Finite Group Databasebeta

← #231900

Subgroups — #231900

Order 512.

653 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23204C211yesmaximal, characteristic
2#23149C211yesmaximal, characteristic, complemented
3C4×#489C211yesmaximal, characteristic
4#9864C211yesmaximal, characteristic, complemented
5#9675C211yesmaximal, characteristic
6#9555C211yesmaximal, characteristic, complemented
7#9554C211yesmaximal, characteristic
8#9544C211yesmaximal, characteristic, complemented
9#9541C211yesmaximal, characteristic
10#8846C211yesmaximal, characteristic, complemented
11#8845C211yesmaximal, characteristic
12#8275C211yesmaximal, characteristic, complemented
13#8262C211yesmaximal, characteristic, complemented
14#8200C211yesmaximal, characteristic
15C8×#103C211yesmaximal, characteristic, complemented
16C2×#489C411yescharacteristic
17C2×C8×Q8—21no—
18C2×C8×Q8C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×#124C2211yescharacteristic
22#1993C411yescharacteristic
23#1984—21no—
24#1984C2211yescharacteristic
25#1959C411yescharacteristic
26#1954C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1936C411yescharacteristic
30#1936C411yescharacteristic
31#1935C411yescharacteristic
32#1935C411yescharacteristic
33#1923—21no—
34#1923—21no—
35#1916—21no—
36#1916—21no—
37#1850—21no—
38#1836C411yescharacteristic
39C8×C4⋊C4C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C8×C22⋊C4—21no—
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×#124C2×C411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#489C818yescomplemented
74C8×Q8—22no—
75C8×Q8—41no—
76C8×Q8—44no—
77C8×D4—22no—
78C8×D4—41no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8D411yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#103C2×C411yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×#103C2×C411yescharacteristic
92C2×#103C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×#97C2×C411yescharacteristic
101C2×#97C2×C411yescharacteristic
102C2×#97C2×C411yescharacteristic
103C2×#97C2×C411yescharacteristic
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391—21no—
116#391C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#389—21no—
119#389C2×C411yescharacteristic
120#389C2×C411yescharacteristic
121#388—21no—
122#388—21no—
123#384C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C43C2311yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—22no—
151#334—22no—
152Q8⋊C8—22no—
153Q8⋊C8—41no—
154D4⋊C8—22no—
155D4⋊C8—41no—
156#322—41no—
157C82—41no—
158C22×Q8—21no—
159C22×Q8C22×C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—81no—
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⋊Q8C2×C814yescomplemented
181#124—22no—
182#124C2×C814yes—
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—22no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—41no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8#4112yes—
233C4⋊C8#4112yes—
234#103—22no—
235#103—41no—
236#103C2×C814yescomplemented
237#103C2×C814yescomplemented
238#102—22no—
239#102—41no—
240#102C2×C814yes—
241#102C2×C814yes—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8—81no—
245#97—22no—
246#97—22no—
247#97—41no—
248#97C2×C814yescomplemented
249C4×C8—21no—
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—21no—
261C4×C8—42no—
262C4×C8—42no—
263C4×C8—42no—
264C4×C8—42no—
265C4×C8—81no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274C24—21no—
275C2×Q8—21no—
276C2×Q8—24no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8C22×C812yes—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4C22×C812yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
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—41no—
334C2×C8—41no—
335C2×C8—41no—
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—84no—
349C2×C8C22×C812yes—
350C2×C8C22×C812yes—
351C2×C8#13112yes—
352C2×C8AGL(1,ℤ/8)12yes—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—24no—
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—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—81no—
370C4⋊C4C22×C812yes—
371C4⋊C4C22×C812yes—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—42no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—81no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42C22×C812yes—
393C42#13112yes—
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C42H(5,𝔽2)11yescharacteristic
397C422−1+411yescharacteristic
398C23—22no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23C2×C4×D411yescharacteristic
406Q8—22no—
407Q8—22no—
408Q8—48no—
409Q8—41no—
410Q8—41no—
411D4—22no—
412D4—22no—
413D4—41no—
414D4—41no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
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—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
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—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4C8×D412yes—
453C2×C4C8×D412yes—
454C2×C4C4×#4111yescharacteristic
455C2×C4#52011yescharacteristic
456C2×C4#54611yescharacteristic
457C2×C4#54811yescharacteristic
458C2×C4#55611yescharacteristic
459C2×C4#55911yescharacteristic
460C2×C4C23×C811yescharacteristic
461C2×C4C2×#13111yescharacteristic
462C2×C4C2×#13111yescharacteristic
463C2×C4#56811yescharacteristic
464C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
465C2×C4#57511yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C8—22no—
469C8—22no—
470C8—22no—
471C8—22no—
472C8—42no—
473C8—42no—
474C8—41no—
475C8—41no—
476C8—41no—
477C8—41no—
478C8—42no—
479C8—81no—
480C8—81no—
481C8—81no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22#236711yescharacteristic
490C22C2×C8×D411yescharacteristic
491C22#293511yescharacteristic
492C22C4×AGL(1,ℤ/8)11yescharacteristic
493C22#294711yescharacteristic
494C22#295511yescharacteristic
495C22#295611yescharacteristic
496C4—22no—
497C4—22no—
498C4—21no—
499C4—21no—
500C4—41no—
501C4—41no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4C8×#4111yescharacteristic
510C4#296911yescharacteristic
511C4#298811yescharacteristic
512C4#299411yescharacteristic
513C4#330811yescharacteristic
514C4#331111yescharacteristic
515C4#337311yescharacteristic
516C4#337711yescharacteristic
517C2—82no—
518C2C8×AGL(1,ℤ/8)11yescharacteristic
519C2#1785311yescharacteristic
520C2#1807811yescharacteristic
521C2#1808311yescharacteristic
522C2#2320411yescharacteristic
523C2#2322811yescharacteristic
524C2#2323611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C2×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
2. #23149C2semidirect
4. #9864C2semidirect
6. #9555C2semidirect
8. #9544C2semidirect
10. #8846C2semidirect
12. #8275C2semidirect
13. #8262C2semidirect
15. C8×#103C2semidirect
73. #489C8semidirect
180. C4⋊Q8C2×C8semidirect
236. #103C2×C8semidirect
237. #103C2×C8semidirect
248. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
#232042
#231492
C4×#4892
#98642
#96752
#95552
#95542
#95442
#95412
#88462
#88452
#82752
#82622
#82002
C8×#1032