Finite Group Databasebeta

← #501310

Subgroups — #501310

Order 512.

867 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2184C211yesmaximal, characteristic
2C2×#2163C211yesmaximal, characteristic, complemented
3#33129C211yesmaximal, characteristic, complemented
4C2×#1970C211yesmaximal, characteristic, complemented
5#13476C212yesmaximal, complemented
6#13448C212yesmaximal, complemented
7#13239C212yesmaximal, complemented
8#13149C212yesmaximal, complemented
9#12991C211yesmaximal, characteristic, complemented
10#12920C211yesmaximal, characteristic, complemented
11#12731C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×D16C2212yes—
14C2×#466C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#102—21no—
17C22×C4×C8C2211yescharacteristic
18#2854—21no—
19C2×#390—21no—
20#2184—24no—
21#2163—24no—
22C2×#367C2212yescomplemented
23C2×C4⋊C16C2212yescomplemented
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×#357C2212yes—
27C2×#357C2212yescomplemented
28C2×C22⋊C16C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×#346C2212yescomplemented
31#1970—21no—
32#1970—21no—
33#1970C2212yes—
34#1970C2212yes—
35#1970C2211yescharacteristic, complemented
36#1970C2211yescharacteristic, complemented
37#1942C2211yescharacteristic, complemented
38#1942C2211yescharacteristic, complemented
39#1934—21no—
40#1934—21no—
41#1893—21no—
42#1808C2211yescharacteristic
43#1380C2212yescomplemented
44#1380C2212yescomplemented
45C23×D4—21no—
46C22×D8—24no—
47C22×D8—21no—
48C22×D8—22no—
49C22×D8—22no—
50C22×D8—42no—
51C22×D8C2311yescharacteristic
52C22×D8C2312yes—
53C23×C8D411yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#123—41no—
56C2×#121—21no—
57C22×C4⋊C4—21no—
58C22×C22⋊C4—41no—
59C22×C22⋊C4—41no—
60C22×C42D411yescharacteristic
61C2×D16—24no—
62C2×D16—24no—
63C2×D16—42no—
64C22×C16C2312yes—
65C22×C16C2312yes—
66#466—22no—
67#466—22no—
68C2×#107—21no—
69C2×#107—42no—
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2312yes—
74C2×#102—22no—
75C2×#102—22no—
76C2×#102—21no—
77C2×#102—21no—
78C2×#102—21no—
79C2×#102—44no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8D411yescharacteristic
83C2×C4×C8D411yescharacteristic
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#390—21no—
87#390—21no—
88#390—22no—
89#390—22no—
90#390—41no—
91#390—41no—
92#389—21no—
93#389—21no—
94#388—41no—
95#388—41no—
96#386—41no—
97#380—21no—
98#367—42no—
99C4⋊C16—24no—
100#357—24no—
101#357—24no—
102#357—42no—
103#357—42no—
104C22⋊C16—22no—
105C22⋊C16D414yes—
106#346—24no—
107#340—21no—
108#340—21no—
109#340—21no—
110C25—41no—
111C22×D4—24no—
112C22×D4—22no—
113C22×D4—21no—
114C22×D4—42no—
115C22×D4—42no—
116C22×D4—42no—
117C22×D4—41no—
118C22×D4—42no—
119C22×D4—41no—
120C22×D4—42no—
121C23×C4—41no—
122C23×C4D811yescharacteristic
123C23×C4D811yescharacteristic
124C23×C4C2×D411yescharacteristic
125C2×D8—24no—
126C2×D8—22no—
127C2×D8—21no—
128C2×D8—22no—
129C2×D8—22no—
130C2×D8—21no—
131C2×D8—22no—
132C2×D8—22no—
133C2×D8—22no—
134C2×D8—22no—
135C2×D8—48no—
136C2×D8—48no—
137C2×D8—42no—
138C2×D8—42no—
139C2×D8#4114yes—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—41no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D412yes—
148C22×C8C2×D411yescharacteristic
149C22×C8#4112yes—
150C22×C8C2411yescharacteristic
151#123—84no—
152#121—42no—
153#121—42no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—84no—
174C2×C22⋊C4—84no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42D811yescharacteristic
178C2×C42D811yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181D16—48no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—22no—
185C2×C16—22no—
186C2×C16—22no—
187C2×C16C2×D414yes—
188#107—21no—
189#107—22no—
190#107—21no—
191#107—22no—
192#107—22no—
193#107—22no—
