Finite Group Databasebeta

← #484097

Subgroups — #484097

Order 512.

491 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2177C211yesmaximal, characteristic
2#18064C211yesmaximal, characteristic, complemented
3#18044C211yesmaximal, characteristic, complemented
4#12089C212yesmaximal, complemented
5#12078C212yesmaximal, complemented
6#12078C212yesmaximal, complemented
7#12067C212yesmaximal, complemented
8#12019C211yesmaximal, characteristic, complemented
9#11667C211yesmaximal, characteristic, complemented
10#11663C211yesmaximal, characteristic, complemented
11#11413C211yesmaximal, characteristic, complemented
12C22×D16C411yescharacteristic
13#2974—21no—
14C2×C4×D8C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16D4×M16—21no—
17#2177C411yescharacteristic
18#2177C411yescharacteristic
19#2177C411yescharacteristic
20#2177C411yescharacteristic
21#2177C2211yescharacteristic
22#2177C2211yescharacteristic
23#2177C2211yescharacteristic
24#2177C2211yescharacteristic
25#2163C2211yescharacteristic
26C2×#367C411yescharacteristic
27C2×#357C411yescharacteristic
28C2×#357C411yescharacteristic
29C2×#346C2211yescharacteristic
30#1594C2212yescomplemented
31#1594C2212yescomplemented
32#1586—22no—
33#1581—22no—
34#1580C2212yescomplemented
35#1580C2212yescomplemented
36#1568C2211yescharacteristic
37#1566C2211yescharacteristic
38#1490—21no—
39#1486—21no—
40#1451C2211yescharacteristic
41#1368C2212yescomplemented
42#1368C2212yescomplemented
43#1333C2212yescomplemented
44#1333C2212yescomplemented
45#1333C2212yescomplemented
46#1333C2212yescomplemented
47#1313C2212yescomplemented
48#1313C2212yescomplemented
49C22×D8C2×C411yescharacteristic
50C22×D8C2×C411yescharacteristic
51C22×M16—41no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×D16—24no—
55C2×D16C2×C411yescharacteristic
56C2×D16C2×C411yescharacteristic
57C2×D16C2×C411yescharacteristic
58C2×D16C2×C411yescharacteristic
59C22×C16C2×C411yescharacteristic
60C22×C16C2×C411yescharacteristic
61C4×D8—21no—
62C4×D8—21no—
63C4×D8—21no—
64C4×D8—21no—
65C4×D8C2×C411yescharacteristic
66C4×D8C2×C411yescharacteristic
67C4×D8C2×C411yescharacteristic
68C4×D8C2×C411yescharacteristic
69C4×D8C2311yescharacteristic
70C4×D8C2311yescharacteristic
71C4×D8C2311yescharacteristic
72C4×D8C2311yescharacteristic
73#436—22no—
74#436—22no—
75#435—42no—
76#435—42no—
77C8×D4—22no—
78C8×D4—22no—
79#432—21no—
80#431D411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83#423—21no—
84C2×#102—21no—
85C2×#102—21no—
86#407—41no—
87#407—41no—
88C2×C22⋊C8—41no—
89C4×M16D411yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#367C2×C411yescharacteristic
92#367C2×C411yescharacteristic
93#367C2×C411yescharacteristic
94#367C2×C411yescharacteristic
95C4⋊C16—21no—
96#357C2×C411yescharacteristic
97#357C2×C411yescharacteristic
98#357C2×C411yescharacteristic
99#357C2×C411yescharacteristic
100#357C2×C411yescharacteristic
101#357C2×C411yescharacteristic
102#357C2×C411yescharacteristic
103#357C2×C411yescharacteristic
104C22⋊C16—21no—
105#346C2×C411yescharacteristic
106#346C2×C411yescharacteristic
107#346C2312yes—
108#346C2311yescharacteristic
109#346C2311yescharacteristic
110#335—21no—
111#334—21no—
112#334C2312yes—
113#334C2312yes—
114D4⋊C8—22no—
115D4⋊C8—22no—
116D4⋊C8—22no—
117D4⋊C8—22no—
118#322D412yes—
119#322C2312yes—
120C22×D4—21no—
121C22×D4—21no—
122C23×C4—41no—
123C23×C4—41no—
124C2×D8—22no—
125C2×D8—22no—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—41no—
129C2×D8—41no—
130C2×D8C22×C411yescharacteristic
131C2×D8C22×C411yescharacteristic
132C2×D8C22×C411yescharacteristic
133C2×D8C22×C411yescharacteristic
134C2×M16—21no—
135C2×M16—42no—
136C2×M16—42no—
137C2×M16—41no—
138C2×M16—84no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8C22×C411yescharacteristic
142C22×C8#4111yescharacteristic
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154C4×D4—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C42C2×D411yescharacteristic
