Finite Group Databasebeta

← #329913

Subgroups — #329913

Order 512.

627 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33210C211yesmaximal, characteristic
2#33148C211yesmaximal, characteristic
3#33108C211yesmaximal, characteristic
4#21146C211yesmaximal, characteristic
5#10652C211yesmaximal, characteristic, complemented
6#10651C211yesmaximal, characteristic, complemented
7#10602C211yesmaximal, characteristic, complemented
8#10601C211yesmaximal, characteristic, complemented
9#10598C211yesmaximal, characteristic, complemented
10#10597C211yesmaximal, characteristic, complemented
11#10544C211yesmaximal, characteristic, complemented
12#10544C211yesmaximal, characteristic, complemented
13#9137C211yesmaximal, characteristic
14#9135C211yesmaximal, characteristic
15#8597C211yesmaximal, characteristic
16C2×#498C2211yescharacteristic
17C2×#483—21no—
18C2×#423—21no—
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2211yescharacteristic
27#1987C2211yescharacteristic
28#1981C2211yescharacteristic
29#1981C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1943—21no—
35#1943—21no—
36#1940—21no—
37#1940—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1939C2211yescharacteristic
42#1927—21no—
43#1927—21no—
44#1920—21no—
45#1920—21no—
46#1872—21no—
47#1853—21no—
48#1845C2211yescharacteristic
49#1823—21no—
50#1809—21no—
51#1809—21no—
52#1797C2211yescharacteristic
53#1797C2211yescharacteristic
54#1728—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C22×M16D411yescharacteristic
69C22×M16D411yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#498—41no—
79#498—41no—
80#483—44no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89#423—41no—
90#423—41no—
91#423—44no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C4×M16—41no—
110C4×M16—41no—
111C2×#97D411yescharacteristic
112C2×#97D411yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391—21no—
116#391—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C2×#95—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#335—41no—
153#334—41no—
154#333—41no—
155#333—41no—
156#332—41no—
157#332—41no—
158C22×Q8—21no—
159C22×Q8—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4#4111yescharacteristic
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—41no—
168C2×M16—41no—
169C2×M16—41no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16—42no—
173C2×M16SD1614yes—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C22⋊Q8—82no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#107—42no—
234#107—42no—
235#106—42no—
236#106—42no—
237#106—42no—
238#106—42no—
239#106—42no—
240#106—42no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258#103—42no—
259#103—42no—
260#103—42no—
261#103—42no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264#97—41no—
265#97—41no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95—41no—
278C24#12111yescharacteristic
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—81no—
288C2×Q8—81no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4C2×SD1612yes—
321C22×C4#13712yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325M16—44no—
326M16—44no—
327M16—84no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8C2×SD1612yes—
356C2×C8C2×SD1612yes—
357C2×C8#13712yes—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—81no—
383C4⋊C4—81no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4AGL(1,ℤ/8)12yes—
387C4⋊C4#13712yes—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—42no—
402C42H(5,𝔽2)11yescharacteristic
403C42H(5,𝔽2)11yescharacteristic
404C422−1+411yescharacteristic
405C422−1+411yescharacteristic
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23#46112yes—
413C23C2×#12111yescharacteristic
414Q8—81no—
415Q8—81no—
416Q8—81no—
417Q8—81no—
418Q8—161no—
419Q8—161no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
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—81no—
451C2×C4—81no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4#46112yes—
467C2×C4#53411yescharacteristic
468C2×C4#53611yescharacteristic
469C2×C4#53711yescharacteristic
470C2×C4#53711yescharacteristic
471C2×C4#53911yescharacteristic
472C2×C4#53911yescharacteristic
473C2×C4C22×SD1611yescharacteristic
474C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
475C2×C4C2×#13711yescharacteristic
476C2×C4C2×#13711yescharacteristic
477C2×C4C2×#13711yescharacteristic
478C2×C4#57511yescharacteristic
479C2×C4#57511yescharacteristic
480C2×C4#57511yescharacteristic
481C8—22no—
482C8—22no—
483C8—22no—
484C8—41no—
485C8—81no—
486C8—81no—
487C8—81no—
488C8—81no—
489C8—81no—
490C8—81no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22#261811yescharacteristic
499C22C2×#46111yescharacteristic
500C22#303711yescharacteristic
501C22#305511yescharacteristic
502C22#305611yescharacteristic
503C22#306811yescharacteristic
504C22#306911yescharacteristic
505C4—22no—
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—42no—
511C4—42no—
512C4—81no—
513C4—81no—
514C4—161no—
515C4—161no—
516C4—161no—
517C4—161no—
518C4#318711yescharacteristic
519C4#319511yescharacteristic
520C4#322911yescharacteristic
521C4#323611yescharacteristic
522C4#325111yescharacteristic
523C4#325211yescharacteristic
524C4#325411yescharacteristic
525C4#325611yescharacteristic
526C2—42no—
527C2#1916411yescharacteristic
528C2#1916511yescharacteristic
529C2#1931011yescharacteristic
530C2#1931111yescharacteristic
531C2#2975611yescharacteristic
532C2#2978411yescharacteristic
533C2#2979311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10652C2semidirect
6. #10651C2semidirect
7. #10602C2semidirect
8. #10601C2semidirect
9. #10598C2semidirect
10. #10597C2semidirect
11. #10544C2semidirect
12. #10544C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#331482
#331082
#211462
#106522
#106512
#106022
#106012
#105982
#105972
#105442
#91372
#91352
#85972