194#107—42no—
195#102—42no—
196#102—44no—
197#102—42no—
198#102—42no—
199#102—44no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—21no—
206#95—41no—
207#95—41no—
208#95—41no—
209C24—44no—
210C24—42no—
211C24—41no—
212C24—82no—
213C24—82no—
214C24—82no—
215C24—82no—
216C24—82no—
217C24—82no—
218C24C2×D811yescharacteristic
219C2×D4—24no—
220C2×D4—48no—
221C2×D4—48no—
222C2×D4—44no—
223C2×D4—42no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—41no—
230C2×D4—42no—
231C2×D4—41no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—82no—
235C2×D4—84no—
236C2×D4—84no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4—82no—
240C2×D4—82no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4#12112yes—
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4#13512yes—
265C22×C4C22×D411yescharacteristic
266D8—24no—
267D8—24no—
268D8—48no—
269D8—42no—
270D8—42no—
271D8—42no—
272D8—42no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—44no—
283C2×C8#12112yes—
284C2×C8AGL(1,ℤ/8)12yes—
285C2×C8C22×D411yescharacteristic
286C2×C8C2×#4112yes—
287C2×C8C2×#4111yescharacteristic
288C2×C8H(5,𝔽2)12yes—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C22⋊C4—84no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—84no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—22no—
315C42—22no—
316C16—24no—
317C16—42no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—44no—
323C23—88no—
324C23—88no—
325C23—88no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—164no—
340C23—164no—
341C23—164no—
342C23—164no—
343C23#46612yes—
344C23#50912yes—
345C23#50912yes—
346C23C22×D811yescharacteristic
347D4—44no—
348D4—44no—
349D4—88no—
350D4—81no—
351D4—81no—
352D4—82no—
353D4—81no—
354D4—82no—
355D4—81no—
356D4—164no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—164no—
389C2×C4#46612yes—
390C2×C4#50912yes—
391C2×C4C2×#12111yescharacteristic
392C2×C4#52511yescharacteristic
393C2×C4#53511yescharacteristic
394C2×C4C2×#13511yescharacteristic
395C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
396C2×C4#57511yescharacteristic
397C8—22no—
398C8—22no—
399C8—41no—
400C8#54612yes—
401C8#55112yes—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—84no—
409C22—84no—
410C22—84no—
411C22—1616no—
412C22—162no—
413C22—162no—
414C22—162no—
415C22—162no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22#221814yes—
424C22C2×#46611yescharacteristic
425C22#310011yescharacteristic
426C22#320511yescharacteristic
427C22C2×#50911yescharacteristic
428C22C2×#50911yescharacteristic
429C22C2×#50911yescharacteristic
430C22#341411yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4—162no—
440C4#241111yescharacteristic
441C4#305511yescharacteristic
442C4#310511yescharacteristic
443C4#321111yescharacteristic
444C2—24no—
445C2—164no—
446C2—321no—
447C2#2450011yescharacteristic
448C2C2×#221811yescharacteristic
449C2#3360511yescharacteristic
450C2#3366911yescharacteristic
451C2#3367411yescharacteristic
452C2#3377911yescharacteristic
453C2#3378311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#2184C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2163C2semidirect
3. #33129C2semidirect
4. C2×#1970C2semidirect
5. #13476C2semidirect
6. #13448C2semidirect
7. #13239C2semidirect
8. #13149C2semidirect
9. #12991C2semidirect
10. #12920C2semidirect
11. #12731C2semidirect
22. C2×#367C22semidirect
23. C2×C4⋊C16C22semidirect
27. C2×#357C22semidirect
30. C2×#346C22semidirect
35. #1970C22semidirect
36. #1970C22semidirect
37. #1942C22semidirect
38. #1942C22semidirect
43. #1380C22semidirect
44. #1380C22semidirect

Maximal subgroups

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

subgroupindex
C2×#21842
C2×#21632
#331292
C2×#19702
#134762
#134482
#132392
#131492
#129912
#129202
#127312