Finite Group Databasebeta

← #497985

Subgroups — #497985

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#509C211yesmaximal, characteristic
2#22813C211yesmaximal, characteristic, complemented
3#22686C211yesmaximal, characteristic, complemented
4#13155C212yesmaximal, complemented
5#13119C212yesmaximal, complemented
6#13040C212yesmaximal
7#12886C211yesmaximal, characteristic, complemented
8#12818C211yesmaximal, characteristic, complemented
9#12816C211yesmaximal, characteristic, complemented
10#12666C211yesmaximal, characteristic, complemented
11#12650C212yesmaximal, complemented
12C2×#509C2211yescharacteristic
13C2×#444C2212yes—
14C4×#135C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16#2906—21no—
17#2354—21no—
18#2177C412yescomplemented
19#2175C412yescomplemented
20C4×SD32C412yescomplemented
21C4×D16C412yescomplemented
22#2159C2211yescharacteristic
23#2150C2211yescharacteristic
24C2×#361C2212yes—
25C2×#360C2212yes—
26#2140C2211yescharacteristic
27#2139C2211yescharacteristic
28C2×#347C2212yes—
29C4×M32C2211yescharacteristic
30#1972C2212yes—
31#1946C2212yes—
32#1926—22no—
33#1866—22no—
34#1848C2211yescharacteristic
35#1844C2211yescharacteristic
36#1806—21no—
37#1802—21no—
38#1776C2212yes—
39#1770C2211yescharacteristic
40#1387C2212yes—
41#1386C2212yes—
42#1383C2212yes—
43#1383C2212yes—
44#1375C2212yes—
45C2×#135C2311yescharacteristic
46C22×D8C2311yescharacteristic
47C2×#131—22no—
48C4×#41—21no—
49C2×C4×D4—21no—
50C2×#117—41no—
51#509—22no—
52#509—22no—
53C2×SD32C2×C412yes—
54C2×D16C2×C412yes—
55C2×M32C2312yes—
56C2×M32C2311yescharacteristic
57C2×M32C2311yescharacteristic
58#444—24no—
59#444—42no—
60C4×Q16C2×C411yescharacteristic
61C4×Q16C2×C411yescharacteristic
62C4×SD16—21no—
63C4×SD16—21no—
64C4×D8—21no—
65C4×D8—21no—
66C4×D8C2×C412yes—
67C4×D8C2×C412yes—
68C4×D8C2×C411yescharacteristic
69C4×D8C2×C411yescharacteristic
70C2×#108—21no—
71C2×#108C2312yes—
72#427C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#106—21no—
75C2×#106C2312yes—
76#424—22no—
77#424—42no—
78#423—21no—
79#423—41no—
80C2×C4⋊C8—41no—
81C2×C4⋊C8D411yescharacteristic
82C2×#104—22no—
83#416—21no—
84C2×#102—21no—
85C2×#102—22no—
86C2×#97C2312yes—
87C2×C4×C8C2311yescharacteristic
88#390—22no—
89#388—42no—
90#384—21no—
91#380—41no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4D411yescharacteristic
94#367C2×C412yes—
95#366C2×C412yes—
96C4⋊C16—21no—
97C4⋊C16—21no—
98#361—42no—
99#360—24no—
100#358C2×C412yes—
101#357C2×C412yes—
102#357C2×C412yes—
103#357C2×C412yes—
104#347—42no—
105#346C2×C412yes—
106C4×C16C2×C412yes—
107#340—22no—
108#339—22no—
109#336D412yes—
110C2×#41—21no—
111C22×D4—21no—
112C23×C4—41no—
113#135—21no—
114#135—21no—
115#135#4112yes—
116#135#4112yes—
117C2×Q16C22×C411yescharacteristic
118C2×SD16—21no—
119C2×D8—21no—
120C2×D8—21no—
121C2×D8—21no—
122C2×D8—21no—
123C2×D8—21no—
124C2×D8C22×C412yes—
125C2×D8C22×C411yescharacteristic
126#131—44no—
127#131—82no—
128C2×M16—22no—
129C2×M16—42no—
130C22×C8—42no—
131C22×C8C2×D412yes—
132C22×C8C2×D411yescharacteristic
133C22×C8C2411yescharacteristic
134C4×Q8—21no—
135C4×Q8—21no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—21no—
139C4×D4—21no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144#117—21no—
145#117—41no—
146#117—84no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C42—41no—
156C2×C42—42no—
157C2×C42C2×D412yes—
158C2×C42C2×D411yescharacteristic
159SD32—24no—
160D16—24no—
161M32—21no—
162M32—21no—
163M32—21no—
164M32—21no—
165M32—22no—
166M32—22no—
167C2×C16—22no—
168C2×C16C22×C412yes—
169C2×C16C22×C412yes—
170#108—22no—
171#108—22no—
172#108—81no—
173#107—21no—
174#107C22×C411yescharacteristic
175#107C22×C411yescharacteristic
176#107C22×C412yes—
177#106—21no—
178#106—22no—
