Finite Group Databasebeta

← #323899

Subgroups — #323899

Order 512.

738 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33172C211yesmaximal, characteristic, complemented
2#24844C212yesmaximal
3C8×#121C211yesmaximal, characteristic
4#10652C211yesmaximal, characteristic, complemented
5#10616C211yesmaximal, characteristic, complemented
6#10538C211yesmaximal, characteristic, complemented
7#10274C211yesmaximal, characteristic, complemented
8#9562C212yesmaximal, complemented
9#9219C212yesmaximal, complemented
10#9137C212yesmaximal, complemented
11#7874C211yesmaximal, characteristic, complemented
12C2×#494C2211yescharacteristic
13C2×#474C2212yes—
14C2×#467—21no—
15C2×#467—21no—
16C2×C8×D4—21no—
17C2×C8×D4C2211yescharacteristic
18C2×#423C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2857C2212yes—
21C4×#121C2211yescharacteristic
22#2087—22no—
23#2065—22no—
24#1988C2212yes—
25#1969—21no—
26#1969—21no—
27#1956C2212yes—
28#1954C2211yescharacteristic
29#1948C2212yes—
30#1939C2211yescharacteristic
31#1923—21no—
32#1895—22no—
33#1869—21no—
34#1845C2211yescharacteristic
35#1809—22no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C8×C22⋊C4—21no—
40C2×#332C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×Q16—21no—
48C22×Q16—21no—
49C22×M16—22no—
50C23×C8—21no—
51C23×C8D411yescharacteristic
52C23×C8D411yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×C22⋊Q8—22no—
55C2×#121D411yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#494—42no—
60#474—24no—
61#474—42no—
62#467—44no—
63#467—44no—
64C8×D4—42no—
65C8×D4—44no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107—22no—
69C2×#106—22no—
70#423—24no—
71#423—42no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×#103—22no—
79C2×#103—22no—
80C2×#102—22no—
81C2×#102—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—42no—
88C2×C4×C8D411yescharacteristic
89C2×C4×C8D411yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#391—22no—
93#390D411yescharacteristic
94#389—22no—
95#389—22no—
96#389D411yescharacteristic
97#388—21no—
98#386—21no—
99#384C2312yes—
100#380—22no—
101#380—22no—
102#380D411yescharacteristic
103C4×C4⋊C4C2311yescharacteristic
104C4×C22⋊C4—21no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#336—21no—
110#336—21no—
111#336—21no—
112#332—24no—
113#332—42no—
114C22×Q8—22no—
115C22×D4—21no—
116C22×D4C2×D411yescharacteristic
117C23×C4—21no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C2×Q16—42no—
122C2×Q16—42no—
123C2×Q16—42no—
124C2×Q16—42no—
125C2×Q16—42no—
126C2×Q16—42no—
127C2×M16—44no—
128C2×M16—44no—
129C2×M16—82no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C22⋊Q8—44no—
152C22⋊Q8—44no—
153#121—22no—
154#121Q1614yescomplemented
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—42no—
158C4×D4—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—42no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42C2411yescharacteristic
182#107—44no—
183#107—42no—
184#107—42no—
185#106—82no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—44no—
192C4⋊C8—42no—
193C4⋊C8#4114yes—
194#103—44no—
195#103—44no—
196#102—44no—
197#102—82no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200C22⋊C8—42no—
201C22⋊C8—42no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—41no—
205C4×C8—41no—
206C4×C8—42no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—42no—
211#95—42no—
212C24—21no—
213C24#12111yescharacteristic
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—21no—
223C2×D4—22no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4C2×Q1612yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4C2×Q1612yes—
253C22×C4#13512yes—
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C2×#4111yescharacteristic
257Q16—84no—
258Q16—84no—
259M16—88no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—24no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—84no—
284C2×C8AGL(1,ℤ/8)12yes—
285C2×C8AGL(1,ℤ/8)12yes—
286C2×C8AGL(1,ℤ/8)12yes—
287C4⋊C4—21no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—82no—
300C4⋊C4—84no—
301C4⋊C4C2×Q1612yes—
302C22⋊C4—24no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—84no—
308C42—21no—
309C42—21no—
310C42—22no—
311C42—22no—
312C42—41no—
313C42—41no—
314C42C2×#4112yes—
315C42H(5,𝔽2)12yes—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23#46712yes—
330C23C2×#12111yescharacteristic
331Q8—82no—
332Q8—82no—
333Q8—82no—
334Q8—82no—
335D4—24no—
336D4—48no—
337D4—42no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4#46712yes—
374C2×C4#47312yes—
375C2×C4#53211yescharacteristic
376C2×C4#53411yescharacteristic
377C2×C4#54612yes—
378C2×C4#55312yes—
379C2×C4C22×Q1611yescharacteristic
380C2×C4C2×#13511yescharacteristic
381C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
382C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
384C2×C4#57511yescharacteristic
385C2×C4#57712yes—
386C8—22no—
387C8—22no—
388C8—22no—
389C8—41no—
390C8—44no—
391C8—81no—
392C8—81no—
393C8—82no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#254311yescharacteristic
409C22#303611yescharacteristic
410C22#303811yescharacteristic
411C22C2×#46711yescharacteristic
412C22#306011yescharacteristic
413C22C2×#47311yescharacteristic
414C22#306711yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—42no—
419C4—41no—
420C4—81no—
421C4—81no—
422C4—162no—
423C4—162no—
424C4#317111yescharacteristic
425C4#318411yescharacteristic
426C4#319211yescharacteristic
427C4#319411yescharacteristic
428C4#329612yes—
429C4#329912yes—
430C2—42no—
431C2—82no—
432C2#1925611yescharacteristic
433C2#1926811yescharacteristic
434C2#1933811yescharacteristic
435C2#1935511yescharacteristic
436C2#2855311yescharacteristic
437C2#2857811yescharacteristic
438C2#2859111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
1. #33172C2semidirect
4. #10652C2semidirect
5. #10616C2semidirect
6. #10538C2semidirect
7. #10274C2semidirect
8. #9562C2semidirect
9. #9219C2semidirect
10. #9137C2semidirect
11. #7874C2semidirect
154. #121Q16semidirect

Maximal subgroups

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

subgroupindex
#331722
#248442
C8×#1212
#106522
#106162
#105382
#102742
#95622
#92192
#91372
#78742