Finite Group Databasebeta

← #203409

Subgroups — #203409

Order 512.

677 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21277C211yesmaximal, characteristic
2#21266C211yesmaximal, characteristic, complemented
3#20485C211yesmaximal, characteristic, complemented
4#9545C211yesmaximal, characteristic, complemented
5#9542C211yesmaximal, characteristic, complemented
6#9539C211yesmaximal, characteristic, complemented
7#9534C211yesmaximal, characteristic, complemented
8#8951C211yesmaximal, characteristic
9#8947C211yesmaximal, characteristic
10#8849C211yesmaximal, characteristic, complemented
11#8843C211yesmaximal, characteristic, complemented
12#8276C211yesmaximal, characteristic, complemented
13#8179C211yesmaximal, characteristic, complemented
14#8179C211yesmaximal, characteristic, complemented
15#8132C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#433—21no—
20C2×#405—21no—
21#2293C2211yescharacteristic
22#2293C2211yescharacteristic
23#1972C411yescharacteristic
24#1972C411yescharacteristic
25#1954C2211yescharacteristic
26#1954C2211yescharacteristic
27#1943C411yescharacteristic
28#1943C411yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1936C411yescharacteristic
32#1936C411yescharacteristic
33#1936C411yescharacteristic
34#1936C411yescharacteristic
35#1923—21no—
36#1923—21no—
37C4×#106C2211yescharacteristic
38#1770C2211yescharacteristic
39C8×C4⋊C4C2211yescharacteristic
40#1767C2211yescharacteristic
41#1767C2211yescharacteristic
42C4×#102C2211yescharacteristic
43C4×#102C2211yescharacteristic
44#1753—21no—
45C8×C22⋊C4—21no—
46#1750—21no—
47#1750—21no—
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C22×M16—21no—
69C23×C8—21no—
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C2×#117—21no—
74C4×D8—22no—
75C4×D8—22no—
76#436—22no—
77#436—22no—
78C8×D4—22no—
79C8×D4—22no—
80#433—44no—
81C2×#107C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102#405—44no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97C2×C411yescharacteristic
107C2×#97C2×C411yescharacteristic
108C2×#97C2×C411yescharacteristic
109C2×#97C2×C411yescharacteristic
110C2×C4×C8—21no—
111C2×C4×C8D411yescharacteristic
112C2×C4×C8D411yescharacteristic
113C2×C4×C8D411yescharacteristic
114C2×C4×C8D411yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#390C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#390C2×C411yescharacteristic
124#390C2×C411yescharacteristic
125#389C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#388—21no—
130#388—21no—
131#388—21no—
132#388—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C43C2311yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#335—22no—
153#334—22no—
154D4⋊C8—22no—
155D4⋊C8—22no—
156D4⋊C8—22no—
157D4⋊C8—22no—
158C22×D4C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C2×D8—22no—
163C2×D8—22no—
164C2×D8—41no—
165C2×M16—24no—
166C2×M16—42no—
167C2×M16—42no—
168C22×C8—24no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4#4112yes—
192C4×D4#4112yes—
193#117—42no—
194#117—42no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
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×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42C2411yescharacteristic
226#107—22no—
227#106—22no—
228#106—41no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8#4112yes—
238C4⋊C8#4112yes—
239#102—22no—
240#102—22no—
241#102—22no—
242#102—22no—
243#102—41no—
244#102—41no—
245C22⋊C8—24no—
246C22⋊C8—24no—
247C22⋊C8—42no—
248C22⋊C8—42no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251#97—22no—
252#97—22no—
253#97—41no—
254#97—41no—
255#97—42no—
256#97—42no—
257#97—42no—
258C4×C8—22no—
259C4×C8—22no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
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—41no—
273C4×C8—41no—
274C4×C8—42no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281C24—21no—
282C24—21no—
283C2×D4—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—21no—
294C2×D4#11712yes—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320D8—44no—
321M16—48no—
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—22no—
335C2×C8—22no—
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—48no—
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#13112yes—
362C2×C8#13112yes—
363C2×C8AGL(1,ℤ/8)12yes—
364C2×C8AGL(1,ℤ/8)12yes—
365C4⋊C4—21no—
366C4⋊C4—21no—
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—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4#11712yes—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42C2×#4111yescharacteristic
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C23—22no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23C2×C4×D411yescharacteristic
426D4—22no—
427D4—22no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—41no—
432D4—41no—
433D4—41no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
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×C4C2×#11711yescharacteristic
475C2×C4C4×#4111yescharacteristic
476C2×C4#52511yescharacteristic
477C2×C4#52511yescharacteristic
478C2×C4#52911yescharacteristic
479C2×C4#52911yescharacteristic
480C2×C4C2×#13111yescharacteristic
481C2×C4C2×#13111yescharacteristic
482C2×C4#56811yescharacteristic
483C2×C4#56811yescharacteristic
484C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
485C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
486C2×C4#57511yescharacteristic
487C2×C4#57511yescharacteristic
488C8—22no—
489C8—22no—
490C8—22no—
491C8—22no—
492C8—42no—
493C8—41no—
494C8—42no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—81no—
499C8—81no—
500C8—81no—
501C8—81no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22#229011yescharacteristic
517C22#293111yescharacteristic
518C22#293211yescharacteristic
519C22C4×AGL(1,ℤ/8)11yescharacteristic
520C22C4×AGL(1,ℤ/8)11yescharacteristic
521C22#295511yescharacteristic
522C22#295511yescharacteristic
523C4—21no—
524C4—21no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4#292411yescharacteristic
536C4#292511yescharacteristic
537C4#296811yescharacteristic
538C4#296911yescharacteristic
539C4#310911yescharacteristic
540C4#311111yescharacteristic
541C4#311411yescharacteristic
542C4#311511yescharacteristic
543C2—82no—
544C2—161no—
545C2#1796411yescharacteristic
546C2#1796611yescharacteristic
547C2#1798511yescharacteristic
548C2#1798511yescharacteristic
549C2#2111011yescharacteristic
550C2#2112211yescharacteristic
551C2#2113811yescharacteristic
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 ◁ C42×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21266C2semidirect
3. #20485C2semidirect
4. #9545C2semidirect
5. #9542C2semidirect
6. #9539C2semidirect
7. #9534C2semidirect
10. #8849C2semidirect
11. #8843C2semidirect
12. #8276C2semidirect
13. #8179C2semidirect
14. #8179C2semidirect
15. #8132C2semidirect

Maximal subgroups

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

subgroupindex
#212772
#212662
#204852
#95452
#95422
#95392
#95342
#89512
#89472
#88492
#88432
#82762
#81792
#81322