Finite Group Databasebeta

← #211797

Subgroups — #211797

Order 512.

633 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21734C211yesmaximal, characteristic, complemented
2#20495C211yesmaximal, characteristic
3#19137C211yesmaximal, characteristic
4#17902C212yesmaximal
5#9749C211yesmaximal, characteristic, complemented
6#9628C211yesmaximal, characteristic, complemented
7#9579C212yesmaximal, complemented
8#9235C212yesmaximal
9#8662C211yesmaximal, characteristic
10#8268C211yesmaximal, characteristic, complemented
11#8256C212yesmaximal, complemented
12#3160C2211yescharacteristic
13#3101C2212yes—
14#2960C2212yes—
15C2×#436C2211yescharacteristic
16C2×C8×D4C2211yescharacteristic
17C2×#405—21no—
18#2293C2211yescharacteristic
19#1994C2212yes—
20#1946—21no—
21#1946—21no—
22#1848C2211yescharacteristic
23#1773—21no—
24#1773—21no—
25#1770C2211yescharacteristic
26C8×C4⋊C4C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28#1753—21no—
29#1752—21no—
30#1750—21no—
31C4×#97C2211yescharacteristic
32#1695C412yescomplemented
33#1693C412yescomplemented
34#1691C414yescomplemented
35#1582—24no—
36C8×SD16—24no—
37#1570C2212yes—
38#1564C2211yescharacteristic
39#1564C2211yescharacteristic
40#1542C2212yes—
41#1503C2212yes—
42#1486C2212yes—
43#1481C2212yes—
44#1450C2212yes—
45#1303C2212yes—
46#1301C2212yes—
47#1276C2212yes—
48C22×M16—21no—
49C23×C8—21no—
50#533C2312yes—
51C2×C4×D4C2311yescharacteristic
52C2×#117—21no—
53#500C2×C411yescharacteristic
54#500C2×C411yescharacteristic
55#499—21no—
56#499—21no—
57#498C2×C412yes—
58#483—22no—
59#476C2×C414yes—
60#461C2×C414yes—
61#450—22no—
62#446—22no—
63#446—22no—
64C8×Q8—22no—
65C8×Q8—22no—
66C4×SD16—24no—
67#436—21no—
68#436—21no—
69#436C2×C412yes—
70#436C2×C412yes—
71C8×D4—22no—
72C8×D4D414yes—
73#431C2312yes—
74#431C2312yes—
75C2×#108—21no—
76C2×#108—21no—
77#427—21no—
78#427—21no—
79#424—22no—
80#423C2312yes—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85#405—22no—
86#405—44no—
87C2×#97—21no—
88C2×#97D411yescharacteristic
89C2×#97D411yescharacteristic
90C2×#97D411yescharacteristic
91C2×#97D411yescharacteristic
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#390—21no—
97#389—21no—
98#388—22no—
99C4×C4⋊C4C2311yescharacteristic
100C4×C22⋊C4—21no—
101C43C2311yescharacteristic
102#339—22no—
103#339—22no—
104#336—21no—
105#336—21no—
106#336—21no—
107#335—21no—
108#335C2×C412yes—
109#334—24no—
110#334—21no—
111#334C2×C412yes—
112#333C2×C414yes—
113#332C2×C414yes—
114#328—24no—
115Q8⋊C8—24no—
116D4⋊C8—24no—
117#322—22no—
118C82—22no—
119C22×D4#4111yescharacteristic
120C23×C4—21no—
121C2×SD16—24no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—42no—
126C2×M16—42no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—24no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C4⋊Q8C22×C412yes—
137C4⋊Q8C22×C412yes—
138#125—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×D4—21no—
144C4×D4—21no—
145C4×D4C22×C412yes—
146C4×D4C2×D412yes—
147#117—22no—
148#117—42no—
149#117—42no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4#4111yescharacteristic
153C2×C22⋊C4—21no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2411yescharacteristic
162#108—42no—
163#108—42no—
164#107—21no—
165#107—21no—
166#107—21no—
167#107—21no—
168#106—22no—
169#106—22no—
170#106—24no—
171C4⋊C8—22no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—42no—
177C4⋊C8—42no—
