Finite Group Databasebeta

← #316220

Subgroups — #316220

Order 512.

844 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27678C211yesmaximal, characteristic
2#24298C211yesmaximal, characteristic
3C8×#121C211yesmaximal, characteristic
4C2×#1956C211yesmaximal, characteristic, complemented
5#10600C212yesmaximal, complemented
6#10485C212yesmaximal, complemented
7#9562C212yesmaximal, complemented
8#9219C212yesmaximal, complemented
9#8725C211yesmaximal, characteristic, complemented
10#8596C211yesmaximal, characteristic, complemented
11#8340C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#474C2212yes—
14C2×C8×D4—21no—
15C2×C8×D4C2211yescharacteristic
16C22×#103—21no—
17C22×#103—21no—
18C22×C4×C8C2211yescharacteristic
19C4×#121C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2100—21no—
22#2094—21no—
23#2057—21no—
24#2023—21no—
25#1988C2212yes—
26#1956—21no—
27#1956—21no—
28#1956C2212yes—
29#1956C2212yes—
30#1954C2211yescharacteristic
31#1948C2212yes—
32#1939C2211yescharacteristic
33#1923—21no—
34#1874—21no—
35#1874—21no—
36#1869—21no—
37#1853—21no—
38#1852—21no—
39#1802C2212yes—
40#1797C2211yescharacteristic
41#1797C2211yescharacteristic
42#1797C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C22⋊C4—21no—
45C2×#335C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×Q8—21no—
52C23×C8—21no—
53C23×C8D411yescharacteristic
54C23×C8D411yescharacteristic
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8C2312yes—
61C2×#121D411yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4C2311yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C42C2311yescharacteristic
70#474—24no—
71#474—42no—
72C8×D4—42no—
73C8×D4—44no—
74C2×#107—22no—
75C2×#106—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—44no—
86C2×#103—44no—
87C2×#102—22no—
88C2×#102—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8—42no—
97C2×C4×C8D411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#390D411yescharacteristic
102#389D411yescharacteristic
103#388—21no—
104#386—21no—
105#384—21no—
106#384—21no—
107#384—21no—
108#384C2312yes—
109#380D411yescharacteristic
110C4×C4⋊C4C2311yescharacteristic
111C4×C22⋊C4—21no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#336—21no—
121#335—42no—
122#332—24no—
123#332—42no—
124C22×Q8—24no—
125C22×Q8—22no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—41no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×D4—21no—
133C22×D4C2×D411yescharacteristic
134C23×C4—21no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4#4111yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—41no—
149C22×C8—41no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8—42no—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C4⋊Q8—24no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—44no—
162C4⋊Q8—44no—
163C4⋊Q8—42no—
164C4⋊Q8#4114yes—
165#121—22no—
166#121Q1614yes—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—42no—
170C4×D4—42no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42C2411yescharacteristic
203#107—44no—
204#106—82no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—42no—
212#103—44no—
213#103—44no—
214#103—44no—
215#103—44no—
216#102—44no—
217#102—82no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—41no—
227C4×C8—41no—
228C4×C8—42no—
229#95—21no—
230#95—21no—
231#95—21no—
232C24—21no—
233C24#12111yescharacteristic
234C2×Q8—24no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—82no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—82no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—22no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4C2×Q1612yes—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4C2×Q1612yes—
284C22×C4#13512yes—
285C22×C4#13512yes—
286C22×C4AGL(1,ℤ/8)12yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—84no—
315C2×C8AGL(1,ℤ/8)12yes—
316C2×C8AGL(1,ℤ/8)12yes—
317C4⋊C4—21no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4C2×Q1612yes—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C42—21no—
342C42—21no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C42C2×#4112yes—
348C42H(5,𝔽2)12yes—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23#46312yes—
361C23#46712yes—
362C23#47112yes—
363C23C2×#12111yescharacteristic
364Q8—44no—
365Q8—44no—
366Q8—88no—
367Q8—82no—
368Q8—82no—
369D4—24no—
370D4—48no—
371D4—42no—
372C2×C4—28no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—44no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—88no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—164no—
408C2×C4#46312yes—
409C2×C4#46712yes—
410C2×C4#47312yes—
411C2×C4#52511yescharacteristic
412C2×C4#53411yescharacteristic
413C2×C4#54612yes—
414C2×C4#55112yes—
415C2×C4C22×Q1611yescharacteristic
416C2×C4C2×#13511yescharacteristic
417C2×C4C2×#13511yescharacteristic
418C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
419C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
420C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
421C2×C4#57712yes—
422C8—22no—
423C8—22no—
424C8—41no—
425C8—41no—
426C8—81no—
427C8—81no—
428C8—82no—
429C8—82no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22#168314yes—
444C22#253411yescharacteristic
445C22C2×#46311yescharacteristic
446C22#303811yescharacteristic
447C22C2×#46711yescharacteristic
448C22#305711yescharacteristic
449C22C2×#47111yescharacteristic
450C22C2×#47311yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—42no—
456C4—81no—
457C4—84no—
458C4—81no—
459C4—162no—
460C4#310511yescharacteristic
461C4#310911yescharacteristic
462C4#318811yescharacteristic
463C4#319211yescharacteristic
464C4#329912yes—
465C4#330112yes—
466C2—24no—
467C2—82no—
468C2C2×#168311yescharacteristic
469C2#1919711yescharacteristic
470C2#1926811yescharacteristic
471C2#1935211yescharacteristic
472C2#2809511yescharacteristic
473C2#2811711yescharacteristic
474C2#2813511yescharacteristic
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
4. C2×#1956C2semidirect
5. #10600C2semidirect
6. #10485C2semidirect
7. #9562C2semidirect
8. #9219C2semidirect
9. #8725C2semidirect
10. #8596C2semidirect
11. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#276782
#242982
C8×#1212
C2×#19562
#106002
#104852
#95622
#92192
#87252
#85962
#83402