Finite Group Databasebeta

← #306139

Subgroups — #306139

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28817C211yesmaximal, characteristic, complemented
2#26090C211yesmaximal, characteristic
3#26044C211yesmaximal, characteristic
4#22663C211yesmaximal, characteristic
5#10654C211yesmaximal, characteristic, complemented
6#10653C211yesmaximal, characteristic, complemented
7#10580C211yesmaximal, characteristic, complemented
8#10574C211yesmaximal, characteristic, complemented
9#9618C211yesmaximal, characteristic, complemented
10#9594C211yesmaximal, characteristic, complemented
11#8889C211yesmaximal, characteristic
12#8886C211yesmaximal, characteristic
13#8813C211yesmaximal, characteristic, complemented
14#8650C211yesmaximal, characteristic
15#8206C211yesmaximal, characteristic
16C2×#501—21no—
17C2×#476—21no—
18C2×#436C2211yescharacteristic
19C2×#428—21no—
20#2552C2211yescharacteristic
21#2551C2211yescharacteristic
22#2354C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#2094—21no—
25#2088—21no—
26#2065—21no—
27#2059—21no—
28#2058—21no—
29#2058—21no—
30#1948—21no—
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1943—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1920—21no—
41#1864—21no—
42#1854—21no—
43#1853—21no—
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1844C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1809—21no—
51#1802C2211yescharacteristic
52#1802C2211yescharacteristic
53C4×#107C2211yescharacteristic
54#1770C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×#125—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C22×C4⋊C4—21no—
75#501—44no—
76#476—44no—
77#436—22no—
78#436Q814yescomplemented
79#428—44no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107D411yescharacteristic
83C2×#107D411yescharacteristic
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—41no—
97C2×#106—41no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#102—21no—
105C2×#102—21no—
106C2×#102D411yescharacteristic
107C2×#102D411yescharacteristic
108C2×C22⋊C8—21no—
109C2×#97—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#398—41no—
114#395—41no—
115#390—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#388—41no—
122#385—41no—
123#384—21no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#380—21no—
129#380—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
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—
150D4⋊C8Q814yescomplemented
151C82—22no—
152C22×Q8—41no—
153C22×D4C2×D411yescharacteristic
154C23×C4—21no—
155C23×C4—41no—
156C2×M16—24no—
157C2×M16—42no—
158C2×M16—42no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—81no—
171C4⋊Q8—81no—
172#125—42no—
173#125—81no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—42no—
177C4×D4—81no—
178C4×D4C2×Q812yes—
179C4×D4C2×Q812yes—
180#117—42no—
181#117—81no—
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—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—81no—
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×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—21no—
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×C42C2411yescharacteristic
218#107—22no—
219#107—42no—
220#107—42no—
221#107SD1614yes—
222#106—42no—
223#106—42no—
224#106—42no—
225#106—42no—
226#106—42no—
227#106—42no—
228#106—42no—
229#106—81no—
230#106—81no—
231#106—81no—
232#106—81no—
233#106—81no—
234#106—82no—
235#106—82no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C4⋊C8C2×Q812yes—
245C4⋊C8C2×Q812yes—
246#102—22no—
247#102—22no—
248#102—42no—
249#102—81no—
250C22⋊C8—24no—
251#97—42no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8C2×Q812yes—
256C4×C8C2×Q812yes—
257C4×C8#4112yes—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265C24—21no—
266C2×Q8—82no—
267C2×Q8—82no—
268C2×Q8—161no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4C4⋊Q812yes—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—161no—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4H(5,𝔽2)11yescharacteristic
298M16—48no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8C22⋊Q812yes—
321C2×C8C22⋊Q812yes—
322C2×C8C2×SD1612yes—
323C2×C8AGL(1,ℤ/8)12yes—
324C2×C8AGL(1,ℤ/8)12yes—
325C2×C8#13712yes—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
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—82no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—161no—
363C4⋊C4C4⋊Q812yes—
364C4⋊C4C2×SD1612yes—
365C4⋊C4C2×SD1612yes—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4—161no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—82no—
378C42C22×Q811yescharacteristic
379C42C22×Q811yescharacteristic
380C42C2×#4111yescharacteristic
381C422−1+411yescharacteristic
382C23—22no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23D4211yescharacteristic
390Q8—164no—
391D4—22no—
392D4—22no—
393D4—41no—
394D4—41no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
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—81no—
423C2×C4—81no—
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—164no—
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×C4⋊Q811yescharacteristic
437C2×C4#54411yescharacteristic
438C2×C4#54711yescharacteristic
439C2×C4D4×Q811yescharacteristic
440C2×C4#55411yescharacteristic
441C2×C4C22×SD1611yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57711yescharacteristic
449C8—24no—
450C8—22no—
451C8—22no—
452C8—22no—
453C8—42no—
454C8—42no—
455C8—81no—
456C8—81no—
457C8#47614yescomplemented
458C22—42no—
459C22—42no—
460C22—42no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22#244011yescharacteristic
466C22#327211yescharacteristic
467C22#327211yescharacteristic
468C22#327811yescharacteristic
469C22D4×SD1611yescharacteristic
470C22#329111yescharacteristic
471C22#329111yescharacteristic
472C4—21no—
473C4—21no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4C2×#47611yescharacteristic
486C4#308411yescharacteristic
487C4#316311yescharacteristic
488C4#316511yescharacteristic
489C4#334411yescharacteristic
490C4#334511yescharacteristic
491C4Q8×SD1611yescharacteristic
492C4#338411yescharacteristic
493C2—82no—
494C2#1989011yescharacteristic
495C2#1989711yescharacteristic
496C2#2031411yescharacteristic
497C2#2031711yescharacteristic
498C2#2604411yescharacteristic
499C2#2607811yescharacteristic
500C2#2608111yescharacteristic
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. #28817C2semidirect
5. #10654C2semidirect
6. #10653C2semidirect
7. #10580C2semidirect
8. #10574C2semidirect
9. #9618C2semidirect
10. #9594C2semidirect
13. #8813C2semidirect
78. #436Q8semidirect
150. D4⋊C8Q8semidirect
457. C8#476semidirect

Maximal subgroups

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

subgroupindex
#288172
#260902
#260442
#226632
#106542
#106532
#105802
#105742
#96182
#95942
#88892
#88862
#88132
#86502
#82062