Finite Group Databasebeta

← #228316

Subgroups — #228316

Order 512.

693 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22985C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic, complemented
3C4×#482C211yesmaximal, characteristic
4#9596C211yesmaximal, characteristic, complemented
5#9554C211yesmaximal, characteristic, complemented
6#9541C211yesmaximal, characteristic, complemented
7#9536C211yesmaximal, characteristic, complemented
8#9535C211yesmaximal, characteristic, complemented
9#9074C211yesmaximal, characteristic
10#8843C211yesmaximal, characteristic, complemented
11#8829C211yesmaximal, characteristic, complemented
12C8×#107C211yesmaximal, characteristic, complemented
13#8200C211yesmaximal, characteristic, complemented
14#8148C211yesmaximal, characteristic, complemented
15#7891C211yesmaximal, characteristic, complemented
16C2×#482C411yescharacteristic
17C2×C8×D4—21no—
18C2×C8×D4C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C4×#121C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1954C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1926C411yescharacteristic
28#1925C411yescharacteristic
29#1923—21no—
30#1923—21no—
31#1923—21no—
32#1921C411yescharacteristic
33#1920C411yescharacteristic
34#1875C411yescharacteristic
35#1869—21no—
36#1854C411yescharacteristic
37#1850—21no—
38#1831—21no—
39#1825C411yescharacteristic
40#1797C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#106C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1744—21no—
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C2×#121C2×C411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2311yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C22×C42C2311yescharacteristic
75#482C818yescomplemented
76C8×D4—22no—
77C8×D4—41no—
78C8×D4—44no—
79#431—22no—
80#431—41no—
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×#106C2×C411yescharacteristic
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—41no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8C2×C411yescharacteristic
106C2×C22⋊C8C2×C411yescharacteristic
107C2×C22⋊C8D411yescharacteristic
108C2×C22⋊C8D411yescharacteristic
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#390C2×C411yescharacteristic
119#389C2×C411yescharacteristic
120#388—21no—
121#386—21no—
122#380C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132C43—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#336C2×C411yescharacteristic
149#336C2×C411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152#335—22no—
153#335—41no—
154D4⋊C8—22no—
155D4⋊C8—41no—
156#322—41no—
157C82—22no—
158C22×D4—21no—
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—81no—
175C22×C8—81no—
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184#121—22no—
185#121C2×C814yes—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—42no—
191C4×D4—42no—
192#117—22no—
193#117—22no—
194#117—41no—
195#117C2×C814yescomplemented
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42C2411yescharacteristic
226#107—22no—
227#107C2×C814yescomplemented
228#106—41no—
229#106C2×C814yescomplemented
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—81no—
241C4⋊C8—81no—
242C4⋊C8#4112yes—
243#102—22no—
244#102—41no—
245#102C2×C814yescomplemented
246#102C2×C814yescomplemented
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—41no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—81no—
254C22⋊C8—81no—
255C22⋊C8C2×C814yescomplemented
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—42no—
268C4×C8—42no—
269C4×C8—42no—
270C4×C8#4112yes—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278C24—21no—
279C24C4×D411yescharacteristic
280C2×D4—24no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4C22×C812yes—
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—41no—
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×C4C22×C812yes—
320C22×C4#13112yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C2×#4111yescharacteristic
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—22no—
335C2×C8—21no—
336C2×C8—22no—
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—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—41no—
362C2×C8—41no—
363C2×C8—84no—
364C2×C8—84no—
365C2×C8C22×C812yes—
366C2×C8C22×C812yes—
367C2×C8AGL(1,ℤ/8)12yes—
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—42no—
379C4⋊C4C22×C812yes—
380C4⋊C4C22×C812yes—
381C4⋊C4C22×C812yes—
382C4⋊C4#13112yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—81no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—24no—
398C42—42no—
399C42—42no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—81no—
407C42C2×#4111yescharacteristic
408C42C2×#4111yescharacteristic
409C42H(5,𝔽2)11yescharacteristic
410C422−1+411yescharacteristic
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—22no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23C8×D412yes—
425C23C2×C4×D411yescharacteristic
426D4—22no—
427D4—22no—
428D4—48no—
429D4—41no—
430D4—41no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
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—41no—
469C2×C4—41no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4C8×D412yes—
477C2×C4C4×#4111yescharacteristic
478C2×C4#52011yescharacteristic
479C2×C4#54611yescharacteristic
480C2×C4#54711yescharacteristic
481C2×C4#55111yescharacteristic
482C2×C4#55211yescharacteristic
483C2×C4C23×C811yescharacteristic
484C2×C4C2×#13111yescharacteristic
485C2×C4C2×#13111yescharacteristic
486C2×C4#56811yescharacteristic
487C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
488C2×C4#57511yescharacteristic
489C2×C4#57611yescharacteristic
490C2×C4#57811yescharacteristic
491C8—22no—
492C8—22no—
493C8—22no—
494C8—42no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—42no—
500C8—41no—
501C8—81no—
502C8—81no—
503C8—81no—
504C8—81no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22#235911yescharacteristic
520C22C2×C8×D411yescharacteristic
521C22#293311yescharacteristic
522C22C4×AGL(1,ℤ/8)11yescharacteristic
523C22#295111yescharacteristic
524C22#295511yescharacteristic
525C22#295711yescharacteristic
526C4—22no—
527C4—21no—
528C4—21no—
529C4—42no—
530C4—41no—
531C4—41no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4C8×#4111yescharacteristic
539C4#296811yescharacteristic
540C4#298811yescharacteristic
541C4#299411yescharacteristic
542C4#332511yescharacteristic
543C4#332711yescharacteristic
544C4#334311yescharacteristic
545C4#334711yescharacteristic
546C2—42no—
547C2—82no—
548C2C8×AGL(1,ℤ/8)11yescharacteristic
549C2#1785711yescharacteristic
550C2#1800711yescharacteristic
551C2#1803311yescharacteristic
552C2#2290011yescharacteristic
553C2#2294611yescharacteristic
554C2#2295111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22632C2semidirect
4. #9596C2semidirect
5. #9554C2semidirect
6. #9541C2semidirect
7. #9536C2semidirect
8. #9535C2semidirect
10. #8843C2semidirect
11. #8829C2semidirect
12. C8×#107C2semidirect
13. #8200C2semidirect
14. #8148C2semidirect
15. #7891C2semidirect
75. #482C8semidirect
195. #117C2×C8semidirect
227. #107C2×C8semidirect
229. #106C2×C8semidirect
245. #102C2×C8semidirect
246. #102C2×C8semidirect
255. C22⋊C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#229852
#226322
C4×#4822
#95962
#95542
#95412
#95362
#95352
#90742
#88432
#88292
C8×#1072
#82002
#81482
#78912