Finite Group Databasebeta

← #241363

Subgroups — #241363

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5#10580C212yesmaximal, complemented
6#10579C212yesmaximal, complemented
7#10574C212yesmaximal, complemented
8#10573C212yesmaximal, complemented
9#8043C211yesmaximal, characteristic
10#7903C211yesmaximal, characteristic
11#7896C211yesmaximal, characteristic
12C2×#544—21no—
13C2×#431C2211yescharacteristic
14C2×#428—22no—
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2868—21no—
18#2859—21no—
19#2660—22no—
20#2552—21no—
21#2552C2212yes—
22#2551—21no—
23#2551C2212yes—
24#2440—22no—
25#2440—22no—
26#2317—21no—
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1948—21no—
32#1948C2211yescharacteristic
33#1948C2212yes—
34#1948C2211yescharacteristic
35#1947—21no—
36#1947C2211yescharacteristic
37#1947C2212yes—
38#1947C2211yescharacteristic
39#1854—21no—
40#1854—21no—
41#1847—22no—
42#1845C2212yes—
43#1844C2212yes—
44#1809—22no—
45C4×#107—21no—
46C4×#106—21no—
47#1296C2212yes—
48#1296C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C22×M16—21no—
54C22×M16—21no—
55#544—44no—
56#544—44no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×#125—22no—
62C2×#125—22no—
63C2×C22⋊Q8—24no—
64C2×#117—21no—
65C2×#117—21no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C42C2311yescharacteristic
72#431—22no—
73#431Q814yescomplemented
74#428—48no—
75C2×#107—24no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—22no—
79C2×#106—24no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—22no—
83#423—22no—
84#423Q814yescomplemented
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C22⋊C8—21no—
90C4×M16—22no—
91C4×M16Q814yescomplemented
92C2×#97—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2312yes—
95#398—24no—
96#398—24no—
97#395—24no—
98#395—24no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#385—24no—
106#384—22no—
107#384—22no—
108#384—22no—
109#384—21no—
110#384—21no—
111#384D411yescharacteristic
112#384D411yescharacteristic
113#384D412yes—
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2312yes—
117#382—22no—
118#382—21no—
119#382—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380D412yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2312yes—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134#376—21no—
135C43—21no—
136#340—22no—
137#340—22no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142C22×Q8—24no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—24no—
149C2×M16—24no—
150C2×M16—24no—
151C2×M16—81no—
152C2×M16—81no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C4⋊Q8—48no—
158C4⋊Q8—44no—
159C4⋊Q8—44no—
160C4⋊Q8—82no—
161C4⋊Q8—82no—
162C4⋊Q8—84no—
163#125—48no—
164#125—44no—
165#125—44no—
166C22⋊Q8—416no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—42no—
172#117—42no—
173#117—44no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
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—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D414yes—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—41no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—41no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×Q812yes—
222C2×C42C2×Q812yes—
223C2×C42C2×Q812yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226#107—48no—
227#107—42no—
228#107—42no—
229#107—82no—
230#106—48no—
231#106—42no—
232#106—42no—
233#106—82no—
234C4⋊C8—24no—
235C4⋊C8—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8C2×Q814yes—
239C4⋊C8C2×Q812yes—
240C4⋊C8C2×Q812yes—
241C22⋊C8—24no—
242#97—24no—
243C4×C8—24no—
244C4×C8—22no—
245C4×C8C2×Q814yes—
246#95—24no—
247#95—24no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95—22no—
254#95—41no—
255#95—41no—
256C24#12011yescharacteristic
257C2×Q8—48no—
258C2×Q8—84no—
259C2×Q8—84no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—44no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4C22⋊Q812yes—
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4C4⋊Q812yes—
289C22×C4C4⋊Q812yes—
290C22×C4C4⋊Q812yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294M16—216no—
295M16—84no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C4⋊C4—48no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—42no—
331C4⋊C4—42no—
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—82no—
341C4⋊C4—84no—
342C4⋊C4—84no—
343C4⋊C4AGL(1,ℤ/8)12yes—
344C4⋊C4AGL(1,ℤ/8)12yes—
345C4⋊C4#13712yes—
346C4⋊C4#13712yes—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—44no—
350C22⋊C4—82no—
351C42—24no—
352C42—24no—
353C42—42no—
354C42—42no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—42no—
360C42—42no—
361C42—82no—
362C42C22⋊Q812yes—
363C42C22⋊Q812yes—
364C42C22⋊Q812yes—
365C42C4⋊Q812yes—
366C42C4⋊Q812yes—
367C42C4⋊Q812yes—
368C42C22×Q811yescharacteristic
369C42C22×Q811yescharacteristic
370C42C22×Q811yescharacteristic
371C42C2×#4111yescharacteristic
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23#39512yes—
379C23C2×#12011yescharacteristic
380Q8—168no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
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—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—164no—
425C2×C4#39512yes—
426C2×C4#39512yes—
427C2×C4#39512yes—
428C2×C4#45312yes—
429C2×C4#45612yes—
430C2×C4C2×C22⋊Q811yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×C4⋊Q811yescharacteristic
434C2×C4C2×C4⋊Q811yescharacteristic
435C2×C4C2×C4⋊Q811yescharacteristic
436C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4C2×#13711yescharacteristic
440C2×C4#57611yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57711yescharacteristic
443C2×C4#57811yescharacteristic
444C8—28no—
445C8—42no—
446C8—42no—
447C8—44no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22C2×#39511yescharacteristic
456C22#300311yescharacteristic
457C22#300411yescharacteristic
458C22#301211yescharacteristic
459C22#301311yescharacteristic
460C22C2×#45311yescharacteristic
461C22C2×#45611yescharacteristic
462C4—22no—
463C4—22no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—84no—
471C4—82no—
472C4—82no—
473C4—161no—
474C4—161no—
475C4#206014yes—
476C4#307911yescharacteristic
477C4#307911yescharacteristic
478C4#308111yescharacteristic
479C4#308211yescharacteristic
480C4#316311yescharacteristic
481C4#316411yescharacteristic
482C4#316511yescharacteristic
483C4#316511yescharacteristic
484C2—42no—
485C2#1856611yescharacteristic
486C2#1856711yescharacteristic
487C2#1859011yescharacteristic
488C2#1859111yescharacteristic
489C2#2355211yescharacteristic
490C2#2355711yescharacteristic
491C2C2×#206011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10580C2semidirect
6. #10579C2semidirect
7. #10574C2semidirect
8. #10573C2semidirect
73. #431Q8semidirect
84. #423Q8semidirect
91. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#219642
#213622
#208002
#105802
#105792
#105742
#105732
#80432
#79032
#78962