Finite Group Databasebeta

← #210595

Subgroups — #210595

Order 512.

688 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23681C211yesmaximal, characteristic
2C8×#124C211yesmaximal, characteristic
3#20516C211yesmaximal, characteristic, complemented
4C2×#1587C211yesmaximal, characteristic, complemented
5C2×#1585C211yesmaximal, characteristic
6#9676C211yesmaximal, characteristic
7#9606C211yesmaximal, characteristic, complemented
8#9544C211yesmaximal, characteristic, complemented
9#9542C211yesmaximal, characteristic
10#8948C211yesmaximal, characteristic
11#8946C211yesmaximal, characteristic, complemented
12#8570C211yesmaximal, characteristic, complemented
13#8261C211yesmaximal, characteristic, complemented
14#8205C211yesmaximal, characteristic, complemented
15#8204C211yesmaximal, characteristic
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C4×#124C2211yescharacteristic
23#2295C2211yescharacteristic
24#2085C411yescharacteristic
25#2074C411yescharacteristic
26#2067C411yescharacteristic
27#2066C411yescharacteristic
28#2056C411yescharacteristic
29#2055C411yescharacteristic
30#2042C411yescharacteristic
31#2015C411yescharacteristic
32#1984C2211yescharacteristic
33#1954C2211yescharacteristic
34#1923—21no—
35#1923—21no—
36#1831C2211yescharacteristic
37#1771—21no—
38#1770C2211yescharacteristic
39C8×C4⋊C4—21no—
40C8×C4⋊C4C2211yescharacteristic
41#1768—21no—
42#1767C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C8×C22⋊C4—21no—
48C8×C22⋊C4—21no—
49#1726—21no—
50C42×C8C2211yescharacteristic
51#1587—24no—
52#1585—24no—
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C23×C8—21no—
72C2×#124C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76#446—22no—
77#446D414yes—
78C8×Q8—22no—
79C8×Q8—22no—
80C8×Q8—22no—
81C8×Q8—22no—
82C4×Q16—22no—
83C4×Q16—22no—
84C4×SD16—22no—
85C4×SD16—22no—
86C8×D4—22no—
87C8×D4—22no—
88C2×#107C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8D411yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#391C2×C411yescharacteristic
128#391C2×C411yescharacteristic
129#391C2×C411yescharacteristic
130#390C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#388—21no—
134#388—21no—
135#387—21no—
136#387—21no—
137#382C2×C411yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#376—21no—
143C43C2311yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336C2×C411yescharacteristic
157#336C2×C411yescharacteristic
158#335—22no—
159#334—22no—
160Q8⋊C8—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166#322—22no—
167#322—22no—
168C22×Q8C22×C411yescharacteristic
169C22×Q8C22×C411yescharacteristic
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C2×Q16—22no—
173C2×Q16—22no—
174C2×Q16—41no—
175C2×SD16—22no—
176C2×SD16—22no—
177C2×SD16—41no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195#124—22no—
196#124—22no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8C2×D412yes—
204C4×Q8#4112yes—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#107—22no—
234#106—22no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C4⋊C8—24no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C4⋊C8—42no—
247C4⋊C8C2×D412yes—
248C4⋊C8#4112yes—
249#103—22no—
250#103—22no—
251#103—22no—
252#103—22no—
253#103—41no—
254#102—22no—
255#102—22no—
256#102—41no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#97—24no—
261#97—42no—
262#97—42no—
263C4×C8—24no—
264C4×C8—22no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—41no—
276C4×C8—41no—
277C4×C8—41no—
278C4×C8—42no—
279C4×C8—42no—
280C4×C8C2×D412yes—
281C4×C8#4112yes—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8C4×D412yes—
305C2×Q8#13112yes—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4#13112yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332Q16—44no—
333SD16—44no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—84no—
374C2×C8#12112yes—
375C2×C8#12112yes—
376C2×C8#13712yes—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4C4×D412yes—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—42no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—42no—
399C42—22no—
400C42—22no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—22no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C42C2×#4111yescharacteristic
418C42C2×#4111yescharacteristic
419C23—22no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23C2×C4×D411yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—22no—
430Q8—22no—
431Q8—41no—
432Q8—41no—
433Q8—41no—
434Q8—41no—
435D4—22no—
436D4—22no—
437D4—41no—
438D4—41no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
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—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4#44312yes—
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C4×#4111yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4#52511yescharacteristic
482C2×C4#52911yescharacteristic
483C2×C4#53211yescharacteristic
484C2×C4C2×#13111yescharacteristic
485C2×C4C2×#13111yescharacteristic
486C2×C4#56811yescharacteristic
487C2×C4#56811yescharacteristic
488C2×C4C2×#13711yescharacteristic
489C2×C4#57511yescharacteristic
490C2×C4#57511yescharacteristic
491C2×C4#57511yescharacteristic
492C8—22no—
493C8—22no—
494C8—22no—
495C8—42no—
496C8—42no—
497C8—41no—
498C8—42no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—42no—
504C8—81no—
505C8—81no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22C4×#12111yescharacteristic
514C22#292911yescharacteristic
515C22#293211yescharacteristic
516C22#294311yescharacteristic
517C22C4×#13711yescharacteristic
518C22C2×#44311yescharacteristic
519C22#295511yescharacteristic
520C4—21no—
521C4—21no—
522C4—22no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—42no—
527C4—41no—
528C4—41no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4#293311yescharacteristic
534C4#293311yescharacteristic
535C4#296811yescharacteristic
536C4#296811yescharacteristic
537C4#303711yescharacteristic
538C4#304011yescharacteristic
539C4#311111yescharacteristic
540C4#311711yescharacteristic
541C2—82no—
542C2#1790911yescharacteristic
543C2#1791311yescharacteristic
544C2#1798211yescharacteristic
545C2#1798411yescharacteristic
546C2#2149211yescharacteristic
547C2#2150811yescharacteristic
548C2#2151711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20516C2semidirect
4. C2×#1587C2semidirect
7. #9606C2semidirect
8. #9544C2semidirect
11. #8946C2semidirect
12. #8570C2semidirect
13. #8261C2semidirect
14. #8205C2semidirect

Maximal subgroups

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

subgroupindex
#236812
C8×#1242
#205162
C2×#15872
C2×#15852
#96762
#96062
#95442
#95422
#89482
#89462
#85702
#82612
#82052
#82042