Finite Group Databasebeta

← #316241

Subgroups — #316241

Order 512.

810 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27644C211yesmaximal, characteristic
2#24299C211yesmaximal, characteristic
3C8×#121C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5#10589C212yesmaximal, complemented
6#10480C212yesmaximal, complemented
7#9568C212yesmaximal, complemented
8#9215C212yesmaximal, complemented
9#8723C211yesmaximal, characteristic, complemented
10#8595C211yesmaximal, characteristic, complemented
11#8342C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#480—21no—
14C2×#476C2212yes—
15C2×C8×D4—21no—
16C2×C8×D4C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C4×#121C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2083—21no—
21#2071—21no—
22#2058—21no—
23#2023—21no—
24#1990—21no—
25#1990—21no—
26#1990C2212yes—
27#1990C2212yes—
28#1987C2212yes—
29#1954C2211yescharacteristic
30#1947C2212yes—
31#1939C2211yescharacteristic
32#1923—21no—
33#1874—21no—
34#1869—21no—
35#1853—21no—
36#1823—21no—
37#1802C2212yes—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C8×C22⋊C4—21no—
43C2×#340—21no—
44C2×#340—21no—
45C2×#334C2212yes—
46C2×#333C2212yes—
47C2×#333C2212yes—
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#480—44no—
71#476—24no—
72#476—42no—
73C8×D4—42no—
74C8×D4—44no—
75C2×#107—22no—
76C2×#106—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2312yes—
82C2×#102—22no—
83C2×#102—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—42no—
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#390D411yescharacteristic
97#389D411yescharacteristic
98#388—21no—
99#386—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384C2312yes—
104#380D411yescharacteristic
105C4×C4⋊C4C2311yescharacteristic
106C4×C22⋊C4—21no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#336—21no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#334—42no—
121#333—24no—
122#333—42no—
123C22×Q8—22no—
124C22×Q8—44no—
125C22×Q8—41no—
126C22×Q8—82no—
127C22×Q8—82no—
128C22×D4—21no—
129C22×D4C2×D411yescharacteristic
130C23×C4—21no—
131C23×C4—41no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4#4111yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C4⋊Q8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159C4⋊Q8—42no—
160C4⋊Q8#4114yes—
161#121—22no—
162#121SD1614yes—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—42no—
166C4×D4—42no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42C2411yescharacteristic
201#107—44no—
202#106—44no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—42no—
210#102—44no—
211#102—44no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215C22⋊C8—42no—
216C22⋊C8—42no—
217C22⋊C8—42no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—41no—
221C4×C8—41no—
222C4×C8—42no—
223#95—21no—
224#95—21no—
225#95—21no—
226C24—21no—
227C24#12111yescharacteristic
228C2×Q8—24no—
229C2×Q8—44no—
230C2×Q8—42no—
231C2×Q8—816no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4C2×SD1612yes—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4C2×SD1612yes—
273C22×C4#13512yes—
274C22×C4#13512yes—
275C22×C4AGL(1,ℤ/8)12yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—84no—
304C2×C8#13712yes—
305C2×C8#13712yes—
306C4⋊C4—21no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4C2×SD1612yes—
325C22⋊C4—24no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C42—21no—
331C42—21no—
332C42—22no—
333C42—22no—
334C42—41no—
335C42—41no—
336C42C2×#4112yes—
337C42H(5,𝔽2)12yes—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23#46312yes—
350C23#46512yes—
351C23#47212yes—
352C23C2×#12111yescharacteristic
353Q8—84no—
354Q8—84no—
355Q8—88no—
356D4—24no—
357D4—48no—
358D4—42no—
359C2×C4—28no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—88no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—164no—
395C2×C4#46312yes—
396C2×C4#46512yes—
397C2×C4#47112yes—
398C2×C4#52511yescharacteristic
399C2×C4#53411yescharacteristic
400C2×C4#54612yes—
401C2×C4#55112yes—
402C2×C4C22×SD1611yescharacteristic
403C2×C4C2×#13511yescharacteristic
404C2×C4C2×#13511yescharacteristic
405C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
406C2×C4C2×#13711yescharacteristic
407C2×C4C2×#13711yescharacteristic
408C2×C4#57812yes—
409C8—22no—
410C8—22no—
411C8—41no—
412C8—41no—
413C8—81no—
414C8—81no—
415C8—82no—
416C8—82no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22#167714yes—
431C22#253411yescharacteristic
432C22C2×#46311yescharacteristic
433C22#303911yescharacteristic
434C22C2×#46511yescharacteristic
435C22#305911yescharacteristic
436C22C2×#47111yescharacteristic
437C22C2×#47211yescharacteristic
438C4—22no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—42no—
443C4—81no—
444C4—84no—
445C4—81no—
446C4—162no—
447C4#310811yescharacteristic
448C4#311211yescharacteristic
449C4#319011yescharacteristic
450C4#319311yescharacteristic
451C4#329812yes—
452C4#330012yes—
453C2—24no—
454C2—82no—
455C2C2×#167711yescharacteristic
456C2#1919911yescharacteristic
457C2#1927011yescharacteristic
458C2#1935911yescharacteristic
459C2#2809811yescharacteristic
460C2#2811811yescharacteristic
461C2#2813411yescharacteristic
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×#1990C2semidirect
5. #10589C2semidirect
6. #10480C2semidirect
7. #9568C2semidirect
8. #9215C2semidirect
9. #8723C2semidirect
10. #8595C2semidirect
11. #8342C2semidirect

Maximal subgroups

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

subgroupindex
#276442
#242992
C8×#1212
C2×#19902
#105892
#104802
#95682
#92152
#87232
#85952
#83422