C411yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C23×C8—22no—
46C2×#123—22no—
47C2×#123—22no—
48C2×#123—24no—
49C2×#123—24no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
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⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78#407—24no—
79#407D418yes—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8C2314yes—
85C2×C22⋊C8C2314yes—
86C2×#97—22no—
87C2×C4×C8—22no—
88#400—24no—
89#399—24no—
90#397—24no—
91#397—24no—
92#395—24no—
93#394—24no—
94#394—24no—
95#393—24no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104C4×C22⋊C4D414yes—
105C4×C22⋊C4D414yes—
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#324—28no—
121#324—28no—
122#324—28no—
123#323—28no—
124C25C22×C411yescharacteristic
125C25C22×C411yescharacteristic
126C22×Q8—22no—
127C22×Q8—24no—
128C22×Q8—81no—
129C22×Q8—81no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—81no—
133C22×D4—81no—
134C23×C4—81no—
135C23×C4—81no—
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×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D414yes—
151C23×C4#4114yes—
152C23×C4C2411yescharacteristic
153C2×M16—28no—
154C2×M16—82no—
155C2×M16—82no—
156C22×C8—24no—
157C22×C8—24no—
158C22×C8—44no—
159C22×C8—82no—
160C22×C8—82no—
161#123—44no—
162#123—44no—
163#123—44no—
164#123—44no—
165C22⋊Q8—44no—
166C22⋊Q8—44no—
167C22⋊Q8—44no—
168C22⋊Q8—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—81no—
192C2×C4⋊C4—81no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—81no—
214C2×C22⋊C4—81no—
215C2×C22⋊C4C22⋊C414yes—
216C2×C22⋊C4C22⋊C414yes—
217C2×C42—81no—
218C2×C42—81no—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C4⋊C8—82no—
224C4⋊C8—82no—
225C22⋊C8—28no—
226C22⋊C8—24no—
227C22⋊C8—24no—
228C22⋊C8—24no—
229C22⋊C8—44no—
230C22⋊C8C2×D418yes—
231#97—82no—
232C4×C8—82no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95C22⋊C414yes—
242#95C22⋊C414yes—
243C24—22no—
244C24—22no—
245C24—21no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24C2×C22⋊C412yes—
258C24C4×D414yes—
259C24C23×C411yescharacteristic
260C2×Q8—44no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—88no—
264C2×D4—44no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—88no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
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—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
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—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C4×D414yes—
323C22×C4#12114yes—
324C22×C4#12114yes—
325C22×C4#12414yes—
326C22×C4#12714yes—
327C22×C4#13114yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D412yes—
331C22×C4C2×#4112yes—
332M16—88no—
333C2×C8—28no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—88no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C22⋊C4—28no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—42no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—28no—
383C23—44no—
384C23—44no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—44no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23#39014yes—
404C23C22×C22⋊C411yescharacteristic
405C23C2×C4×D412yes—
406C23C2×#13112yes—
407C23#56812yes—
408Q8—88no—
409D4—88no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
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—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4#39014yes—
467C2×C4#40814yes—
468C2×C4C2×#12112yes—
469C2×C4C2×#12411yescharacteristic
470C2×C4C2×#12711yescharacteristic
471C8—48no—
472C8—84no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—24no—
480C22—24no—
481C22—24no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—88no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—164no—
495C22C2×#39011yescharacteristic
496C22C2×#40811yescharacteristic
497C22#287911yescharacteristic
498C22#288311yescharacteristic
499C22#288311yescharacteristic
500C22#288911yescharacteristic
501C22#288911yescharacteristic
502C22#293514yes—
503C4—22no—
504C4—21no—
505C4—21no—
506C4—24no—
507C4—24no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—81no—
518C2—24no—
519C2—44no—
520C2—161no—
521C2#1738711yescharacteristic
522C2#1738811yescharacteristic
523C2#1740611yescharacteristic
524C2#1740711yescharacteristic
525C2#2118911yescharacteristic
526C2#2121311yescharacteristic
527C2#2122011yescharacteristic
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. #21930C2semidirect
3. #20524C2semidirect
4. C2×#1472C2semidirect
6. #9654C2semidirect
7. #9652C2semidirect
8. #9651C2semidirect
9. #9650C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26632
#219302
#205242
C2×#14722
C2×#14682
#96542
#96522
#96512
#96502