Finite Group Databasebeta

← #308063

Subgroups — #308063

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26507C211yesmaximal, characteristic, complemented
2#26087C211yesmaximal, characteristic
3#23061C211yesmaximal, characteristic
4#22658C211yesmaximal, characteristic
5#10674C211yesmaximal, characteristic, complemented
6#10652C211yesmaximal, characteristic, complemented
7#10601C211yesmaximal, characteristic, complemented
8#10576C211yesmaximal, characteristic, complemented
9#9615C211yesmaximal, characteristic, complemented
10#9533C211yesmaximal, characteristic, complemented
11#9215C211yesmaximal, characteristic
12#8976C211yesmaximal, characteristic
13#8928C211yesmaximal, characteristic, complemented
14#8885C211yesmaximal, characteristic
15#8843C211yesmaximal, characteristic
16C2×#476C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×#427—21no—
19#2571C2211yescharacteristic
20#2369C2211yescharacteristic
21#2354C2211yescharacteristic
22#2354C2211yescharacteristic
23#2101—21no—
24#2092—21no—
25#2086—21no—
26#2067—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1987C2211yescharacteristic
30#1956C2211yescharacteristic
31#1949—21no—
32#1949—21no—
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1934—21no—
41#1926—21no—
42#1921—21no—
43#1920—21no—
44#1920—21no—
45#1854—21no—
46#1854—21no—
47#1845C2211yescharacteristic
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
50#1808—21no—
51#1802C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C4×#106—21no—
54C8×C4⋊C4C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117—21no—
72#476—22no—
73#476Q814yes—
74C8×D4—22no—
75C8×D4—22no—
76#427—44no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—41no—
82C2×#107D411yescharacteristic
83C2×#107D411yescharacteristic
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103—41no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102D411yescharacteristic
100C2×#102D411yescharacteristic
101C2×C22⋊C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#400—41no—
107#391—41no—
108#390—21no—
109#390—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#388—41no—
115#388—41no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120#382—21no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130#340—21no—
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—21no—
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#340D411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#335—22no—
150#333—22no—
151D4⋊C8—22no—
152C22×Q8—21no—
153C22×D4C2×D411yescharacteristic
154C23×C4—21no—
155C22×C8—24no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—42no—
162C22×C8—42no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C2×Q812yes—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4C2×Q812yes—
174C4×D4#4112yes—
175#117—42no—
176#117—42no—
177#117—81no—
178#117—81no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42C2411yescharacteristic
211#107—22no—
212#107—22no—
213#107—42no—
214#107—42no—
215#107—81no—
216#107—81no—
217#107—82no—
218#106—24no—
219#106—42no—
220#106—42no—
221#106—42no—
222#106—42no—
223#106—81no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8C2×Q812yes—
233C4⋊C8#4112yes—
234#103—82no—
235#102—24no—
236#102—22no—
237#102—41no—
238#102—81no—
239C22⋊C8—24no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—42no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257C24—21no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—81no—
261C2×Q8—81no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4C22⋊Q812yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4H(5,𝔽2)11yescharacteristic
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—22no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—48no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8#13512yes—
315C2×C8#13512yes—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—161no—
348C4⋊C4—161no—
349C4⋊C4C22⋊Q812yes—
350C4⋊C4AGL(1,ℤ/8)12yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—161no—
356C22⋊C4—161no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—82no—
364C42—82no—
365C42—161no—
366C42C22×Q811yescharacteristic
367C42C2×#4111yescharacteristic
368C42C2×#4111yescharacteristic
369C422−1+411yescharacteristic
370C23—22no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23D4211yescharacteristic
378Q8—82no—
379Q8—161no—
380D4—22no—
381D4—22no—
382D4—41no—
383D4—41no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4C2×C22⋊Q811yescharacteristic
424C2×C4#52511yescharacteristic
425C2×C4#53911yescharacteristic
426C2×C4#54711yescharacteristic
427C2×C4D4×Q811yescharacteristic
428C2×C4#55411yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
432C2×C4#57511yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57711yescharacteristic
435C2×C4#57811yescharacteristic
436C2×C4#57811yescharacteristic
437C8—22no—
438C8—22no—
439C8—42no—
440C8—42no—
441C8—41no—
442C8—42no—
443C8—41no—
444C8—81no—
445C8—81no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22#244911yescharacteristic
454C22#326511yescharacteristic
455C22#327111yescharacteristic
456C22#328311yescharacteristic
457C22#328311yescharacteristic
458C22#328911yescharacteristic
459C22#329211yescharacteristic
460C4—21no—
461C4—21no—
462C4—42no—
463C4—42no—
464C4—82no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—161no—
471C4—161no—
472C4—161no—
473C4#307611yescharacteristic
474C4#308211yescharacteristic
475C4#310511yescharacteristic
476C4#312011yescharacteristic
477C4#333711yescharacteristic
478C4#334911yescharacteristic
479C4#338111yescharacteristic
480C4#338611yescharacteristic
481C2—82no—
482C2#1986011yescharacteristic
483C2#1986611yescharacteristic
484C2#2001911yescharacteristic
485C2#2007911yescharacteristic
486C2#2634911yescharacteristic
487C2#2637411yescharacteristic
488C2#2640011yescharacteristic
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
1. #26507C2semidirect
5. #10674C2semidirect
6. #10652C2semidirect
7. #10601C2semidirect
8. #10576C2semidirect
9. #9615C2semidirect
10. #9533C2semidirect
13. #8928C2semidirect

Maximal subgroups

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

subgroupindex
#265072
#260872
#230612
#226582
#106742
#106522
#106012
#105762
#96152
#95332
#92152
#89762
#89282
#88852
#88432