Finite Group Databasebeta

← #226646

Subgroups — #226646

Order 512.

691 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22796C211yesmaximal, characteristic, complemented
2D4×C4⋊C8C211yesmaximal, characteristic, complemented
3#21113C211yesmaximal, characteristic
4#9560C211yesmaximal, characteristic, complemented
5#9556C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9536C211yesmaximal, characteristic, complemented
8#8947C211yesmaximal, characteristic
9#8947C211yesmaximal, characteristic
10#8787C211yesmaximal, characteristic, complemented
11#8567C211yesmaximal, characteristic, complemented
12#8563C211yesmaximal, characteristic, complemented
13#8265C211yesmaximal, characteristic, complemented
14#8204C211yesmaximal, characteristic, complemented
15#8200C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#433—21no—
20C22×C4⋊C8—21no—
21#2293C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1954C2211yescharacteristic
24#1954C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1925C411yescharacteristic
28#1925C411yescharacteristic
29#1924—21no—
30#1923—21no—
31#1918C411yescharacteristic
32#1918C411yescharacteristic
33#1879C411yescharacteristic
34#1879C411yescharacteristic
35#1862C411yescharacteristic
36#1862C411yescharacteristic
37#1850—21no—
38#1850—21no—
39#1842C2211yescharacteristic
40#1842C2211yescharacteristic
41#1798—21no—
42#1797—21no—
43C4×#107C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×D4⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C23×C8—21no—
69C23×C8—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C22×C42—21no—
76C4×D8—22no—
77C4×D8—22no—
78C8×D4—22no—
79C8×D4—22no—
80C8×D4—41no—
81C8×D4—41no—
82#433—44no—
83C2×#107C2×C411yescharacteristic
84C2×#107C2×C411yescharacteristic
85C2×#107C2×C411yescharacteristic
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—44no—
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#390C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#389C2×C411yescharacteristic
124#388—21no—
125#388—21no—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C43C2311yescharacteristic
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#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#336C2×C411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152D4⋊C8—22no—
153D4⋊C8—22no—
154D4⋊C8—41no—
155D4⋊C8—41no—
156#322—41no—
157#322—41no—
158C22×D4C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C2×D8—22no—
165C2×D8—22no—
166C2×D8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—81no—
178C22×C8—81no—
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—41no—
197C4×D4#4112yes—
198C4×D4#4112yes—
199#117—42no—
200#117—42no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230#107—22no—
231#107—22no—
232#107—41no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—81no—
249C4⋊C8—81no—
250#102—22no—
251#102—22no—
252#102—22no—
253#102—22no—
254#102—22no—
255#102—41no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260C22⋊C8—81no—
261C22⋊C8—81no—
262#97—42no—
263C4×C8—21no—
264C4×C8—21no—
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—42no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282C24—21no—
283C24—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4#11712yes—
294C2×D4#13112yes—
295C2×D4#13112yes—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—81no—
320C22×C4C4×D411yescharacteristic
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327D8—44no—
328C2×C8—22no—
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—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
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—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—84no—
371C2×C8—84no—
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—21no—
381C4⋊C4—21no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4#11712yes—
385C4⋊C4#13112yes—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—81no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—42no—
409C42—42no—
410C42AGL(1,ℤ/8)12yes—
411C42AGL(1,ℤ/8)12yes—
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C42H(5,𝔽2)11yescharacteristic
415C422−1+411yescharacteristic
416C23—22no—
417C23—22no—
418C23—22no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23C2×C4×D411yescharacteristic
431D4—22no—
432D4—22no—
433D4—22no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4#43314yescomplemented
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—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—48no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
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—41no—
474C2×C4—84no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4#44212yes—
480C2×C4#44212yes—
481C2×C4C2×#11711yescharacteristic
482C2×C4#52011yescharacteristic
483C2×C4#54611yescharacteristic
484C2×C4#54811yescharacteristic
485C2×C4#55211yescharacteristic
486C2×C4#55311yescharacteristic
487C2×C4C2×#13111yescharacteristic
488C2×C4C2×#13111yescharacteristic
489C2×C4C2×#13111yescharacteristic
490C2×C4#56811yescharacteristic
491C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
492C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
493C2×C4#57611yescharacteristic
494C2×C4#57811yescharacteristic
495C8—42no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—81no—
505C8—81no—
506C8—81no—
507C8—81no—
508C8—81no—
509C22—24no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22#235211yescharacteristic
524C22#292911yescharacteristic
525C22#293311yescharacteristic
526C22C2×#44211yescharacteristic
527C22#295111yescharacteristic
528C22#295511yescharacteristic
529C22#295611yescharacteristic
530C4—21no—
531C4—21no—
532C4—22no—
533C4—22no—
534C4—41no—
535C4—41no—
536C4—42no—
537C4—42no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4C2×#43311yescharacteristic
543C4#292511yescharacteristic
544C4#299211yescharacteristic
545C4#299411yescharacteristic
546C4#332511yescharacteristic
547C4#333011yescharacteristic
548C4#336611yescharacteristic
549C4#336711yescharacteristic
550C2—44no—
551C2—161no—
552C2#1790711yescharacteristic
553C2#1792611yescharacteristic
554C2#1800111yescharacteristic
555C2#1802711yescharacteristic
556C2#2275411yescharacteristic
557C2#2275811yescharacteristic
558C2#2278411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22796C2semidirect
2. D4×C4⋊C8C2semidirect
4. #9560C2semidirect
5. #9556C2semidirect
6. #9540C2semidirect
7. #9536C2semidirect
10. #8787C2semidirect
11. #8567C2semidirect
12. #8563C2semidirect
13. #8265C2semidirect
14. #8204C2semidirect
15. #8200C2semidirect
438. D4#433semidirect

Maximal subgroups

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

subgroupindex
#227962
D4×C4⋊C82
#211132
#95602
#95562
#95402
#95362
#89472
#87872
#85672
#85632
#82652
#82042
#82002