179#106—22no—
180#106—81no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—42no—
184C4⋊C8—42no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187#104—44no—
188#103—21no—
189#103—21no—
190#102—22no—
191#102—22no—
192#102—21no—
193#102—21no—
194#102—82no—
195#97—22no—
196#97—22no—
197C4×C8C22×C411yescharacteristic
198C4×C8C22×C411yescharacteristic
199C4×C8#4111yescharacteristic
200C4×C8#4111yescharacteristic
201#95—21no—
202#95—42no—
203C24—41no—
204#41—22no—
205#41—22no—
206#41—41no—
207#41—41no—
208C2×Q8—21no—
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4AGL(1,ℤ/8)11yescharacteristic
228C22×C4AGL(1,ℤ/8)11yescharacteristic
229C22×C4C22×D411yescharacteristic
230Q16—21no—
231Q16#11712yescomplemented
232SD16—22no—
233SD16—22no—
234D8—22no—
235D8—21no—
236D8—22no—
237D8—41no—
238D8—41no—
239D8#11712yescomplemented
240M16—42no—
241M16—42no—
242M16—84no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—42no—
249C2×C8—84no—
250C2×C8C4×D411yescharacteristic
251C2×C8C4×D411yescharacteristic
252C2×C8C23×C411yescharacteristic
253C2×C8C2×#4111yescharacteristic
254C2×C8C2×#4111yescharacteristic
255C2×C8C2×#4111yescharacteristic
256C2×C8H(5,𝔽2)11yescharacteristic
257C2×C82−1+411yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—82no—
268C4⋊C4—161no—
269C22⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—84no—
275C22⋊C4—162no—
276C42—22no—
277C42—22no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—84no—
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42AGL(1,ℤ/8)11yescharacteristic
285C42AGL(1,ℤ/8)11yescharacteristic
286C16—22no—
287C16—22no—
288C16—42no—
289C23—41no—
290C23—42no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23C2×AGL(1,ℤ/8)11yescharacteristic
298Q8—22no—
299Q8—41no—
300D4—22no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—41no—
305D4—42no—
306D4—81no—
307D4—81no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—84no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—161no—
328C2×C4#44211yescharacteristic
329C2×C4#44211yescharacteristic
330C2×C4#44211yescharacteristic
331C2×C4#44211yescharacteristic
332C2×C4C2×C4×D411yescharacteristic
333C2×C4#54611yescharacteristic
334C2×C4#54811yescharacteristic
335C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
336C2×C4#57611yescharacteristic
337C2×C4#57811yescharacteristic
338C8—21no—
339C8—21no—
340C8—42no—
341C8—82no—
342C8C2×#11711yescharacteristic
343C8#52011yescharacteristic
344C8#55211yescharacteristic
345C8#55311yescharacteristic
346C22—21no—
347C22—21no—
348C22—42no—
349C22—81no—
350C22—81no—
351C22—82no—
352C22—82no—
353C22—82no—
354C22—161no—
355C22—161no—
356C22—161no—
357C22—161no—
358C22C2×#44211yescharacteristic
359C22#333011yescharacteristic
360C22#336611yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—82no—
366C4—81no—
367C4—82no—
368C4—81no—
369C4—162no—
370C4AGL(1,ℤ/16)12yes—
371C4#235211yescharacteristic
372C4#295111yescharacteristic
373C4#332511yescharacteristic
374C4#336711yescharacteristic
375C2—21no—
376C2—21no—
377C2—82no—
378C2—162no—
379C2#2275811yescharacteristic
380C2C2×AGL(1,ℤ/16)11yescharacteristic
381C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#509C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22813C2semidirect
3. #22686C2semidirect
4. #13155C2semidirect
5. #13119C2semidirect
7. #12886C2semidirect
8. #12818C2semidirect
9. #12816C2semidirect
10. #12666C2semidirect
11. #12650C2semidirect
18. #2177C4semidirect
19. #2175C4semidirect
20. C4×SD32C4semidirect
21. C4×D16C4semidirect
231. Q16#117semidirect
239. D8#117semidirect

Maximal subgroups

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

subgroupindex
C4×#5092
#228132
#226862
#131552
#131192
#130402
#128862
#128182
#128162
#126662
#126502