Finite Group Databasebeta

← #308459

Subgroups — #308459

Order 512.

617 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26462C211yesmaximal, characteristic, complemented
2#26024C211yesmaximal, characteristic
3#23066C211yesmaximal, characteristic
4#22657C211yesmaximal, characteristic
5#10686C211yesmaximal, characteristic, complemented
6#10654C211yesmaximal, characteristic, complemented
7#10594C211yesmaximal, characteristic, complemented
8#10575C211yesmaximal, characteristic, complemented
9#9619C211yesmaximal, characteristic, complemented
10#9538C211yesmaximal, characteristic, complemented
11#9219C211yesmaximal, characteristic
12#8984C211yesmaximal, characteristic
13#8926C211yesmaximal, characteristic, complemented
14#8893C211yesmaximal, characteristic
15#8843C211yesmaximal, characteristic
16C2×#475—21no—
17C2×#474C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#427—21no—
20#2551C2211yescharacteristic
21#2365C2211yescharacteristic
22#2354C2211yescharacteristic
23#2354C2211yescharacteristic
24#2066—21no—
25#2065—21no—
26#2024—21no—
27#2023—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1987—21no—
31#1981C2211yescharacteristic
32#1956C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1936—21no—
41#1926—21no—
42#1921—21no—
43#1920—21no—
44#1865—21no—
45#1854—21no—
46#1854—21no—
47#1844C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1807—21no—
51#1802C2211yescharacteristic
52#1776—21no—
53C4×#106C2211yescharacteristic
54#1770C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117—21no—
75#475—44no—
76#474—22no—
77#474Q814yescomplemented
78C8×D4—22no—
79C8×D4—22no—
80#427—44no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106D411yescharacteristic
90C2×#106D411yescharacteristic
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102D411yescharacteristic
104C2×#102D411yescharacteristic
105C2×C22⋊C8—21no—
106C2×#97—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#395—41no—
111#391—21no—
112#390—21no—
113#390—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#388—41no—
118#388—41no—
119#387—41no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
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#340—41no—
143#340D411yescharacteristic
144#340D411yescharacteristic
145#340D411yescharacteristic
146#340D411yescharacteristic
147#336—21no—
148#336—21no—
149#336—21no—
150#335—22no—
151#332—22no—
152D4⋊C8—22no—
153C22×Q8—21no—
154C22×Q8—41no—
155C22×D4C2×D411yescharacteristic
156C23×C4—21no—
157C22×C8—24no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—42no—
164C22×C8—42no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—81no—
175C4⋊Q8—81no—
176C4⋊Q8C2×Q812yes—
177C4×Q8—42no—
178C4×Q8—81no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4C2×Q812yes—
182C4×D4#4112yes—
183#117—42no—
184#117—42no—
185#117—81no—
186#117—81no—
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—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42C2411yescharacteristic
220#107—24no—
221#107—42no—
222#107—42no—
223#106—22no—
224#106—22no—
225#106—42no—
226#106—42no—
227#106—42no—
228#106—42no—
229#106—81no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—42no—
237C4⋊C8—81no—
238C4⋊C8C2×Q812yes—
239C4⋊C8#4112yes—
240#103—42no—
241#103—42no—
242#103—42no—
243#103—81no—
244#102—24no—
245#102—22no—
246#102—41no—
247#102—81no—
248C22⋊C8—24no—
249#97—42no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264C24—21no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×Q8—161no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4C22⋊Q812yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—48no—
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×C8—41no—
327C2×C8AGL(1,ℤ/8)12yes—
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—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4C22⋊Q812yes—
362C4⋊C4#13512yes—
363C4⋊C4AGL(1,ℤ/8)12yes—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—161no—
369C22⋊C4—161no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—82no—
377C42—82no—
378C42—82no—
379C42C22×Q811yescharacteristic
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C422−1+411yescharacteristic
383C23—22no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23D4211yescharacteristic
391Q8—42no—
392Q8—42no—
393Q8—81no—
394Q8—81no—
395Q8—164no—
396D4—22no—
397D4—22no—
398D4—41no—
399D4—41no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
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—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4C2×C22⋊Q811yescharacteristic
440C2×C4#52511yescharacteristic
441C2×C4#53911yescharacteristic
442C2×C4#54711yescharacteristic
443C2×C4D4×Q811yescharacteristic
444C2×C4#55411yescharacteristic
445C2×C4C2×#13511yescharacteristic
446C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
447C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
448C2×C4#57511yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57711yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57811yescharacteristic
453C8—22no—
454C8—42no—
455C8—41no—
456C8—41no—
457C8—42no—
458C8—41no—
459C8—81no—
460C8—81no—
461C8—81no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22#244911yescharacteristic
470C22#326511yescharacteristic
471C22#326911yescharacteristic
472C22#327211yescharacteristic
473C22#328211yescharacteristic
474C22#328911yescharacteristic
475C22#329011yescharacteristic
476C4—21no—
477C4—21no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—82no—
484C4—81no—
485C4—81no—
486C4—161no—
487C4—161no—
488C4—161no—
489C4#307611yescharacteristic
490C4#308211yescharacteristic
491C4#310911yescharacteristic
492C4#312511yescharacteristic
493C4#334211yescharacteristic
494C4#334811yescharacteristic
495C4#338411yescharacteristic
496C4#338411yescharacteristic
497C2—82no—
498C2#1979511yescharacteristic
499C2#1991011yescharacteristic
500C2#2007711yescharacteristic
501C2#2022711yescharacteristic
502C2#2633711yescharacteristic
503C2#2639211yescharacteristic
504C2#2640411yescharacteristic
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. #26462C2semidirect
5. #10686C2semidirect
6. #10654C2semidirect
7. #10594C2semidirect
8. #10575C2semidirect
9. #9619C2semidirect
10. #9538C2semidirect
13. #8926C2semidirect
77. #474Q8semidirect

Maximal subgroups

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

subgroupindex
#264622
#260242
#230662
#226572
#106862
#106542
#105942
#105752
#96192
#95382
#92192
#89842
#89262
#88932
#88432