Finite Group Databasebeta

← #219202

Subgroups — #219202

Order 512.

601 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21713C211yesmaximal, characteristic, complemented
2C4×#489C211yesmaximal, characteristic
3#21634C211yesmaximal, characteristic
4#9877C211yesmaximal, characteristic, complemented
5#9646C211yesmaximal, characteristic
6#9547C211yesmaximal, characteristic, complemented
7#9545C211yesmaximal, characteristic
8#9543C211yesmaximal, characteristic, complemented
9#9541C211yesmaximal, characteristic
10#8849C211yesmaximal, characteristic
11#8849C211yesmaximal, characteristic
12#8844C211yesmaximal, characteristic, complemented
13#8844C211yesmaximal, characteristic, complemented
14#8808C211yesmaximal, characteristic, complemented
15#8136C211yesmaximal, characteristic, complemented
16C2×#489C411yescharacteristic
17C2×#446—21no—
18C2×C8×Q8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×#124C2211yescharacteristic
22#1993C411yescharacteristic
23#1985—21no—
24#1984C2211yescharacteristic
25#1959C411yescharacteristic
26#1954C2211yescharacteristic
27#1940—21no—
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1936C411yescharacteristic
31#1936C411yescharacteristic
32#1935C411yescharacteristic
33#1935C411yescharacteristic
34#1924—21no—
35#1924—21no—
36#1842C2211yescharacteristic
37#1836C411yescharacteristic
38#1831C2211yescharacteristic
39#1771—21no—
40#1770C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C4×#103C2211yescharacteristic
43C4×#103C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46#1753—21no—
47#1753—21no—
48C4×#97C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×#124C2×C411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#489—24no—
74#446—44no—
75C8×Q8—41no—
76C8×Q8—41no—
77#436—41no—
78#436—41no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103C2×C411yescharacteristic
89C2×#103C2×C411yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×#103C2×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×#97—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97C2×C411yescharacteristic
103C2×#97C2×C411yescharacteristic
104C2×#97C2×C411yescharacteristic
105C2×#97C2×C411yescharacteristic
106C2×#97D411yescharacteristic
107C2×#97D411yescharacteristic
108C2×#97D411yescharacteristic
109C2×#97D411yescharacteristic
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#391—21no—
119#391C2×C411yescharacteristic
120#390C2×C411yescharacteristic
121#389—21no—
122#389C2×C411yescharacteristic
123#389C2×C411yescharacteristic
124#388—21no—
125#388—21no—
126#384C2×C411yescharacteristic
127#382C2×C411yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C43C2311yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#335—41no—
151#335—41no—
152#334—41no—
153#334—41no—
154Q8⋊C8—41no—
155Q8⋊C8—41no—
156D4⋊C8—41no—
157D4⋊C8—41no—
158C22×Q8—21no—
159C22×Q8C22×C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C2×M16—42no—
163C2×M16—42no—
164C2×M16—42no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C4⋊Q8—22no—
180C4⋊Q8—41no—
181#124—22no—
182#124—41no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233#103—22no—
234#103—22no—
235#103—41no—
236#103M1614yescomplemented
237#102—22no—
238#102—22no—
239#102—22no—
240#102—41no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244#97—22no—
245#97—22no—
246#97—22no—
247#97—22no—
248#97—41no—
249#97—41no—
250#97—41no—
251#97—41no—
252#97—42no—
253#97—42no—
254#97—42no—
255C4×C8—21no—
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—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277C24—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—81no—
286C2×Q8C2×M1612yes—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4C4×D411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4111yescharacteristic
313M16—84no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
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—21no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8C2×M1612yes—
353C2×C8#13112yes—
354C2×C8AGL(1,ℤ/8)12yes—
355C2×C8#13712yes—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—81no—
374C4⋊C4C2×M1612yes—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—81no—
380C22⋊C4—81no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42H(5,𝔽2)11yescharacteristic
398C42H(5,𝔽2)11yescharacteristic
399C422−1+411yescharacteristic
400C422−1+411yescharacteristic
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23C2×C4×D411yescharacteristic
409Q8—22no—
410Q8—22no—
411Q8—41no—
412Q8—41no—
413Q8—84no—
414D4—41no—
415D4—41no—
416D4—41no—
417D4—41no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
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—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4#52011yescharacteristic
458C2×C4#52011yescharacteristic
459C2×C4#53511yescharacteristic
460C2×C4#53611yescharacteristic
461C2×C4#54211yescharacteristic
462C2×C4#54211yescharacteristic
463C2×C4C22×M1611yescharacteristic
464C2×C4C2×#13111yescharacteristic
465C2×C4#56811yescharacteristic
466C2×C4#56811yescharacteristic
467C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
468C2×C4C2×#13711yescharacteristic
469C2×C4#57511yescharacteristic
470C2×C4#57511yescharacteristic
471C8—22no—
472C8—22no—
473C8—22no—
474C8—22no—
475C8—42no—
476C8—42no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—81no—
482C8—81no—
483C8—81no—
484C8—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22#232611yescharacteristic
493C22D4×M1611yescharacteristic
494C22#293511yescharacteristic
495C22C4×AGL(1,ℤ/8)11yescharacteristic
496C22C4×#13711yescharacteristic
497C22#295611yescharacteristic
498C22#295611yescharacteristic
499C4—21no—
500C4—21no—
501C4—41no—
502C4—42no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—42no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4#298911yescharacteristic
513C4#299011yescharacteristic
514C4#299211yescharacteristic
515C4#299511yescharacteristic
516C4#321411yescharacteristic
517C4#321511yescharacteristic
518C4#323811yescharacteristic
519C4#324011yescharacteristic
520C2—161no—
521C2#1804811yescharacteristic
522C2#1804911yescharacteristic
523C2#1807611yescharacteristic
524C2#1807911yescharacteristic
525C2#2221611yescharacteristic
526C2#2222411yescharacteristic
527C2#2223711yescharacteristic
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 ◁ C4×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21713C2semidirect
4. #9877C2semidirect
6. #9547C2semidirect
8. #9543C2semidirect
12. #8844C2semidirect
13. #8844C2semidirect
14. #8808C2semidirect
15. #8136C2semidirect
236. #103M16semidirect

Maximal subgroups

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

subgroupindex
#217132
C4×#4892
#216342
#98772
#96462
#95472
#95452
#95432
#95412
#88492
#88442
#88082
#81362