Finite Group Databasebeta

← #212166

Subgroups — #212166

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758C211yesmaximal, characteristic
2C8×#121C211yesmaximal, characteristic
3#20476C211yesmaximal, characteristic, complemented
4#9546C211yesmaximal, characteristic, complemented
5#9545C211yesmaximal, characteristic, complemented
6#9538C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#8943C211yesmaximal, characteristic
9#8834C211yesmaximal, characteristic, complemented
10#8711C211yesmaximal, characteristic, complemented
11#8285C211yesmaximal, characteristic, complemented
12#8284C211yesmaximal, characteristic, complemented
13#8200C211yesmaximal, characteristic, complemented
14#8200C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic, complemented
16C2×C8×D4—21no—
17C2×C8×D4C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C4×#121C2211yescharacteristic
21#2283C2211yescharacteristic
22#2101C411yescharacteristic
23#2092C411yescharacteristic
24#2085C411yescharacteristic
25#2077C411yescharacteristic
26#2076C411yescharacteristic
27#2073C411yescharacteristic
28#2034C411yescharacteristic
29#2033C411yescharacteristic
30#1954C2211yescharacteristic
31#1939C2211yescharacteristic
32#1923—21no—
33#1869—21no—
34#1797C2211yescharacteristic
35C4×#107C2211yescharacteristic
36C4×#106C2211yescharacteristic
37#1770C2211yescharacteristic
38#1770C2211yescharacteristic
39C8×C4⋊C4C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C4×#102C2211yescharacteristic
43C4×#102C2211yescharacteristic
44#1752—21no—
45C8×C22⋊C4—21no—
46C8×C22⋊C4—21no—
47C4×C22⋊C8C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49#1746—21no—
50#1726—21no—
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C23×C8D411yescharacteristic
69C23×C8D411yescharacteristic
70C2×#121C2×C411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C8×D4—22no—
75C8×D4—22no—
76C8×D4—44no—
77#432—22no—
78#432—22no—
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#102C2×C411yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8C2×C411yescharacteristic
99C2×C22⋊C8C2×C411yescharacteristic
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8D411yescharacteristic
109C2×C4×C8D411yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#390C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#388—21no—
119#386—21no—
120#383—21no—
121#382C2×C411yescharacteristic
122#382C2×C411yescharacteristic
123#381—21no—
124#380C2×C411yescharacteristic
125#380C2×C411yescharacteristic
126#380C2×C411yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132#376—21no—
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336C2×C411yescharacteristic
150#336C2×C411yescharacteristic
151#335—22no—
152#334—22no—
153D4⋊C8—22no—
154D4⋊C8—22no—
155#322—22no—
156#322—22no—
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—41no—
173C22×C8—41no—
174C22×C8—41no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186#121—22no—
187#121—22no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—42no—
193C4×D4—42no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#107—22no—
221#107—22no—
222#106—22no—
223#106—22no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—42no—
231C4⋊C8#4112yes—
232C4⋊C8#4112yes—
233#102—22no—
234#102—22no—
235#102—22no—
236#102—22no—
237C22⋊C8—24no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8—42no—
245#97—24no—
246#97—42no—
247#97—42no—
248C4×C8—22no—
249C4×C8—22no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—42no—
261C4×C8—42no—
262C4×C8—42no—
263C4×C8#4112yes—
264C4×C8#4112yes—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276C24—21no—
277C24C4×D411yescharacteristic
278C2×D4—24no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4#13112yes—
287C22×C4—21no—
288C22×C4—22no—
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—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—44no—
344C2×C8—44no—
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×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—84no—
362C2×C8#12312yes—
363C2×C8#12312yes—
364C2×C8#13112yes—
365C2×C8#13112yes—
366C2×C8AGL(1,ℤ/8)12yes—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—42no—
381C4⋊C4#13112yes—
382C22⋊C4—24no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—22no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23#44412yes—
420C23C2×C4×D411yescharacteristic
421D4—22no—
422D4—22no—
423D4—48no—
424D4—41no—
425D4—41no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—42no—
439C2×C4—42no—
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—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
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—41no—
467C2×C4—41no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4C4×#4111yescharacteristic
473C2×C4C4×#4111yescharacteristic
474C2×C4C2×#12311yescharacteristic
475C2×C4#52511yescharacteristic
476C2×C4#52911yescharacteristic
477C2×C4#52911yescharacteristic
478C2×C4C2×#13111yescharacteristic
479C2×C4C2×#13111yescharacteristic
480C2×C4C2×#13111yescharacteristic
481C2×C4C2×#13111yescharacteristic
482C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57511yescharacteristic
485C2×C4#57511yescharacteristic
486C8—22no—
487C8—22no—
488C8—22no—
489C8—22no—
490C8—22no—
491C8—41no—
492C8—42no—
493C8—42no—
494C8—41no—
495C8—41no—
496C8—81no—
497C8—81no—
498C8—81no—
499C8—81no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22C4×#12311yescharacteristic
515C22#292911yescharacteristic
516C22#293111yescharacteristic
517C22#294311yescharacteristic
518C22C4×AGL(1,ℤ/8)11yescharacteristic
519C22C2×#44411yescharacteristic
520C22#295511yescharacteristic
521C4—21no—
522C4—21no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—42no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4#296611yescharacteristic
534C4#296611yescharacteristic
535C4#296611yescharacteristic
536C4#296611yescharacteristic
537C4#309311yescharacteristic
538C4#309411yescharacteristic
539C4#311111yescharacteristic
540C4#311511yescharacteristic
541C2—42no—
542C2—82no—
543C2#1790911yescharacteristic
544C2#1791411yescharacteristic
545C2#1796311yescharacteristic
546C2#1796611yescharacteristic
547C2#2156611yescharacteristic
548C2#2158211yescharacteristic
549C2#2159311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20476C2semidirect
4. #9546C2semidirect
5. #9545C2semidirect
6. #9538C2semidirect
7. #9537C2semidirect
9. #8834C2semidirect
10. #8711C2semidirect
11. #8285C2semidirect
12. #8284C2semidirect
13. #8200C2semidirect
14. #8200C2semidirect
15. #8145C2semidirect

Maximal subgroups

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

subgroupindex
#237582
C8×#1212
#204762
#95462
#95452
#95382
#95372
#89432
#88342
#87112
#82852
#82842
#82002
#81452