160D16—22no—
161D16—22no—
162D16—22no—
163D16M1614yescomplemented
164C2×C16—22no—
165C2×C16—22no—
166C2×C16—41no—
167C2×C16C22×C411yescharacteristic
168C2×C16C22×C411yescharacteristic
169C2×C16C22×C411yescharacteristic
170C2×C16C22×C411yescharacteristic
171#107—21no—
172#107—21no—
173#107C22×C411yescharacteristic
174#107C22×C411yescharacteristic
175#107C22×C411yescharacteristic
176#107C22×C411yescharacteristic
177C4⋊C8—21no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—41no—
181#102—21no—
182#102—21no—
183#102—21no—
184#102—21no—
185#102—21no—
186#102—21no—
187#102—21no—
188#102—21no—
189C22⋊C8—21no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—41no—
195#97—21no—
196#97—41no—
197C4×C8C22×C411yescharacteristic
198C4×C8C2×D412yes—
199C4×C8C2×D412yes—
200C4×C8C2×D411yescharacteristic
201C4×C8#4111yescharacteristic
202C4×C8C2411yescharacteristic
203C24—41no—
204C24—41no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—81no—
214C2×D4—81no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—162no—
228C22×C4C4×D411yescharacteristic
229C22×C4AGL(1,ℤ/8)11yescharacteristic
230C22×C4AGL(1,ℤ/8)11yescharacteristic
231D8—21no—
232D8—21no—
233D8—21no—
234D8—21no—
235D8—21no—
236D8—21no—
237D8—42no—
238D8—42no—
239D8C2×M1612yes—
240D8C2×M1612yescomplemented
241M16—42no—
242M16—84no—
243M16—82no—
244M16—162no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—84no—
253C2×C8—162no—
254C2×C8C4×D411yescharacteristic
255C2×C8C4×D411yescharacteristic
256C2×C8C23×C411yescharacteristic
257C2×C8C2×#4111yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C42C4×D411yescharacteristic
273C42AGL(1,ℤ/8)11yescharacteristic
274C42AGL(1,ℤ/8)11yescharacteristic
275C42C22×D411yescharacteristic
276C16—21no—
277C16—21no—
278C16—21no—
279C16—42no—
280C16C2×M1612yescomplemented
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—82no—
286C23—82no—
287C23—82no—
288C23—82no—
289C23—161no—
290C23—161no—
291C23#44211yescharacteristic
292D4—22no—
293D4—22no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—82no—
301D4—82no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—161no—
320C2×C4—161no—
321C2×C4—161no—
322C2×C4#44211yescharacteristic
323C2×C4#44211yescharacteristic
324C2×C4#44211yescharacteristic
325C2×C4C2×C4×D411yescharacteristic
326C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
327C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
328C8—21no—
329C8—21no—
330C8—21no—
331C8—42no—
332C8—42no—
333C8—82no—
334C8—82no—
335C8C22×M1611yescharacteristic
336C8#56811yescharacteristic
337C22—21no—
338C22—21no—
339C22—42no—
340C22—42no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—164no—
352C22—164no—
353C22AGL(1,ℤ/16)12yes—
354C22C2×#44211yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4D4×M1611yescharacteristic
365C4#295711yescharacteristic
366C2—22no—
367C2—82no—
368C2—82no—
369C2—161no—
370C2—161no—
371C2#1804411yescharacteristic
372C2C2×AGL(1,ℤ/16)11yescharacteristic
373C2#3347111yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radicalC22×D16C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18064C2semidirect
3. #18044C2semidirect
4. #12089C2semidirect
5. #12078C2semidirect
6. #12078C2semidirect
7. #12067C2semidirect
8. #12019C2semidirect
9. #11667C2semidirect
10. #11663C2semidirect
11. #11413C2semidirect
30. #1594C22semidirect
31. #1594C22semidirect
34. #1580C22semidirect
35. #1580C22semidirect
41. #1368C22semidirect
42. #1368C22semidirect
43. #1333C22semidirect
44. #1333C22semidirect
45. #1333C22semidirect
46. #1333C22semidirect
47. #1313C22semidirect
48. #1313C22semidirect
163. D16M16semidirect
240. D8C2×M16semidirect
280. C16C2×M16semidirect

Maximal subgroups

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

subgroupindex
C2×#21772
#180642
#180442
#120892
#120782
#120672
#120192
#116672
#116632
#114132