178C4⋊C8C22×C412yes—
179C4⋊C8C22×C414yes—
180C4⋊C8C2×D412yes—
181#103—24no—
182#102—24no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—24no—
187C22⋊C8—22no—
188#97—22no—
189#97—22no—
190#97—41no—
191#97—42no—
192#97—42no—
193#97SD1614yescomplemented
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—42no—
197C4×C8—42no—
198C4×C8—42no—
199C4×C8C22×C411yescharacteristic
200C4×C8C22×C411yescharacteristic
201C4×C8C2×D412yes—
202C4×C8#4111yescharacteristic
203C4×C8#4111yescharacteristic
204C4×C8#4112yes—
205#95—22no—
206#95#4112yes—
207C24—21no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×D4—21no—
211C2×D4—42no—
212C2×D4C4×D412yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4C22×D411yescharacteristic
224C22×C4C2×#4111yescharacteristic
225C22×C4C2×#4111yescharacteristic
226SD16—48no—
227M16—44no—
228M16—44no—
229M16—42no—
230M16—42no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—24no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—48no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8C4×D411yescharacteristic
246C2×C8C4×D411yescharacteristic
247C2×C8C4×D411yescharacteristic
248C2×C8C4×D411yescharacteristic
249C2×C8#12112yes—
250C2×C8#12112yes—
251C2×C8C2×SD1612yes—
252C2×C8C2×SD1612yes—
253C2×C8#13512yes—
254C2×C8#13712yes—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4C4×D412yes—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—82no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—42no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—81no—
278C42C2×SD1612yes—
279C42C23×C411yescharacteristic
280C42C22×D411yescharacteristic
281C42C2×#4111yescharacteristic
282C42C2×#4111yescharacteristic
283C23—22no—
284C23—42no—
285C23—42no—
286C23—41no—
287C23—41no—
288C23#52511yescharacteristic
289Q8—44no—
290Q8—82no—
291D4—24no—
292D4—42no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4C2×C4×D411yescharacteristic
314C2×C4C2×C4×D411yescharacteristic
315C2×C4C4×#4111yescharacteristic
316C2×C4C2×#12111yescharacteristic
317C2×C4#52911yescharacteristic
318C2×C4#53211yescharacteristic
319C2×C4C22×SD1611yescharacteristic
320C2×C4C2×#13511yescharacteristic
321C2×C4C2×#13711yescharacteristic
322C2×C4#57511yescharacteristic
323C8—21no—
324C8—21no—
325C8—22no—
326C8—42no—
327C8—44no—
328C8—42no—
329C8—82no—
330C8—82no—
331C8C4×SD1612yes—
332C8C4×SD1612yes—
333C8#46114yes—
334C22—21no—
335C22—21no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22C4×#12111yescharacteristic
342C22#310111yescharacteristic
343C22#310711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—42no—
349C4—82no—
350C4—81no—
351C4—82no—
352C4—82no—
353C4C2×C4×SD1611yescharacteristic
354C4C4×#13711yescharacteristic
355C4#295512yes—
356C4C2×#46111yescharacteristic
357C4#303511yescharacteristic
358C2—21no—
359C2—21no—
360C2—82no—
361C2C4×#46111yescharacteristic
362C2#2155011yescharacteristic
363C2#2155911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC22×D4#41
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21734C2semidirect
5. #9749C2semidirect
6. #9628C2semidirect
7. #9579C2semidirect
10. #8268C2semidirect
11. #8256C2semidirect
32. #1695C4semidirect
33. #1693C4semidirect
34. #1691C4semidirect
193. #97SD16semidirect

Maximal subgroups

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

subgroupindex
#217342
#204952
#191372
#179022
#97492
#96282
#95792
#92352
#86622
#82682
#82562