Finite Group Databasebeta

← #307933

Subgroups — #307933

Order 512.

637 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26508C211yesmaximal, characteristic, complemented
2#26041C211yesmaximal, characteristic
3#22963C211yesmaximal, characteristic
4D4×#106C211yesmaximal, characteristic
5#10680C211yesmaximal, characteristic, complemented
6#10654C211yesmaximal, characteristic, complemented
7#10601C211yesmaximal, characteristic, complemented
8#10582C211yesmaximal, characteristic, complemented
9#9615C211yesmaximal, characteristic, complemented
10#9534C211yesmaximal, characteristic, complemented
11#9215C211yesmaximal, characteristic
12#8973C211yesmaximal, characteristic
13#8928C211yesmaximal, characteristic, complemented
14#8894C211yesmaximal, characteristic
15#8843C211yesmaximal, characteristic
16C2×#478—21no—
17C2×#476C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C22×#106—21no—
20#2552C2211yescharacteristic
21#2369C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#2099—21no—
25#2088—21no—
26#2087—21no—
27#2059—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1987C2211yescharacteristic
31#1956C2211yescharacteristic
32#1948—21no—
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1941—21no—
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1919—21no—
42#1919—21no—
43#1864—21no—
44#1862—21no—
45#1853—21no—
46#1853—21no—
47#1845C2211yescharacteristic
48#1844C2211yescharacteristic
49#1834C2211yescharacteristic
50#1807—21no—
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#102—21no—
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×#125—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75#478—44no—
76#476—22no—
77#476Q814yes—
78C8×D4—22no—
79C8×D4—22no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—44no—
91C2×#106D411yescharacteristic
92C2×#106D411yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103—41no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102D411yescharacteristic
103C2×#102D411yescharacteristic
104C2×C22⋊C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#398—41no—
110#391—41no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#385—41no—
118#385—41no—
119#384—21no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#380—21no—
124#380—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#340—41no—
143#340D411yescharacteristic
144#340D411yescharacteristic
145#340D411yescharacteristic
146#340D411yescharacteristic
147#336—21no—
148#336—21no—
149#336—21no—
150#335—22no—
151#333—22no—
152D4⋊C8—22no—
153C22×Q8—21no—
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C23×C4—41no—
157C23×C4—41no—
158C22×C8—24no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—42no—
165C22×C8—42no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C2×Q812yes—
174#125—42no—
175#125—81no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—81no—
181C4×D4—81no—
182C4×D4C2×Q812yes—
183C4×D4#4112yes—
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—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4—81no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—41no—
219C2×C42C2411yescharacteristic
220#107—42no—
221#107—81no—
222#106—22no—
223#106—24no—
224#106—22no—
225#106—42no—
226#106—42no—
227#106—42no—
228#106—42no—
229#106—42no—
230#106—81no—
231#106—81no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8C2×Q812yes—
241C4⋊C8#4112yes—
242#103—82no—
243#102—24no—
244#102—22no—
245#102—42no—
246#102SD1614yes—
247C22⋊C8—24no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—81no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263C24—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—81no—
267C2×Q8—81no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4C22⋊Q812yes—
272C2×D4C2×SD1612yes—
273C2×D4#13712yes—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—161no—
296C22×C4—161no—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—48no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8C2×SD1612yes—
327C2×C8#13512yes—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
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—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—82no—
362C4⋊C4—161no—
363C4⋊C4—161no—
364C4⋊C4C22⋊Q812yes—
365C4⋊C4C2×SD1612yes—
366C4⋊C4AGL(1,ℤ/8)12yes—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—161no—
378C42C22×Q811yescharacteristic
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C422−1+411yescharacteristic
382C23—22no—
383C23—22no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23D4211yescharacteristic
390Q8—82no—
391Q8—161no—
392D4—22no—
393D4—22no—
394D4—22no—
395D4#47514yes—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—164no—
436C2×C4—164no—
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4#52511yescharacteristic
439C2×C4#53911yescharacteristic
440C2×C4#54711yescharacteristic
441C2×C4D4×Q811yescharacteristic
442C2×C4#55411yescharacteristic
443C2×C4C22×SD1611yescharacteristic
444C2×C4C2×#13511yescharacteristic
445C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57811yescharacteristic
451C8—22no—
452C8—22no—
453C8—42no—
454C8—41no—
455C8—42no—
456C8—41no—
457C8—81no—
458C8—81no—
459C8—81no—
460C22—24no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22#244911yescharacteristic
468C22#326511yescharacteristic
469C22#326611yescharacteristic
470C22D4×SD1611yescharacteristic
471C22#328311yescharacteristic
472C22#328811yescharacteristic
473C22#329011yescharacteristic
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—82no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4C2×#47511yescharacteristic
488C4#308411yescharacteristic
489C4#310511yescharacteristic
490C4#312011yescharacteristic
491C4#333511yescharacteristic
492C4#334411yescharacteristic
493C4#338111yescharacteristic
494C4#338611yescharacteristic
495C2—44no—
496C2#1980411yescharacteristic
497C2#1986111yescharacteristic
498C2#1998011yescharacteristic
499C2#2003911yescharacteristic
500C2#2634011yescharacteristic
501C2#2637411yescharacteristic
502C2#2640211yescharacteristic
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. #26508C2semidirect
5. #10680C2semidirect
6. #10654C2semidirect
7. #10601C2semidirect
8. #10582C2semidirect
9. #9615C2semidirect
10. #9534C2semidirect
13. #8928C2semidirect

Maximal subgroups

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

subgroupindex
#265082
#260412
#229632
D4×#1062
#106802
#106542
#106012
#105822
#96152
#95342
#92152
#89732
#89282
#88942
#88432