Finite Group Databasebeta

← #281138

Subgroups — #281138

Order 512.

609 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28142C211yesmaximal, characteristic
2#25058C211yesmaximal, characteristic
3#25056C211yesmaximal, characteristic
4C4⋊C4×Q16C211yesmaximal, characteristic, complemented
5#10620C211yesmaximal, characteristic, complemented
6#10618C211yesmaximal, characteristic, complemented
7#10436C211yesmaximal, characteristic
8#10434C211yesmaximal, characteristic
9#9662C211yesmaximal, characteristic, complemented
10#9616C211yesmaximal, characteristic, complemented
11#8896C211yesmaximal, characteristic, complemented
12#8884C211yesmaximal, characteristic, complemented
13#8820C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic
15#8200C211yesmaximal, characteristic
16C2×C4×Q16—21no—
17C2×C4×Q16C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2538C2211yescharacteristic
20#2538C2211yescharacteristic
21Q8×C4⋊C4C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23#2094—21no—
24#2093—21no—
25#2084—21no—
26#2083—21no—
27#2076—21no—
28#2001—21no—
29#1984C2211yescharacteristic
30#1984C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947C2211yescharacteristic
33#1942—21no—
34#1926—21no—
35#1925—21no—
36#1919—21no—
37#1918—21no—
38#1892—21no—
39#1892—21no—
40#1863—21no—
41#1863—21no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1844C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1802C2211yescharacteristic
53#1802C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×Q16D411yescharacteristic
68C23×C8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C4×Q16—22no—
77C4×Q16—22no—
78C4×Q16—44no—
79C8×D4—22no—
80C8×D4—22no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—41no—
87C2×#107D411yescharacteristic
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103D411yescharacteristic
100C2×#103D411yescharacteristic
101C2×#102—21no—
102C2×#102—21no—
103C2×#102D411yescharacteristic
104C2×#102D411yescharacteristic
105C2×C22⋊C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#396—41no—
111#396—41no—
112#390—21no—
113#390—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—41no—
119#389—41no—
120#384—21no—
121#384—21no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#380—21no—
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2311yescharacteristic
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—41no—
142#340D411yescharacteristic
143#340D411yescharacteristic
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#335—22no—
150Q8⋊C8—22no—
151Q8⋊C8—22no—
152D4⋊C8—22no—
153#322—22no—
154C22×Q8C2×D411yescharacteristic
155C22×Q8C2×D411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C2×Q16—22no—
159C2×Q16—22no—
160C2×Q16—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—81no—
168C22×C8—81no—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175C22⋊Q8—81no—
176C22⋊Q8—81no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—81no—
188C4×Q8—81no—
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⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C2411yescharacteristic
229#107—22no—
230#107—42no—
231#107—42no—
232#107—81no—
233#107—81no—
234#107—81no—
235#107—82no—
236#106—42no—
237#106—81no—
238#106—81no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247#103—22no—
248#103—22no—
249#103—42no—
250#103—42no—
251#103—81no—
252#103—81no—
253#102—22no—
254#102—81no—
255#102—81no—
256#102Q1614yescomplemented
257C22⋊C8—42no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—42no—
262C4×C8#4112yes—
263C4×C8#4112yes—
264C4×C8#4112yes—
265#95—21no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8AGL(1,ℤ/8)12yes—
282C2×Q8#13712yes—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4C2×Q1612yes—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4H(5,𝔽2)11yescharacteristic
309Q16—28no—
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—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—84no—
334C2×C8#12312yes—
335C2×C8#12312yes—
336C2×C8C2×Q1612yes—
337C2×C8#13512yes—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4—161no—
376C4⋊C4—161no—
377C4⋊C4C2×Q1612yes—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—161no—
383C22⋊C4—161no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—82no—
391C42—82no—
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C422−1+411yescharacteristic
396C23—22no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23D4211yescharacteristic
404Q8—22no—
405Q8—22no—
406Q8—22no—
407Q8—22no—
408Q8—41no—
409Q8—41no—
410Q8—41no—
411Q8—41no—
412D4—22no—
413D4—22no—
414D4—41no—
415D4—41no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
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—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4C2×#12311yescharacteristic
456C2×C4#53211yescharacteristic
457C2×C4#54211yescharacteristic
458C2×C4#54711yescharacteristic
459C2×C4#54711yescharacteristic
460C2×C4#55311yescharacteristic
461C2×C4C22×Q1611yescharacteristic
462C2×C4C2×#13511yescharacteristic
463C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
464C2×C4C2×#13711yescharacteristic
465C2×C4#57611yescharacteristic
466C2×C4#57611yescharacteristic
467C2×C4#57711yescharacteristic
468C2×C4#57811yescharacteristic
469C8—22no—
470C8—22no—
471C8—22no—
472C8—22no—
473C8—42no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—81no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22#242211yescharacteristic
486C22#326411yescharacteristic
487C22#326611yescharacteristic
488C22D4×Q1611yescharacteristic
489C22#328711yescharacteristic
490C22#328811yescharacteristic
491C22#328911yescharacteristic
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—161no—
502C4—161no—
503C4—161no—
504C4—161no—
505C4#309211yescharacteristic
506C4#309611yescharacteristic
507C4#317011yescharacteristic
508C4#317511yescharacteristic
509C4#333611yescharacteristic
510C4#333811yescharacteristic
511C4#334511yescharacteristic
512C4#334611yescharacteristic
513C2—82no—
514C2#1981111yescharacteristic
515C2#1981311yescharacteristic
516C2#1986711yescharacteristic
517C2#1986911yescharacteristic
518C2#2502211yescharacteristic
519C2#2503511yescharacteristic
520C2#2505511yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×Q16C2semidirect
5. #10620C2semidirect
6. #10618C2semidirect
9. #9662C2semidirect
10. #9616C2semidirect
11. #8896C2semidirect
12. #8884C2semidirect
13. #8820C2semidirect
256. #102Q16semidirect

Maximal subgroups

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

subgroupindex
#281422
#250582
#250562
C4⋊C4×Q162
#106202
#106182
#104362
#104342
#96622
#96162
#88962
#88842
#88202
#86442
#82002